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Probability density functions.
This module defines L{AbstractDensity}: a common interface for all PDFs.
Each L{AbstractDensity} describes a specific type of probability distribution,
for example L{Normal} is an implementation of the Gaussian distribution:
>>> pdf = Normal(mu=10, sigma=1.1)
>>> pdf.mu, pdf['sigma']
10.0, 1.1
Every PDF provides an implementation of the L{AbstractDensity.evaluate}
method, which evaluates the PDF for a list of input data points:
>>> pdf.evaluate([10, 9, 11, 12])
array([ 0.3626748 , 0.2399147 , 0.2399147 , 0.06945048])
PDF instances also behave like functions:
>>> pdf(data) # the same as pdf.evaluate(data)
Some L{AbstractDensity} implementations may support drawing random numbers from
the distribution (or raise an exception otherwise):
>>> pdf.random(2)
array([ 9.86257083, 9.73760515])
Each implementation of L{AbstractDensity} may support infinite number of estimators,
used to estimate and re-initialize the PDF parameters from a set of observed data
points:
>>> pdf.estimate([5, 5, 10, 10])
>>> pdf.mu, pdf.sigma
(7.5, 2.5)
>>> pdf.estimator
<csb.statistics.pdf.GaussianMLEstimator>
Estimators implement the L{AbstractEstimator} interface. They are treated as
pluggable tools, which can be exchanged through the L{AbstractDensity.estimator}
property (you could create, initialize and plug your own estimator as well).
This is a classic Strategy pattern.
"""
import numpy.random
import scipy.special
import csb.core
from abc import ABCMeta, abstractmethod
from csb.core import OrderedDict
from csb.numeric import log, exp, psi, inv_psi, EULER_MASCHERONI
from scipy.special import gammaln
from numpy import array, fabs, power, sqrt, pi, mean, median, clip
class IncompatibleEstimatorError(TypeError):
pass
class ParameterNotFoundError(AttributeError):
pass
class ParameterValueError(ValueError):
def __init__(self, param, value):
self.param = param
self.value = value
super(ParameterValueError, self).__init__(param, value)
def __str__(self):
return '{0} = {1}'.format(self.param, self.value)
class EstimationFailureError(ParameterValueError):
pass
class AbstractEstimator(object):
"""
Density parameter estimation strategy.
"""
__metaclass__ = ABCMeta
@abstractmethod
def estimate(self, context, data):
"""
Estimate the parameters of the distribution from same {data}.
@param context: context distribution
@type context: L{AbstractDensity}
@param data: sample values
@type data: array
@return: a new distribution, initialized with the estimated parameters
@rtype: L{AbstractDensity}
@raise EstimationFailureError: if estimation is not possible
"""
pass
class NullEstimator(AbstractEstimator):
"""
Does not estimate anything.
"""
def estimate(self, context, data):
raise NotImplementedError()
class LaplaceMLEstimator(AbstractEstimator):
def estimate(self, context, data):
x = array(data)
mu = median(x)
b = mean(fabs(x - mu))
return Laplace(mu, b)
class GaussianMLEstimator(AbstractEstimator):
def estimate(self, context, data):
x = array(data)
mu = mean(x)
sigma = sqrt(mean((x - mu) ** 2))
return Normal(mu, sigma)
class InverseGaussianMLEstimator(AbstractEstimator):
def estimate(self, context, data):
x = array(data)
mu = mean(x)
il = mean((1.0 / x) - (1.0 / mu))
if il == 0:
raise EstimationFailureError('lambda', float('inf'))
return InverseGaussian(mu, 1.0 / il)
class GammaMLEstimator(AbstractEstimator):
def __init__(self):
super(GammaMLEstimator, self).__init__()
self.n_iter = 1000
def estimate(self, context, data):
mu = mean(data)
logmean = mean(log(data))
a = 0.5 / (log(mu) - logmean)
for dummy in range(self.n_iter):
a = inv_psi(logmean - log(mu) + log(a))
return Gamma(a, a / mu)
class GenNormalBruteForceEstimator(AbstractEstimator):
def __init__(self, minbeta=0.5, maxbeta=8.0, step=0.1):
