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API Reference

This reference is generated automatically from the package docstrings via mkdocstrings.

Top-level package

CircularDiffusionModel(threshold_dynamic='fixed')

Circular Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Circular.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Circular Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angle

Parameters:

Name Type Description Default
rt array - like

Response times

 required
theta array - like

Choice angles in radians

 required
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The initial decision threshold

 required
decay float

The decay rate of the threshold (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Type Description
array - like

The joint log-probability density evaluated at (rt, theta) with same shape as rt and theta
Source code in src/jeam/Models/Circular.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angle

    Parameters
    ----------
    rt : array-like
        Response times
    theta : array-like
        Choice angles in radians
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The initial decision threshold
    decay : float
        The decay rate of the threshold (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    array-like
        The joint log-probability density evaluated at (rt, theta) with same shape as rt and theta
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 2))

    if drift_vec.shape[1] != 2 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (2,) or (n_samples, 2)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = cdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * cdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = cdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * cdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0] * np.cos(theta)
        mu_dot_x1 = drift_vec[:, 1] * np.sin(theta)

        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1) / sigma**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density =  np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi

                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability     
            s_v2 = s_v**2
            x0 =  a * np.cos(theta)
            x1 =  a * np.sin(theta)
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))

            log_density = 2*np.log(fixed) + exponent0 + exponent1 + np.log(fpt_z) - np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 =  threshold * np.cos(theta[i])
                        x1 =  threshold * np.sin(theta[i])
                    elif self.threshold_dynamic == 'linear':
                        x0 =  (threshold - decay * (tt[i]-eps)) * np.cos(theta[i])
                        x1 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'exponential':
                        x0 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i])
                        x1 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 =  (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i])
                        x1 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'custom':
                        x0 =  threshold_function(tt[i]-eps) * np.cos(theta[i])
                        x1 =  threshold_function(tt[i]-eps) * np.sin(theta[i])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i]-eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i]-eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i]-eps) + 1))

                    integrand = fixed**2 * np.exp(exponent0 + exponent1) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * 0.5/np.pi
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt - s_t <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Circular Diffusion Model with collapsing boundaries

Parameters:

Name Type Description Default
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The initial decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The decay rate of the threshold (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

The time step for simulation (default is 0.001)

 0.001
n_sample int

The number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Circular.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Circular Diffusion Model with collapsing boundaries

    Parameters
    ----------
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The initial decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The decay rate of the threshold (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        The time step for simulation (default is 0.001)
    n_sample : int, optional
        The number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample,))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 2))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_CDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n], 
                                                  threshold_dynamic=self.threshold_dynamic,
                                                  decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_custom_threshold_CDM_trial(threshold_function,
                                                                   drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                   s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response'])

HyperSphericalDiffusionModel(threshold_dynamic='fixed')

Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/HyperSpherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Hyper-Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 3))

The choice angles in spherical coordinates (theta1, theta2, theta3)

 required
drift_vec (array - like, shape(4) or (n_samples, 4))

The drift rates in each dimension

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 3)
        The choice angles in spherical coordinates (theta1, theta2, theta3)
    drift_vec : array-like, shape (4,) or (n_samples, 4)
        The drift rates in each dimension
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 4))

    if drift_vec.shape[1] != 4 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (4,) or (n_samples, 4)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * hsdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1) 
            fpt_lt = hsdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * hsdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0]*np.cos(theta[:, 0])
        mu_dot_x1 = drift_vec[:, 1]*np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
        mu_dot_x2 = drift_vec[:, 2]*np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.cos(theta[:, 2])
        mu_dot_x3 = drift_vec[:, 3]*np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.sin(theta[:, 2])
        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1 + mu_dot_x2 + mu_dot_x3) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2 + drift_vec[:, 3]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - 2*np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2 + drift_vec[:, 3]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1 + mu_dot_x2 + mu_dot_x3) / sigma**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2

                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            s_v2 = s_v**2
            x0 =  a * np.cos(theta[:, 0])
            x1 =  a * np.sin(theta[:, 0])*np.cos(theta[:, 1])
            x2 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.cos(theta[:, 2])
            x3 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.sin(theta[:, 2])
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent2 = -0.5*drift_vec[:, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[:, 2])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent3 = -0.5*drift_vec[:, 3]**2/s_v2 + 0.5*(x3 * s_v2/sigma**2 + drift_vec[:, 3])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))

            # the joint density of choice and RT for the full process
            log_density = 4*np.log(fixed) + exponent0 + exponent1 + exponent2 + exponent3 + np.log(fpt_z) - 2*np.log(2*np.pi)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 =  threshold * np.cos(theta[i, 0])
                        x1 =  threshold * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'linear':
                        x0 =  (threshold - decay * (tt[i]-eps)) * np.cos(theta[i, 0])
                        x1 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'exponential':
                        x0 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 =  (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'custom':
                        x0 =  threshold_function(tt[i]-eps) * np.cos(theta[i, 0])
                        x1 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent2 = -0.5*drift_vec[i, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[i, 2])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent3 = -0.5*drift_vec[i, 3]**2/s_v2 + 0.5*(x3 * s_v2/sigma**2 + drift_vec[i, 3])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))

                    integrand = fixed**4 * np.exp(exponent0 + exponent1 + exponent2 + exponent3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(4) or (n_samples, 4))

The drift rates in each dimension

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Hyper-Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (4,) or (n_samples, 4)
        The drift rates in each dimension
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 3))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 4))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_HSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                      threshold_dynamic=self.threshold_dynamic, 
                                                      decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_HSDM_trial(threshold_function,
                                                                       drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                       s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2', 'response3'])

ProjectedHyperSphericalDiffusionModel(threshold_dynamic='fixed')

Projected Hyper-Spherical Diffusion Model

Source code in src/jeam/Models/HyperSpherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    self.name = 'Projected Hyper-Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles for the Projected Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 2))

The choice angles in spherical coordinates (theta1, theta2)

 required
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift rates in each dimension

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles for the Projected Hyper-Spherical Diffusion Model

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 2)
        The choice angles in spherical coordinates (theta1, theta2)
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift rates in each dimension
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''
    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 3))

    if drift_vec.shape[1] != 3 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (3,) or (n_samples, 3)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = 1/threshold**2 * hsdm_short_t_fpt_z(tt/threshold**2, 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = 1/threshold**2 * hsdm_short_t_fpt_z(T/threshold**2, 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)
    norm_mu = np.sqrt(drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2)

    theta1_mu = np.arctan2(drift_vec[:, 2], drift_vec[:, 0])
    theta2_mu = np.arctan2(drift_vec[:, 2], drift_vec[:, 1])

    # Girsanov:
    if s_v == 0:
        # No drift variability
        if s_t == 0:
            # No non-decision time variability
            x0 = np.cos(theta1_mu) * np.cos(theta[:, 0])
            x1 = np.sin(theta1_mu) * np.sin(theta[:, 0]) * np.cos(theta2_mu) * np.cos(theta[:, 1])
            term1 = np.exp(a * norm_mu * (x0 + x1) / sigma**2)
            term2 = iv(0, a * norm_mu * np.sin(theta1_mu) * np.sin(theta[:, 0]) * np.sin(theta2_mu) * np.sin(theta[:, 1])/sigma**2)
            term3 = -0.5 * norm_mu**2 * tt

            log_density = np.log(2*np.pi) + np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * term1 * term2 * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'linear':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp((threshold - decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, (threshold - decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'exponential':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold*np.exp(-decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold/(1  + decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold/(1  + decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'custom':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold_function(tt[i] - eps) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold_function(tt[i] - eps) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            c1 = a * np.sin(theta[:, 0]) * np.sin(theta[:, 1]) * s_v2
            c2 = 2*s_v2 * (s_v2 * tt + 1)
            term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 2]/c2)
            term2 = 1/(s_v2 * tt + 1)**2
            p1 = (c1**2 + drift_vec[:, 2]**2)/c2
            p2 = (a * np.cos(theta[:, 0]) * s_v2 + drift_vec[:, 0])**2 / c2
            p3 = (a * np.sin(theta[:, 0]) * np.cos(theta[:, 1]) * s_v2 + drift_vec[:, 1])**2 / c2
            p4 = (norm_mu**2)/(2*s_v2)

            log_density = np.log(term1) + np.log(term2) + (p1 + p2 + p3 - p4) + np.log(fpt_z)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    c2 = 2*s_v2 * (s_v2 * (tt[i] - eps) + 1)
                    if self.threshold_dynamic == 'fixed':
                        c1 = threshold * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = (threshold * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = (threshold * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'linear':
                        c1 = (threshold - decay*(tt[i]-eps)) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold - decay*(tt[i]-eps)) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold - decay*(tt[i]-eps)) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'exponential':
                        c1 = (threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'hyperbolic':
                        c1 = (threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold / (1 + decay*(tt[i]-eps))) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'custom':
                        c1 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = (threshold_function(tt[i]-eps) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = (threshold_function(tt[i]-eps) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2

                    term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 2]/c2)
                    term2 = 1/(s_v2 * (tt[i] - eps) + 1)**2
                    p4 = (norm_mu**2)/(2*s_v2)
                    term3 = np.exp(p1 + p2 + p3 - p4)
                    integrand = term1 * term2 * term3 * np.interp(tt[i]-eps, T, fpt_z)/s_t

                    density = trapz_1d(integrand, eps)

                    if density > 0.1**14:
                            log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Projected Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift vector [drift_x, drift_y, drift_z]. Note that drift_z must be non-negative, as it represents the projection onto the upper hemisphere.

