"""Created on Aug 03 17:13:21 2024"""
__all__ = ['_beta_masking', '_pdf_scaling', '_remove_nans',
'arc_sine_pdf_',
'beta_pdf_', 'beta_cdf_',
'chi_square_pdf_', 'chi_square_cdf_',
'exponential_pdf_', 'exponential_cdf_',
'folded_normal_pdf_', 'folded_normal_cdf_',
'gamma_sr_pdf_', 'gamma_sr_cdf_',
'gamma_ss_pdf_',
'gaussian_pdf_', 'gaussian_cdf_',
'half_normal_pdf_', 'half_normal_cdf_',
'laplace_pdf_', 'laplace_cdf_',
'log_normal_pdf_', 'log_normal_cdf_',
'skew_normal_pdf_', 'skew_normal_cdf_',
'uniform_pdf_', 'uniform_cdf_']
from typing import Union
import numpy as np
from custom_inherit import doc_inherit
from scipy.special import betainc, erf, gamma, gammainc, gammaln, owens_t
from .. import doc_style
[docs]
def arc_sine_pdf_(x: np.ndarray,
amplitude: float = 1.0,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF of :class:`~pymultifit.distributions.arcSine_d.ArcSineDistribution`.
Parameters
----------
x : np.ndarray
Input array of values where PDF is evaluated.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
loc : float, optional
The location parameter specifying the lower bound of the distribution.
Defaults to 0.0.
scale : float, optional
The scale parameter, specifying the width of the distribution.
Defaults to 1.0.
normalize : bool, optional
If True, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated PDF values.
Notes
-----
The ArcSine PDF is defined as:
.. math:: f(y) = \frac{1}{\pi \sqrt{y(1-y)}}
where, :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = \frac{x - \text{loc}}{\text{scale}}
The final PDF is expressed as :math:`f(y)/\text{scale}`.
"""
return beta_pdf_(x=x, amplitude=amplitude, alpha=0.5, beta=0.5, loc=loc, scale=scale, normalize=normalize)
[docs]
def beta_pdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, beta: float = 1.0,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF of :class:`~pymultifit.distributions.beta_d.BetaDistribution`.
Parameters
----------
x : np.ndarray
Input array of values where PDF is evaluated.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
alpha : float, optional
The :math:`\alpha` parameter.
Default is 1.0.
beta : float, optional
The :math:`\beta` parameter.
Default is 1.0.
loc : float, optional
The location parameter, for shifting.
Default is 0.0.
scale : float, optional
The scale parameter, for scaling.
Default is 1.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as `x`, containing the evaluated PDF values.
Notes
-----
The Beta PDF is defined as:
.. math:: f(y; \alpha, \beta) = \frac{y^{\alpha - 1} (1 - y)^{\beta - 1}}{B(\alpha, \beta)}
where :math:`B(\alpha, \beta)` is the Beta function (see, :obj:`scipy.special.beta`), and :math:`y` is the transformed value of :math:`x` such that:
.. math:: y = \frac{x - \text{loc}}{\text{scale}}
The final PDF is expressed as :math:`f(y)/\text{scale}`.
"""
if x.size == 0:
return np.array([])
y = (x - loc) / scale
pdf_ = np.zeros_like(a=y, dtype=float)
numerator = y**(alpha - 1) * (1 - y)**(beta - 1)
normalization_factor = gamma(alpha) * gamma(beta) / gamma(alpha + beta)
mask_ = _beta_masking(y=y, alpha=alpha, beta=beta)
pdf_[~mask_] = numerator[~mask_] / normalization_factor
if alpha <= 1:
pdf_[y == 0] = np.inf
if beta <= 1:
pdf_[y == 1] = np.inf
if alpha == 1 and beta == 1:
pdf_[y == 1] = 1
# handle the cases where nans can occur with nan_to_num
# np.inf and -np.inf to not affect the infinite values
pdf_ = _remove_nans(pdf_ / scale)
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return pdf_
@doc_inherit(parent=beta_pdf_, style=doc_style)
def beta_logpdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, beta: float = 1.0,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
r"""Compute logPDF for :class:`~pymultifit.distributions.beta_d.BetaDistribution`."""
