Source code for GPy.util.univariate_Gaussian

# Copyright (c) 2012, 2013 Ricardo Andrade
# Copyright (c) 2015 James Hensman
# Licensed under the BSD 3-clause license (see LICENSE.txt)

import numpy as np
from scipy.special import ndtr as std_norm_cdf

#define a standard normal pdf
_sqrt_2pi = np.sqrt(2*np.pi)
[docs]def std_norm_pdf(x): x = np.clip(x,-1e150,1e150) return np.exp(-np.square(x)/2)/_sqrt_2pi
[docs]def inv_std_norm_cdf(x): """ Inverse cumulative standard Gaussian distribution Based on Winitzki, S. (2008) """ z = 2*x -1 ln1z2 = np.log(1-z**2) a = 8*(np.pi -3)/(3*np.pi*(4-np.pi)) b = 2/(np.pi * a) + ln1z2/2 inv_erf = np.sign(z) * np.sqrt( np.sqrt(b**2 - ln1z2/a) - b ) return np.sqrt(2) * inv_erf
[docs]def logPdfNormal(z): """ Robust implementations of log pdf of a standard normal. @see [[https://github.com/mseeger/apbsint/blob/master/src/eptools/potentials/SpecfunServices.h original implementation]] in C from Matthias Seeger. """ return -0.5 * (M_LN2PI + z * z)
[docs]def cdfNormal(z): """ Robust implementations of cdf of a standard normal. @see [[https://github.com/mseeger/apbsint/blob/master/src/eptools/potentials/SpecfunServices.h original implementation]] in C from Matthias Seeger. */ """ if (abs(z) < ERF_CODY_LIMIT1): # Phi(z) approx (1+y R_3(y^2))/2, y=z/sqrt(2) return 0.5 * (1.0 + (z / M_SQRT2) * _erfRationalHelperR3(0.5 * z * z)) elif (z < 0.0): # Phi(z) approx N(z)Q(-z)/(-z), z<0 return np.exp(logPdfNormal(z)) * _erfRationalHelper(-z) / (-z) else: return 1.0 - np.exp(logPdfNormal(z)) * _erfRationalHelper(z) / z
[docs]def logCdfNormal(z): """ Robust implementations of log cdf of a standard normal. @see [[https://github.com/mseeger/apbsint/blob/master/src/eptools/potentials/SpecfunServices.h original implementation]] in C from Matthias Seeger. """ if (abs(z) < ERF_CODY_LIMIT1): # Phi(z) approx (1+y R_3(y^2))/2, y=z/sqrt(2) return np.log1p((z / M_SQRT2) * _erfRationalHelperR3(0.5 * z * z)) - M_LN2 elif (z < 0.0): # Phi(z) approx N(z)Q(-z)/(-z), z<0 return logPdfNormal(z) - np.log(-z) + np.log(_erfRationalHelper(-z)) else: return np.log1p(-(np.exp(logPdfNormal(z))) * _erfRationalHelper(z) / z)
[docs]def derivLogCdfNormal(z): """ Robust implementations of derivative of the log cdf of a standard normal. @see [[https://github.com/mseeger/apbsint/blob/master/src/eptools/potentials/SpecfunServices.h original implementation]] in C from Matthias Seeger. """ if (abs(z) < ERF_CODY_LIMIT1): # Phi(z) approx (1 + y R_3(y^2))/2, y = z/sqrt(2) return 2.0 * np.exp(logPdfNormal(z)) / (1.0 + (z / M_SQRT2) * _erfRationalHelperR3(0.5 * z * z)) elif (z < 0.0): # Phi(z) approx N(z) Q(-z)/(-z), z<0 return -z / _erfRationalHelper(-z) else: t = np.exp(logPdfNormal(z)) return t / (1.0 - t * _erfRationalHelper(z) / z)
def _erfRationalHelper(x): assert x > 0.0, "Arg of erfRationalHelper should be >0.0; was {}".format(x) if (x >= ERF_CODY_LIMIT2): """ x/sqrt(2) >= 4 Q(x) = 1 + sqrt(pi) y R_1(y), R_1(y) = poly(p_j,y) / poly(q_j,y), where y = 2/(x*x) Ordering of arrays: 4,3,2,1,0,5 (only for numerator p_j; q_5=1) ATTENTION: The p_j are negative of the entries here p (see P1_ERF) q (see Q1_ERF) """ y = 2.0 / (x * x) res = y * P1_ERF[5] den = y i = 0 while (i <= 3): res = (res + P1_ERF[i]) * y den = (den + Q1_ERF[i]) * y i += 1 # Minus, because p(j) values have to be negated return 1.0 - M_SQRTPI * y * (res + P1_ERF[4]) / (den + Q1_ERF[4]) else: """ x/sqrt(2) < 4, x/sqrt(2) >= 0.469 Q(x) = sqrt(pi) y R_2(y), R_2(y) = poly(p_j,y) / poly(q_j,y), y = x/sqrt(2) Ordering of arrays: 7,6,5,4,3,2,1,0,8 (only p_8; q_8=1) p (see P2_ERF) q (see Q2_ERF """ y = x / M_SQRT2 res = y * P2_ERF[8] den = y i = 0 while (i <= 6): res = (res + P2_ERF[i]) * y den = (den + Q2_ERF[i]) * y i += 1 return M_SQRTPI * y * (res + P2_ERF[7]) / (den + Q2_ERF[7]) def _erfRationalHelperR3(y): assert y >= 0.0, "Arg of erfRationalHelperR3 should be >=0.0; was {}".format(y) nom = y * P3_ERF[4] den = y i = 0 while (i <= 2): nom = (nom + P3_ERF[i]) * y den = (den + Q3_ERF[i]) * y i += 1 return (nom + P3_ERF[3]) / (den + Q3_ERF[3]) ERF_CODY_LIMIT1 = 0.6629 ERF_CODY_LIMIT2 = 5.6569 M_LN2PI = 1.83787706640934533908193770913 M_LN2 = 0.69314718055994530941723212146 M_SQRTPI = 1.77245385090551602729816748334 M_SQRT2 = 1.41421356237309504880168872421 #weights for the erfHelpers (defined here to avoid redefinitions at every call) P1_ERF = [ 3.05326634961232344e-1, 3.60344899949804439e-1, 1.25781726111229246e-1, 1.60837851487422766e-2, 6.58749161529837803e-4, 1.63153871373020978e-2] Q1_ERF = [ 2.56852019228982242e+0, 1.87295284992346047e+0, 5.27905102951428412e-1, 6.05183413124413191e-2, 2.33520497626869185e-3] P2_ERF = [ 5.64188496988670089e-1, 8.88314979438837594e+0, 6.61191906371416295e+1, 2.98635138197400131e+2, 8.81952221241769090e+2, 1.71204761263407058e+3, 2.05107837782607147e+3, 1.23033935479799725e+3, 2.15311535474403846e-8] Q2_ERF = [ 1.57449261107098347e+1, 1.17693950891312499e+2, 5.37181101862009858e+2, 1.62138957456669019e+3, 3.29079923573345963e+3, 4.36261909014324716e+3, 3.43936767414372164e+3, 1.23033935480374942e+3] P3_ERF = [ 3.16112374387056560e+0, 1.13864154151050156e+2, 3.77485237685302021e+2, 3.20937758913846947e+3, 1.85777706184603153e-1] Q3_ERF = [ 2.36012909523441209e+1, 2.44024637934444173e+2, 1.28261652607737228e+3, 2.84423683343917062e+3]