/usr/lib/python3/dist-packages/csb/statistics/scalemixture.py is in python3-csb 1.2.2+dfsg-2ubuntu1.
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Approximation of a distribution as a mixture of gaussians with a zero mean but different sigmas
"""
import numpy.random
import csb.core
import csb.statistics.ars
import csb.statistics.rand
from csb.core import typedproperty
from abc import abstractmethod, ABCMeta
from csb.numeric import log, exp, approx_psi, inv_psi, d_approx_psi
from scipy.special import psi, kve
from csb.statistics import harmonic_mean, geometric_mean
from csb.statistics.pdf import AbstractEstimator, AbstractDensity, BaseDensity
from csb.statistics.pdf import Gamma, InverseGamma, NullEstimator
def inv_digamma_minus_log(y, tol=1e-10, n_iter=100):
"""
Solve y = psi(alpha) - log(alpha) for alpha by fixed point
integration.
"""
if y >= -log(6.):
x = 1 / (2 * (1 - exp(y)))
else:
x = 1.e-10
for _i in range(n_iter):
z = approx_psi(x) - log(x) - y
if abs(z) < tol:
break
x -= x * z / (x * d_approx_psi(x) - 1)
x = abs(x)
return x
class ScaleMixturePriorEstimator(AbstractEstimator):
"""
Prior on the scales of a L{ScaleMixture}, which determines how the scales
are estimated.
"""
__metaclass__ = ABCMeta
@abstractmethod
def get_scale_estimator(self):
"""
Return an appropriate estimator for the scales of the mixture distribution
under this prior.
"""
pass
class ScaleMixturePrior(object):
"""
Prior on the scales of a L{ScaleMixture}, which determines how the scales
are estimated.
"""
def __init__(self, *args):
super(ScaleMixturePrior, self).__init__(*args)
self._scale = None
self._scale_estimator = NullEstimator()
@property
def scale_estimator(self):
return self._scale_estimator
@property
def estimator(self):
return self._estimator
@estimator.setter
def estimator(self, strategy):
if not isinstance(strategy, AbstractEstimator):
raise TypeError(strategy)
self._estimator = strategy
if isinstance(strategy, ScaleMixturePriorEstimator):
self._scale_estimator = strategy.get_scale_estimator()
else:
self._scale_estimator = NullEstimator()
class ScaleMixture(BaseDensity):
"""
Robust probabilistic superposition and comparison of protein structures
Martin Mechelke and Michael Habeck
Represenation of a distribution as a mixture of gaussians with a mean of
zero and different inverse variances/scales. The number of scales equals
the number of datapoints.
The underlying family is determined by a prior L{ScaleMixturePrior} on the
scales. Choosing a L{GammaPrior} results in Stundent-t posterior, wheras a
L{InvGammaPrior} leads to a K-Distribution as posterior.
"""
def __init__(self, scales=numpy.array([1., 1.]), prior=None, d=3):
super(ScaleMixture, self).__init__()
self._register('scales')
self._d = d
self.set_params(scales=scales)
self._prior = prior
if self._prior is not None:
self.estimator = self._prior.scale_estimator
@property
def scales(self):
return numpy.squeeze(self['scales'])
@scales.setter
def scales(self, value):
if not isinstance(value, csb.core.string) and \
isinstance(value, (numpy.ndarray, list, tuple)):
self['scales'] = numpy.array(value)
else:
raise ValueError("numpy array expected")
@property
def prior(self):
return self._prior
@prior.setter
def prior(self, value):
if not isinstance(value, ScaleMixturePrior):
raise TypeError(value)
self._prior = value
self.estimator = self._prior.scale_estimator
def log_prob(self, x):
from csb.numeric import log_sum_exp
dim = self._d
s = self.scales
log_p = numpy.squeeze(-numpy.multiply.outer(x * x, 0.5 * s)) + \
numpy.squeeze(dim * 0.5 * (log(s) - log(2 * numpy.pi)))
if self._prior is not None:
log_p += numpy.squeeze(self._prior.log_prob(s))
return log_sum_exp(log_p.T, 0)
def random(self, shape=None):
s = self.scales
if shape is None:
return numpy.random.normal() * s[numpy.random.randint(len(s))]
else:
#n = s.shape[0]
nrv = numpy.random.normal(size=shape)
indices = numpy.random.randint(len(s), size=shape)
return s[indices] * nrv
class ARSPosteriorAlpha(csb.statistics.ars.LogProb):
"""
This class represents the posterior distribution of the alpha parameter
of the Gamma and Inverse Gamma prior, and allows sampling using adaptive
rejection sampling L{ARS}.