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Projected Hyper-Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift vector [drift_x, drift_y, drift_z]. Note that drift_z must be non-negative, as it represents the projection onto the upper hemisphere.
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 2))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 3))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_PHSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                       threshold_dynamic=self.threshold_dynamic, 
                                                       decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_PHSDM_trial(threshold_function,
                                                                        drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                        s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2'])

ProjectedSphericalDiffusionModel(threshold_dynamic='fixed')

Projected Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Spherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Projected Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angle

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples))

The choice angles in radians

 required
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angle
Source code in src/jeam/Models/Spherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angle

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples,)
        The choice angles in radians
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angle
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 2))

    if drift_vec.shape[1] != 2 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (2,) or (n_samples, 2)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    norm_mu = np.sqrt(drift_vec[:, 0]**2 + drift_vec[:, 1]**2)
    theta_mu = np.arctan2(drift_vec[:, 1], drift_vec[:, 0])

    # Girsanov:
    if s_v == 0:
        # No drift variability
        if s_t == 0:
            # No non-decision time variability
            term1 = np.exp(a/sigma**2 * norm_mu * np.cos(theta_mu) * np.cos(theta))
            term2 = iv(0, a/sigma**2 * norm_mu * np.sin(theta_mu) * np.sin(theta))
            term3 = -0.5 * norm_mu**2 * tt / sigma**2
            log_density = np.log(2*np.pi) + np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        term1 = np.exp(threshold/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * term1 * term2 * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'linear':
                        term1 = np.exp((threshold - decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, (threshold - decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'exponential':
                        term1 = np.exp(threshold*np.exp(-decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold*np.exp(-decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'hyperbolic':
                        term1 = np.exp(threshold/(1  + decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold/(1  + decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'custom':
                        term1 = np.exp(threshold_function(tt[i] - eps)/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold_function(tt[i] - eps)/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            c1 = a * np.sin(theta) * s_v2/sigma**2
            c2 = 2*s_v2 * (s_v2/sigma**2 * tt + 1)
            term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 1]/c2)
            term2 = (1/(np.sqrt(s_v2/sigma**2 * tt + 1)))**3
            p1 = (c1**2 + drift_vec[:, 1]**2)/c2
            p2 = (a * np.cos(theta) * s_v2/sigma**2 + drift_vec[:, 0])**2 / c2
            p3 = (norm_mu**2)/(2*s_v2)
            # term3 = np.exp(p1 + p2 - p3)
            # log_density = np.log(term1) + np.log(term2) + np.log(term3) + np.log(fpt_z)

            term3 = p1 + p2 - p3
            log_density = np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    c2 = 2*s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1)
                    if self.threshold_dynamic == 'fixed':
                        c1 = threshold * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = (threshold * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'linear':
                        c1 = (threshold - decay*(tt[i]-eps)) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold - decay*(tt[i]-eps)) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'exponential':
                        c1 = (threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'hyperbolic':
                        c1 = (threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold / (1 + decay*(tt[i]-eps))) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'custom':
                        c1 = threshold_function(tt[i]-eps) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = (threshold_function(tt[i]-eps) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2

                    term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[i, 1]/c2)
                    term2 = (1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1)))**3
                    p1 = (c1**2 + drift_vec[i, 1]**2)/c2
                    p3 = (norm_mu[i]**2)/(2*s_v2)
                    term3 = np.exp(p1 + p2 - p3)
                    integrand = term1 * term2 * term3 * np.interp(tt[i]-eps, T, fpt_z)/s_t

                    density = trapz_1d(integrand, eps)

                    if density > 0.1**14:
                            log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate response times and choices from the Projected Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]. Note that drift_y must be non-negative, as it represents the projection onto the upper half-plane.

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Spherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate response times and choices from the Projected Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]. Note that drift_y must be non-negative, as it represents the projection onto the upper half-plane.
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''    
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample,))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 2))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_PSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                   threshold_dynamic=self.threshold_dynamic,
                                                   decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_custom_threshold_PSDM_trial(threshold_function,
                                                                    drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                    s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response'])

SphericalDiffusionModel(threshold_dynamic='fixed')

Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Spherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 2))

The choice angles in spherical coordinates (theta1, theta2)

 required
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift rates in each dimension

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/Spherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 2)
        The choice angles in spherical coordinates (theta1, theta2)
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift rates in each dimension
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 3))

    if drift_vec.shape[1] != 3 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (3,) or (n_samples, 3)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0]*np.cos(theta[:, 0])
        mu_dot_x1 = drift_vec[:, 1]*np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
        mu_dot_x2 = drift_vec[:, 2]*np.sin(theta[:, 0])*np.sin(theta[:, 1])

        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1 + mu_dot_x2) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - 1.5*np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1 + mu_dot_x2) / sigma**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5

                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            x0 =  a * np.cos(theta[:, 0])
            x1 =  a * np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
            x2 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent2 = -0.5*drift_vec[:, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[:, 2])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            log_density = 3*np.log(fixed) + exponent0 + exponent1 + exponent2 + np.log(fpt_z) - 1.5*np.log(2*np.pi)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 = threshold * np.cos(theta[i, 0])
                        x1 = threshold * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'linear':
                        x0 = (threshold - decay * (tt[i]-eps)) * np.cos(theta[i, 0])
                        x1 = (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'exponential':
                        x0 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 = (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 = (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'custom':
                        x0 = threshold_function(tt[i]-eps) * np.cos(theta[i, 0])
                        x1 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent2 = -0.5*drift_vec[i, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[i, 2])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))

                    integrand = fixed**3 * np.exp(exponent0 + exponent1 + exponent2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(3) or (n_samples, 3))

Drift vector; a three-dimensional array

 required
ndt float or array-like with the shape of (n_samples,)

Non-decision time; a positive floating number

 required
threshold float or array-like with the shape of (n_samples,)

Decision threshold; a positive floating number (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

Decay rate of the collapsing boundary (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Spherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        Drift vector; a three-dimensional array
    ndt : float or array-like with the shape of (n_samples,)
        Non-decision time; a positive floating number
    threshold : float or array-like with the shape of (n_samples,)
        Decision threshold; a positive floating number (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        Decay rate of the collapsing boundary (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 2))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 3))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_SDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n], 
                                                     threshold_dynamic=self.threshold_dynamic, 
                                                     decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_SDM_trial(threshold_function,
                                                                      drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                      s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2'])

iv_numba(v, x)

Modified Bessel function of the first kind I_v(x) for real v and real x. Uses series for small x and asymptotic for large x.

WARNING: I_v(x) grows like exp(x)/sqrt(x), so iv_numba will overflow to inf for sufficiently large x (around x > ~709 in float64).

Source code in src/jeam/utility/helpers.py 
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@jit(nopython=False)
def iv_numba(v, x):
    """
    Modified Bessel function of the first kind I_v(x) for real v and real x.
    Uses series for small x and asymptotic for large x.

    WARNING: I_v(x) grows like exp(x)/sqrt(x), so iv_numba will overflow to inf
    for sufficiently large x (around x > ~709 in float64).
    """
    ax = np.abs(x)
    if ax == 0.0:
        if v == 0.0:
            return 1.0
        return 0.0

    if ax < 18.0:
        return _iv_series(v, ax)
    else:
        # Use ive asymptotic then multiply by exp(ax) if safe
        scaled = _ive_asymptotic(v, ax)
        # exp(709.78...) is near float64 max
        if ax > 709.0:
            return np.inf
        return scaled * np.exp(ax)

ive_numba(v, x)

Exponentially scaled modified Bessel I: ive(v,x) = exp(-abs(x)) * I_v(x) for real v and real x.

