# Copyright (c) 2010 Matej Laitl <matej@laitl.cz>
# Distributed under the terms of the GNU General Public License v2 or any
# later version of the license, at your option.
"""
This module contains models of common probability density functions, abbreviated
as pdfs.
All classes from this module are currently imported to top-level pybayes module,
so instead of ``from pybayes.pdfs import Pdf`` you can type ``from pybayes import
Pdf``.
"""
from copy import deepcopy
import math
from numpy import random, empty
from .wrappers import _linalg as linalg
from .wrappers import _numpy as np
[docs]class RVComp(object):
"""Atomic component of a random variable.
:var int dimension: dimension; do not change unless you know what you are doing
:var str name: name; can be changed as long as it remains a string (warning:
parent RVs are not updated)
"""
[docs] def __init__(self, dimension, name = None):
"""Initialise new component of a random variable :class:`RV`.
:param dimension: number of vector components this component occupies
:type dimension: positive integer
:param name: name of the component; default: None for anonymous component
:type name: string or None
:raises TypeError: non-integer dimension or non-string name
:raises ValueError: non-positive dimension
"""
if name is not None and not isinstance(name, str):
raise TypeError("name must be either None or a string")
self.name = name
if not isinstance(dimension, int):
raise TypeError("dimension must be integer (int)")
if dimension < 1:
raise ValueError("dimension must be non-zero positive")
self.dimension = dimension
#def __eq__(self, other):
#"""We want RVComp have to be hashable
#(http://docs.python.org/glossary.html#term-hashable), but default __eq__()
#and __hash__() implementations suffice, as they are instance-based.
#"""
[docs]class RV(object):
"""Representation of a random variable made of one or more components. Each component is
represented by :class:`RVComp` class.
:var int dimension: cummulative dimension; do not change
:var str name: pretty name, can be changed but needs to be a string
:var list components: list of RVComps; do not change
*Please take into account that all RVComp comparisons inside RV are
instance-based and component names are purely informational. To demonstrate:*
>>> rv = RV(RVComp(1, "a"))
>>> ...
>>> rv.contains(RVComp(1, "a"))
False
Right way to do this would be:
>>> a = RVComp(1, "arbitrary pretty name for a")
>>> rv = RV(a)
>>> ...
>>> rv.contains(a)
True
"""
[docs] def __init__(self, *components):
"""Initialise random variable meta-representation.
:param \*components: components that should form the random variable. You may
also pass another RVs which is a shotrcut for adding all their components.
:type \*components: :class:`RV`, :class:`RVComp` or a sequence of :class:`RVComp` items
:raises TypeError: invalid object passed (neither a :class:`RV` or a :class:`RVComp`)
Usual way of creating a RV could be:
>>> x = RV(RVComp(1, 'x_1'), RVComp(1, 'x_2'))
>>> x.name
'[x_1, x_2]'
>>> xy = RV(x, RVComp(2, 'y'))
>>> xy.name
'[x_1, x_2, y]'
"""
self.dimension = 0
self.components = []
if len(components) is 0:
self.name = '[]'
return
self.name = '['
for component in components:
if isinstance(component, RVComp):
self._add_component(component)
elif isinstance(component, RV):
for subcomp in component.components:
self._add_component(subcomp)
else:
try:
for subcomp in component:
self._add_component(subcomp)
except TypeError:
raise TypeError('component ' + str(component) + ' is neither an instance '
+ 'of RVComp or RV and is not iterable of RVComps')
self.name = self.name[:-2] + ']'
def __copy__(self):
ret = type(self).__new__(type(self))
ret.name = self.name
ret.dimension = self.dimension
ret.components = self.components
return ret
def __deepcopy__(self, memo):
ret = type(self).__new__(type(self))
ret.name = self.name # no need to deepcopy - string is immutable
ret.dimension = self.dimension # ditto
# Following shallow copy is special behaviour of RV:
ret.components = self.components[:]
return ret
def __str__(self):
return "<pybayes.pdfs.RV '{0}' dim={1} {2}>".format(self.name, self.dimension, self.components)
def _add_component(self, component):
"""Add new component to this random variable.
Internal function, do not use outside of RV."""
if not isinstance(component, RVComp):
raise TypeError("component is not of type RVComp")
self.components.append(component)
self.dimension += component.dimension
self.name = '{0}{1}, '.format(self.name, component.name)
return True
[docs] def contains(self, component):
"""Return True if this random variable contains the exact same instance of
the **component**.
:param component: component whose presence is tested
:type component: :class:`RVComp`
:rtype: bool
"""
return component in self.components
[docs] def contains_all(self, test_components):
"""Return True if this RV contains all RVComps from sequence
**test_components**.
:param test_components: list of components whose presence is checked
:type test_components: sequence of :class:`RVComp` items
"""
for test_comp in test_components:
if not self.contains(test_comp):
return False
return True;
[docs] def contains_any(self, test_components):
"""Return True if this RV contains any of **test_components**.
:param test_components: sequence of components whose presence is tested
:type test_components: sequence of :class:`RVComp` items
"""
for test_comp in test_components:
if self.contains(test_comp):
return True
return False
[docs] def contained_in(self, test_components):
"""Return True if sequence **test_components** contains all components
from this RV (and perhaps more).
:param test_components: set of components whose presence is checked
:type test_components: sequence of :class:`RVComp` items
"""
for component in self.components:
if component not in test_components:
return False
return True
[docs] def indexed_in(self, super_rv):
"""Return index array such that this rv is indexed in **super_rv**, which
must be a superset of this rv. Resulting array can be used with :func:`numpy.take`
and :func:`numpy.put`.
:param super_rv: returned indices apply to this rv
:type super_rv: :class:`RV`
:rtype: 1D :class:`numpy.ndarray` of ints with dimension = self.dimension
"""
ret = np.index_vector(self.dimension)
ret_ind = 0 # current index in returned index array
# process each component from target rv
for comp in self.components:
# find associated component in source_rv components:
src_ind = 0 # index in source vector
for source_comp in super_rv.components:
if source_comp is comp:
ret[ret_ind:] = np.index_range(src_ind, src_ind + comp.dimension)
ret_ind += comp.dimension
break;
src_ind += source_comp.dimension
else:
raise AttributeError("Cannont find component "+str(comp)+" in source_rv.components.")
return ret
[docs]class CPdf(object):
r"""Base class for all Conditional (in general) Probability Density Functions.
When you evaluate a CPdf the result generally also depends on a condition
(vector) named `cond` in PyBayes. For a CPdf that is a :class:`Pdf` this is
not the case, the result is unconditional.
Every CPdf takes (apart from others) 2 optional arguments to constructor:
**rv** and **cond_rv**. (both :class:`RV` or a sequence of :class:`RVComp` objects) When
specified, they denote that the CPdf is associated with a particular random variable
(respectively its condition is associated with a particular random variable); when unspecified,
*anonymous* random variable is assumed (exceptions exist, see :class:`ProdPdf`).
It is an error to pass RV whose dimension is not same as CPdf's dimension
(or cond dimension respectively).
