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from typing import Union, Iterable
import numpy as np
import scipy.stats
import xarray as xr
def weibull(*,
intensity: Union[float, Iterable[float]],
threshold: Union[float, Iterable[float]],
slope: Union[float, Iterable[float]] = 3.5,
lower_asymptote: Union[float, Iterable[float]] = 0.01,
lapse_rate: Union[float, Iterable[float]] = 0.01,
scale: str = 'log10') -> xr.DataArray:
"""
A Weibull psychometric function.
Parameters
----------
intensity
Stimulus values on the abscissa, :math:`x`.
threshold
The threshold parameter, :math:`\\alpha`.
slope
The slope parameter, :math:`\\beta`.
lower_asymptote
The lower asymptote, :math:`\\gamma`, which is equivalent to the
false-alarm rate in a yes-no task, or :math:`\\frac{1}{n}` in an
:math:`n`-AFC task.
lapse_rate
The lapse rate, :math:`\\delta`. The upper asymptote of the psychometric
function will be :math:`1-\\delta`.
scale
The scale of the stimulus parameters. Possible values are ``log10``,
``dB``, and ``linear``.
Returns
-------
p
The psychometric function evaluated at the specified intensities for
all parameters combinations.
Notes
-----
An appropriate parametrization of the function is chosen based on the
`scale` keyword argument. Specifically, the following parametrizations
are used:
scale='linear'
:math:`p = 1 - \delta - (1 - \gamma - \delta)\\, e^{-\\left (\\frac{x}{t} \\right )^\\beta}`
scale='log10'
:math:`p = 1 - \delta - (1 - \gamma - \delta)\\, e^{-10^{\\beta (x - t)}}`
scale='dB':
:math:`p = 1 - \delta - (1 - \gamma - \delta)\\, e^{-10^{\\frac{\\beta}{20} (x - t)}}`
"""
intensity = np.atleast_1d(intensity)
threshold = np.atleast_1d(threshold)
slope = np.atleast_1d(slope)
lower_asymptote = np.atleast_1d(lower_asymptote)
lapse_rate = np.atleast_1d(lapse_rate)
# Implementation using NumPy. Leave it here for reference.
#
# x, t, beta, gamma, delta = np.meshgrid(intensity,
# threshold,
# slope,
# lower_asymptote,
# lapse_rate,
# indexing='ij', sparse=True)
x = xr.DataArray(data=intensity, dims=['intensity'],
coords=dict(intensity=intensity))
t = xr.DataArray(data=threshold, dims=['threshold'],
coords=dict(threshold=threshold))
beta = xr.DataArray(data=slope, dims=['slope'],
coords=dict(slope=slope))
gamma = xr.DataArray(data=lower_asymptote, dims=['lower_asymptote'],
coords=dict(lower_asymptote=lower_asymptote))
delta = xr.DataArray(data=lapse_rate, dims=['lapse_rate'],
coords=dict(lapse_rate=lapse_rate))
assert np.atleast_1d(x.squeeze()).shape == np.atleast_1d(intensity).shape
assert np.atleast_1d(t.squeeze()).shape == np.atleast_1d(threshold).shape
assert np.atleast_1d(beta.squeeze()).shape == np.atleast_1d(slope).shape
assert np.atleast_1d(gamma.squeeze()).shape == np.atleast_1d(lower_asymptote).shape
assert np.atleast_1d(delta.squeeze()).shape == np.atleast_1d(lapse_rate).shape
if scale == 'linear':
p = 1 - delta - (1 - gamma - delta) * np.exp(-(x / t)**beta)
elif scale == 'log10':
p = 1 - delta - (1 - gamma - delta) * np.exp(-10 ** (beta * (x - t)))
elif scale == 'dB':
p = 1 - delta - (1 - gamma - delta) * np.exp(-10 ** (beta * (x - t) / 20))
else:
raise ValueError('Invalid scale specified.')
return p
def csf(*,
contrast: Union[float, Iterable[float]],
spatial_freq: Union[float, Iterable[float]],
temporal_freq: Union[float, Iterable[float]],
c0: Union[float, Iterable[float]],
cf: Union[float, Iterable[float]],
cw: Union[float, Iterable[float]],
min_thresh: Union[float, Iterable[float]],
slope: Union[float, Iterable[float]] = 3.5,
lower_asymptote: Union[float, Iterable[float]] = 0.01,
lapse_rate: Union[float, Iterable[float]] = 0.01,
scale: str = 'log10') -> np.ndarray:
"""
The spatio-temporal contrast sensitivity function.
Parameters
----------
contrast
spatial_freq
temporal_freq
c0
cf
cw
min_thresh
slope
lower_asymptote
lapse_rate
scale
Returns
-------
"""
contrast = np.atleast_1d(contrast)
spatial_freq = np.atleast_1d(spatial_freq)
temporal_freq = np.atleast_1d(temporal_freq)
c0 = np.atleast_1d(c0)
cf = np.atleast_1d(cf)
cw = np.atleast_1d(cw)
min_thresh = np.atleast_1d(min_thresh)
slope = np.atleast_1d(slope)
lower_asymptote = np.atleast_1d(lower_asymptote)
lapse_rate = np.atleast_1d(lapse_rate)