"""
def __init__(self, a, b, n):
self.a = float(a)
self.b = float(b)
self.n = float(n)
def __call__(self, x):
from scipy.special import gammaln
return self.a * x - \
self.n * gammaln(numpy.clip(x, 1e-308, 1e308)) + \
self.b * log(x), \
self.a - self.n * psi(x) + self.b / x
def initial_values(self, tol=1e-10):
"""
Generate initial values by doing fixed point
iterations to solve for alpha
"""
n, a, b = self.n, self.a, self.b
if abs(b) < 1e-10:
alpha = inv_psi(a / n)
else:
alpha = 1.
z = tol + 1.
while abs(z) > tol:
z = n * psi(alpha) - \
b / numpy.clip(alpha, 1e-300, 1e300) - a
alpha -= z / (n * d_approx_psi(alpha) - b
/ (alpha ** 2 + 1e-300))
alpha = numpy.clip(alpha, 1e-100, 1e300)
return numpy.clip(alpha - 1 / (n + 1e-300), 1e-100, 1e300), \
alpha + 1 / (n + 1e-300), alpha
class GammaPosteriorSampler(ScaleMixturePriorEstimator):
def __init__(self):
super(GammaPosteriorSampler, self).__init__()
self.n_samples = 2
def get_scale_estimator(self):
return GammaScaleSampler()
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
## sufficient statistics
a = numpy.mean(data)
b = exp(numpy.mean(log(data)))
v = numpy.std(data) ** 2
n = len(data)
beta = a / v
alpha = beta * a
samples = []
for _i in range(self.n_samples):
## sample beta from Gamma distribution
beta = numpy.random.gamma(n * alpha + context._hyper_beta.alpha,
1 / (n * a + context._hyper_beta.beta))
## sample alpha with ARS
logp = ARSPosteriorAlpha(n * log(beta * b)\
- context.hyper_alpha.beta,
context.hyper_alpha.alpha - 1., n)
ars = csb.statistics.ars.ARS(logp)
ars.initialize(logp.initial_values()[:2], z0=0.)
alpha = ars.sample()
if alpha is None:
raise ValueError("ARS failed")
samples.append((alpha, beta))
pdf.alpha, pdf.beta = samples[-1]
return pdf
class GammaPosteriorMAP(ScaleMixturePriorEstimator):
def __init__(self):
super(GammaPosteriorMAP, self).__init__()
def get_scale_estimator(self):
return GammaScaleMAP()
def estimate(self, context, data):
"""
Estimate alpha and beta from their posterior
"""
pdf = GammaPrior()
s = data[0].mean()
y = data[1].mean() - log(s) - 1.
alpha = abs(inv_digamma_minus_log(numpy.clip(y,
- 1e308,
- 1e-300)))
beta = alpha / s
pdf.alpha, pdf.beta = alpha, beta
return pdf
class InvGammaPosteriorSampler(ScaleMixturePriorEstimator):
def __init__(self):
super(InvGammaPosteriorSampler, self).__init__()
self.n_samples = 2
def get_scale_estimator(self):
return InvGammaScaleSampler()
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
## sufficient statistics
h = harmonic_mean(numpy.clip(data, 1e-308, 1e308))
g = geometric_mean(numpy.clip(data, 1e-308, 1e308))
n = len(data)
samples = []
a = numpy.mean(1 / data)
v = numpy.std(1 / data) ** 2
beta = a / v
alpha = beta * a
for i in range(self.n_samples):
## sample alpha with ARS
logp = ARSPosteriorAlpha(n * (log(beta) - log(g)) - context.hyper_alpha.beta,
context.hyper_alpha.alpha - 1., n)
ars = csb.statistics.ars.ARS(logp)
ars.initialize(logp.initial_values()[:2], z0=0.)
