# scipy-yli: Helpful SciPy utilities and recipes
# Copyright © 2022 Lee Yingtong Li (RunasSudo)
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Affero General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Affero General Public License for more details.
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# You should have received a copy of the GNU Affero General Public License
# along with this program. If not, see .
import numpy as np
from scipy import integrate, optimize, stats
from .config import config
from .utils import Interval
class betarat_gen(stats.rv_continuous):
"""
Ratio of 2 independent beta-distributed variables
This is a SciPy *stats.rv_continuous* distribution which takes 4 parameters, *a1*, *b1*, *a2* and *b2*,
and gives the distribution of Beta(*a1*, *b1*) / Beta(*a2*, *b2*).
**References:**
1. Pham-Gia T. Distributions of the ratios of independent beta variables and applications. *Communications in Statistics: Theory and Methods*. 2000;29(12):2693–715. `doi: 10.1080/03610920008832632 `_
2. Weekend Editor. On the ratio of Beta-distributed random variables. *Some Weekend Reading*. 2021 Sep 13. https://www.someweekendreading.blog/beta-ratios/
"""
# TODO: _rvs based on stats.beta
def from_scipy(self, beta1, beta2):
"""
Construct a new *beta_ratio* distribution from two SciPy beta distributions
:param beta1: Distribution for the numerator of the ratio
:type beta1: frozen SciPy stats.beta
:param beta2: Distribution for the denominator of the ratio
:type beta2: frozen SciPy stats.beta
:rtype: Frozen *beta_ratio* distribution
"""
return self(*beta1.args, *beta2.args)
def _do_vectorized(self, func, x, a1, b1, a2, b2):
"""Helper function to call the implementation over potentially multiple values"""
x = np.atleast_1d(x)
result = np.zeros(x.size)
for i, (x_, a1_, b1_, a2_, b2_) in enumerate(zip(x, np.pad(a1, x.size, 'edge'), np.pad(b1, x.size, 'edge'), np.pad(a2, x.size, 'edge'), np.pad(b2, x.size, 'edge'))):
result[i] = func(x_, a1_, b1_, a2_, b2_)
return result
def _pdf_one(self, w, a1, b1, a2, b2):
"""PDF for the distribution, given by Pham-Gia"""
from mpmath import beta, hyp2f1, power
if w <= 0:
return 0
elif w < 1:
term1 = beta(a1 + a2, b2) / (beta(a1, b1) * beta(a2, b2))
term2 = power(w, a1 - 1)
term3 = hyp2f1(a1 + a2, 1 - b1, a1 + a2 + b2, w)
else:
term1 = beta(a1 + a2, b1) / (beta(a1, b1) * beta(a2, b2))
term2 = 1 / power(w, a2 + 1)
term3 = hyp2f1(a1 + a2, 1 - b2, a1 + a2 + b1, 1/w)
return float(term1 * term2 * term3)
def _pdf(self, w, a1, b1, a2, b2):
return self._do_vectorized(self._pdf_one, w, a1, b1, a2, b2)
def _cdf_one(self, w, a1, b1, a2, b2):
"""PDF for the distribution, given by Pham-Gia"""
from mpmath import beta, hyper, power
if w <= 0:
return 0
elif w < 1:
term1 = beta(a1 + a2, b2) / (beta(a1, b1) * beta(a2, b2))
term2 = power(w, a1) / a1
term3 = hyper([a1, a1 + a2, 1 - b1], [a1 + 1, a1 + a2 + b2], w)
return float(term1 * term2 * term3)
else:
term1 = beta(a1 + a2, b1) / (beta(a1, b1) * beta(a2, b2))
term2 = 1 / (a2 * power(w, a2))
term3 = hyper([a2, a1 + a2, 1 - b2], [a2 + 1, a1 + a2 + b1], 1/w)
return 1 - float(term1 * term2 * term3)
def _cdf(self, w, a1, b1, a2, b2):
return self._do_vectorized(self._cdf_one, w, a1, b1, a2, b2)
def _munp_one(self, k, a1, b1, a2, b2):
"""Moments of the distribution, given by Weekend Editor"""
from mpmath import rf
term1 = rf(a1, k) * rf(a2 + b2 - k, k)
term2 = rf(a1 + b1, k) * rf(a2 - k, k)
return float(term1 / term2)
def _munp(self, k, a1, b1, a2, b2):
return self._do_vectorized(self._munp_one, k, a1, b1, a2, b2)
beta_ratio = betarat_gen(name='beta_ratio', a=0) # a = lower bound of support
class betaoddsrat_gen(stats.rv_continuous):
"""
Ratio of the odds of 2 independent beta-distributed variables
This is a SciPy *stats.rv_continuous* distribution which takes 4 parameters, *a1*, *b1*, *a2* and *b2*,
and gives the distribution of (*X*/(1-*X*)) / (*Y*/(1-*Y*)), where *X* ~ Beta(*a1*, *b1*), and *Y* ~ Beta(*a2*, *b2*).
