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cauchy.py
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cauchy.py
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# Copyright 2018 The JAX Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import numpy as np
from jax import lax
from jax._src.lax.lax import _const as _lax_const
from jax._src.numpy.util import promote_args_inexact
from jax.numpy import arctan
from jax._src.typing import Array, ArrayLike
def logpdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy log probability distribution function.
JAX implementation of :obj:`scipy.stats.cauchy` ``logpdf``.
The Cauchy probability distribution function is
.. math::
f(x) = \frac{1}{\pi(1 + x^2)}
Args:
x: arraylike, value at which to evaluate the PDF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of logpdf values
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
x, loc, scale = promote_args_inexact("cauchy.logpdf", x, loc, scale)
pi = _lax_const(x, np.pi)
scaled_x = lax.div(lax.sub(x, loc), scale)
normalize_term = lax.log(lax.mul(pi, scale))
return lax.neg(lax.add(normalize_term, lax.log1p(lax.mul(scaled_x, scaled_x))))
def pdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy probability distribution function.
JAX implementation of :obj:`scipy.stats.cauchy` ``pdf``.
The Cauchy probability distribution function is
.. math::
f(x) = \frac{1}{\pi(1 + x^2)}
Args:
x: arraylike, value at which to evaluate the PDF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of pdf values
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
return lax.exp(logpdf(x, loc, scale))
def cdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy cumulative distribution function.
JAX implementation of :obj:`scipy.stats.cauchy` ``cdf``.
The cdf is defined as
.. math::
f_{cdf} = \int_{-\infty}^x f_{pdf}(y) \mathrm{d}y
where here :math:`f_{pdf}` is the Cauchy probability distribution function,
:func:`jax.scipy.stats.cauchy.pdf`.
Args:
x: arraylike, value at which to evaluate the CDF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of cdf values.
See Also:
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
x, loc, scale = promote_args_inexact("cauchy.cdf", x, loc, scale)
pi = _lax_const(x, np.pi)
scaled_x = lax.div(lax.sub(x, loc), scale)
return lax.add(_lax_const(x, 0.5), lax.mul(lax.div(_lax_const(x, 1.), pi), arctan(scaled_x)))
def logcdf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy log cumulative distribution function.
JAX implementation of :obj:`scipy.stats.cauchy` ``logcdf``
The cdf is defined as
.. math::
f_{cdf} = \int_{-\infty}^x f_{pdf}(y) \mathrm{d}y
where here :math:`f_{pdf}` is the Cauchy probability distribution function,
:func:`jax.scipy.stats.cauchy.pdf`.
Args:
x: arraylike, value at which to evaluate the CDF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of logcdf values.
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
return lax.log(cdf(x, loc, scale))
def sf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy distribution log survival function.
JAX implementation of :obj:`scipy.stats.cauchy` ``sf``.
The survival function is defined as
.. math::
f_{sf}(x) = 1 - f_{cdf}(x)
where :math:`f_{cdf}(x)` is the cumulative distribution function,
:func:`jax.scipy.stats.cauchy.cdf`.
Args:
x: arraylike, value at which to evaluate the SF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of sf values
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
x, loc, scale = promote_args_inexact("cauchy.sf", x, loc, scale)
return cdf(-x, -loc, scale)
def logsf(x: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy distribution log survival function.
JAX implementation of :obj:`scipy.stats.cauchy` ``logsf``
The survival function is defined as
.. math::
f_{sf}(x) = 1 - f_{cdf}(x)
where :math:`f_{cdf}(x)` is the cumulative distribution function,
:func:`jax.scipy.stats.cauchy.cdf`.
Args:
x: arraylike, value at which to evaluate the SF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of logsf values.
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.isf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
x, loc, scale = promote_args_inexact("cauchy.logsf", x, loc, scale)
return logcdf(-x, -loc, scale)
def isf(q: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy distribution inverse survival function.
JAX implementation of :obj:`scipy.stats.cauchy` ``isf``.
Returns the inverse of the survival function,
:func:`jax.scipy.stats.cauchy.sf`.
Args:
q: arraylike, value at which to evaluate the ISF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of isf values.
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.ppf`
"""
q, loc, scale = promote_args_inexact("cauchy.isf", q, loc, scale)
pi = _lax_const(q, np.pi)
half_pi = _lax_const(q, np.pi / 2)
unscaled = lax.tan(lax.sub(half_pi, lax.mul(pi, q)))
return lax.add(lax.mul(unscaled, scale), loc)
def ppf(q: ArrayLike, loc: ArrayLike = 0, scale: ArrayLike = 1) -> Array:
r"""Cauchy distribution percent point function.
JAX implementation of :obj:`scipy.stats.cauchy` ``ppf``.
The percent point function is defined as the inverse of the
cumulative distribution function, :func:`jax.scipy.stats.cauchy.cdf`.
Args:
q: arraylike, value at which to evaluate the PPF
loc: arraylike, distribution offset parameter
scale: arraylike, distribution scale parameter
Returns:
array of ppf values.
See Also:
- :func:`jax.scipy.stats.cauchy.cdf`
- :func:`jax.scipy.stats.cauchy.pdf`
- :func:`jax.scipy.stats.cauchy.sf`
- :func:`jax.scipy.stats.cauchy.logcdf`
- :func:`jax.scipy.stats.cauchy.logpdf`
- :func:`jax.scipy.stats.cauchy.logsf`
- :func:`jax.scipy.stats.cauchy.isf`
"""
q, loc, scale = promote_args_inexact("cauchy.ppf", q, loc, scale)
pi = _lax_const(q, np.pi)
half_pi = _lax_const(q, np.pi / 2)
unscaled = lax.tan(lax.sub(lax.mul(pi, q), half_pi))
return lax.add(lax.mul(unscaled, scale), loc)