if x.size == 0:
return np.array([])
y = (x - loc) / scale
mask_ = _beta_masking(y=y, alpha=alpha, beta=beta)
logpdf_ = np.full_like(a=y, fill_value=-np.inf, dtype=float)
log_numerator = (alpha - 1) * np.log(y) + (beta - 1) * np.log(1 - y)
if normalize:
log_normalization = gammaln(alpha) + gammaln(beta) - gammaln(alpha + beta)
amplitude = 1.0
else:
log_normalization = 0.0
logpdf_[~mask_] = np.log(amplitude) + log_numerator[~mask_] - log_normalization
if alpha <= 1:
logpdf_[y == 0] = np.inf
if beta <= 1:
logpdf_[y == 1] = np.inf
return logpdf_ - np.log(scale)
[docs]
@doc_inherit(parent=beta_pdf_, style=doc_style)
def beta_cdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, beta: float = 1.0,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for :class:`~pymultifit.distributions.beta_d.BetaDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
For API consistency only.
normalize : bool, optional
For API consistency only.
Notes
-----
The Beta CDF is defined as:
.. math:: I_x(\alpha, \beta)
where :math:`I_x(\alpha, \beta)` is the regularized incomplete Beta function, see :obj:`~scipy.special.betainc`.
"""
if x.size == 0:
return np.array([])
y = (x - loc) / scale
cdf_ = np.zeros_like(a=y, dtype=float)
mask_ = np.logical_and(y > 0, y < 1)
cdf_[mask_] = betainc(alpha, beta, y[mask_])
cdf_[y >= 1] = 1
return cdf_
[docs]
def chi_square_pdf_(x: np.ndarray,
amplitude: float = 1.0, degree_of_freedom: Union[int, float] = 1,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :mod:`~pymultifit.distributions.chiSquare_d.ChiSquareDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
degree_of_freedom : int, optional
The degrees of freedom parameter.
Defaults to 1.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
scale: float, optional
The scale parameter, for scaling.
Defaults to 1.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The ChiSquare PDF is defined as:
.. math:: f(y\ |\ k) = \dfrac{y^{(k/2) - 1} e^{-y/2}}{2^{k/2} \Gamma(k/2)}
where :math:`\Gamma(k)` is the :obj:`~scipy.special.gamma` function,
and :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = x - \text{loc}
The final PDF is expressed as :math:`f(y)`.
"""
y = (x - loc) / scale
pdf_ = np.zeros_like(x, dtype=float)
mask_ = y > 0
pdf_[mask_] = gamma_sr_pdf_(y[mask_], amplitude=amplitude, alpha=degree_of_freedom / 2, lambda_=0.5, loc=0,
normalize=normalize)
return pdf_ / scale
@doc_inherit(parent=chi_square_pdf_, style=doc_style)
def chi_square_cdf_(x: np.ndarray,
amplitude: float = 1.0, degree_of_freedom: Union[int, float] = 1,
loc: float = 0.0, scale: float = 1.0, normalize: bool = False) -> np.ndarray:
"""
Compute PDF for :mod:`~pymultifit.distributions.chiSquare_d.ChiSquareDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: bool, optional
For API consistency only.
Notes
-----
The ChiSquare CDF is defined as:
"""
y = (x - loc) / scale
cdf_ = np.zeros_like(x, dtype=float)
mask_ = y >= 0
cdf_[mask_] = gammainc(degree_of_freedom / 2, y[mask_] / 2)
return cdf_
[docs]
def exponential_pdf_(x: np.ndarray,
amplitude: float = 1., lambda_: float = 1., loc: float = 0.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.exponential_d.ExponentialDistribution`.
.. note::
This function uses :func:`~pymultifit.distributions.utilities_d.gamma_sr_pdf_` to calculate the PDF with
:math:`\alpha = 1` and :math:`\lambda_\text{gammaSR} = \lambda_\text{expon}`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
lambda_ : float, optional
The scale parameter, :math:`\lambda`.