This implementation assumes x >= 0 for best behavior.

Source code in src/jeam/utility/helpers.py 
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@jit(nopython=False)
def ive_numba(v, x):
    """
    Exponentially scaled modified Bessel I:
      ive(v,x) = exp(-abs(x)) * I_v(x)
    for real v and real x.

    This implementation assumes x >= 0 for best behavior.
    """
    ax = np.abs(x)
    if ax == 0.0:
        # ive(v,0) = I_v(0)
        if v == 0.0:
            return 1.0
        return 0.0

    # Threshold where asymptotic is usually accurate and faster than the series
    if ax < 18.0:
        # compute I_v(ax) then scale
        iv = _iv_series(v, ax)
        return iv * np.exp(-ax)
    else:
        return _ive_asymptotic(v, ax)

load_fennell2023()

Load data from experiment one in Fennell and Ratcliff (2023).

Returns:

Type Description
DataFrame

A dataframe containing the data with columns: 'subjectNumber', 'blockNumber', 'trialNumber', 'rt', 'numberOfStimulus', 'responseError'
Source code in src/jeam/utility/datasets.py 
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def load_fennell2023():
    '''
    Load data from experiment one in Fennell and Ratcliff (2023).

    Returns
    -------
    pd.DataFrame
        A dataframe containing the data with columns: 'subjectNumber', 'blockNumber', 'trialNumber', 'rt', 'numberOfStimulus', 'responseError'
    '''
    return _read_csv("Fennell2023_exp4.csv", index_col=0)

load_kvam2019()

Load data from Kvam et al. (2019).

Returns:

Type Description
DataFrame

A dataframe containing the data with columns:
Source code in src/jeam/utility/datasets.py 
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def load_kvam2019():
    '''
    Load data from Kvam et al. (2019).

    Returns
    -------
    pd.DataFrame
        A dataframe containing the data with columns: 
    '''
    data = _read_csv("Kvam2019.csv")
    data = data.drop(
        columns=[
            "isCued",
            "cueOrientation",
            "cueDeflections",
            "points",
            "jitterLevel",
            "targetOrientation",
        ]
    )
    data = data.rename(columns={"RT": "rt"})
    data["deviation"] *= 2
    data["absoluteDeviation"] = np.abs(data["deviation"])
    return data

simulate_CDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the circular diffusion model (CDM).

Parameters:

Name Type Description Default
threshold float

A positive floating number representing the decision threshold.

 required
drift_vec (array_like, shape(2))

A two-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
threshold_dynamic (fixed, linear, exponential, hyperbolic)

Type of threshold collapse. Default is 'fixed'.

 'fixed'
decay float

Decay rate of the collapsing boundary. Default is 0.

 0
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta float

Response angle between [-pi, pi].
Source code in src/jeam/utility/simulators.py 
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@jit(nopython=True)
def simulate_CDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the circular diffusion model (CDM).

    Parameters
    ----------
    threshold : float
        A positive floating number representing the decision threshold.
    drift_vec : array_like, shape (2,)
        A two-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    threshold_dynamic : {'fixed', 'linear', 'exponential', 'hyperbolic'}, optional
        Type of threshold collapse. Default is 'fixed'.
    decay : float, optional
        Decay rate of the collapsing boundary. Default is 0.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : float
        Response angle between [-pi, pi].
    '''
    x = np.zeros((2,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(2)
    else:
        mu_t = drift_vec

    if threshold_dynamic == 'fixed':
        while np.linalg.norm(x) < threshold:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(2)
            rt += dt
    elif threshold_dynamic == 'linear':
        while np.linalg.norm(x) < threshold - decay*rt:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(2)
            rt += dt
    elif threshold_dynamic == 'exponential':
        while np.linalg.norm(x) < threshold * np.exp(-decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(2)
            rt += dt
    elif threshold_dynamic == 'hyperbolic':
        while np.linalg.norm(x) < threshold / (1 + decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(2)
            rt += dt
    theta = np.arctan2(x[1], x[0]) 
    return ndt_t+rt, theta

simulate_HSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the hyper spherical diffusion model (HSDM).

Parameters:

Name Type Description Default
threshold float

A positive floating number representing the decision threshold.

 required
drift_vec (array_like, shape(4))

A four-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
threshold_dynamic (fixed, linear, exponential, hyperbolic)

Type of threshold collapse. Default is 'fixed'.

 'fixed'
decay float

Decay rate of the collapsing boundary. Default is 0.

 0
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2, theta3); theta[0] and theta[1] between [0, pi], and theta[2] between [-pi, pi]
Source code in src/jeam/utility/simulators.py 
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@jit(nopython=True)
def simulate_HSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the hyper spherical diffusion model (HSDM).

    Parameters
    ----------
    threshold : float
        A positive floating number representing the decision threshold.
    drift_vec : array_like, shape (4,)
        A four-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    threshold_dynamic : {'fixed', 'linear', 'exponential', 'hyperbolic'}, optional
        Type of threshold collapse. Default is 'fixed'.
    decay : float, optional
        Decay rate of the collapsing boundary. Default is 0.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2, theta3); theta[0] and theta[1] between [0, pi], and theta[2] between [-pi, pi]
    '''
    x = np.zeros((4,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(4)
    else:
        mu_t = drift_vec

    if threshold_dynamic == 'fixed':
        while np.linalg.norm(x) < threshold:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'linear':
        while np.linalg.norm(x) < threshold - decay*rt:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'exponential':
        while np.linalg.norm(x) < threshold * np.exp(-decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'hyperbolic':
        while np.linalg.norm(x) < threshold / (1 + decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt

    theta1 = np.arctan2(np.sqrt(x[3]**2 + x[2]**2 + x[1]**2), x[0])
    theta2 = np.arctan2(np.sqrt(x[3]**2 + x[2]**2), x[1])
    theta3 = np.arctan2(x[3], x[2])

    return ndt_t+rt, (theta1, theta2, theta3)

simulate_PHSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the projected hyper spherical diffusion model (PHSDM).

Parameters:

Name Type Description Default
threshold float

A positive floating number representing the decision threshold.

 required
drift_vec (array_like, shape(3))

A three-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
threshold_dynamic (fixed, linear, exponential, hyperbolic)

Type of threshold collapse. Default is 'fixed'.

 'fixed'
decay float

Decay rate of the collapsing boundary. Default is 0.

 0
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2); theta[0] and theta[1] between [0, pi]
Source code in src/jeam/utility/simulators.py 
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@jit(nopython=True)
def simulate_PHSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the projected hyper spherical diffusion model (PHSDM).

    Parameters
    ----------
    threshold : float
        A positive floating number representing the decision threshold.
    drift_vec : array_like, shape (3,)
        A three-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    threshold_dynamic : {'fixed', 'linear', 'exponential', 'hyperbolic'}, optional
        Type of threshold collapse. Default is 'fixed'.
    decay : float, optional
        Decay rate of the collapsing boundary. Default is 0.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2); theta[0] and theta[1] between [0, pi]
    '''

    x = np.zeros((4,))
    muw = drift_vec[0]
    muz = drift_vec[1]
    eta = drift_vec[2]

    norm_mu = np.sqrt(eta**2 + muz**2 + muw**2)
    theta1_mu = np.arctan2(eta, muw)
    theta2_mu = np.arctan2(eta, muz)

    rt = 0
    rphi = np.pi/4 # it is not important (just a dummpy value)
    mux = norm_mu * np.sin(theta1_mu) * np.sin(theta2_mu) * np.cos(rphi)
    muy = norm_mu * np.sin(theta1_mu) * np.sin(theta2_mu) * np.sin(rphi)

    mu = np.array([mux, muy, muz, muw])

    if s_v>0:
        mu_t = mu + s_v*np.random.randn(4)
    else:
        mu_t = mu

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if threshold_dynamic == 'fixed':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) < threshold:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'linear':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) < threshold - decay*rt:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'exponential':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) < threshold * np.exp(-decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt
    elif threshold_dynamic == 'hyperbolic':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) < threshold / (1 + decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
            rt += dt

    theta1 = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[3])
    theta2 = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[2])   

    return ndt_t+rt, (theta1, theta2)

simulate_PSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the projected spherical diffusion model (PSDM).

Parameters:

Name Type Description Default
threshold float

A positive floating number representing the decision threshold.

 required
drift_vec (array_like, shape(2))

A two-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
threshold_dynamic (fixed, linear, exponential, hyperbolic)

Type of threshold collapse. Default is 'fixed'.