:var RV rv: associated random variable (always set in constructor, contains
at least one RVComp)
:var RV cond_rv: associated condition random variable (set in constructor to
potentially empty RV)
*While you can assign different rv and cond_rv to a CPdf, you should be
cautious because sanity checks are only performed in constructor.*
While entire idea of random variable associations may not be needed in simple
cases, it allows you to express more complicated situations. Assume the state
variable is composed of 2 components :math:`x_t = [a_t, b_t]` and following
probability density function has to be modelled:
.. math::
p(x_t|x_{t-1}) &= p_1(a_t|a_{t-1}, b_t) p_2(b_t|b_{t-1}) \\
p_1(a_t|a_{t-1}, b_t) &= \mathcal{N}(a_{t-1}, b_t) \\
p_2(b_t|b_{t-1}) &= \mathcal{N}(b_{t-1}, 0.0001)
This is done in PyBayes with associated RVs:
>>> a_t, b_t = RVComp(1, 'a_t'), RVComp(1, 'b_t') # create RV components
>>> a_tp, b_tp = RVComp(1, 'a_{t-1}'), RVComp(1, 'b_{t-1}') # t-1 case
>>> p1 = LinGaussCPdf(1., 0., 1., 0., [a_t], [a_tp, b_t])
>>> # params for p2:
>>> cov, A, b = np.array([[0.0001]]), np.array([[1.]]), np.array([0.])
>>> p2 = MLinGaussCPdf(cov, A, b, [b_t], [b_tp])
>>> p = ProdCPdf((p1, p2), [a_t, b_t], [a_tp, b_tp])
>>> p.sample(np.array([1., 2.]))
>>> p.eval_log()
"""
[docs] def shape(self):
"""Return pdf shape = number of dimensions of the random variable.
:meth:`mean` and :meth:`variance` methods must return arrays of this shape.
Default implementation (which you should not override) just returns
self.rv.dimension.
:rtype: int
"""
return self.rv.dimension
[docs] def cond_shape(self):
"""Return condition shape = number of dimensions of the conditioning variable.
Default implementation (which you should not override) just returns
self.cond_rv.dimension.
:rtype: int
"""
return self.cond_rv.dimension
[docs] def mean(self, cond = None):
"""Return (conditional) mean value of the pdf.
:rtype: :class:`numpy.ndarray`
"""
raise NotImplementedError("Derived classes must implement this function")
[docs] def variance(self, cond = None):
"""Return (conditional) variance (diagonal elements of covariance).
:rtype: :class:`numpy.ndarray`
"""
raise NotImplementedError("Derived classes must implement this function")
[docs] def eval_log(self, x, cond = None):
"""Return logarithm of (conditional) likelihood function in point x.
:param x: point which to evaluate the function in
:type x: :class:`numpy.ndarray`
:rtype: double
"""
raise NotImplementedError("Derived classes must implement this function")
[docs] def sample(self, cond = None):
"""Return one random (conditional) sample from this distribution
:rtype: :class:`numpy.ndarray`"""
raise NotImplementedError("Derived classes must implement this function")
[docs] def samples(self, n, cond = None):
"""Return n samples in an array. A convenience function that just calls
:meth:`shape` multiple times.
:param int n: number of samples to return
:rtype: 2D :class:`numpy.ndarray` of shape (*n*, m) where m is pdf
dimension"""
ret = np.matrix(n, self.shape())
for i in range(n):
ret[i, :] = self.sample(cond)
return ret
def _check_cond(self, cond):
"""Return True if cond has correct type and shape, raise Error otherwise.
:raises TypeError: cond is not of correct type
:raises ValueError: cond doesn't have appropriate shape
:rtype: bool
"""
expected_cond_shape = self.cond_shape()
if expected_cond_shape == 0:
return True # ignore cond for condition-free pdfs
if cond is None: # cython-specific
raise TypeError("cond must be an array of doubles")
if cond.ndim != 1:
raise ValueError("cond must be 1D numpy array (a vector)")
if cond.shape[0] != expected_cond_shape:
raise ValueError("cond must be of shape ({0},) array of shape ({1},) given".format(self.cond_shape(), cond.shape[0]))
return True
def _check_x(self, x):
"""Return True if x has correct type and shape (determined by shape()),
raise Error otherwise.
:raises TypeError: cond is not of correct type
:raises ValueError: cond doesn't have appropriate shape
:rtype: bool
"""
if x is None: # cython-specific
raise TypeError("x must be an array of doubles")
if x.ndim != 1:
raise ValueError("x must be 1D numpy array (a vector)")
if x.shape[0] != self.shape():
raise ValueError("x must be of shape ({0},) array of shape ({1},) given".format(self.shape(), x.shape[0]))
return True
def _set_rvs(self, exp_shape, rv, exp_cond_shape, cond_rv):
"""Internal heper to check and set rv and cond_rv.
:param int exp_shape: expected random variable shape
:param rv: associated random variable or None to have it auto-created
:type rv: :class:`RV` or a sequence of :class:`RVComp` objects
:param int exp_cond_shape: expected conditioning variable shape
:param cond_rv: associated conditioning variable or None to have it auto-created
:type cond_rv: :class:`RV` or a sequence of :class:`RVComp` objects
:raises TypeError: rv or cond_rv doesnt have right type
:raises ValueError: dimensions do not match
"""
if rv is None:
self.rv = RV(RVComp(exp_shape)) # create RV with one anonymous component
else:
if isinstance(rv, RV):
self.rv = rv
else:
self.rv = RV(rv) # assume that rv is a sequence of components
if self.rv.dimension != exp_shape:
raise ValueError("rv has wrong dimension {0}, {1} expected".format(rv.dimension, exp_shape))
if cond_rv is None:
if exp_cond_shape is 0:
self.cond_rv = RV() # create empty RV to denote empty condition
else:
self.cond_rv = RV(RVComp(exp_cond_shape)) # create RV with one anonymous component
else:
if isinstance(cond_rv, RV):
self.cond_rv = cond_rv
else:
self.cond_rv = RV(cond_rv)
if self.cond_rv.dimension is not exp_cond_shape:
raise ValueError("cond_rv has wrong dimension {0}, {1} expected".format(cond_rv.dimension, exp_cond_shape))
return True
[docs]class Pdf(CPdf):
"""Base class for all unconditional (static) multivariate Probability Density
Functions. Subclass of :class:`CPdf`.
As in CPdf, constructor of every Pdf takes optional **rv** (:class:`RV`)
keyword argument (and no *cond_rv* argument as it would make no sense). For
discussion about associated random variables see :class:`CPdf`.
"""
[docs] def cond_shape(self):
"""Return zero as Pdfs have no condition."""
return 0
def _set_rv(self, exp_shape, rv):
"""Internal helper - shortcut to :meth:`~CPdf._set_rvs`"""
return self._set_rvs(exp_shape, rv, 0, None)
[docs]class UniPdf(Pdf):
r"""Simple uniform multivariate probability density function. Extends
:class:`Pdf`.
.. math:: f(x) = \Theta(x - a) \Theta(b - x) \prod_{i=1}^n \frac{1}{b_i-a_i}
:var a: left border
:type a: 1D :class:`numpy.ndarray`
:var b: right border
:type b: 1D :class:`numpy.ndarray`
You may modify these attributes as long as you don't change their shape and
assumption **a** < **b** still holds.
"""
[docs] def __init__(self, a, b, rv = None):
"""Initialise uniform distribution.