Defaults to 1.0.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The Exponential PDF is defined as:
.. math::
f(y, \lambda) =
\begin{cases}
\lambda \exp\left[-\lambda y\right] &; y \geq 0, \\
0 &; y < 0.
\end{cases}
where, :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = x - \text{loc}
The final PDF is expressed as :math:`f(y)`.
"""
return gamma_sr_pdf_(x, amplitude=amplitude, alpha=1., lambda_=lambda_, loc=loc, normalize=normalize)
[docs]
@doc_inherit(parent=exponential_pdf_, style=doc_style)
def exponential_cdf_(x: np.ndarray,
amplitude: float = 1., scale: float = 1., loc: float = 0.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute CDF of :class:`~pymultifit.distributions.exponential_d.ExponentialDistribution`.
.. note::
This function uses :func:`~pymultifit.distributions.utilities_d.gamma_sr_cdf_` to calculate the CDF with
:math:`\alpha = 1` and :math:`\lambda_\text{gammaSR} = \lambda_\text{expon}`.
Parameters
----------
amplitude : float, optional
For API consistency only.
normalize : bool, optional
For API consistency only.
Notes
-----
The Exponential CDF is defined as:
.. math:: F(x) = 1 - \exp\left[-\lambda x\right].
"""
y = x - loc
pdf_ = np.zeros_like(a=y, dtype=float)
mask_ = y > 0
pdf_[mask_] = 1 - np.exp(-scale * y[mask_])
return pdf_
[docs]
def folded_normal_pdf_(x: np.ndarray,
amplitude: float = 1., mean: float = 0.0, sigma: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.foldedNormal_d.FoldedNormalDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
mean : float, optional
The mean parameter, :math:`\mu`.
Defaults to 0.0.
sigma : float, optional
The standard deviation parameter, :math:`\sigma`.
Defaults to 1.0.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The FoldedNormal PDF is defined as:
.. math:: f(y\ |\ \mu, \sigma) = \phi(y\ |\ \mu, 1) + \phi(y\ |\ -\mu, 1),
where :math:`\phi` is the PDF of :class:`~pymultifit.distributions.gaussian_d.GaussianDistribution`,
and :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = \dfrac{x - \text{loc}}{\sigma}
The final PDF is expressed as :math:`f(y)/\sigma`.
"""
if x.size == 0:
return np.array([])
_, pdf_ = _folded(x=x, mean=mean, sigma=sigma, loc=loc, g_func=gaussian_pdf_)
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return pdf_ / sigma
def _folded(x, mean, sigma, loc, g_func):
if sigma <= 0 or mean < 0:
return np.full(shape=x.size, fill_value=np.nan)
y = (x - loc) / sigma
temp_ = np.zeros_like(a=x, dtype=float)
mask = y >= 0
g1 = g_func(x=y[mask], mean=mean, normalize=True)
g2 = g_func(x=y[mask], mean=-mean, normalize=True)
temp_[mask] = g1 + g2
return mask, temp_
[docs]
@doc_inherit(parent=folded_normal_pdf_, style=doc_style)
def folded_normal_cdf_(x: np.ndarray,
amplitude: float = 1., mean: float = 0.0, sigma: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for :class:`~pymultifit.distributions.foldedNormal_d.FoldedNormalDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: bool, optional
For API consistency only.
Notes
-----
The FoldedNormal CDF is defined as:
.. math::
F(y) = \Phi(y\ | \mu, 1) + \Phi(y\ | -\mu, 1) - 1
where :math:`\Phi` is the CDF of :class:`~pymultifit.distributions.gaussian_d.GaussianDistribution`,
and :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = \dfrac{x - \text{loc}}{\sigma}
"""
if x.size == 0:
return np.array([])
mask_, cdf_ = _folded(x=x, mean=mean, sigma=sigma, loc=loc, g_func=gaussian_cdf_)
cdf_[mask_] -= 1
return cdf_
[docs]
def gamma_sr_pdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, lambda_: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.gamma_d.GammaDistributionSR` with :math:`\alpha` (shape)
and :math:`\lambda` (rate) parameters.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
alpha : float, optional
The shape parameter, :math:`\alpha`.
Defaults to 1.0.
lambda_ : float, optional
The rate parameter, :math:`\lambda`.