 'fixed'
decay float

Decay rate of the collapsing boundary. Default is 0.

 0
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta float

Response angle between [0, pi]
Source code in src/jeam/utility/simulators.py 
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@jit(nopython=True)
def simulate_PSDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the projected spherical diffusion model (PSDM).

    Parameters
    ----------
    threshold : float
        A positive floating number representing the decision threshold.
    drift_vec : array_like, shape (2,)
        A two-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    threshold_dynamic : {'fixed', 'linear', 'exponential', 'hyperbolic'}, optional
        Type of threshold collapse. Default is 'fixed'.
    decay : float, optional
        Decay rate of the collapsing boundary. Default is 0.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : float
        Response angle between [0, pi]
    '''
    x = np.zeros((3,))
    muz = drift_vec[0]
    eta = drift_vec[1]

    norm_mu = np.sqrt(eta**2 + muz**2)
    theta_mu = np.arctan2(eta, muz)

    rt = 0
    rphi = np.pi/4 # it is not important (just a dummpy value)
    mux = norm_mu * np.sin(theta_mu) * np.cos(rphi)
    muy = norm_mu * np.sin(theta_mu) * np.sin(rphi)

    mu = np.array([mux, muy, muz])

    if s_v>0:
        mu_t = mu + s_v*np.random.randn(3)
    else:
        mu_t = mu

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if threshold_dynamic == 'fixed':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2) < threshold:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'linear':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2) < threshold - decay*rt:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'exponential':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2) < threshold * np.exp(-decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'hyperbolic':
        while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2) < threshold / (1 + decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt

    theta = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[2])    

    return ndt_t+rt, theta

simulate_SDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the spherical diffusion model (SDM).

Parameters:

Name Type Description Default
threshold float

A positive floating number representing the decision threshold.

 required
drift_vec (array_like, shape(3))

A three-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
threshold_dynamic (fixed, linear, exponential, hyperbolic)

Type of threshold collapse. Default is 'fixed'.

 'fixed'
decay float

Decay rate of the collapsing boundary. Default is 0.

 0
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2); theta[0] between [0, pi] and theta[1] between [-pi, pi]
Source code in src/jeam/utility/simulators.py 
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@jit(nopython=True)
def simulate_SDM_trial(threshold, drift_vec, ndt, threshold_dynamic='fixed', decay=0, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the spherical diffusion model (SDM).

    Parameters
    ----------
    threshold : float
        A positive floating number representing the decision threshold.
    drift_vec : array_like, shape (3,)
        A three-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    threshold_dynamic : {'fixed', 'linear', 'exponential', 'hyperbolic'}, optional
        Type of threshold collapse. Default is 'fixed'.
    decay : float, optional
        Decay rate of the collapsing boundary. Default is 0.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2); theta[0] between [0, pi] and theta[1] between [-pi, pi]
    '''
    x = np.zeros((3,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(3)
    else:
        mu_t = drift_vec

    if threshold_dynamic == 'fixed':
        while np.linalg.norm(x) < threshold:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'linear':
        while np.linalg.norm(x) < threshold - decay*rt:
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'exponential':
        while np.linalg.norm(x) < threshold * np.exp(-decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt
    elif threshold_dynamic == 'hyperbolic':
        while np.linalg.norm(x) < threshold / (1 + decay*rt):
            x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
            rt += dt

    theta1 = np.arctan2(np.sqrt(x[2]**2 + x[1]**2), x[0])
    theta2 = np.arctan2(x[2], x[1])

    return ndt_t+rt, (theta1, theta2)

simulate_custom_threshold_CDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the circular diffusion model (CDM) with a user defined threshold function.

Parameters:

Name Type Description Default
threshold_function callable

A function that takes time t and returns the threshold at time t.

 required
drift_vec (array_like, shape(2))

A two-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
drift_vec (array_like, shape(2))

A two-dimensional array representing the drift vector.

 required
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta float

Response angle between [-pi, pi].
Source code in src/jeam/utility/simulators.py 
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def simulate_custom_threshold_CDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the circular diffusion model (CDM) with a user defined threshold function.

    Parameters
    ----------
    threshold_function : callable
        A function that takes time t and returns the threshold at time t.
    drift_vec : array_like, shape (2,)
        A two-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    drift_vec : array_like, shape (2,)
        A two-dimensional array representing the drift vector.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : float
        Response angle between [-pi, pi].
    '''
    x = np.zeros((2,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand() - s_t
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(2)
    else:
        mu_t = drift_vec

    while np.linalg.norm(x) < threshold_function(rt):
        x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(2)
        rt += dt

    theta = np.arctan2(x[1], x[0]) 

    return ndt_t+rt, theta

simulate_custom_threshold_HSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the hyper spherical diffusion model (HSDM) with a user defined threshold function.

Parameters:

Name Type Description Default
threshold_function callable

A function that takes time and returns the threshold at that time.

 required
drift_vec (array_like, shape(4))

A four-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2, theta3); theta[0] and theta[1] between [0, pi], and theta[2] between [-pi, pi]
Source code in src/jeam/utility/simulators.py 
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def simulate_custom_threshold_HSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the hyper spherical diffusion model (HSDM) with a user defined threshold function.

    Parameters
    ----------
    threshold_function : callable
        A function that takes time and returns the threshold at that time.
    drift_vec : array_like, shape (4,)
        A four-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2, theta3); theta[0] and theta[1] between [0, pi], and theta[2] between [-pi, pi]
    '''
    x = np.zeros((4,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(4)
    else:
        mu_t = drift_vec

    while np.linalg.norm(x) < threshold_function(rt):
        x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
        rt += dt

    theta1 = np.arctan2(np.sqrt(x[3]**2 + x[2]**2 + x[1]**2), x[0])
    theta2 = np.arctan2(np.sqrt(x[3]**2 + x[2]**2), x[1])
    theta3 = np.arctan2(x[3], x[2])

    return ndt_t+rt, (theta1, theta2, theta3)

simulate_custom_threshold_PHSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the projected hyper spherical diffusion model (PHSDM) with a user defined threshold function.

Parameters:

Name Type Description Default
threshold_function callable

A function that takes a time value and returns the threshold at that time.

 required
drift_vec (array_like, shape(3))

A three-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2); theta[0] and theta[1] between [0, pi]
Source code in src/jeam/utility/simulators.py 
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def simulate_custom_threshold_PHSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the projected hyper spherical diffusion model (PHSDM) with a user defined threshold function.

    Parameters
    ----------
    threshold_function : callable
        A function that takes a time value and returns the threshold at that time.
    drift_vec : array_like, shape (3,)
        A three-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2); theta[0] and theta[1] between [0, pi]
    '''
    x = np.zeros((4,))
    muw = drift_vec[0]
    muz = drift_vec[1]
    eta = drift_vec[2]

    norm_mu = np.sqrt(eta**2 + muz**2 + muw**2)
    theta1_mu = np.arctan2(eta, muw)
    theta2_mu = np.arctan2(eta, muz)

    rt = 0
    rphi = np.pi/4 # it is not important (just a dummpy value)
    mux = norm_mu * np.sin(theta1_mu) * np.sin(theta2_mu) * np.cos(rphi)
    muy = norm_mu * np.sin(theta1_mu) * np.sin(theta2_mu) * np.sin(rphi)

    mu = np.array([mux, muy, muz, muw])

    if s_v>0:
        mu_t = mu + s_v*np.random.randn(4)
    else:
        mu_t = mu

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt


    while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2 + x[3]**2) < threshold_function(rt):
        x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(4)
        rt += dt


    theta1 = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[3])
    theta2 = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[2])   

    return ndt_t+rt, (theta1, theta2)

simulate_custom_threshold_PSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the projected spherical diffusion model (PSDM) with a user defined threshold function.

Parameters:

Name Type Description Default
threshold_function callable

A function that takes time and returns threshold.

 required
drift_vec (array_like, shape(2))

A two-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta float

Response angle between [0, pi]
Source code in src/jeam/utility/simulators.py 
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def simulate_custom_threshold_PSDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the projected spherical diffusion model (PSDM) with a user defined threshold function.

    Parameters
    ----------
    threshold_function : callable
        A function that takes time and returns threshold.
    drift_vec : array_like, shape (2,)
        A two-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001. 