:param a: left border
:type a: 1D :class:`numpy.ndarray`
:param b: right border
:type b: 1D :class:`numpy.ndarray`
**b** must be greater (in each dimension) than **a**.
To construct uniform distribution on interval [0,1]:
>>> uni = UniPdf(np.array([0.]), np.array([1.]), rv)
"""
self.a = np.asarray(a)
self.b = np.asarray(b)
if a.ndim != 1 or b.ndim != 1:
raise ValueError("both a and b must be 1D numpy arrays (vectors)")
if a.shape[0] != b.shape[0]:
raise ValueError("a must have same shape as b")
if np.any(self.b <= self.a):
raise ValueError("b must be greater than a in each dimension")
self._set_rv(a.shape[0], rv)
def mean(self, cond = None):
return (self.a+self.b)/2. # element-wise division
def variance(self, cond = None):
return ((self.b-self.a)**2)/12. # element-wise power and division
def eval_log(self, x, cond = None):
self._check_x(x)
if np.any(x <= self.a) or np.any(x >= self.b):
return float('-inf')
return -math.log(np.prod(self.b-self.a))
def sample(self, cond = None):
return random.uniform(-0.5, 0.5, self.shape()) * (self.b-self.a) + self.mean()
[docs]class AbstractGaussPdf(Pdf):
r"""Abstract base for all Gaussian-like pdfs - the ones that take vector mean
and matrix covariance parameters. Extends :class:`Pdf`.
:var mu: mean value
:type mu: 1D :class:`numpy.ndarray`
:var R: covariance matrix
:type R: 2D :class:`numpy.ndarray`
You can modify object parameters only if you are absolutely sure that you
pass allowable values - parameters are only checked once in constructor.
"""
def __copy__(self):
"""Make a shallow copy of AbstractGaussPdf (or its derivative provided
that is doesn't add class variables)"""
# we cannont use AbstractGaussPdf statically - this method may be called
# by derived class
ret = type(self).__new__(type(self)) # optimisation TODO: currently slower than PY_NEW()
ret.mu = self.mu
ret.R = self.R
ret.rv = self.rv
ret.cond_rv = self.cond_rv
return ret
def __deepcopy__(self, memo):
"""Make a deep copy of AbstractGaussPdf (or its derivative provided
that is doesn't add class variables)"""
# we cannont use AbstractGaussPdf statically - this method may be called
# by derived class
ret = type(self).__new__(type(self)) # optimisation TODO: currently slower than PY_NEW()
ret.mu = self.mu.copy()
ret.R = self.R.copy()
ret.rv = deepcopy(self.rv, memo)
ret.cond_rv = deepcopy(self.cond_rv, memo)
return ret
[docs]class GaussPdf(AbstractGaussPdf):
r"""Unconditional Gaussian (normal) probability density function. Extends
:class:`AbstractGaussPdf`.
.. math:: f(x) \propto \exp \left( - \left( x-\mu \right)' R^{-1} \left( x-\mu \right) \right)
"""
[docs] def __init__(self, mean, cov, rv = None):
r"""Initialise Gaussian pdf.
:param mean: mean value; stored in **mu** attribute
:type mean: 1D :class:`numpy.ndarray`
:param cov: covariance matrix; stored in **R** arrtibute
:type cov: 2D :class:`numpy.ndarray`
Covariance matrix **cov** must be *positive definite*. This is not checked during
initialisation; it fail or give incorrect results in :meth:`~CPdf.eval_log` or
:meth:`~CPdf.sample`. To create standard normal distribution:
>>> # note that cov is a matrix because of the double [[ and ]]
>>> norm = GaussPdf(np.array([0.]), np.array([[1.]]))
"""
if mean.ndim != 1:
raise ValueError("mean must be one-dimensional (" + str(mean.ndim) + " dimensions encountered)")
n = mean.shape[0]
if cov.ndim != 2:
raise ValueError("cov must be two-dimensional")
if cov.shape[0] != n or cov.shape[1] != n:
raise ValueError("cov must have shape (" + str(n) + ", " + str(n) + "), " +
str(cov.shape) + " given")
if np.any(cov != cov.T):
raise ValueError("cov must be symmetric (complex covariance not supported)")
self.mu = mean
self.R = cov
self._set_rv(mean.shape[0], rv)
def __str__(self):
return "<pybayes.pdfs.GaussPdf mu={0} R={1}>".format(np.asarray(self.mu), np.asarray(self.R))
def mean(self, cond = None):
return self.mu
def variance(self, cond = None):
return np.diag(self.R)
def eval_log(self, x, cond = None):
self._check_x(x)
# compute logarithm of normalization constant (can be cached somehow in future)
# log(2*Pi) = 1.83787706640935
log_norm = -1/2. * (self.mu.shape[0]*1.83787706640935 + math.log(linalg.det(self.R)))
# part that actually depends on x
distance = np.subtract_vv(x, self.mu)
log_val = -1/2. * np.dot_vv(distance, np.dot_mv(linalg.inv(self.R), distance))
return log_norm + log_val # = log(norm*val)
def sample(self, cond = None):
# in univariate case, random.normal() can be used directly:
if self.mu.shape[0] == 1:
return random.normal(loc=self.mu[0], scale=math.sqrt(self.R[0,0]), size=1)
z = random.normal(size=self.mu.shape[0]);
# NumPy's cholesky(R) is equivalent to Matlab's chol(R).transpose()
return np.add_vv(self.mu, np.dot_mv(linalg.cholesky(self.R), z), z);
[docs]class LogNormPdf(AbstractGaussPdf):
r"""Unconditional log-normal probability density function. Extends
:class:`AbstractGaussPdf`.
More precisely, the density of random variable :math:`Y` where
:math:`Y = exp(X); ~ X \sim \mathcal{N}(\mu, R)`
"""
[docs] def __init__(self, mean, cov, rv = None):
r"""Initialise log-normal pdf.
:param mean: mean value of the **logarithm** of the associated random variable
:type mean: 1D :class:`numpy.ndarray`
:param cov: covariance matrix of the **logarithm** of the associated random variable
:type cov: 2D :class:`numpy.ndarray`
A current limitation is that LogNormPdf is only univariate. To create
standard log-normal distribution:
>>> lognorm = LogNormPdf(np.array([0.]), np.array([[1.]])) # note the shape of covariance
"""
if mean.ndim != 1:
raise ValueError("mean must be one-dimensional (" + str(mean.ndim) + " dimensions encountered)")
n = mean.shape[0]
if n != 1:
raise ValueError("LogNormPdf is currently limited to univariate random variables")
if cov.ndim != 2:
raise ValueError("cov must be two-dimensional")
if cov.shape[0] != n or cov.shape[1] != n:
raise ValueError("cov must have shape (" + str(n) + ", " + str(n) + "), " +
str(cov.shape) + " given")
if cov[0,0] <= 0.:
raise ValueError("cov must be positive")
self.mu = mean
self.R = cov
self._set_rv(1, rv)
def mean(self, cond = None):
ret = np.vector(1)
ret[0] = math.exp(self.mu[0] + self.R[0, 0]/2.)
return ret
def variance(self, cond = None):
ret = np.vector(1)
ret[0] = (math.exp(self.R[0, 0]) - 1.) * math.exp(2*self.mu[0]+ self.R[0, 0])
return ret
def eval_log(self, x, cond = None):
self._check_x(x)
if x[0] <= 0.: # log-normal pdf support = (0, +inf)
return float('-inf')
# 1/2.*log(2*pi) = 0.91893853320467
return -((math.log(x[0]) - self.mu[0])**2)/(2.*self.R[0, 0]) - math.log(x[0]*math.sqrt(self.R[0, 0])) - 0.91893853320467
def sample(self, cond = None):
# size parameter ( = 1) makes lognormal() return an array
return random.lognormal(self.mu[0], math.sqrt(self.R[0,0]), 1)
[docs]class TruncatedNormPdf(Pdf):
r"""One-dimensional Truncated Normal distribution.