Defaults to 1.0.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated PDF values.
Notes
-----
The Gamma SR PDF is defined as:
.. math::
f(y; \alpha, \lambda) =
\begin{cases}
\dfrac{\lambda^\alpha}{\Gamma(\alpha)} y^{\alpha - 1} \exp\left[-\lambda y\right], & y > \text{loc}, \\
0, & y \leq \text{loc}.
\end{cases}
where :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = x - \text{loc}
The final PDF is expressed as :math:`f(y)`.
"""
if x.size == 0:
return np.array([])
y = x - loc
numerator = y**(alpha - 1) * np.exp(-lambda_ * y)
normalization_factor = gamma(alpha) / lambda_**alpha
pdf_ = numerator / normalization_factor
pdf_[x < loc] = 0
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return pdf_
[docs]
@doc_inherit(parent=gamma_sr_pdf_, style=doc_style)
def gamma_sr_cdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, lambda_: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for :class:`~pymultifit.distributions.gamma_d.GammaDistributionSR` with :math:`\alpha`
and :math:`\lambda` parameters.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: float, optional
For API consistency only.
Notes
-----
The Gamma CDF is defined as:
.. math::
F(x) = \dfrac{1}{\Gamma(\alpha)}\gamma(\alpha, \lambda x)
where, :math:`\gamma(\alpha, \lambda x)` is the lower incomplete gamma function, see :obj:`~scipy.special.gammainc`.
"""
if x.size == 0:
return np.array([])
y = x - loc
y = np.maximum(y, 0)
return gammainc(alpha, lambda_ * y)
[docs]
def gamma_ss_pdf_(x: np.ndarray,
amplitude: float = 1.0, alpha: float = 1.0, theta: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.gamma_d.GammaDistributionSS` with :math:`\alpha` (shape)
and :math:`\theta` (scale) parameters.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
alpha : float, optional
The shape parameter, :math:`\alpha`.
Defaults to 1.0.
theta : float, optional
The scale parameter, :math:`\lambda`.
Defaults to 1.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The Gamma SS PDF is defined as:
.. math::
f(y\ |\ \alpha, \theta) = \dfrac{1}{\Gamma(\alpha)\theta^\alpha}y^{\alpha-1}\exp\left[-\dfrac{y}{\theta}\right]
where :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = x - \text{loc}
The final PDF is expressed as :math:`f(y)`.
"""
return gamma_sr_pdf_(x=x, amplitude=amplitude, alpha=alpha, lambda_=1 / theta, normalize=normalize)
@doc_inherit(parent=gamma_ss_pdf_, style=doc_style)
def gamma_ss_cdf_(x: np.ndarray,
amplitude: float = 1., alpha: float = 1.0, theta: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for :class:`~pymultifit.distributions.gamma_d.GammaDistributionSS` with :math:`\alpha` (shape)
and :math:`\theta` (scale) parameters.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: bool, optional
For API consistency only.
Notes
-----
The Gamma CDF is defined as:
.. math:: F(y) = \dfrac{1}{\Gamma(\alpha)}\gamma\left(\alpha, \dfrac{y}{\theta}\right)
where, :math:`\gamma(\alpha, \lambda y)` is the lower incomplete gamma function, see :obj:`~scipy.special.gammainc`.
"""
return gamma_sr_cdf_(x=x, amplitude=amplitude, alpha=alpha, lambda_=1 / theta, normalize=normalize)
[docs]
def gaussian_pdf_(x: np.ndarray,
amplitude: float = 1.0, mean: float = 0.0, std: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for the :mod:`~pymultifit.distributions.gaussian_d.GaussianDistribution`
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
mean : float, optional
The mean parameter, :math:`\mu`.
Defaults to 0.0.
std : float, optional
The standard deviation parameter, :math:`\sigma`.
Defaults to 1.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The Gaussian PDF is defined as:
.. math::
f(x; \mu, \sigma) = \phi\left(\dfrac{x-\mu}{\sigma}\right) =
\dfrac{1}{\sqrt{2\pi\sigma}}\exp\left[-\dfrac{1}{2}\left(\dfrac{x-\mu}{\sigma}\right)^2\right]
The final PDF is expressed as :math:`f(x)`.