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : float
        Response angle between [0, pi]
    '''
    x = np.zeros((3,))
    muz = drift_vec[0]
    eta = drift_vec[1]

    norm_mu = np.sqrt(eta**2 + muz**2)
    theta_mu = np.arctan2(eta, muz)

    rt = 0
    rphi = np.pi/4 # it is not important (just a dummpy value)
    mux = norm_mu * np.sin(theta_mu) * np.cos(rphi)
    muy = norm_mu * np.sin(theta_mu) * np.sin(rphi)

    mu = np.array([mux, muy, muz])

    if s_v>0:
        mu_t = mu + s_v*np.random.randn(3)
    else:
        mu_t = mu

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    while np.sqrt(x[0]**2 + x[1]**2 + x[2]**2) < threshold_function(rt):
        x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)

        rt += dt

    theta = np.arctan2(np.sqrt(x[0]**2 + x[1]**2), x[2])    

    return ndt_t+rt, theta

simulate_custom_threshold_SDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001)

Simulate a single trial of the spherical diffusion model (SDM) with a user defined threshold function.

Parameters:

Name Type Description Default
threshold_function callable

A function that takes time t and returns the threshold at time t.

 required
drift_vec (array_like, shape(3))

A three-dimensional array representing the drift vector.

 required
ndt float

A positive floating number representing the non-decision time.

 required
s_v float

Standard deviation of drift rate variability. Default is 0.

 0
s_t float

Range of non-decision time variability. Default is 0.

 0
sigma float

Diffusion coefficient (standard deviation of the diffusion process). Default is 1.

 1
dt float

Time step for the simulation. Default is 0.001.

 0.001

Returns:

Name Type Description
rt float

Response time in seconds.
theta tuple

A tuple of response angles (theta1, theta2); theta[0] between [0, pi] and theta[1] between [-pi, pi]
Source code in src/jeam/utility/simulators.py 
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def simulate_custom_threshold_SDM_trial(threshold_function, drift_vec, ndt, s_v=0, s_t=0, sigma=1, dt=0.001):
    '''
    Simulate a single trial of the spherical diffusion model (SDM) with a user defined threshold function.

    Parameters
    ----------
    threshold_function : callable
        A function that takes time t and returns the threshold at time t.
    drift_vec : array_like, shape (3,)
        A three-dimensional array representing the drift vector.
    ndt : float
        A positive floating number representing the non-decision time.
    s_v : float, optional
        Standard deviation of drift rate variability. Default is 0.
    s_t : float, optional
        Range of non-decision time variability. Default is 0.
    sigma : float, optional
        Diffusion coefficient (standard deviation of the diffusion process). Default is 1.
    dt : float, optional
        Time step for the simulation. Default is 0.001.

    Returns
    -------
    rt : float
        Response time in seconds.
    theta : tuple
        A tuple of response angles (theta1, theta2); theta[0] between [0, pi] and theta[1] between [-pi, pi]
    '''

    x = np.zeros((3,))

    rt = 0

    if s_t>0:
        ndt_t = ndt + s_t*np.random.rand()
    else:
        ndt_t = ndt

    if s_v>0:
        mu_t = drift_vec + s_v*np.random.randn(3)
    else:
        mu_t = drift_vec

    while np.linalg.norm(x) < threshold_function(rt):
        x += mu_t*dt + sigma*np.sqrt(dt)*np.random.randn(3)
        rt += dt

    theta1 = np.arctan2(np.sqrt(x[2]**2 + x[1]**2), x[0])
    theta2 = np.arctan2(x[2], x[1])

    return ndt_t+rt, (theta1, theta2)

trapz_1d(y, x)

Trapezoidal integral of y over x. Both 1D, same length.

Source code in src/jeam/utility/helpers.py 
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@njit(cache=True, fastmath=True)
def trapz_1d(y, x):
    """
    Trapezoidal integral of y over x. Both 1D, same length.
    """
    s = 0.0
    for i in range(x.shape[0] - 1):
        dx = x[i + 1] - x[i]
        s += 0.5 * (y[i] + y[i + 1]) * dx
    return s

Models

Circular diffusion model

CircularDiffusionModel(threshold_dynamic='fixed')

Circular Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Circular.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Circular Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angle

Parameters:

Name Type Description Default
rt array - like

Response times

 required
theta array - like

Choice angles in radians

 required
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The initial decision threshold

 required
decay float

The decay rate of the threshold (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Type Description
array - like

The joint log-probability density evaluated at (rt, theta) with same shape as rt and theta
Source code in src/jeam/Models/Circular.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angle

    Parameters
    ----------
    rt : array-like
        Response times
    theta : array-like
        Choice angles in radians
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The initial decision threshold
    decay : float
        The decay rate of the threshold (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    array-like
        The joint log-probability density evaluated at (rt, theta) with same shape as rt and theta
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 2))

    if drift_vec.shape[1] != 2 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (2,) or (n_samples, 2)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = cdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * cdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = cdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * cdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 2*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0] * np.cos(theta)
        mu_dot_x1 = drift_vec[:, 1] * np.sin(theta)

        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1) / sigma**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density =  np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * 0.5/np.pi

                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability     
            s_v2 = s_v**2
            x0 =  a * np.cos(theta)
            x1 =  a * np.sin(theta)
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))

            log_density = 2*np.log(fixed) + exponent0 + exponent1 + np.log(fpt_z) - np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 =  threshold * np.cos(theta[i])
                        x1 =  threshold * np.sin(theta[i])
                    elif self.threshold_dynamic == 'linear':
                        x0 =  (threshold - decay * (tt[i]-eps)) * np.cos(theta[i])
                        x1 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'exponential':
                        x0 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i])
                        x1 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 =  (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i])
                        x1 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i])
                    elif self.threshold_dynamic == 'custom':
                        x0 =  threshold_function(tt[i]-eps) * np.cos(theta[i])
                        x1 =  threshold_function(tt[i]-eps) * np.sin(theta[i])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i]-eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i]-eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i]-eps) + 1))

                    integrand = fixed**2 * np.exp(exponent0 + exponent1) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * 0.5/np.pi
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt - s_t <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Circular Diffusion Model with collapsing boundaries

Parameters:

Name Type Description Default
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The initial decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The decay rate of the threshold (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

The time step for simulation (default is 0.001)

 0.001
n_sample int

The number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Circular.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Circular Diffusion Model with collapsing boundaries

    Parameters
    ----------
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The initial decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The decay rate of the threshold (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        The time step for simulation (default is 0.001)
    n_sample : int, optional
        The number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample,))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 2))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_CDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n], 
                                                  threshold_dynamic=self.threshold_dynamic,
                                                  decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_custom_threshold_CDM_trial(threshold_function,
                                                                   drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                   s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response'])

Spherical diffusion models

ProjectedSphericalDiffusionModel(threshold_dynamic='fixed')

Projected Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Spherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Projected Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angle

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples))

The choice angles in radians

 required
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angle
Source code in src/jeam/Models/Spherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angle

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples,)
        The choice angles in radians
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angle
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 2))

    if drift_vec.shape[1] != 2 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (2,) or (n_samples, 2)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    norm_mu = np.sqrt(drift_vec[:, 0]**2 + drift_vec[:, 1]**2)
    theta_mu = np.arctan2(drift_vec[:, 1], drift_vec[:, 0])

    # Girsanov:
    if s_v == 0:
        # No drift variability
        if s_t == 0:
            # No non-decision time variability
            term1 = np.exp(a/sigma**2 * norm_mu * np.cos(theta_mu) * np.cos(theta))
            term2 = iv(0, a/sigma**2 * norm_mu * np.sin(theta_mu) * np.sin(theta))
            term3 = -0.5 * norm_mu**2 * tt / sigma**2
            log_density = np.log(2*np.pi) + np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        term1 = np.exp(threshold/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * term1 * term2 * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'linear':
                        term1 = np.exp((threshold - decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, (threshold - decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'exponential':
                        term1 = np.exp(threshold*np.exp(-decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold*np.exp(-decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'hyperbolic':
                        term1 = np.exp(threshold/(1  + decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold/(1  + decay * (tt[i] - eps))/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'custom':
                        term1 = np.exp(threshold_function(tt[i] - eps)/sigma**2 * norm_mu[i] * np.cos(theta_mu[i]) * np.cos(theta[i]))
                        term2 = iv(0, threshold_function(tt[i] - eps)/sigma**2 * norm_mu[i] * np.sin(theta_mu[i]) * np.sin(theta[i]))
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps) / sigma**2
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            c1 = a * np.sin(theta) * s_v2/sigma**2
            c2 = 2*s_v2 * (s_v2/sigma**2 * tt + 1)
            term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 1]/c2)
            term2 = (1/(np.sqrt(s_v2/sigma**2 * tt + 1)))**3
            p1 = (c1**2 + drift_vec[:, 1]**2)/c2
            p2 = (a * np.cos(theta) * s_v2/sigma**2 + drift_vec[:, 0])**2 / c2
            p3 = (norm_mu**2)/(2*s_v2)
            # term3 = np.exp(p1 + p2 - p3)
            # log_density = np.log(term1) + np.log(term2) + np.log(term3) + np.log(fpt_z)