Suppose :math:`X \sim \mathcal{N}(\mu, \sigma^2) ~ \mathrm{and} ~ Y = X | a \leq x \leq b.`
Then :math:`Y \sim t\mathcal{N}(\mu, \sigma^2, a, b).` :math:`a` may be
:math:`-\infty` and :math:`b` may be :math:`+\infty.`
"""
[docs] def __init__(self, mean, sigma_sq, a = float('-inf'), b = float('+inf'), rv=None):
r"""Initialise Truncated Normal distribution.
:param double mean: :math:`\mu`
:param double sigma_sq: :math:`\sigma^2`
:param double a: :math:`a,` defaults to :math:`-\infty`
:param double b: :math:`b,` defaults to :math:`+\infty`
To create Truncated Normal distribution constrained to :math:`[0, +\infty)`:
>>> tnorm = TruncatedNormPdf(0., 1., a=0.)
To create Truncated Normal distribution constrained to :math:`[-1, 1]`:
>>> tnorm = TruncatedNormPdf(0., 1., a=-1., b=1.)
"""
assert a < float('+inf')
assert b > float('-inf')
assert a < b
self.mu = mean
self.sigma_sq = sigma_sq
self.a = a
self.b = b
self._set_rv(1, rv)
def mean(self, cond = None):
ret = np.vector(1)
Z = self._cdf(self.b) - self._cdf(self.a)
ret[0] = self.mu + (self._pdf(self.a) - self._pdf(self.b)) / Z
return ret
def variance(self, cond = None):
ret = np.vector(1)
Z = self._cdf(self.b) - self._cdf(self.a)
pdf_a = self._pdf(self.a)
pdf_b = self._pdf(self.b)
ret[0] = 1. + (((self.a - self.mu) * pdf_a if pdf_a else 0.) - ((self.b - self.mu) * pdf_b if pdf_b else 0.)) / Z
ret[0] -= self.sigma_sq * ((pdf_a - pdf_b) / Z)**2.
ret[0] *= self.sigma_sq
return ret
def eval_log(self, x, cond = None):
self._check_x(x)
if x[0] < self.a or x[0] > self.b:
return float('-inf')
Z = self._cdf(self.b) - self._cdf(self.a)
return math.log(self._pdf(x[0]) / Z)
def sample(self, cond = None):
# TODO: more efficient algo, SciPy?
for i in range(100):
ret = random.normal(loc=self.mu, scale=math.sqrt(self.sigma_sq), size=1)
if self.a <= ret[0] <= self.b:
return ret;
raise AttributeError("Failed to reach interval within 100 rejection sampling " +
"iterations, more efficient algo needed. mu={0}; ".format(self.mu) +
"sigma_sq={0}; a={1}; b={2}".format(self.sigma_sq, self.a, self.b))
def _pdf(self, x):
"""Return value of the pdf of the Normal distribution with self.mu and self.sigma_sq
parameters in point x"""
if x == float('-inf') or x == float('+inf'):
return 0.
return 1. / math.sqrt(2. * math.pi * self.sigma_sq) * \
math.exp( -(x - self.mu)**2. / (2. * self.sigma_sq))
def _cdf(self, x):
"""Return value of the cdf of the Normal distribution with self.mu and self.sigma_sq
parameters in point x"""
if x == float('+inf'):
return 1.
if x == float('-inf'):
return 0.
return 1. / 2. + math.erf((x - self.mu) / math.sqrt(2. * self.sigma_sq)) / 2.
[docs]class GammaPdf(Pdf):
r"""Gamma distribution with shape parameter :math:`k` and scale parameter
:math:`\theta`. Extends :class:`Pdf`.
.. math:: f(x) = \frac{1}{\Gamma(k)\theta^k} x^{k-1} e^{\frac{-x}{\theta}}
"""
[docs] def __init__(self, k, theta, rv = None):
r"""Initialise Gamma pdf.
:param double k: :math:`k` shape parameter above
:param double theta: :math:`\theta` scale parameter above
"""
assert k > 0.
assert theta > 0.
self.k = k
self.theta = theta
self._set_rv(1, rv)
def mean(self, cond = None):
return np.array([self.k * self.theta])
def variance(self, cond = None):
return np.array([self.k * self.theta**2])
def eval_log(self, x, cond = None):
self._check_x(x)
if x[0] <= 0.:
return float('-inf')
return -math.lgamma(self.k) - self.k*math.log(self.theta) + (self.k - 1)*math.log(x[0]) - x[0]/self.theta
def sample(self, cond = None):
return random.gamma(self.k, self.theta, size=(1,))
[docs]class InverseGammaPdf(Pdf):
r"""Inverse gamma distribution with shape parameter :math:`\alpha` and scale parameter
:math:`\beta`. Extends :class:`Pdf`.
If random variable :math:`X \sim \text{Gamma}(k, \theta)` then :math:`Y = 1/X` will
have distribution :math:`\text{InverseGamma}(k, 1/\theta)` i.e.
:math:`\alpha = k, \beta = 1/\theta`
.. math:: f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha-1} e^{\frac{-\beta}{x}}
"""
[docs] def __init__(self, alpha, beta, rv = None):
r"""Initialise Inverse gamma pdf.
:param double alpha: :math:`\alpha` shape parameter above
:param double beta: :math:`\beta` scale parameter above
"""
assert alpha > 0.
assert beta > 0.
self.alpha = alpha
self.beta = beta
self._set_rv(1, rv)
def mean(self, cond = None):
if self.alpha <= 1.0:
raise NotImplementedError("Indeterminate form")
return np.array([self.beta / (self.alpha - 1.0)])
def variance(self, cond = None):
if self.alpha <= 2.0:
raise NotImplementedError("Indeterminate form")
return np.array([self.beta**2 / ((self.alpha - 2.0)*(self.alpha - 1.0)**2)])
def eval_log(self, x, cond = None):
self._check_x(x)
if x[0] <= 0.:
return float('-inf')
return self.alpha*math.log(self.beta) - math.lgamma(self.alpha) \
+ (-self.alpha - 1.0)*math.log(x[0]) - self.beta/x[0]
def sample(self, cond = None):
return 1.0 / random.gamma(self.alpha, 1.0/self.beta, size=(1,))
[docs]class AbstractEmpPdf(Pdf):
r"""An abstraction of empirical probability density functions that provides common methods such
as weight normalisation. Extends :class:`Pdf`.
:var numpy.ndarray weights: 1D array of particle weights
:math:`\omega_i >= 0 \forall i; \quad \sum \omega_i = 1`
"""
[docs] def normalise_weights(self):
r"""Multiply weights by appropriate constant so that :math:`\sum \omega_i = 1`
:raise AttributeError: when :math:`\exists i: \omega_i < 0` or
:math:`\forall i: \omega_i = 0`
"""
wsum = np.sum_v(self.weights)
if wsum == 0:
raise AttributeError("Sum of weights == 0: weights cannot be normalised")
# self.weights *= 1./wsum
np.multiply_vs(self.weights, 1./wsum, self.weights)
return True
[docs] def get_resample_indices(self):
r"""Calculate first step of resampling process (dropping low-weight particles and
replacing them with more weighted ones.