"""
if x.size == 0:
return np.array([])
exponent_factor = (x - mean)**2 / (2 * std**2)
exponent_factor = np.exp(-exponent_factor)
normalization_factor = std * np.sqrt(2 * np.pi)
pdf_ = exponent_factor / normalization_factor
if not normalize:
pdf_ = pdf_ / np.max(pdf_)
pdf_ *= amplitude
return pdf_
[docs]
@doc_inherit(parent=gaussian_pdf_, style=doc_style)
def gaussian_cdf_(x: np.ndarray,
amplitude: float = 1.0, mean: float = 0.0, std: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for the :mod:`~pymultifit.distributions.gaussian_d.GaussianDistribution`
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: float, optional
For API consistency only.
Notes
-----
The Gaussian CDF is defined as:
.. math::
F(x) = \Phi\left(\dfrac{x-\mu}{\sigma}\right) =
\dfrac{1}{2} \left[1 + \text{erf}\left(\dfrac{x - \mu}{\sigma\sqrt{2}}\right)\right]
"""
if x.size == 0:
return np.array([])
num_ = x - mean
den_ = std * np.sqrt(2)
return 0.5 * (1 + erf(num_ / den_))
[docs]
def half_normal_pdf_(x: np.ndarray,
amplitude: float = 1.0, sigma: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for the :class:`~pymultifit.distributions.halfNormal_d.HalfNormalDistribution`.
.. note::
The :class:`~pymultifit.distributions.halfNormal_d.HalfNormalDistribution`. is a special case of the
:class:`~pymultifit.distributions.foldedNormal_d.FoldedNormalDistribution` with :math:`\mu = 0`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF. Defaults to 1.0.
Ignored if **normalize** is ``True``.
sigma : float, optional
The standard deviation :math:`\sigma`.
Defaults to 1.0.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The HalfNormal PDF is defined as:
.. math::
f(x\ |\ \sigma) = \sqrt{\dfrac{2}{\pi\sigma^2}}\exp\left[-\dfrac{x^2}{2\sigma^2}\right]
where :math:`x >= 0`.
"""
return folded_normal_pdf_(x, amplitude=amplitude, mean=0, sigma=sigma, loc=loc, normalize=normalize)
[docs]
@doc_inherit(parent=half_normal_pdf_, style=doc_style)
def half_normal_cdf_(x: np.ndarray,
amplitude: float = 1.0, scale: float = 1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute the CDF for :class:`~pymultifit.distributions.halfNormal_d.HalfNormalDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: float, optional
For API consistency only.
Notes
-----
The HalfNormal CDF is defined as:
.. math:: F(x) = \text{erf}\left( \frac{x}{\sqrt{2\sigma^2}}\right)
"""
return folded_normal_cdf_(x=x, amplitude=amplitude, normalize=normalize)
[docs]
def laplace_pdf_(x: np.ndarray,
amplitude: float = 1., mean: float = 0., diversity: float = 1.,
normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.laplace_d.LaplaceDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF.
Defaults to 1.0.
Ignored if **normalize** is ``True``.
mean : float, optional
The mean parameter, :math:`\mu`.
Defaults to 0.0.
diversity : float, optional
The diversity parameter, :math:`b`.
Defaults to 1.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The Laplace PDF is defined as:
.. math:: f(x\ |\ \mu, b) = \dfrac{1}{2b}\exp\left(-\dfrac{|x - \mu|}{b}\right)
The final PDF is expressed as :math:`f(x)`.