            term3 = p1 + p2 - p3
            log_density = np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    c2 = 2*s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1)
                    if self.threshold_dynamic == 'fixed':
                        c1 = threshold * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = (threshold * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'linear':
                        c1 = (threshold - decay*(tt[i]-eps)) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold - decay*(tt[i]-eps)) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'exponential':
                        c1 = (threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'hyperbolic':
                        c1 = (threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = ((threshold / (1 + decay*(tt[i]-eps))) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2
                    elif self.threshold_dynamic == 'custom':
                        c1 = threshold_function(tt[i]-eps) * np.sin(theta[i]) * s_v2/sigma**2
                        p2 = (threshold_function(tt[i]-eps) * np.cos(theta[i]) * s_v2/sigma**2 + drift_vec[i, 0])**2 / c2

                    term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[i, 1]/c2)
                    term2 = (1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1)))**3
                    p1 = (c1**2 + drift_vec[i, 1]**2)/c2
                    p3 = (norm_mu[i]**2)/(2*s_v2)
                    term3 = np.exp(p1 + p2 - p3)
                    integrand = term1 * term2 * term3 * np.interp(tt[i]-eps, T, fpt_z)/s_t

                    density = trapz_1d(integrand, eps)

                    if density > 0.1**14:
                            log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate response times and choices from the Projected Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(2) or (n_samples, 2))

The drift vector [drift_x, drift_y]. Note that drift_y must be non-negative, as it represents the projection onto the upper half-plane.

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Spherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate response times and choices from the Projected Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (2,) or (n_samples, 2)
        The drift vector [drift_x, drift_y]. Note that drift_y must be non-negative, as it represents the projection onto the upper half-plane.
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''    
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample,))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 2))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_PSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                   threshold_dynamic=self.threshold_dynamic,
                                                   decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n] = simulate_custom_threshold_PSDM_trial(threshold_function,
                                                                    drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                    s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response'])

SphericalDiffusionModel(threshold_dynamic='fixed')

Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/Spherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 2))

The choice angles in spherical coordinates (theta1, theta2)

 required
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift rates in each dimension

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/Spherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 2)
        The choice angles in spherical coordinates (theta1, theta2)
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift rates in each dimension
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 3))

    if drift_vec.shape[1] != 3 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (3,) or (n_samples, 3)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = sdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * sdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 3*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0]*np.cos(theta[:, 0])
        mu_dot_x1 = drift_vec[:, 1]*np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
        mu_dot_x2 = drift_vec[:, 2]*np.sin(theta[:, 0])*np.sin(theta[:, 1])

        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1 + mu_dot_x2) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - 1.5*np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1 + mu_dot_x2) / sigma**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5

                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            x0 =  a * np.cos(theta[:, 0])
            x1 =  a * np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
            x2 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent2 = -0.5*drift_vec[:, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[:, 2])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            log_density = 3*np.log(fixed) + exponent0 + exponent1 + exponent2 + np.log(fpt_z) - 1.5*np.log(2*np.pi)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 = threshold * np.cos(theta[i, 0])
                        x1 = threshold * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'linear':
                        x0 = (threshold - decay * (tt[i]-eps)) * np.cos(theta[i, 0])
                        x1 = (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'exponential':
                        x0 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 = (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 = (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    elif self.threshold_dynamic == 'custom':
                        x0 = threshold_function(tt[i]-eps) * np.cos(theta[i, 0])
                        x1 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.cos(theta[i, 1]) 
                        x2 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent2 = -0.5*drift_vec[i, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[i, 2])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))

                    integrand = fixed**3 * np.exp(exponent0 + exponent1 + exponent2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * (0.5/np.pi)**1.5
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(3) or (n_samples, 3))

Drift vector; a three-dimensional array

 required
ndt float or array-like with the shape of (n_samples,)

Non-decision time; a positive floating number

 required
threshold float or array-like with the shape of (n_samples,)

Decision threshold; a positive floating number (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

Decay rate of the collapsing boundary (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/Spherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        Drift vector; a three-dimensional array
    ndt : float or array-like with the shape of (n_samples,)
        Non-decision time; a positive floating number
    threshold : float or array-like with the shape of (n_samples,)
        Decision threshold; a positive floating number (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        Decay rate of the collapsing boundary (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 2))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 3))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_SDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n], 
                                                     threshold_dynamic=self.threshold_dynamic, 
                                                     decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_SDM_trial(threshold_function,
                                                                      drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                      s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)

    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2'])

Hyper-spherical diffusion models

HyperSphericalDiffusionModel(threshold_dynamic='fixed')

Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
threshold_dynamic str

The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'

 'fixed'
Source code in src/jeam/Models/HyperSpherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    '''
    Parameters
    ----------
    threshold_dynamic : str, optional
        The type of threshold collapse ('fixed', 'linear', 'exponential', 'hyperbolic', or 'custom'), default is 'fixed'
    '''
    self.name = 'Hyper-Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 3))

The choice angles in spherical coordinates (theta1, theta2, theta3)

 required
drift_vec (array - like, shape(4) or (n_samples, 4))

The drift rates in each dimension

 required
ndt (float or array - like, shape(n_samples))

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 3)
        The choice angles in spherical coordinates (theta1, theta2, theta3)
    drift_vec : array-like, shape (4,) or (n_samples, 4)
        The drift rates in each dimension
    ndt : float or array-like, shape (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 4))

    if drift_vec.shape[1] != 4 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (4,) or (n_samples, 4)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * hsdm_short_t_fpt_z(sigma**2 * tt/threshold**2, sigma**2 * 0.1**8/threshold**2)
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1) 
            fpt_lt = hsdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = sigma**2/threshold**2 * hsdm_short_t_fpt_z(sigma**2 * T/threshold**2, sigma**2 * 0.1**8/threshold**2)
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)

    # Girsanov:
    if s_v == 0:
        # No drift variability
        mu_dot_x0 = drift_vec[:, 0]*np.cos(theta[:, 0])
        mu_dot_x1 = drift_vec[:, 1]*np.sin(theta[:, 0])*np.cos(theta[:, 1]) 
        mu_dot_x2 = drift_vec[:, 2]*np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.cos(theta[:, 2])
        mu_dot_x3 = drift_vec[:, 3]*np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.sin(theta[:, 2])
        if s_t == 0:
            # No non-decision time variability
            term1 = a * (mu_dot_x0 + mu_dot_x1 + mu_dot_x2 + mu_dot_x3) / sigma**2
            term2 = 0.5 * (drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2 + drift_vec[:, 3]**2) * tt / sigma**2
            log_density = term1 - term2 + np.log(fpt_z) - 2*np.log(2*np.pi)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            norm2_drift = drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2 + drift_vec[:, 3]**2
            mu_dot_x = (mu_dot_x0 + mu_dot_x1 + mu_dot_x2 + mu_dot_x3) / sigma**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        integrand = np.exp(- 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = np.exp(threshold * mu_dot_x[i]) * trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'linear':
                        integrand = np.exp((threshold - decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'exponential':
                        integrand = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'hyperbolic':
                        integrand = np.exp(threshold/(1  + decay * (tt[i] - eps)) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    elif self.threshold_dynamic == 'custom':
                        integrand = np.exp(threshold_function(tt[i] - eps) * mu_dot_x[i] - 0.5 * norm2_drift[i] * (tt[i] - eps)/sigma**2) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = trapz_1d(integrand, eps) * (0.5/np.pi)**2

                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            s_v2 = s_v**2
            x0 =  a * np.cos(theta[:, 0])
            x1 =  a * np.sin(theta[:, 0])*np.cos(theta[:, 1])
            x2 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.cos(theta[:, 2])
            x3 =  a * np.sin(theta[:, 0])*np.sin(theta[:, 1])*np.sin(theta[:, 2])
            fixed = 1/(np.sqrt(s_v2/sigma**2 * tt + 1))
            exponent0 = -0.5*drift_vec[:, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[:, 0])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent1 = -0.5*drift_vec[:, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[:, 1])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent2 = -0.5*drift_vec[:, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[:, 2])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))
            exponent3 = -0.5*drift_vec[:, 3]**2/s_v2 + 0.5*(x3 * s_v2/sigma**2 + drift_vec[:, 3])**2 / (s_v2 * (s_v2/sigma**2 * tt + 1))