alpha = numpy.abs(ars.sample())
if alpha is None:
raise ValueError("Sampling failed")
## sample beta from Gamma distribution
beta = numpy.random.gamma(n * alpha + context.hyper_beta.alpha, \
1 / (n / h + context.hyper_beta.beta))
samples.append((alpha, beta))
pdf.alpha, pdf.beta = samples[-1]
return pdf
class InvGammaPosteriorMAP(ScaleMixturePriorEstimator):
def __init__(self):
super(InvGammaPosteriorMAP, self).__init__()
def get_scale_estimator(self):
return InvGammaScaleMAP()
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
y = log(data).mean() - log((data ** -1).mean())
alpha = inv_digamma_minus_log(numpy.clip(y,
- 1e308,
- 1e-300))
alpha = abs(alpha)
beta = numpy.clip(alpha /
(data ** (-1)).mean(),
1e-100, 1e100)
pdf.alpha, pdf.beta = alpha, beta
return pdf
class GammaScaleSampler(AbstractEstimator):
"""
Sample the scalies given the data
"""
def estimate(self, context, data):
pdf = ScaleMixture()
alpha = context.prior.alpha
beta = context.prior.beta
d = context._d
if len(data.shape) == 1:
data = data[:, numpy.newaxis]
a = alpha + 0.5 * d * len(data.shape)
b = beta + 0.5 * data.sum(-1) ** 2
s = numpy.clip(numpy.random.gamma(a, 1. / b), 1e-20, 1e10)
pdf.scales = s
context.prior.estimate(s)
pdf.prior = context.prior
return pdf
class GammaScaleMAP(AbstractEstimator):
"""
MAP estimator of the scales
"""
def estimate(self, context, data):
pdf = ScaleMixture()
alpha = context.prior.alpha
beta = context.prior.beta
d = context._d
if len(data.shape) == 1:
data = data[:, numpy.newaxis]
a = alpha + 0.5 * d * len(data.shape)
b = beta + 0.5 * data.sum(-1) ** 2
s = a / b
log_s = psi(a) - log(b)
pdf.scales = s
context.prior.estimate([s, log_s])
pdf.prior = context.prior
return pdf
class InvGammaScaleSampler(AbstractEstimator):
"""
Sample the scales given the data
"""
def estimate(self, context, data):
pdf = ScaleMixture()
alpha = context.prior.alpha
beta = context.prior.beta
d = context._d
if len(data.shape) == 1:
data = data[:, numpy.newaxis]
p = -alpha + 0.5 * d
b = 2 * beta
a = 1e-5 + data.sum(-1) ** 2
s = csb.statistics.rand.gen_inv_gaussian(a, b, p)
s = numpy.clip(s, 1e-300, 1e300)
pdf.scales = s
context.prior.estimate(s)
pdf.prior = context.prior
return pdf
class InvGammaScaleMAP(AbstractEstimator):
"""
MAP estimator of the scales
"""
def estimate(self, context, data):
pdf = ScaleMixture()
alpha = context.prior.alpha
beta = context.prior.beta
d = context._d
if len(data.shape) == 1:
data = data[:, numpy.newaxis]
p = -alpha + 0.5 * d
b = 2 * beta
a = 1e-5 + data.sum(-1) ** 2
s = numpy.clip((numpy.sqrt(b) * kve(p + 1, numpy.sqrt(a * b)))
/ (numpy.sqrt(a) * kve(p, numpy.sqrt(a * b))),
1e-10, 1e10)
pdf.scales = s
context.prior.estimate(s)
pdf.prior = context.prior
return pdf
class GammaPrior(ScaleMixturePrior, Gamma):
"""
Gamma prior on mixture weights including a weak gamma prior on its parameter.
"""
def __init__(self, alpha=1., beta=1., hyper_alpha=(4., 1.),
hyper_beta=(2., 1.)):
super(GammaPrior, self).__init__(alpha, beta)
self._hyper_alpha = Gamma(*hyper_alpha)
self._hyper_beta = Gamma(*hyper_beta)
self.estimator = GammaPosteriorSampler()
@typedproperty(AbstractDensity)
def hyper_beta():
pass
@typedproperty(AbstractDensity)
def hyper_alpha():
pass
def log_prob(self, x):
a, b = self['alpha'], self['beta']
l_a = self._hyper_alpha(a)
l_b = self._hyper_beta(b)
return super(GammaPrior, self).log_prob(x) + l_a + l_b
class InvGammaPrior(ScaleMixturePrior, InverseGamma):
"""
Inverse Gamma prior on mixture weights including a weak gamma
prior on its parameter.
"""
def __init__(self, alpha=1., beta=1., hyper_alpha=(4., 1.),
hyper_beta=(2., 1.)):
super(InvGammaPrior, self).__init__(alpha, beta)
self._hyper_alpha = Gamma(*hyper_alpha)
self._hyper_beta = Gamma(*hyper_beta)
self.estimator = InvGammaPosteriorSampler()
@typedproperty(AbstractDensity)
def hyper_beta():
pass
@typedproperty(AbstractDensity)
def hyper_alpha():
pass
def log_prob(self, x):
a, b = self['alpha'], self['beta']
l_a = self._hyper_alpha(a)
l_b = self._hyper_beta(b)
return super(InvGammaPrior, self).log_prob(x) + l_a + l_b
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