**Reference:** Hora SC, Kelley GD. Bayesian inference on the odds and risk ratios. *Communications in Statistics: Theory and Methods*. 1983;12(6):725–38. `doi: 10.1080/03610928308828491 `_
"""
# TODO: _rvs based on stats.beta
def from_scipy(self, beta1, beta2):
"""
Construct a new *beta_oddsratio* distribution from two SciPy beta distributions
:param beta1: Distribution for the numerator of the ratio
:type beta1: frozen SciPy stats.beta
:param beta2: Distribution for the denominator of the ratio
:type beta2: frozen SciPy stats.beta
:rtype: Frozen *beta_oddsratio* distribution
"""
return self(*beta1.args, *beta2.args)
def set_cdf_terms(self, cdf_terms):
"""
Set the number of terms to use when calculating the CDF
If *cdf_terms* = *np.inf* (default), the CDF will be computed by numerical integration of the PDF.
Otherwise, the CDF will be computed by truncating the infinite sum given by Hora & Kelley to the specified number of terms.
"""
BETA_ODDSRATIO_CDF_TERMS[0] = cdf_terms
def _do_vectorized(self, func, x, a1, b1, a2, b2):
"""Helper function to call the implementation over potentially multiple values"""
x = np.atleast_1d(x)
result = np.zeros(x.size)
for i, (x_, a1_, b1_, a2_, b2_) in enumerate(zip(x, np.pad(np.atleast_1d(a1), x.size, 'edge'), np.pad(np.atleast_1d(b1), x.size, 'edge'), np.pad(np.atleast_1d(a2), x.size, 'edge'), np.pad(np.atleast_1d(b2), x.size, 'edge'))):
result[i] = func(x_, a1_, b1_, a2_, b2_)
return result
def _pdf_one(self, w, a1, b1, a2, b2):
"""PDF for the distribution, given by Hora & Kelley"""
from mpmath import beta, hyp2f1, power
if w <= 0:
return 0
elif w < 1:
term1 = beta(a1 + a2, b1 + b2) / (beta(a1, b1) * beta(a2, b2))
term2 = power(w, a1 - 1)
term3 = hyp2f1(a1 + b1, a1 + a2, a1 + a2 + b1 + b2, 1 - w)
else:
term1 = beta(a1 + a2, b1 + b2) / (beta(a1, b1) * beta(a2, b2))
term2 = power(w, -a2 - 1)
term3 = hyp2f1(a2 + b2, a1 + a2, a1 + a2 + b1 + b2, 1 - power(w, -1))
return float(term1 * term2 * term3)
def _pdf(self, w, a1, b1, a2, b2):
return self._do_vectorized(self._pdf_one, w, a1, b1, a2, b2)
def _cdf_one_infsum(self, w, a1, b1, a2, b2):
"""CDF for the distribution, by truncating infinite sum given by Hora & Kelly"""
from mpmath import beta, betainc, factorial, power, rf
if w <= 0:
return 0
elif w < 1:
k = beta(a1 + a2, b1 + b2) / (beta(a1, b1) * beta(a2, b2))
inf_sum = 0
for j in range(0, BETA_ODDSRATIO_CDF_TERMS[0]):
kj1 = rf(a1 + b1, j) * rf(a1 + a2, j)
kj2 = rf(a1 + a2 + b1 + b2, j) * factorial(j)
inf_sum += kj1/kj2 * betainc(a1, j + 1, 0, w)
return k * inf_sum
else:
k = beta(a1 + a2, b1 + b2) / (beta(a1, b1) * beta(a2, b2))
inf_sum = 0
for j in range(0, BETA_ODDSRATIO_CDF_TERMS[0]):
kj1 = rf(a2 + b2, j) * rf(a1 + a2, j)
kj2 = rf(a1 + a2 + b1 + b2, j) * factorial(j)
inf_sum += kj1/kj2 * betainc(a2, j + 1, 0, power(w, -1))
return 1 - k * inf_sum
def _cdf(self, w, a1, b1, a2, b2):
"""CDF for the distribution"""
w = np.atleast_1d(w)
a1 = np.atleast_1d(a1)
b1 = np.atleast_1d(b1)
a2 = np.atleast_1d(a2)
b2 = np.atleast_1d(b2)
if not ((a1 == a1[0]).all() and (b1 == b1[0]).all() and (a2 == a2[0]).all() and (b2 == b2[0]).all()):
raise ValueError('Cannot compute CDF from different distributions')
if BETA_ODDSRATIO_CDF_TERMS[0] == np.inf:
# Exact solution requested
if w.size == 1:
if w <= 0:
return 0
# Just compute an integral
if w < self.mean(a1, b1, a2, b2):
# Integrate normally
return integrate.quad(lambda x: self._pdf_one(x, a1[0], b1[0], a2[0], b2[0]), 0, w)[0]
else:
# Integrate on the distribution of 1/w (much faster)
return 1 - integrate.quad(lambda x: self._pdf_one(x, a2[0], b2[0], a1[0], b1[0]), 0, 1/w)[0]
else:
# Multiple points requested: use ODE solver
solution = integrate.solve_ivp(lambda x, _: self._pdf_one(x, a1[0], b1[0], a2[0], b2[0]), (0, w.max()), [0], t_eval=w, method='LSODA', rtol=1.5e-8, atol=1.5e-8)
return solution.y
else:
# Truncate infinite sum
return self._do_vectorized(self._cdf_one_infsum, w, a1, b1, a2, b2)
def _ppf_one(self, p, a1, b1, a2, b2):