            # the joint density of choice and RT for the full process
            log_density = 4*np.log(fixed) + exponent0 + exponent1 + exponent2 + exponent3 + np.log(fpt_z) - 2*np.log(2*np.pi)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 =  threshold * np.cos(theta[i, 0])
                        x1 =  threshold * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  threshold * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'linear':
                        x0 =  (threshold - decay * (tt[i]-eps)) * np.cos(theta[i, 0])
                        x1 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold - decay * (tt[i]-eps)) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'exponential':
                        x0 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold * np.exp(-decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 =  (threshold / (1 + decay * (tt[i]-eps))) * np.cos(theta[i, 0])
                        x1 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  (threshold / (1 + decay * (tt[i]-eps))) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    elif self.threshold_dynamic == 'custom':
                        x0 =  threshold_function(tt[i]-eps) * np.cos(theta[i, 0])
                        x1 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.cos(theta[i, 1])
                        x2 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.cos(theta[i, 2])
                        x3 =  threshold_function(tt[i]-eps) * np.sin(theta[i, 0])*np.sin(theta[i, 1])*np.sin(theta[i, 2])
                    fixed = 1/(np.sqrt(s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent0 = -0.5*drift_vec[i, 0]**2/s_v2 + 0.5*(x0 * s_v2/sigma**2 + drift_vec[i, 0])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent1 = -0.5*drift_vec[i, 1]**2/s_v2 + 0.5*(x1 * s_v2/sigma**2 + drift_vec[i, 1])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent2 = -0.5*drift_vec[i, 2]**2/s_v2 + 0.5*(x2 * s_v2/sigma**2 + drift_vec[i, 2])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))
                    exponent3 = -0.5*drift_vec[i, 3]**2/s_v2 + 0.5*(x3 * s_v2/sigma**2 + drift_vec[i, 3])**2 / (s_v2 * (s_v2/sigma**2 * (tt[i] - eps) + 1))

                    integrand = fixed**4 * np.exp(exponent0 + exponent1 + exponent2 + exponent3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                    density = trapz_1d(integrand, eps) * (0.5/np.pi)**2
                    if density > 0.1**14:
                        log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(4) or (n_samples, 4))

The drift rates in each dimension

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Hyper-Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (4,) or (n_samples, 4)
        The drift rates in each dimension
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 3))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 4))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_HSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                      threshold_dynamic=self.threshold_dynamic, 
                                                      decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_HSDM_trial(threshold_function,
                                                                       drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                       s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2', 'response3'])

ProjectedHyperSphericalDiffusionModel(threshold_dynamic='fixed')

Projected Hyper-Spherical Diffusion Model

Source code in src/jeam/Models/HyperSpherical.py 
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def __init__(self, threshold_dynamic='fixed'):
    self.name = 'Projected Hyper-Spherical Diffusion Model'

    if threshold_dynamic in ['fixed', 'linear', 'exponential', 'hyperbolic', 'custom']:
        self.threshold_dynamic = threshold_dynamic
    else:
        raise ValueError("\'threshold_dynamic\' must be one of \'fixed\', \'linear\', \'exponential\', \'hyperbolic\', or \'custom\'. However, got \'{}\'".format(threshold_dynamic))

joint_lpdf(rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01)

Compute the joint log-probability density function of response time and choice angles for the Projected Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
rt (array - like, shape(n_samples))

The response times

 required
theta (array - like, shape(n_samples, 2))

The choice angles in spherical coordinates (theta1, theta2)

 required
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift rates in each dimension

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float

The decision threshold (default is 1)

 required
decay float

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
dt_threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the derivative of the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
approximation_step float

The time step for numerical estimation of first-passage time densities (default is 0.01)

 0.01

Returns:

Name Type Description
log_density (array - like, shape(n_samples))

The joint log-probability density of response time and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def joint_lpdf(self, rt, theta, drift_vec, ndt, threshold, decay=0, threshold_function=None, dt_threshold_function=None, s_v=0, s_t=0, sigma=1, approximation_step=0.01):
    '''
    Compute the joint log-probability density function of response time and choice angles for the Projected Hyper-Spherical Diffusion Model

    Parameters
    ----------
    rt : array-like, shape (n_samples,)
        The response times
    theta : array-like, shape (n_samples, 2)
        The choice angles in spherical coordinates (theta1, theta2)
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift rates in each dimension
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float
        The decision threshold (default is 1)
    decay : float, optional
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    dt_threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the derivative of the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    approximation_step : float, optional
        The time step for numerical estimation of first-passage time densities (default is 0.01)

    Returns
    -------
    log_density : array-like, shape (n_samples,)
        The joint log-probability density of response time and choice angles
    '''
    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((rt.shape[0], 3))

    if drift_vec.shape[1] != 3 or drift_vec.ndim != 2:
        raise ValueError("drift_vec must have shape (3,) or (n_samples, 3)")

    tt = np.maximum(rt - ndt, 0)

    # first-passage time density of zero drift process
    if self.threshold_dynamic == 'fixed':
        a = threshold
        s0 = 0.002
        s1 = 0.02
        if s_t == 0:
            s = tt/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(tt, threshold, sigma=sigma)
            fpt_st = 1/threshold**2 * hsdm_short_t_fpt_z(tt/threshold**2, 0.1**8/threshold**2)   
        else:
            T = np.arange(0, tt.max()+0.05, 0.05)
            s = T/threshold**2
            w = np.minimum(np.maximum((s - s0) / (s1 - s0), 0), 1)
            fpt_lt = hsdm_long_t_fpt_z(T, threshold, sigma=sigma)
            fpt_st = 1/threshold**2 * hsdm_short_t_fpt_z(T/threshold**2, 0.1**8/threshold**2)   
        fpt_z =  (1 - w) * fpt_st + w * fpt_lt
    elif self.threshold_dynamic == 'linear':
        a = threshold - decay*tt
        T_max = min(rt.max(), threshold/decay)
        g_z, T = ie_fpt_linear(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=T_max)
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'exponential':
        a = threshold * np.exp(-decay*tt)
        g_z, T = ie_fpt_exponential(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'hyperbolic':
        a = threshold / (1 + decay*tt)
        g_z, T = ie_fpt_hyperbolic(threshold, decay, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)
    elif self.threshold_dynamic == 'custom':
        threshold_function2 = lambda t: threshold_function(t)**2
        dt_threshold_function2 = lambda t: 2 * dt_threshold_function(t) * threshold_function(t)
        a = threshold_function(tt)
        g_z, T = ie_fpt_custom(threshold_function2, dt_threshold_function2, 4*sigma**2, 0.000001, sigma=2*sigma**2, dt=approximation_step, T_max=rt.max())
        fpt_z = np.interp(tt, T, g_z)

    fpt_z = np.maximum(fpt_z, 0.1**14)
    norm_mu = np.sqrt(drift_vec[:, 0]**2 + drift_vec[:, 1]**2 + drift_vec[:, 2]**2)

    theta1_mu = np.arctan2(drift_vec[:, 2], drift_vec[:, 0])
    theta2_mu = np.arctan2(drift_vec[:, 2], drift_vec[:, 1])

