Implement beta_oddsratio

tests/test_beta_oddsratio.py

@@ -0,0 +1,94 @@
+#   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.
+#
+#   You should have received a copy of the GNU Affero General Public License
+#   along with this program.  If not, see <https://www.gnu.org/licenses/>.
+
+import numpy as np
+from scipy import stats
+
+import yli
+
+def test_beta_ratio_cdf_vs_empirical():
+	"""Compare beta_ratio.cdf with empirical distribution"""
+	
+	# Define the example beta distribution
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_oddsratio.from_scipy(beta1, beta2)
+	
+	# Compute empirical distribution
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = (samples_p1 / (1 - samples_p1)) / (samples_p2 / (1 - samples_p2))
+	
+	# Values to check
+	x = np.linspace(0, 2, 10)
+	y1 = dist.cdf(x)
+	y2 = [(sample < xx).sum()/sample.size for xx in x]
+	
+	# Allow 0.01 tolerance
+	assert (np.abs(y1 - y2) < 0.01).all()
+
+def test_beta_ratio_ppf_vs_empirical():
+	"""Compare beta_ratio.ppf with empirical distribution"""
+	
+	# Define the example beta distribution
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_oddsratio.from_scipy(beta1, beta2)
+	
+	# Compute empirical distribution
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = (samples_p1 / (1 - samples_p1)) / (samples_p2 / (1 - samples_p2))
+	
+	# Values to check
+	x = np.linspace(0, 0.9, 10)
+	y1 = dist.ppf(x)
+	y2 = np.quantile(sample, x)
+	
+	# Allow 0.01 tolerance
+	assert (np.abs(y1 - y2) < 0.01).all()
+
+def test_beta_ratio_mean_vs_empirical():
+	"""Compare beta_ratio.mean (vs _munp) with empirical mean"""
+	
+	# Define the example beta distribution
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_oddsratio.from_scipy(beta1, beta2)
+	
+	# Compute empirical mean
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = (samples_p1 / (1 - samples_p1)) / (samples_p2 / (1 - samples_p2))
+	
+	# Allow 0.01 tolerance
+	assert np.abs(dist.mean() - sample.mean()) < 0.01
+
+def test_beta_ratio_var_vs_empirical():
+	"""Compare beta_ratio.var (vs _munp) with empirical variance"""
+	
+	# Define the example beta distribution
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_oddsratio.from_scipy(beta1, beta2)
+	
+	# Compute empirical mean
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = (samples_p1 / (1 - samples_p1)) / (samples_p2 / (1 - samples_p2))
+	
+	# Allow 0.01 tolerance
+	assert np.abs(dist.var() - sample.var()) < 0.01

tests/test_beta_ratio.py

@@ -15,6 +15,7 @@
 #   along with this program.  If not, see <https://www.gnu.org/licenses/>.
 
 import numpy as np
+from scipy import stats
 
 import yli
 
@@ -30,7 +31,7 @@ def test_beta_ratio_vs_jsaffer_pdf():
 	y = dist.pdf(x)
 	
 	# Compare with expected values from jsaffer implementation
-	expected = np.load('beta_ratio_vs_jsaffer.npy', allow_pickle=False)[0]
+	expected = np.load('tests/beta_ratio_vs_jsaffer.npy', allow_pickle=False)[0]
 	
 	# Allow 1e-10 tolerance
 	diff = np.abs(y - expected)
@@ -48,7 +49,7 @@ def test_beta_ratio_vs_jsaffer_cdf():
 	y = dist.cdf(x)
 	
 	# Compare with expected values from jsaffer implementation
-	expected = np.load('beta_ratio_vs_jsaffer.npy', allow_pickle=False)[1]
+	expected = np.load('tests/beta_ratio_vs_jsaffer.npy', allow_pickle=False)[1]
 	
 	# Allow 1e-10 tolerance
 	diff = np.abs(y - expected)
@@ -65,16 +66,36 @@ def _gen_beta_ratio_vs_jsaffer():
 	y1 = np.vectorize(lambda w: float(beta_quotient_distribution.pdf_bb_ratio(a1, a2, b1, b2, w)))(x)
 	y2 = np.vectorize(lambda w: float(beta_quotient_distribution.cdf_bb_ratio(a1, a2, b1, b2, w)))(x)
 	