"""
if x.size == 0:
return np.array([])
exponent_factor = abs(x - mean) / diversity
exponent_factor = np.exp(-exponent_factor)
normalization_factor = 2 * diversity
pdf_ = exponent_factor / normalization_factor
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return pdf_
[docs]
@doc_inherit(parent=laplace_pdf_, style=doc_style)
def laplace_cdf_(x: np.ndarray,
amplitude: float = 1.0, mean: float = 0.0, diversity: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute CDF for :class:`~pymultifit.distributions.laplace_d.LaplaceDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: bool, optional
For API consistency only.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The Laplace CDF is defined as:
.. math:: F(x) =
\begin{cases}
\dfrac{1}{2}\exp\left(\dfrac{x-\mu}{b}\right) &,&x\leq\mu\\
1 - \dfrac{1}{2}\exp\left(-\dfrac{x-\mu}{b}\right) &,&x\geq\mu
\end{cases}
"""
if x.size == 0:
return np.array([])
def _cdf1(x_):
return 0.5 * np.exp((x_ - mean) / diversity)
def _cdf2(x_):
return 1 - 0.5 * np.exp(-(x_ - mean) / diversity)
# to ensure equality with scipy, had to break down the output with empty array so that sorting is not needed.
result = np.zeros_like(a=x, dtype=np.float64)
mask_leq = x <= mean
result[mask_leq] += _cdf1(x[mask_leq])
result[~mask_leq] += _cdf2(x[~mask_leq])
return result
[docs]
def log_normal_pdf_(x: np.ndarray,
amplitude: float = 1., mean: float = 0., std: float = 1.,
loc: float = 0., normalize: bool = False) -> np.ndarray:
r"""
Compute PDF for :class:`~pymultifit.distributions.logNormal_d.LogNormalDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF.
Defaults to 1.0.
Ignored if **normalize** is ``True``.
mean : float, optional
The mean parameter, :math:`\mu`.
Defaults to 0.0.
std : float, optional
The standard deviation parameter, :math:`\sigma`.
Defaults to 1.0.
loc : float, optional
The location parameter, for shifting.
Defaults to 0.0.
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The LogNormal PDF is defined as:
.. math::
f(y\ |\ \mu, \sigma) = \dfrac{1}{\sigma y\sqrt{2\pi}}\exp\left(-\dfrac{(\ln y - \mu)^2}{2\sigma^2}\right)
where, :math:`y` is the transformed value of :math:`x`, defined as:
.. math::
y = x - \text{loc}
The final PDF is expressed as :math:`f(y)`.
"""
if x.size == 0:
return np.array([])
y = x - loc
exponent_factor = (np.log(y) - mean)**2 / (2 * std**2)
exponent_factor = np.exp(-exponent_factor)
normalization_factor = std * y * np.sqrt(2 * np.pi)
pdf_ = exponent_factor / normalization_factor
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return _remove_nans(pdf_)
[docs]
@doc_inherit(parent=log_normal_pdf_, style=doc_style)
def log_normal_cdf_(x: np.ndarray,
amplitude: float = 1.0, mean: float = 0.0, std=1.0,
loc: float = 0.0, normalize: bool = False) -> np.ndarray:
r"""
Compute CDF of :class:`~pymultifit.distributions.logNormal_d.LogNormalDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: bool, optional
For API consistency only
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The LogNormal CDF is defined as:
.. math::
F(x) = \Phi\left(\dfrac{\ln x - \mu}{\sigma}\right)
"""
return _remove_nans(gaussian_cdf_(x=np.log(x - loc), mean=mean, std=std))
[docs]
def skew_normal_pdf_(x: np.ndarray,
amplitude: float = 1.0, shape: float = 0.0, loc: float = 0.0, scale: float = 1.0,
normalize: bool = False) -> np.ndarray:
r"""
Compute PDF of :class:`~pymultifit.distributions.skewNormal_d.SkewNormalDistribution`.
Parameters
----------
x : np.ndarray
Input array of values.
amplitude : float, optional
The amplitude of the PDF.
Defaults to 1.0.
Ignored if **normalize** is ``True``.
shape : float, optional
The shape parameter, :math:`\alpha`.
Defaults to 0.0.
loc : float, optional
The location parameter, :math:`\xi`.
Defaults to 0.0.
scale: float, optional
The scale parameter, :math:`\omega`
Defaults to 1.0,
normalize : bool, optional
If ``True``, the distribution is normalized so that the total area under the PDF equals 1.
Defaults to ``False``.
Returns
-------
np.ndarray
Array of the same shape as :math:`x`, containing the evaluated values.