self._minbeta = minbeta
self._maxbeta = maxbeta
self._step = step
super(GenNormalBruteForceEstimator, self).__init__()
def estimate(self, context, data):
pdf = GeneralizedNormal(1, 1, 1)
data = array(data)
logl = []
for beta in numpy.arange(self._minbeta, self._maxbeta, self._step):
self.update(pdf, data, beta)
l = pdf.log_prob(data).sum()
logl.append([beta, l])
logl = numpy.array(logl)
# optimal parameters:
beta = logl[ numpy.argmax(logl[:, 1]) ][0]
self.update(pdf, data, beta)
return pdf
def estimate_with_fixed_beta(self, data, beta):
mu = median(data)
v = mean((data - mu) ** 2)
alpha = sqrt(v * exp(gammaln(1. / beta) - gammaln(3. / beta)))
return mu, alpha
def update(self, pdf, data, beta):
mu, alpha = self.estimate_with_fixed_beta(data, beta)
pdf.mu = mu
pdf.alpha = alpha
pdf.beta = beta
return pdf
class MultivariateGaussianMLEstimator(AbstractEstimator):
def __init__(self):
super(MultivariateGaussianMLEstimator, self).__init__()
def estimate(self, context, data):
return MultivariateGaussian(numpy.mean(data, 0), numpy.cov(data.T))
class DirichletEstimator(AbstractEstimator):
def __init__(self):
super(DirichletEstimator, self).__init__()
self.n_iter = 1000
self.tol = 1e-5
def estimate(self, context, data):
log_p = numpy.mean(log(data), 0)
e = numpy.mean(data, 0)
v = numpy.mean(data ** 2, 0)
q = (e[0] - v[0]) / (v[0] - e[0] ** 2)
a = e * q
y = a * 0
k = 0
while(sum(abs(y - a)) > self.tol and k < self.n_iter):
y = psi(sum(a)) + log_p
a = numpy.array(list(map(inv_psi, y)))
k += 1
return Dirichlet(a)
class GumbelMinMomentsEstimator(AbstractEstimator):
def estimate(self, context, data):
x = array(data)
beta = sqrt(6 * numpy.var(x)) / pi
mu = mean(x) + EULER_MASCHERONI * beta
return GumbelMinimum(mu, beta)
class GumbelMaxMomentsEstimator(AbstractEstimator):
def estimate(self, context, data):
x = array(data)
beta = sqrt(6 * numpy.var(x)) / pi
mu = mean(x) - EULER_MASCHERONI * beta
return GumbelMaximum(mu, beta)
class AbstractDensity(object):
"""
Defines the interface and common operations for all probability density
functions. This is a generic class which can operate on parameters of
any type (e.g. simple floats or custom parameter objects).
Subclasses must complete the implementation by implementing the
L{AbstractDensity.log_prob} method. Subclasses could also consider--but
are not obliged to--override the L{AbstractDensity.random} method. If
any of the density parameters need validation, subclasses are expected to
override the L{AbstractDensity._validate} method and raise
L{ParameterValueError} on validation failure. Note that implementing
parameter validation in property setters has almost no effect and is
discouraged.
"""
__metaclass__ = ABCMeta
def __init__(self):
self._params = OrderedDict()
self._estimator = None
self.estimator = NullEstimator()
def __getitem__(self, param):
if param in self._params:
return self._params[param]
else:
raise ParameterNotFoundError(param)
def __setitem__(self, param, value):
if param in self._params:
self._validate(param, value)
self._set(param, value)
else:
raise ParameterNotFoundError(param)
def _set(self, param, value):
"""
Update the C{value} of C{param}.
"""
self._params[param] = value
@property
def estimator(self):
return self._estimator
@estimator.setter
def estimator(self, strategy):
if not isinstance(strategy, AbstractEstimator):
raise TypeError(strategy)
self._estimator = strategy
def __call__(self, x):
return self.evaluate(x)
def __str__(self):
name = self.__class__.__name__
params = ', '.join([ '{0}={1}'.format(p, v) for p, v in self._params.items() ])
return '{0}({1})'.format(name, params)
def _register(self, name):
"""
Register a new parameter name.
"""
if name not in self._params:
self._params[name] = None
def _validate(self, param, value):
"""
Parameter value validation hook.