:return: integer array of length n: :math:`(a_1, a_2 \dots a_n)` where
:math:`a_i` means that particle at ith place should be replaced with particle
number :math:`a_i`
:rtype: :class:`numpy.ndarray` of ints
*This method doesnt modify underlying pdf in any way - it merely calculates how
particles should be replaced.*
"""
n = self.weights.shape[0]
cum_weights = np.cumsum(self.weights)
u = np.vector(n)
fuzz = random.uniform()
for i in range(n):
u[i] = (i + fuzz) / n
# calculate number of babies for each particle
baby_indices = np.index_vector(n) # index array: a[i] contains index of
# original particle that should be at i-th place in new particle array
j = 0
for i in range(n):
while u[i] > cum_weights[j]:
j += 1
baby_indices[i] = j
return baby_indices
[docs]class EmpPdf(AbstractEmpPdf):
r"""Weighted empirical probability density function. Extends :class:`AbstractEmpPdf`.
.. math::
p(x) &= \sum_{i=1}^n \omega_i \delta(x - x^{(i)}) \\
\text{where} \quad x^{(i)} &\text{ is value of the i}^{th} \text{ particle} \\
\omega_i \geq 0 &\text{ is weight of the i}^{th} \text{ particle} \quad \sum \omega_i = 1
:var numpy.ndarray particles: 2D array of particles; shape: (n, m) where n
is the number of particles, m dimension of this pdf
You may alter particles and weights, but you must ensure that their shapes
match and that weight constraints still hold. You can use
:meth:`~AbstractEmpPdf.normalise_weights` to do some work for you.
"""
[docs] def __init__(self, init_particles, rv = None):
r"""Initialise empirical pdf.
:param init_particles: 2D array of initial particles; shape (*n*, *m*)
determines that *n* *m*-dimensioned particles will be used. *Warning:
EmpPdf does not copy the particles - it rather uses passed array
through its lifetime, so it is not safe to reuse it for other
purposes.*
:type init_particles: :class:`numpy.ndarray`
"""
if init_particles.ndim != 2:
raise TypeError("init_particles must be a 2D array")
self.particles = init_particles
# set n weights to 1/n
self.weights = np.ones(self.particles.shape[0]) / self.particles.shape[0]
self._set_rv(init_particles.shape[1], rv)
def mean(self, cond = None):
ret = np.zeros(self.particles.shape[1])
for i in range(self.particles.shape[0]):
# ret += self.weights[i] * self.particles[i]:
np.add_vv(ret, np.multiply_vs(self.particles[i], self.weights[i]), ret)
return ret
def variance(self, cond = None):
ret = np.zeros(self.particles.shape[1])
for i in range(self.particles.shape[0]):
# ret += self.weights[i] * self.particles[i]**2
np.add_vv(ret, np.multiply_vs(np.power_vs(self.particles[i], 2.), self.weights[i]), ret)
# return ret - self.mean()**2
return np.subtract_vv(ret, np.power_vs(self.mean(), 2.), ret)
def eval_log(self, x, cond = None):
raise NotImplementedError("eval_log doesn't make sense for discrete distribution")
def sample(self, cond = None):
raise NotImplementedError("Sample for empirical pdf not (yet?) implemented")
[docs] def resample(self):
"""Drop low-weight particles, replace them with copies of more weighted
ones. Also reset weights to uniform."""
np.reindex_mv(self.particles, self.get_resample_indices())
self.weights[:] = 1./self.weights.shape[0]
return True
[docs] def transition_using(self, i, transition_cpdf):
r"""Transition *i*-th particle from :math:`t-1` to :math:`t` by sampling from
*transition_cpdf* :math:`p(x_t|x_{t-1})`
"""
self.particles[i] = transition_cpdf.sample(self.particles[i])
return True
[docs]class MarginalizedEmpPdf(AbstractEmpPdf):
r"""An extension to empirical pdf (:class:`EmpPdf`) used as posterior density by
:class:`~pybayes.filters.MarginalizedParticleFilter`. Extends :class:`AbstractEmpPdf`.
Assume that random variable :math:`x` can be divided into 2 independent
parts :math:`x = [a, b]`, then probability density function can be written as
.. math::
p(a, b) &= \sum_{i=1}^n \omega_i \Big[ \mathcal{N}\left(\hat{a}^{(i)}, P^{(i)}\right) \Big]_a
\delta(b - b^{(i)}) \\
\text{where } \quad \hat{a}^{(i)} &\text{ and } P^{(i)} \text{ is mean and
covariance of i}^{th} \text{ gauss pdf} \\
b^{(i)} &\text{ is value of the (second part of the) i}^{th} \text{ particle} \\
\omega_i \geq 0 &\text{ is weight of the i}^{th} \text{ particle} \quad \sum \omega_i = 1
:var numpy.ndarray gausses: 1D array that holds :class:`GaussPdf`
for each particle; shape: (n) where n is the number of particles
:var numpy.ndarray particles: 2D array of particles; shape: (n, m) where n
is the number of particles, m dimension of the "empirical" part of random variable
You may alter particles and weights, but you must ensure that their shapes
match and that weight constraints still hold. You can use
:meth:`~AbstractEmpPdf.normalise_weights` to do some work for you.
*Note: this pdf could have been coded as ProdPdf of EmpPdf and a mixture of GaussPdfs. However
it is implemented explicitly for simplicity and speed reasons.*
"""
[docs] def __init__(self, init_gausses, init_particles, rv = None):
r"""Initialise marginalized empirical pdf.
:param init_gausses: 1D array of :class:`GaussPdf` objects, all must have
the dimension
:type init_gausses: :class:`numpy.ndarray`
:param init_particles: 2D array of initial particles; shape (*n*, *m*)
determines that *n* particles whose *empirical* part will have dimension *m*
:type init_particles: :class:`numpy.ndarray`
*Warning: MarginalizedEmpPdf does not copy the particles - it rather uses
both passed arrays through its lifetime, so it is not safe to reuse them
for other purposes.*
"""
if init_gausses.ndim != 1:
raise TypeError("init_gausses must be 1D array")
if init_particles.ndim != 2:
raise TypeError("init_particles must be 2D array")
if init_gausses.shape[0] != init_particles.shape[0] or init_gausses.shape[0] < 1:
raise ValueError("init_gausses count must be same as init_particles count and both must be positive")
gauss_shape = 0
for gauss in init_gausses:
if not isinstance(gauss, GaussPdf):
raise TypeError("all init_gausses items must be (subclasses of) GaussPdf")
if gauss_shape == 0:
gauss_shape = gauss.shape() # guaranteed to be non-zero
elif gauss.shape() != gauss_shape:
raise ValueError("all init_gausses items must have same shape")
self.gausses = init_gausses
self.particles = init_particles
# set n weights to 1/n
self.weights = np.ones(self.particles.shape[0]) / self.particles.shape[0]
self._gauss_shape = gauss_shape # shape of the gaussian component
part_shape = self.particles.shape[1] # shape of the empirical component
self._set_rv(self._gauss_shape + part_shape, rv)
def mean(self, cond = None):
ret = np.zeros(self.shape())
temp = np.vector(self.shape())
for i in range(self.particles.shape[0]):
gauss = self.gausses[i] # work-around Cython bug
temp[0:self._gauss_shape] = gauss.mean()
temp[self._gauss_shape:] = self.particles[i]
np.add_vv(ret, np.multiply_vs(temp, self.weights[i], temp), ret) # ret += self.weights[i] * temp
return ret
def variance(self, cond = None):
# first, compute 2nd non-central moment
mom2 = np.zeros(self.shape())
temp = np.vector(self.shape())
for i in range(self.particles.shape[0]):
gauss = self.gausses[i] # work-around Cython bug
# set gauss part of temp to \mu_i^2 + \sigma_i^2
mean = gauss.mean()
var = gauss.variance()
np.add_vv(np.power_vs(mean, 2.), np.power_vs(var, 2.), temp[0:self._gauss_shape]) # temp[0:self._gauss_shape] = mean**2 + var**2
# set empirical part of temp to x_i^2
np.power_vs(self.particles[i], 2., temp[self._gauss_shape:]) # temp[self._gauss_shape:] = self.particles[i]**2
# finaly scale by \omega_i and add to 2nd non-central moment we are computing
np.add_vv(mom2, np.multiply_vs(temp, self.weights[i], temp), mom2) # mom2 += self.weights[i] * temp
# return 2nd central moment by subtracting square of mean value
return np.subtract_vv(mom2, np.power_vs(self.mean(), 2., temp), mom2) # return mom2 - self.mean()**2
def eval_log(self, x, cond = None):
raise NotImplementedError("eval_log doesn't make sense for (partially) discrete distribution")
def sample(self, cond = None):
raise NotImplementedError("Drawing samples from MarginalizesEmpPdf is not supported")
[docs]class ProdPdf(Pdf):
r"""Unconditional product of multiple unconditional pdfs.