Notes
-----
The SkewNormal PDF is defined as:
.. math:: f(y\ |\ \alpha, \xi, \omega) =
2\phi(y)\Phi(\alpha y)
where, :math:`\phi(y)` and :math:`\Phi(\alpha y)` are the :class:`~pymultifit.distributions.gaussian_d.GaussianDistribution`
PDF and CDF defined at :math:`y` and :math:`\alpha y` respectively. Additionally, :math:`y` is the transformed value of :math:`x`, defined as:
.. math:: y = \dfrac{x - \xi}{\omega}
The final PDF is expressed as :math:`f(y)/\omega`.
"""
if x.size == 0:
return np.array([])
y = (x - loc) / scale
g_pdf_ = gaussian_pdf_(x=y, normalize=True)
g_cdf_ = gaussian_cdf_(x=shape * y, normalize=True)
pdf_ = (2 / scale) * g_pdf_ * g_cdf_
if not normalize:
pdf_ = _pdf_scaling(pdf_=pdf_, amplitude=amplitude)
return _remove_nans(pdf_)
[docs]
@doc_inherit(parent=skew_normal_pdf_, style=doc_style)
def skew_normal_cdf_(x: np.ndarray,
amplitude: float = 1.0, shape: float = 0.0, loc: float = 0.0, scale: float = 1.0,
normalize: bool = False):
r"""
Compute CDF of :class:`~pymultifit.distributions.skewNormal_d.SkewNormalDistribution`.
Parameters
----------
amplitude: float, optional
For API consistency only.
normalize: float, optional
For API consistency only.
Notes
------
The SkewNormal CDF is defined as:
.. math:: F(x) = \Phi\left(\dfrac{x - \xi}{\omega}\right) - 2T\left(\dfrac{x - \xi}{\omega}, \alpha\right)
where, :math:`T` is the Owen's T function, see :obj:`scipy.special.owens_t`, and
:math:`\Phi(\cdot)` is the :class:`~pymultifit.distributions.gaussian_d.GaussianDistribution` CDF function.
"""
if x.size == 0:
return np.array([])
y = (x - loc) / scale
return gaussian_cdf_(x=y, normalize=True) - 2 * owens_t(y, shape)
[docs]
def _beta_masking(y: np.ndarray, alpha: float, beta: float) -> np.ndarray:
"""
Creates a mask for beta distributions to identify out-of-range or undefined values.
Parameters
----------
y : np.ndarray
Array of values to check, typically in the range [0, 1].
alpha : float
Alpha parameter of the beta distribution. Determines the shape of the distribution.
beta : float
Beta parameter of the beta distribution. Determines the shape of the distribution.
Returns
-------
np.ndarray
A boolean mask array where `True` indicates out-of-range or undefined values.
"""
out_of_range_mask = np.logical_or(y < 0, y > 1)
undefined_mask = np.zeros_like(a=y, dtype=bool)
if alpha <= 1:
undefined_mask = np.logical_or(undefined_mask, y == 0)
if beta <= 1:
undefined_mask = np.logical_or(undefined_mask, y == 1)
mask_ = np.logical_or(out_of_range_mask, undefined_mask)
return mask_
[docs]
def _pdf_scaling(pdf_: np.ndarray, amplitude: float) -> np.ndarray:
"""
Scales a probability density function (PDF) by a given amplitude.
Parameters
----------
pdf_ : np.ndarray
The input PDF array to be scaled.
amplitude : float
The amplitude to scale the PDF.
Returns
-------
np.ndarray
The scaled PDF array.
"""
return amplitude * (pdf_ / np.max(pdf_))
[docs]
def _remove_nans(x: np.ndarray) -> np.ndarray:
"""
Replaces NaN, positive infinity, and negative infinity values in an array.
Parameters
----------
x : np.ndarray
Input array that may contain NaN, positive infinity, or negative infinity values.
Returns
-------
np.ndarray
Array with NaN replaced by 0, positive infinity replaced by `np.inf`, and negative infinity replaced by `-np.inf`.
"""
return np.nan_to_num(x=x, copy=False, nan=0, posinf=np.inf, neginf=-np.inf)