-	np.save('beta_ratio_vs_jsaffer.npy', np.array([y1, y2]), allow_pickle=False)
+	np.save('tests/beta_ratio_vs_jsaffer.npy', np.array([y1, y2]), allow_pickle=False)
 
 def test_beta_ratio_mean_vs_empirical():
 	"""Compare beta_ratio.mean (via beta_ratio._munp) with empirical mean"""
 	
 	# Define the example beta distribution
-	dist = yli.beta_ratio(3, 6, 12, 7)
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_ratio.from_scipy(beta1, beta2)
 	
-	# Compute empirical mean
-	sample = dist.rvs(1000, random_state=31415)
+	# Compute empirical distribution
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = samples_p1 / samples_p2
 	
 	# Allow 0.01 tolerance
 	assert np.abs(dist.mean() - sample.mean()) < 0.01
+
+def test_beta_ratio_var_vs_empirical():
+	"""Compare beta_ratio.var (via beta_ratio._munp) with empirical variance"""
+	
+	# Define the example beta distribution
+	beta1 = stats.beta(3, 6)
+	beta2 = stats.beta(12, 7)
+	dist = yli.beta_ratio.from_scipy(beta1, beta2)
+	
+	# Compute empirical distribution
+	samples_p1 = beta1.rvs(10_000, random_state=31415)
+	samples_p2 = beta2.rvs(10_000, random_state=92653)
+	sample = samples_p1 / samples_p2
+	
+	# Allow 0.01 tolerance
+	assert np.abs(dist.var() - sample.var()) < 0.01

yli/distributions.py

@@ -16,7 +16,7 @@
 
 import mpmath
 import numpy as np
-from scipy import optimize, stats
+from scipy import integrate, optimize, stats
 
 import collections
 
@@ -30,6 +30,8 @@ class betarat_gen(stats.rv_continuous):
 	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"""
 		return self(*beta1.args, *beta2.args)
@@ -84,14 +86,121 @@ class betarat_gen(stats.rv_continuous):
 	def _munp_one(self, k, a1, b1, a2, b2):
 		"""Moments of the distribution, given by Weekend Editor"""
 		
-		term1 = mpmath.rf(a1, k) / mpmath.rf(a1 + b1, k)
-		term2 = mpmath.rf(a2 + b2 - k, k) / mpmath.rf(a2 - k, k)
-		return float(term1 * term2)
+		term1 = mpmath.rf(a1, k) * mpmath.rf(a2 + b2 - k, k)
+		term2 = mpmath.rf(a1 + b1, k) * mpmath.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
+	Ratio of (X/(1-X)) / (Y/(1-Y)), where X ~ Beta(a1, b1), 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_ratio distribution from two SciPy beta distributions"""
+		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(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"""
+		
+		if w <= 0:
+			return 0
+		elif w < 1:
+			term1 = mpmath.beta(a1 + a2, b1 + b2) / (mpmath.beta(a1, b1) * mpmath.beta(a2, b2))
+			term2 = mpmath.power(w, a1 - 1)
+			term3 = mpmath.hyp2f1(a1 + b1, a1 + a2, a1 + a2 + b1 + b2, 1 - w)
+		else:
+			term1 = mpmath.beta(a1 + a2, b1 + b2) / (mpmath.beta(a1, b1) * mpmath.beta(a2, b2))
+			term2 = mpmath.power(w, -a2 - 1)
+			term3 = mpmath.hyp2f1(a2 + b2, a1 + a2, a1 + a2 + b1 + b2, 1 - mpmath.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(self, w, a1, b1, a2, b2):
+		"""CDF for the distribution, computed by integrating the PDF"""
+		
+		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 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
+	
+	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"""
+		
+		term1 = mpmath.rf(a1, k) * mpmath.rf(b2, k)
+		term2 = mpmath.rf(b1 - k, k) * mpmath.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)
+beta_oddsratio = betaoddsrat_gen(name='beta_oddsratio', a=0)  # a = lower bound of support
 
 ConfidenceInterval = collections.namedtuple('ConfidenceInterval', ['lower', 'upper'])