You can for example create a pdf that has uniform distribution with regards
to x-axis and normal distribution along y-axis. The caller (you) must ensure
that individial random variables are independent, otherwise their product may
have no mathematical sense. Extends :class:`Pdf`.
.. math:: f(x_1 x_2 x_3) = f_1(x_1) f_2(x_2) f_3(x_3)
"""
[docs] def __init__(self, factors, rv = None):
r"""Initialise product of unconditional pdfs.
:param factors: sequence of sub-distributions
:type factors: sequence of :class:`Pdf`
As an exception from the general rule, ProdPdf does not create anonymous
associated random variable if you do not supply it in constructor - it
rather reuses components of underlying factor pdfs. (You can of course
override this behaviour by bassing custom **rv**.)
Usual way of creating ProdPdf could be:
>>> prod = ProdPdf((UniPdf(...), GaussPdf(...))) # note the double (( and ))
"""
if rv is None:
rv_comps = [] # prepare to construnct associated rv
else:
rv_comps = None
if len(factors) is 0:
raise ValueError("at least one factor must be passed")
self.factors = np.array(factors, dtype=Pdf)
self.shapes = np.index_vector(self.factors.shape[0]) # array of factor shapes
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
if not isinstance(factor, Pdf):
raise TypeError("all records in factors must be (subclasses of) Pdf")
self.shapes[i] = factor.shape()
if rv_comps is not None:
rv_comps.extend(factor.rv.components) # add components of child rvs
# pre-calclate shape
shape = self._calculate_shape()
# associate with a rv (needs to be after _shape calculation)
if rv_comps is None:
self._set_rv(shape, rv)
else:
self._set_rv(shape, RV(*rv_comps))
def mean(self, cond = None):
curr = 0
ret = np.zeros(self.shape())
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
ret[curr:curr + self.shapes[i]] = factor.mean()
curr += self.shapes[i]
return ret;
def variance(self, cond = None):
curr = 0
ret = np.zeros(self.shape())
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
ret[curr:curr + self.shapes[i]] = factor.variance()
curr += self.shapes[i]
return ret;
def eval_log(self, x, cond = None):
self._check_x(x)
curr = 0
ret = 0. # 1 is neutral element in multiplication; log(1) = 0
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
ret += factor.eval_log(x[curr:curr + self.shapes[i]]) # log(x*y) = log(x) + log(y)
curr += self.shapes[i]
return ret;
def sample(self, cond = None):
curr = 0
ret = np.zeros(self.shape())
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
ret[curr:curr + self.shapes[i]] = factor.sample()
curr += self.shapes[i]
return ret;
def _calculate_shape(self):
"""Work-around for Cython"""
return np.sum_v(self.shapes)
[docs]class MLinGaussCPdf(CPdf):
r"""Conditional Gaussian pdf whose mean is a linear function of condition.
Extends :class:`CPdf`.
.. math::
f(x|c) \propto \exp \left( - \left( x-\mu \right)' R^{-1} \left( x-\mu \right) \right)
\quad \quad \text{where} ~ \mu = A c + b
"""
[docs] def __init__(self, cov, A, b, rv = None, cond_rv = None, base_class = None):
r"""Initialise Mean-Linear Gaussian conditional pdf.
:param cov: covariance of underlying Gaussian pdf
:type cov: 2D :class:`numpy.ndarray`
:param A: given condition :math:`c`, :math:`\mu = Ac + b`
:type A: 2D :class:`numpy.ndarray`
:param b: see above
:type b: 1D :class:`numpy.ndarray`
:param class base_class: class whose instance is created as a base pdf for this
cpdf. Must be a subclass of :class:`AbstractGaussPdf` and the default is
:class:`GaussPdf`. One alternative is :class:`LogNormPdf` for example.
"""
if base_class is None:
self.gauss = GaussPdf(np.zeros(cov.shape[0]), cov)
else:
if not issubclass(base_class, AbstractGaussPdf):
raise TypeError("base_class must be a class (not an instance) and subclass of AbstractGaussPdf")
self.gauss = base_class(np.zeros(cov.shape[0]), cov)
self.A = np.asarray(A)
self.b = np.asarray(b)
if self.A.ndim != 2:
raise ValueError("A must be 2D array (matrix)")
if self.b.ndim != 1:
raise ValueError("b must be 1D array (vector)")
if self.b.shape[0] != self.gauss.shape():
raise ValueError("b must have same number of cols as covariance")
if self.A.shape[0] != self.b.shape[0]:
raise ValueError("A must have same number of rows as covariance")
self._set_rvs(self.b.shape[0], rv, self.A.shape[1], cond_rv)
def mean(self, cond = None):
# note: it may not be true that gauss.mu == gauss.mean() for all AbstractGaussPdf
# classes. One such example is LogNormPdf
self._set_mean(cond)
return self.gauss.mean()
def variance(self, cond = None):
# note: for some AbstractGaussPdf variance may depend on mu
self._set_mean(cond)
return self.gauss.variance()
def eval_log(self, x, cond = None):
# x is checked in self.gauss
self._set_mean(cond)
return self.gauss.eval_log(x)
def sample(self, cond = None):
self._set_mean(cond)
return self.gauss.sample()
def _set_mean(self, cond):
self._check_cond(cond)
# self.gauss.mu = self.A * cond
np.dot_mv(self.A, cond, self.gauss.mu)
# self.gauss.mu += self.b
np.add_vv(self.gauss.mu, self.b, self.gauss.mu)
return True
[docs]class LinGaussCPdf(CPdf):
r"""Conditional one-dimensional Gaussian pdf whose mean and covariance are
linear functions of condition. Extends :class:`CPdf`.
.. math::
f(x|c_1 c_2) \propto \exp \left( - \frac{\left( x-\mu \right)^2}{2\sigma^2} \right)
\quad \quad \text{where} \quad \mu = a c_1 + b \quad \text{and}
\quad \sigma^2 = c c_2 + d
"""
[docs] def __init__(self, a, b, c, d, rv = None, cond_rv = None, base_class = None):
r"""Initialise Linear Gaussian conditional pdf.