@raise ParameterValueError: on failed validation (value not accepted)
"""
pass
def get_params(self):
return [self._params[name] for name in self.parameters]
def set_params(self, *values, **named_params):
for p, v in zip(self.parameters, values):
self[p] = v
for p in named_params:
self[p] = named_params[p]
@property
def parameters(self):
"""
Get a list of all distribution parameter names.
"""
return tuple(self._params)
@abstractmethod
def log_prob(self, x):
"""
Evaluate the logarithm of the probability of observing values C{x}.
@param x: values
@type x: array
@rtype: array
"""
pass
def evaluate(self, x):
"""
Evaluate the probability of observing values C{x}.
@param x: values
@type x: array
@rtype: array
"""
x = numpy.array(x)
return exp(self.log_prob(x))
def random(self, size=None):
"""
Generate random samples from the probability distribution.
@param size: number of values to sample
@type size: int
"""
raise NotImplementedError()
def estimate(self, data):
"""
Estimate and load the parameters of the distribution from sample C{data}
using the current L{AbstractEstimator} strategy.
@param data: sample values
@type data: array
@raise NotImplementedError: when no estimator is available for this
distribution
@raise IncompatibleEstimatorError: when the current estimator is not
compatible with this pdf
"""
try:
pdf = self.estimator.estimate(self, data)
for param in pdf.parameters:
self[param] = pdf[param]
except ParameterNotFoundError as e:
raise IncompatibleEstimatorError(self.estimator)
except ParameterValueError as e:
raise EstimationFailureError(e.param, e.value)
class BaseDensity(AbstractDensity):
"""
Base abstract class for all PDFs, which operate on simple float
or array-of-float parameters. Parameters of any other type will trigger
TypeError-s.
"""
def _set(self, param, value):
try:
if csb.core.iterable(value):
value = array(value)
else:
value = float(value)
except ValueError:
raise TypeError(value)
super(BaseDensity, self)._set(param, value)
class Laplace(BaseDensity):
def __init__(self, mu=0, b=1):
super(Laplace, self).__init__()
self._register('mu')
self._register('b')
self.set_params(b=b, mu=mu)
self.estimator = LaplaceMLEstimator()
def _validate(self, param, value):
if param == 'b' and value <= 0:
raise ParameterValueError(param, value)
@property
def b(self):
return self['b']
@b.setter
def b(self, value):
self['b'] = value
@property
def mu(self):
return self['mu']
@mu.setter
def mu(self, value):
self['mu'] = value
def log_prob(self, x):
b = self.b
mu = self.mu
return log(1 / (2. * b)) - fabs(x - mu) / b
def random(self, size=None):
loc = self.mu
scale = self.b
return numpy.random.laplace(loc, scale, size)
class Normal(BaseDensity):
def __init__(self, mu=0, sigma=1):
super(Normal, self).__init__()
self._register('mu')
self._register('sigma')
self.set_params(mu=mu, sigma=sigma)
self.estimator = GaussianMLEstimator()
@property
def mu(self):
return self['mu']
@mu.setter
def mu(self, value):
self['mu'] = value
@property
def sigma(self):
return self['sigma']
@sigma.setter
def sigma(self, value):
self['sigma'] = value
def log_prob(self, x):
mu = self.mu
sigma = self.sigma
return log(1.0 / sqrt(2 * pi * sigma ** 2)) - (x - mu) ** 2 / (2 * sigma ** 2)
def random(self, size=None):
mu = self.mu
sigma = self.sigma
return numpy.random.normal(mu, sigma, size)
class InverseGaussian(BaseDensity):
def __init__(self, mu=1, shape=1):
super(InverseGaussian, self).__init__()
self._register('mu')
self._register('shape')
self.set_params(mu=mu, shape=shape)
self.estimator = InverseGaussianMLEstimator()
def _validate(self, param, value):
if value <= 0:
raise ParameterValueError(param, value)
@property
def mu(self):
return self['mu']
@mu.setter
def mu(self, value):
self['mu'] = value
@property
def shape(self):
return self['shape']
@shape.setter
def shape(self, value):
self['shape'] = value
def log_prob(self, x):
mu = self.mu
scale = self.shape
x = numpy.array(x)
if numpy.min(x) <= 0:
raise ValueError('InverseGaussian is defined for x > 0')
y = -0.5 * scale * (x - mu) ** 2 / (mu ** 2 * x)
z = 0.5 * (log(scale) - log(2 * pi * x ** 3))
return z + y
def random(self, size=None):
mu = self.mu
shape = self.shape
mu_2l = mu / shape / 2.