# Implementation using NumPy. Leave it here for reference.
#
# c, f, w, c0_, cf_, cw_, t, beta, gamma, delta = np.meshgrid(
# contrast, spatial_freq, temporal_freq, c0, cf, cw, min_thresh,
# slope, lower_asymptote, lapse_rate,
# indexing='ij', sparse=True)
x = xr.DataArray(data=contrast, dims=['contrast'],
coords=dict(contrast=contrast))
f = xr.DataArray(data=spatial_freq, dims=['spatial_freq'],
coords=dict(spatial_freq=spatial_freq))
w = xr.DataArray(data=temporal_freq, dims=['temporal_freq'],
coords=dict(temporal_freq=temporal_freq))
c0_ = xr.DataArray(data=c0, dims=['c0'],
coords=dict(c0=c0))
cf_ = xr.DataArray(data=cf, dims=['cf'],
coords=dict(cf=cf))
cw_ = xr.DataArray(data=cw, dims=['cw'],
coords=dict(cw=cw))
min_t = xr.DataArray(data=min_thresh, dims=['min_thresh'],
coords=dict(min_thresh=min_thresh))
beta = xr.DataArray(data=slope, dims=['slope'],
coords=dict(slope=slope))
gamma = xr.DataArray(data=lower_asymptote, dims=['lower_asymptote'],
coords=dict(lower_asymptote=lower_asymptote))
delta = xr.DataArray(data=lapse_rate, dims=['lapse_rate'],
coords=dict(lapse_rate=lapse_rate))
t = np.maximum(min_t, c0_ + cf_ * f + cw_ * w)
# p = weibull(intensity=contrast,
# threshold=threshold,
# slope=slope,
# lower_asymptote=lower_asymptote,
# lapse_rate=lapse_rate,
# scale=scale)
if scale == 'linear':
p = 1 - delta - (1 - gamma - delta) * np.exp(-(x / t)**beta)
elif scale == 'log10':
p = 1 - delta - (1 - gamma - delta) * np.exp(-10 ** (beta * (x - t)))
elif scale == 'dB':
p = 1 - delta - (1 - gamma - delta) * np.exp(-10 ** (beta * (x - t) / 20))
else:
raise ValueError('Invalid scale specified.')
return p
def norm_cdf(*,
intensity: Union[float, Iterable[float]],
mean: Union[float, Iterable[float]],
sd: Union[float, Iterable[float]],
lower_asymptote: Union[float, Iterable[float]] = 0.01,
lapse_rate: Union[float, Iterable[float]] = 0.01,
scale: str = 'linear'):
"""
The cumulate normal distribution.
Parameters
----------
intensity
mean
sd
lower_asymptote
lapse_rate
scale
Returns
-------
"""
if scale != 'linear':
msg = ('Currently, only linear stimulus scaling is supported for this '
'psychometric function.')
raise ValueError(msg)
intensity = np.atleast_1d(intensity)
mean = np.atleast_1d(mean)
sd = np.atleast_1d(sd)
lower_asymptote = np.atleast_1d(lower_asymptote)
lapse_rate = np.atleast_1d(lapse_rate)
x = xr.DataArray(data=intensity, dims=['intensity'],
coords=dict(intensity=intensity))
mu = xr.DataArray(data=mean, dims=['mean'],
coords=dict(mean=mean))
sd_ = xr.DataArray(data=sd, dims=['sd'],
coords=dict(sd=sd))
gamma = xr.DataArray(data=lower_asymptote, dims=['lower_asymptote'],
coords=dict(lower_asymptote=lower_asymptote))
delta = xr.DataArray(data=lapse_rate, dims=['lapse_rate'],
coords=dict(lapse_rate=lapse_rate))
# x, mu, sd_, delta = np.meshgrid(intensity,
# mean,
# sd,
# lapse_rate,
# indexing='ij', sparse=True)
#
# assert np.atleast_1d(intensity.squeeze()).shape == np.atleast_1d(intensity).shape
# assert np.atleast_1d(x.squeeze()).shape == np.atleast_1d(intensity).shape
# assert np.atleast_1d(sd_.squeeze()).shape == np.atleast_1d(sd).shape
# assert np.atleast_1d(delta.squeeze()).shape == np.atleast_1d(lapse_rate).shape
# p = delta + (1 - 2*delta) * scipy.stats.norm.cdf(x, mu, sd_)
def _mu_func(x, mu, sd_, gamma, delta):
norm = scipy.stats.norm(loc=mu, scale=sd_)
return delta + (1 - gamma - delta) * norm.cdf(x)
p = xr.apply_ufunc(_mu_func, x, mu, sd_, gamma, delta)
return p
def norm_cdf_2(*,
intensity: Union[float, Iterable[float]],
mean: Union[float, Iterable[float]],
sd: Union[float, Iterable[float]],
lapse_rate: Union[float, Iterable[float]] = 0.01,
scale: str = 'linear'):
"""
The cumulative normal distribution with lapse rate equal to lower
asymptote.
Parameters
----------
intensity
mean
sd
lapse_rate
scale
Returns
-------
"""
if scale != 'linear':
msg = ('Currently, only linear stimulus scaling is supported for this '
'psychometric function.')
raise ValueError(msg)
intensity = np.atleast_1d(intensity)
mean = np.atleast_1d(mean)
sd = np.atleast_1d(sd)
lapse_rate = np.atleast_1d(lapse_rate)
x = xr.DataArray(data=intensity, dims=['intensity'],
coords=dict(intensity=intensity))
mu = xr.DataArray(data=mean, dims=['mean'],
coords=dict(mean=mean))
sd_ = xr.DataArray(data=sd, dims=['sd'],
coords=dict(sd=sd))
delta = xr.DataArray(data=lapse_rate, dims=['lapse_rate'],
coords=dict(lapse_rate=lapse_rate))
def _mu_func(x, mu, sd_, delta):
norm = scipy.stats.norm(loc=mu, scale=sd_)
return delta + (1 - 2*delta) * norm.cdf(x)
p = xr.apply_ufunc(_mu_func, x, mu, sd_, delta)
return p
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