:param double a, b: mean = a*cond_1 + b
:param double c, d: covariance = c*cond_2 + d
:param class base_class: class whose instance is created as a base pdf for this
cpdf. Must be a subclass of :class:`AbstractGaussPdf` and the default is
:class:`GaussPdf`. One alternative is :class:`LogNormPdf` for example.
"""
if not isinstance(a, float):
raise TypeError("all parameters must be floats")
self.a = a
if not isinstance(b, float):
raise TypeError("all parameters must be floats")
self.b = b
if not isinstance(c, float):
raise TypeError("all parameters must be floats")
self.c = c
if not isinstance(d, float):
raise TypeError("all parameters must be floats")
self.d = d
if base_class is None:
self.gauss = GaussPdf(np.zeros(1), np.array([[1.]]))
else:
if not issubclass(base_class, AbstractGaussPdf):
raise TypeError("base_class must be a class (not an instance) and subclass of AbstractGaussPdf")
self.gauss = base_class(np.zeros(1), np.array([[1.]]))
self._set_rvs(1, rv, 2, cond_rv)
def mean(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.mean()
def variance(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.variance()
def eval_log(self, x, cond = None):
self._set_gauss_params(cond)
# x is checked in self.gauss.eval_log()
return self.gauss.eval_log(x)
def sample(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.sample()
def _set_gauss_params(self, cond):
self._check_cond(cond)
c0 = cond[0] # workaround for cython limitation: no buffer type in pure python mode
c1 = cond[1]
self.gauss.mu[0] = self.a*c0 + self.b
self.gauss.R[0,0] = self.c*c1 + self.d
return True
[docs]class GaussCPdf(CPdf):
r"""The most general normal conditional pdf. Use it only if you cannot use
:class:`MLinGaussCPdf` or :class:`LinGaussCPdf` as this cpdf is least
optimised. Extends :class:`CPdf`.
.. math::
f(x|c) &\propto \exp \left( - \left( x-\mu \right)' R^{-1} \left( x-\mu \right) \right) \\
\text{where} \quad \mu &= f(c) \text{ (interpreted as n-dimensional vector)} \\
R &= g(c) \text{ (interpreted as n*n matrix)}
"""
[docs] def __init__(self, shape, cond_shape, f, g, rv = None, cond_rv = None, base_class = None):
r"""Initialise general gauss cpdf.
:param int shape: dimension of random variable
:param int cond_shape: dimension of conditioning variable
:param callable f: :math:`\mu = f(c)` where c = condition
:param callable g: :math:`R = g(c)` where c = condition
:param class base_class: class whose instance is created as a base pdf for this
cpdf. Must be a subclass of :class:`AbstractGaussPdf` and the default is
:class:`GaussPdf`. One alternative is :class:`LogNormPdf` for example.
*Please note that the way of specifying callback function f and g is not yet fixed and may
be changed in future.*
"""
self.f = f
self.g = g
if base_class is None:
self.gauss = GaussPdf(np.zeros(shape), np.eye(shape))
else:
if not issubclass(base_class, AbstractGaussPdf):
raise TypeError("base_class must be a class (not an instance) and subclass of AbstractGaussPdf")
self.gauss = base_class(np.zeros(shape), np.ones((shape, shape)))
self._set_rvs(shape, rv, cond_shape, cond_rv)
def mean(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.mean()
def variance(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.variance()
def eval_log(self, x, cond = None):
self._set_gauss_params(cond)
# x is checked in self.gauss
return self.gauss.eval_log(x)
def sample(self, cond = None):
self._set_gauss_params(cond)
return self.gauss.sample()
def _set_gauss_params(self, cond):
self._check_cond(cond)
self.gauss.mu = self.f(cond).reshape(self.shape())
self.gauss.R = self.g(cond).reshape((self.shape(), self.shape()))
return True
[docs]class GammaCPdf(CPdf):
r"""Conditional pdf based on :class:`GammaPdf` tuned in a way to have mean
:math:`\mu` and standard deviation :math:`\gamma \mu`. In other words,
:math:`\text{GammaCpdf}(\mu, \gamma) = \text{GammaPdf}\left(
k = \gamma^{-2}, \theta = \gamma^2 \mu \right)`
The :math:`\gamma` parameter is specified in constructor and the :math:`\mu`
parameter is the conditioning variable.
"""
[docs] def __init__(self, gamma, rv = None, cond_rv = None):
"""Initialise conditional gamma pdf.
:param float gamma: :math:`\gamma` parameter above
"""
self.gamma = gamma
self.gamma_pdf = GammaPdf(self.gamma**(-2.), 1.) # theta is set later
self._set_rvs(1, rv, 1, cond_rv)
def mean(self, cond = None):
self._set_cond(cond)
return self.gamma_pdf.mean()
def variance(self, cond = None):
self._set_cond(cond)
return self.gamma_pdf.variance()
def eval_log(self, x, cond = None):
self._set_cond(cond)
# x is checked in self.igamma_pdf
return self.gamma_pdf.eval_log(x)
def sample(self, cond = None):
self._set_cond(cond)
return self.gamma_pdf.sample()
def _set_cond(self, cond):
self._check_cond(cond)
self.gamma_pdf.theta = self.gamma**2. * cond[0]
[docs]class InverseGammaCPdf(CPdf):
r"""Conditional pdf based on :class:`InverseGammaPdf` tuned in a way to have mean
:math:`\mu` and standard deviation :math:`\gamma \mu`. In other words,
:math:`\text{InverseGammaCpdf}(\mu, \gamma) = \text{InverseGammaPdf}\left(
\alpha = \gamma^{-2} + 2, \beta = \left( \gamma^{-2} + 1 \right) \mu \right)`
The :math:`\gamma` parameter is specified in constructor and the :math:`\mu`
parameter is the conditioning variable.
"""
[docs] def __init__(self, gamma, rv = None, cond_rv = None):
"""Initialise conditional inverse gamma pdf.
:param float gamma: :math:`\gamma` parameter above
"""
self.gamma = gamma
self.igamma_pdf = InverseGammaPdf(self.gamma**(-2.) + 2., 1.) # beta is set later
self._set_rvs(1, rv, 1, cond_rv)
def mean(self, cond = None):
self._set_cond(cond)
return self.igamma_pdf.mean()
def variance(self, cond = None):
self._set_cond(cond)
return self.igamma_pdf.variance()
def eval_log(self, x, cond = None):
self._set_cond(cond)
# x is checked in self.igamma_pdf
return self.igamma_pdf.eval_log(x)
def sample(self, cond = None):
self._set_cond(cond)
return self.igamma_pdf.sample()
def _set_cond(self, cond):
self._check_cond(cond)
self.igamma_pdf.beta = (self.gamma**(-2.) + 1.)*cond[0]
[docs]class ProdCPdf(CPdf):
r"""Pdf that is formed as a chain rule of multiple conditional pdfs.
Extends :class:`CPdf`.