    # Girsanov:
    if s_v == 0:
        # No drift variability
        if s_t == 0:
            # No non-decision time variability
            x0 = np.cos(theta1_mu) * np.cos(theta[:, 0])
            x1 = np.sin(theta1_mu) * np.sin(theta[:, 0]) * np.cos(theta2_mu) * np.cos(theta[:, 1])
            term1 = np.exp(a * norm_mu * (x0 + x1) / sigma**2)
            term2 = iv(0, a * norm_mu * np.sin(theta1_mu) * np.sin(theta[:, 0]) * np.sin(theta2_mu) * np.sin(theta[:, 1])/sigma**2)
            term3 = -0.5 * norm_mu**2 * tt

            log_density = np.log(2*np.pi) + np.log(term1) + np.log(term2) + term3 + np.log(fpt_z)
        else:
            # With non-decision time variability
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    if self.threshold_dynamic == 'fixed':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * term1 * term2 * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'linear':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp((threshold - decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, (threshold - decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'exponential':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold*np.exp(-decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold*np.exp(-decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'hyperbolic':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold/(1  + decay * (tt[i] - eps)) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold/(1  + decay * (tt[i] - eps)) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    elif self.threshold_dynamic == 'custom':
                        x0 = np.cos(theta1_mu[i]) * np.cos(theta[i, 0])
                        x1 = np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.cos(theta2_mu[i]) * np.cos(theta[i, 1])
                        term1 = np.exp(threshold_function(tt[i] - eps) * norm_mu[i] * (x0 + x1) / sigma**2)
                        term2 = iv(0, threshold_function(tt[i] - eps) * norm_mu[i] * np.sin(theta1_mu[i]) * np.sin(theta[i, 0]) * np.sin(theta2_mu[i]) * np.sin(theta[i, 1])/sigma**2)
                        term3 = -0.5 * norm_mu[i]**2 * (tt[i] - eps)
                        integrand = term1 * term2 * np.exp(term3) * np.interp(tt[i]-eps, T, fpt_z)/s_t
                        density = 2*np.pi * trapz_1d(integrand, eps)
                    if density > 0.1**14:
                        log_density[i] = np.log(density)
    else:
        # With drift variability
        if s_t == 0:
            # No non-decision time variability
            s_v2 = s_v**2
            c1 = a * np.sin(theta[:, 0]) * np.sin(theta[:, 1]) * s_v2
            c2 = 2*s_v2 * (s_v2 * tt + 1)
            term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 2]/c2)
            term2 = 1/(s_v2 * tt + 1)**2
            p1 = (c1**2 + drift_vec[:, 2]**2)/c2
            p2 = (a * np.cos(theta[:, 0]) * s_v2 + drift_vec[:, 0])**2 / c2
            p3 = (a * np.sin(theta[:, 0]) * np.cos(theta[:, 1]) * s_v2 + drift_vec[:, 1])**2 / c2
            p4 = (norm_mu**2)/(2*s_v2)

            log_density = np.log(term1) + np.log(term2) + (p1 + p2 + p3 - p4) + np.log(fpt_z)
        else:
            log_density = np.log(0.1**14) * np.ones(rt.shape[0])
            eps = np.linspace(0, s_t, max(2, int(s_t//0.02)))
            s_v2 = s_v**2

            for i in range(rt.shape[0]):
                if tt[i] - s_t > 0:
                    c2 = 2*s_v2 * (s_v2 * (tt[i] - eps) + 1)
                    if self.threshold_dynamic == 'fixed':
                        c1 = threshold * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = (threshold * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = (threshold * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'linear':
                        c1 = (threshold - decay*(tt[i]-eps)) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold - decay*(tt[i]-eps)) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold - decay*(tt[i]-eps)) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'exponential':
                        c1 = (threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold * np.exp(-decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'hyperbolic':
                        c1 = (threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = ((threshold / (1 + decay*(tt[i]-eps))) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = ((threshold / (1 + decay*(tt[i]-eps))) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2
                    elif self.threshold_dynamic == 'custom':
                        c1 = threshold_function(tt[i]-eps) * np.sin(theta[i, 0]) * np.sin(theta[i, 1]) * s_v2
                        p2 = (threshold_function(tt[i]-eps) * np.cos(theta[i, 0]) * s_v2 + drift_vec[i, 0])**2 / c2
                        p3 = (threshold_function(tt[i]-eps) * np.sin(theta[i, 0]) * np.cos(theta[i, 1]) * s_v2 + drift_vec[i, 1])**2 / c2

                    term1 = 2*np.pi * iv(0, 2*c1 * drift_vec[:, 2]/c2)
                    term2 = 1/(s_v2 * (tt[i] - eps) + 1)**2
                    p4 = (norm_mu**2)/(2*s_v2)
                    term3 = np.exp(p1 + p2 + p3 - p4)
                    integrand = term1 * term2 * term3 * np.interp(tt[i]-eps, T, fpt_z)/s_t

                    density = trapz_1d(integrand, eps)

                    if density > 0.1**14:
                            log_density[i] = np.log(density)

    log_density[rt - ndt <= 0] = np.log(0.1**14)
    log_density = np.maximum(log_density, np.log(0.1**14))

    return log_density

simulate(drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1)

Simulate data from the Projected Hyper-Spherical Diffusion Model

Parameters:

Name Type Description Default
drift_vec (array - like, shape(3) or (n_samples, 3))

The drift vector [drift_x, drift_y, drift_z]. Note that drift_z must be non-negative, as it represents the projection onto the upper hemisphere.

 required
ndt float or array-like with the shape of (n_samples,)

The non-decision time

 required
threshold float or array-like with the shape of (n_samples,)

The decision threshold (default is 1)

 1
decay float or array-like with the shape of (n_samples,)

The threshold decay rate (default is 0)

 0
threshold_function callable, if threshold_dynamic is 'custom'

A function that takes time t and returns the threshold at time t

 None
s_v float

The standard deviation of drift variability (default is 0)

 0
s_t float

The standard deviation of non-decision time variability (default is 0)

 0
sigma float

The diffusion coefficient (default is 1)

 1
dt float

Time step for the simulation (default is 0.001)

 0.001
n_sample int

Number of samples to simulate (default is 1)

 1

Returns:

Type Description
DataFrame

A DataFrame containing simulated response times and choice angles
Source code in src/jeam/Models/HyperSpherical.py 
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def simulate(self, drift_vec, ndt, threshold=1, decay=0, threshold_function=None, s_v=0, s_t=0, sigma=1, dt=0.001, n_sample=1):
    '''
    Simulate data from the Projected Hyper-Spherical Diffusion Model

    Parameters
    ----------
    drift_vec : array-like, shape (3,) or (n_samples, 3)
        The drift vector [drift_x, drift_y, drift_z]. Note that drift_z must be non-negative, as it represents the projection onto the upper hemisphere.
    ndt : float or array-like with the shape of (n_samples,)
        The non-decision time
    threshold : float or array-like with the shape of (n_samples,)
        The decision threshold (default is 1)
    decay : float or array-like with the shape of (n_samples,)
        The threshold decay rate (default is 0)
    threshold_function : callable, if threshold_dynamic is 'custom'
        A function that takes time t and returns the threshold at time t
    s_v : float, optional
        The standard deviation of drift variability (default is 0)
    s_t : float, optional
        The standard deviation of non-decision time variability (default is 0)
    sigma : float, optional
        The diffusion coefficient (default is 1)
    dt : float, optional
        Time step for the simulation (default is 0.001)
    n_sample : int, optional
        Number of samples to simulate (default is 1)

    Returns
    -------
    pd.DataFrame
        A DataFrame containing simulated response times and choice angles
    '''
    RT = np.empty((n_sample,))
    Choice = np.empty((n_sample, 2))

    if drift_vec.ndim == 1:
        drift_vec = drift_vec * np.ones((n_sample, 3))
    elif drift_vec.shape[0] != n_sample:
        raise ValueError("Number of rows in drift_vec must be equal to n_sample")

    if isinstance(ndt, (float, np.floating)) or isinstance(ndt, (int, np.integer)):
        ndt = np.full((n_sample,), ndt)
    elif len(ndt) != n_sample:
        raise ValueError("Length of ndt must be equal to n_sample")

    if isinstance(threshold, (float, np.floating)) or isinstance(threshold, (int, np.integer)):
        threshold = np.full((n_sample,), threshold)
    elif len(threshold) != n_sample:
        raise ValueError("Length of threshold must be equal to n_sample")

    if isinstance(decay, (float, np.floating)) or isinstance(decay, (int, np.integer)):
        decay = np.full((n_sample,), decay)
    elif len(decay) != n_sample:
        raise ValueError("Length of decay must be equal to n_sample")

    if threshold_function is None and self.threshold_dynamic == 'custom':
        raise ValueError("threshold_function must be provided when threshold_dynamic is 'custom'")

    if threshold_function is not None and self.threshold_dynamic != 'custom':
        raise ValueError("threshold_function should be None when threshold_dynamic is not 'custom'")

    if s_v < 0:
        raise ValueError("s_v must be non-negative")
    if s_t < 0:
        raise ValueError("s_t must be non-negative")

    if self.threshold_dynamic != 'custom':
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_PHSDM_trial(threshold[n], drift_vec[n, :].astype(np.float64), ndt[n],
                                                       threshold_dynamic=self.threshold_dynamic, 
                                                       decay=decay[n], s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    else:
        for n in range(n_sample):
            RT[n], Choice[n, :] = simulate_custom_threshold_PHSDM_trial(threshold_function,
                                                                        drift_vec[n, :].astype(np.float64), ndt[n], 
                                                                        s_v=s_v, s_t=s_t, sigma=sigma, dt=dt)
    return pd.DataFrame(np.c_[RT, Choice], columns=['rt', 'response1', 'response2'])