"""PPF for the distribution, using Newton's method"""
# Default SciPy implementation uses Brent's method
# This one is a bit faster
if p <= 0:
return 0
# Initial guess based on log-normal approximation
mean = self.mean(a1, b1, a2, b2)
se_log = self.std(a1, b1, a2, b2)/mean
initial_guess = np.exp(np.log(mean) + stats.norm.ppf(p) * se_log)
try:
return optimize.newton(lambda x: self.cdf(x, a1, b1, a2, b2) - p, x0=initial_guess, fprime=lambda x: self.pdf(x, a1, b1, a2, b2))
except RuntimeError:
# Failed to converge with Newton's method :(
# Use Brent's method instead
return super()._ppf(p, a1, b1, a2, b2)
def _ppf(self, p, a1, b1, a2, b2):
return self._do_vectorized(self._ppf_one, p, a1, b1, a2, b2)
def _munp_one(self, k, a1, b1, a2, b2):
"""Moments of the distribution, given by Hora & Kelley"""
from mpmath import rf
term1 = rf(a1, k) * rf(b2, k)
term2 = rf(b1 - k, k) * rf(a2 - k, k)
return float(term1 / term2)
def _munp(self, k, a1, b1, a2, b2):
return self._do_vectorized(self._munp_one, k, a1, b1, a2, b2)
# See beta_oddsratio.cdf
BETA_ODDSRATIO_CDF_TERMS = [100]
beta_oddsratio = betaoddsrat_gen(name='beta_oddsratio', a=0) # a = lower bound of support
class transformed_gen(stats.rv_continuous):
"""
Represents a transformation, *Y*, of a "base" random variable, *X*
This is a SciPy *stats.rv_continuous* distribution which takes parameters as described below.
The transformation is *f*\ (*Y*) = *X*.
Hence *Y*.pdf\ (*x*) = *X*.pdf\ (*f*\ (*x*)) \* *f*'\ (*x*).
For example, if *X* is a model parameter, then ``transformed_dist(X, f=np.exp, fprime=np.exp, finv=np.log)`` is the distribution of the log-parameter.
**Parameters:**
* **base** (*frozen SciPy distribution*) – Distribution of the base random variable
* **f** (*callable*) – Function *f* representing the transformation, which takes values of *Y* to values of *X*
* **fprime** (*callable*) – Derivative of the function *f*
* **finv** (*callable*) – Inverse of the function *f*, which takes values of *X* to values of *Y*
"""
def _pdf(self, x, base, f, fprime, finv):
return base.pdf(np.vectorize(f)(x)) * np.vectorize(fprime)(x)
def _cdf(self, x, base, f, fprime, finv):
return base.cdf(np.vectorize(f)(x))
def _ppf(self, p, base, f, fprime, finv):
return np.vectorize(finv)(base.ppf(p))
# Call implementations directly to bypass SciPy args checking, which chokes on "base" parameter
def pdf(self, x, base, f, fprime, finv):
if np.isscalar(x):
return np.atleast_1d(self._pdf(x, base, f, fprime, finv))[0]
return self._pdf(x, base, f, fprime, finv)
def cdf(self, x, base, f, fprime, finv):
if np.isscalar(x):
return np.atleast_1d(self._cdf(x, base, f, fprime, finv))[0]
return self._cdf(x, base, f, fprime, finv)
def ppf(self, x, base, f, fprime, finv):
if np.isscalar(x):
return np.atleast_1d(self._ppf(x, base, f, fprime, finv))[0]
return self._ppf(x, base, f, fprime, finv)
transformed_dist = transformed_gen(name='transformed')
def hdi(distribution, level=None):
"""
Get the highest density interval for the distribution, e.g. for a Bayesian posterior, the highest posterior density interval (HPD/HDI)
:param distribution: Distribution to compute the interval for
:type distribution: frozen SciPy distribution
:param level: Coverage/confidence probability, default (*None*) is 1 − *config.alpha*
:type level: float
:rtype: :class:`yli.utils.Interval`
"""
if level is None:
level = 1 - config.alpha
# For a given lower limit, we can compute the corresponding level*100% interval
def interval_width(lower):
upper = distribution.ppf(distribution.cdf(lower) + level)
return upper - lower
# Find such interval which has the smallest width
# Use equal-tailed interval as initial guess
initial_guess = distribution.ppf((1-level)/2)
optimize_result = optimize.minimize(interval_width, initial_guess) # TODO: Allow customising parameters
lower_limit = optimize_result.x[0]
width = optimize_result.fun
upper_limit = lower_limit + width
return Interval(lower_limit, upper_limit)