In a
simple textbook case denoted below it isn't needed to specify random variables
at all. In this case when no random variable associations are passed,
ProdCPdf ignores rv associations of its factors and everything is determined
from their order. (:math:`x_i` are arbitrary vectors)
.. math::
p(x_1 x_2 x_3 | c) &= p(x_1 | x_2 x_3 c) p(x_2 | x_3 c) p(x_3 | c) \\
\text{or} \quad p(x_1 x_2 x_3) &= p(x_1 | x_2 x_3) p(x_2 | x_3) p(x_3)
>>> f = ProdCPdf((f1, f2, f3))
For less simple situations, specifiying random value associations is needed
to estabilish data-flow:
.. math:: p(x_1 x_2 | y_1 y_2) = p_1(x_1 | y_1) p_2(x_2 | y_2 y_1)
>>> # prepare random variable components:
>>> x_1, x_2 = RVComp(1), RVComp(1, "name is optional")
>>> y_1, y_2 = RVComp(1), RVComp(1, "but recommended")
>>> p_1 = SomePdf(..., rv=[x_1], cond_rv=[x_2])
>>> p_2 = SomePdf(..., rv=[x_2], cond_rv=[y_2, y_1])
>>> p = ProdCPdf((p_2, p_1), rv=[x_1, x_2], cond_rv=[y_1, y_2]) # order of
>>> # pdfs is insignificant - order of rv components determines data flow
*Please note: this will change in near future in following way: it will be always required to
specify rvs and cond_rvs of factor pdfs (at least ones that are shared), but product rv and
cond_rv will be inferred automatically when not specified.*
"""
[docs] def __init__(self, factors, rv = None, cond_rv = None):
r"""Construct chain rule of multiple cpdfs.
:param factors: sequence of densities that will form the product
:type factors: sequence of :class:`CPdf`
Usual way of creating ProdCPdf could be:
>>> prod = ProdCPdf([MLinGaussCPdf(..), UniPdf(..)], rv=[..], cond_rv=[..])
"""
if len(factors) is 0:
raise ValueError("at least one factor must be passed")
self.in_indeces = [] # data link representations
self.out_indeces = []
if rv is None and cond_rv is None:
(shape, cond_shape) = self._init_anonymous(factors)
elif rv is not None and cond_rv is not None:
# needs factors as list
(shape, cond_shape) = self._init_with_rvs(list(factors), rv, cond_rv)
else:
raise AttributeError("Please pass both rv and cond_rv or none of them, other combinations not (yet) supported")
self._set_rvs(shape, rv, cond_shape, cond_rv)
def _init_anonymous(self, factors):
self.factors = np.array(factors, dtype=CPdf)
# overall cond shape equals last factor cond shape:
cum_cond_shape = factors[-1].cond_shape()
cum_shape = factors[0].shape() + factors[0].cond_shape() - cum_cond_shape
start_ind = 0 # current start index in cummulate rv and cond_rv data array
for i in range(self.factors.shape[0]):
factor = self.factors[i]
if not isinstance(factor, CPdf):
raise TypeError("all records in factors must be (subclasses of) CPdf")
shape = factor.shape()
cond_shape = factor.cond_shape()
# expected (normal + cond) shape:
exp_shape = cum_shape + cum_cond_shape - start_ind
if shape + cond_shape != exp_shape:
raise ValueError("Expected that pdf {0} will have shape (={1}) + ".
format(factor, shape) + "cond_shape (={0}) == {1}".
format(cond_shape, exp_shape))
if cond_shape > 0:
self.in_indeces.append(np.index_range(start_ind + shape, start_ind + shape + cond_shape))
else:
self.in_indeces.append(None)
self.out_indeces.append(np.index_range(start_ind, start_ind + shape))
start_ind += shape
if start_ind != cum_shape:
raise ValueError("Shapes do not match")
return (cum_shape, cum_cond_shape)
def _init_with_rvs(self, factors, rv, cond_rv):
"""Initialise ProdCPdf using rv components for data chain construction.
:param factors: factor pdfs that will form the product
:type factors: :class:`list` of :class:`CPdf` items
"""
# rv, cond_rv could be lists, so convert them if needed
rv = RV(rv)
cond_rv = RV(cond_rv)
# gradually filled set of components that would be available in e.g.
# sample() computation:
avail_rvcomps = set(cond_rv.components)
self.factors = empty(len(factors), dtype=CPdf) # initialise factor array
i = self.factors.shape[0] - 1 # factors are filled from right to left
# iterate until all input pdfs are processed
while len(factors) > 0:
# find next pdf that can be added to data chain (all its cond
# components can be already computed)
for j in range(len(factors)):
factor = factors[j]
if not isinstance(factor, CPdf):
raise TypeError("all records in factors must be (subclasses of) CPdf")
if factor.cond_rv.contained_in(avail_rvcomps):
# one such pdf found
#DEBUG: print "Appropriate pdf found:", factor, "with rv:", factor.rv, "and cond_rv:", factor.cond_rv
if not rv.contains_all(factor.rv.components):
raise AttributeError(("Some of {0}'s associated rv components "
+ "({1}) aren't present in rv ({2})").format(factor, factor.rv, rv))
avail_rvcomps.update(factor.rv.components)
self.factors[i] = factor
i += -1
del factors[j]
break;
else:
# we are stuck somewhere in data chain
print "Appropriate pdf not found. avail_rvcomps:", avail_rvcomps, "candidates:"
for factor in factors:
print " ", factor, "with cond_rv:", factor.cond_rv
raise AttributeError("Cannont construct data chain. This means "
+ "that it is impossible to arrange factor pdfs into a DAG "
+ "that starts with ProdCPdf's cond_rv components. Please "
+ "check cond_rv and factor rvs and cond_rvs.")
if not rv.contained_in(avail_rvcomps):
print "These components can be computed:", avail_rvcomps
print "... but we have to fill following rv:", rv
raise AttributeError("Data chain built, some components cannot be "
+ "computed with it.")
cummulate_rv = RV(rv, cond_rv)
for i in range(self.factors.shape[0]):
factor = self.factors[i]
if factor.cond_rv.dimension > 0:
self.in_indeces.append(factor.cond_rv.indexed_in(cummulate_rv))
else:
self.in_indeces.append(None)
self.out_indeces.append(factor.rv.indexed_in(cummulate_rv))
return(rv.dimension, cond_rv.dimension) # in fact no-op, but we already check RV dimensions
def mean(self, cond = None):
raise NotImplementedError("Not yet implemented")
def variance(self, cond = None):
raise NotImplementedError("Not yet implemented")
def eval_log(self, x, cond = None):
self._check_x(x)
self._check_cond(cond)
# combination of evaluation point and condition:
data = np.vector(self.rv.dimension + self.cond_rv.dimension)
data[0:self.rv.dimension] = x
if self.cond_rv.dimension > 0:
data[self.rv.dimension:] = cond
ret = 0.
for i in range(self.factors.shape[0]):
factor = self.factors[i] # work-around Cython bug
# ret += factor.eval_log(data[self.out_indeces[i]], data[self.in_indeces[i]]):
cond = None
if self.in_indeces[i] is not None:
cond = np.take_vv(data, self.in_indeces[i])
ret += factor.eval_log(np.take_vv(data, self.out_indeces[i]), cond)
return ret
def sample(self, cond = None):
self._check_cond(cond)
# combination of sampled variables and condition:
data = np.vector(self.rv.dimension + self.cond_rv.dimension)
if self.cond_rv.dimension > 0:
data[self.rv.dimension:] = cond # rest is undefined
# process pdfs from right to left (they are arranged so that data flow
# is well defined in this case):
for i in range(self.factors.shape[0] -1, -1, -1):
factor = self.factors[i] # work-around Cython bug
# data[self.out_indeces[i]] = factor.sample(data[self.in_indeces[i]]):
cond = None
if self.in_indeces[i] is not None:
cond = np.take_vv(data, self.in_indeces[i])
np.put_vv(data, self.out_indeces[i], factor.sample(cond))
return data[:self.rv.dimension] # return right portion of data