Y = numpy.random.standard_normal(size)
Y = mu * Y ** 2
X = mu + mu_2l * (Y - sqrt(4 * shape * Y + Y ** 2))
U = numpy.random.random(size)
m = numpy.less_equal(U, mu / (mu + X))
return m * X + (1 - m) * mu ** 2 / X
class GeneralizedNormal(BaseDensity):
def __init__(self, mu=0, alpha=1, beta=1):
super(GeneralizedNormal, self).__init__()
self._register('mu')
self._register('alpha')
self._register('beta')
self.set_params(mu=mu, alpha=alpha, beta=beta)
self.estimator = GenNormalBruteForceEstimator()
def _validate(self, param, value):
if param in ('alpha, beta') and value <= 0:
raise ParameterValueError(param, value)
@property
def mu(self):
return self['mu']
@mu.setter
def mu(self, value):
self['mu'] = value
@property
def alpha(self):
return self['alpha']
@alpha.setter
def alpha(self, value):
self['alpha'] = value
@property
def beta(self):
return self['beta']
@beta.setter
def beta(self, value):
self['beta'] = value
def log_prob(self, x):
mu = self.mu
alpha = self.alpha
beta = self.beta
return log(beta / (2.0 * alpha)) - gammaln(1. / beta) - power(fabs(x - mu) / alpha, beta)
class GeneralizedInverseGaussian(BaseDensity):
def __init__(self, a=1, b=1, p=1):
super(GeneralizedInverseGaussian, self).__init__()
self._register('a')
self._register('b')
self._register('p')
self.set_params(a=a, b=b, p=p)
self.estimator = NullEstimator()
def _validate(self, param, value):
if value <= 0:
raise ParameterValueError(param, value)
@property
def a(self):
return self['a']
@a.setter
def a(self, value):
self['a'] = value
@property
def b(self):
return self['b']
@b.setter
def b(self, value):
self['b'] = value
@property
def p(self):
return self['p']
@p.setter
def p(self, value):
self['p'] = value
def log_prob(self, x):
a = self['a']
b = self['b']
p = self['p']
lz = 0.5 * p * (log(a) - log(b)) - log(2 * scipy.special.kv(p, sqrt(a * b)))
return lz + (p - 1) * log(x) - 0.5 * (a * x + b / x)
def random(self, size=None):
from csb.statistics.rand import inv_gaussian
rvs = []
burnin = 10
a = self['a']
b = self['b']
p = self['p']
s = a * 0. + 1.
if p < 0:
a, b = b, a
if size == None:
size = 1
for i in range(int(size)):
for j in range(burnin):
l = b + 2 * s
m = sqrt(l / a)
x = inv_gaussian(m, l, shape=m.shape)
s = numpy.random.gamma(abs(p) + 0.5, x)
if p >= 0:
rvs.append(x)
else:
rvs.append(1 / x)
return numpy.array(rvs)
class Gamma(BaseDensity):
def __init__(self, alpha=1, beta=1):
super(Gamma, self).__init__()
self._register('alpha')
self._register('beta')
self.set_params(alpha=alpha, beta=beta)
self.estimator = GammaMLEstimator()
def _validate(self, param, value):
if value <= 0:
raise ParameterValueError(param, value)
@property
def alpha(self):
return self['alpha']
@alpha.setter
def alpha(self, value):
self['alpha'] = value
@property
def beta(self):
return self['beta']
@beta.setter
def beta(self, value):
self['beta'] = value
def log_prob(self, x):
a, b = self['alpha'], self['beta']
return a * log(b) - gammaln(clip(a, 1e-308, 1e308)) + \
(a - 1) * log(clip(x, 1e-308, 1e308)) - b * x
def random(self, size=None):
return numpy.random.gamma(self['alpha'], 1 / self['beta'], size)
class InverseGamma(BaseDensity):
def __init__(self, alpha=1, beta=1):
super(InverseGamma, self).__init__()
self._register('alpha')
self._register('beta')
self.set_params(alpha=alpha, beta=beta)
self.estimator = NullEstimator()
def _validate(self, param, value):
if value <= 0:
raise ParameterValueError(param, value)
@property
def alpha(self):
return self['alpha']
@alpha.setter
def alpha(self, value):
self['alpha'] = value
@property
def beta(self):
return self['beta']
@beta.setter
def beta(self, value):
self['beta'] = value
def log_prob(self, x):
a, b = self['alpha'], self['beta']
return a * log(b) - gammaln(a) - (a + 1) * log(x) - b / x
def random(self, size=None):
return 1. / numpy.random.gamma(self['alpha'], 1 / self['beta'], size)
class MultivariateGaussian(Normal):
def __init__(self, mu=numpy.zeros(2), sigma=numpy.eye(2)):
super(MultivariateGaussian, self).__init__(mu, sigma)
self.estimator = MultivariateGaussianMLEstimator()
def random(self, size=None):
return numpy.random.multivariate_normal(self.mu, self.sigma, size)
def log_prob(self, x):
from numpy.linalg import det
mu = self.mu
S = self.sigma
D = len(mu)
q = self.__q(x)
return -0.5 * (D * log(2 * pi) + log(abs(det(S)))) - 0.5 * q ** 2
def __q(self, x):
from numpy import sum, dot, reshape
from numpy.linalg import inv
mu = self.mu
S = self.sigma
return sqrt(clip(sum(reshape((x - mu) * dot(x - mu, inv(S).T.squeeze()), (-1, len(mu))), -1), 0., 1e308))
def conditional(self, x, dims):
"""
Return the distribution along the dimensions
dims conditioned on x
@param x: conditional values
@param dims: new dimensions
"""
from numpy import take, dot
from numpy.linalg import inv
dims2 = [i for i in range(self['mu'].shape[0]) if not i in dims]
mu1 = take(self['mu'], dims)
mu2 = take(self['mu'], dims2)
# x1 = take(x, dims)
x2 = take(x, dims2)
A = take(take(self['Sigma'], dims, 0), dims, 1)
B = take(take(self['Sigma'], dims2, 0), dims2, 1)
C = take(take(self['Sigma'], dims, 0), dims2, 1)
mu = mu1 + dot(C, dot(inv(B), x2 - mu2))
Sigma = A - dot(C, dot(inv(B), C.T))
return MultivariateGaussian((mu, Sigma))
class Dirichlet(BaseDensity):
def __init__(self, alpha):
super(Dirichlet, self).__init__()
self._register('alpha')
self.set_params(alpha=alpha)
self.estimator = DirichletEstimator()
@property
def alpha(self):
return self['alpha']
@alpha.setter
def alpha(self, value):
self['alpha'] = numpy.ravel(value)
def log_prob(self, x):
#TODO check wether x is in the probability simplex
alpha = self.alpha
return gammaln(sum(alpha)) - sum(gammaln(alpha)) \
+ numpy.dot((alpha - 1).T, log(x).T)
def random(self, size=None):
return numpy.random.mtrand.dirichlet(self.alpha, size)
class GumbelMinimum(BaseDensity):
def __init__(self, mu=0, beta=1):
super(GumbelMinimum, self).__init__()
self._register('mu')
self._register('beta')
self.set_params(mu=mu, beta=beta)
self.estimator = GumbelMinMomentsEstimator()
def _validate(self, param, value):
if param == 'beta' and value <= 0:
raise ParameterValueError(param, value)
@property
def mu(self):
return self['mu']
@mu.setter
def mu(self, value):
self['mu'] = value
@property
def beta(self):
return self['beta']
@beta.setter
def beta(self, value):
self['beta'] = value
def log_prob(self, x):
mu = self.mu
beta = self.beta
z = (x - mu) / beta
return log(1. / beta) + z - exp(z)
def random(self, size=None):
mu = self.mu
beta = self.beta
return -numpy.random.gumbel(-mu, beta, size)
class GumbelMaximum(GumbelMinimum):
def __init__(self, mu=0, beta=1):
super(GumbelMaximum, self).__init__(mu=mu, beta=beta)
self.estimator = GumbelMaxMomentsEstimator()
def log_prob(self, x):
mu = self.mu
beta = self.beta
z = (x - mu) / beta
return log(1. / beta) - z - exp(-z)
def random(self, size=None):
mu = self.mu
beta = self.beta
return numpy.random.gumbel(mu, beta, size)
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