Distributions used in implementing Hidden Markov Models¶
These distribution classes are designed specifically for HMM’s and not for general use in statistics. For example, they have fixed or non-fixed status, which only make sense relative to being used in a hidden Markov model.
AUTHOR:
William Stein, 2010-03
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class
sage.stats.hmm.distributions.
DiscreteDistribution
¶
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class
sage.stats.hmm.distributions.
Distribution
¶ Bases:
object
A distribution.
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plot
(*args, **kwds)¶ Return a plot of the probability density function.
INPUT:
args and kwds, passed to the Sage plot function
OUTPUT:
a Graphics object
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.plot(-10,30) Graphics object consisting of 1 graphics primitive
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prob
(x)¶ The probability density function evaluated at x.
INPUT:
x – object
OUTPUT:
float
EXAMPLES:
This method must be defined in a derived class:
sage: import sage.stats.hmm.distributions sage: sage.stats.hmm.distributions.Distribution().prob(0) Traceback (most recent call last): ... NotImplementedError
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sample
(n=None)¶ Return either a single sample (the default) or n samples from this probability distribution.
INPUT:
n – None or a positive integer
OUTPUT:
a single sample if n is 1; otherwise many samples
EXAMPLES:
This method must be defined in a derived class:
sage: import sage.stats.hmm.distributions sage: sage.stats.hmm.distributions.Distribution().sample() Traceback (most recent call last): ... NotImplementedError
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class
sage.stats.hmm.distributions.
GaussianDistribution
¶
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class
sage.stats.hmm.distributions.
GaussianMixtureDistribution
¶ Bases:
sage.stats.hmm.distributions.Distribution
A probability distribution defined by taking a weighted linear combination of Gaussian distributions.
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.3,1,2),(.7,-1,1)]); P 0.3*N(1.0,2.0) + 0.7*N(-1.0,1.0) sage: P[0] (0.3, 1.0, 2.0) sage: P.is_fixed() False sage: P.fix(1) sage: P.is_fixed(0) False sage: P.is_fixed(1) True sage: P.unfix(1) sage: P.is_fixed(1) False
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fix
(i=None)¶ Set that this GaussianMixtureDistribution (or its ith component) is fixed when using Baum-Welch to update the corresponding HMM.
INPUT:
i – None (default) or integer; if given, only fix the i-th component
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.fix(1); P.is_fixed() False sage: P.is_fixed(1) True sage: P.fix(); P.is_fixed() True
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is_fixed
(i=None)¶ Return whether or not this GaussianMixtureDistribution is fixed when using Baum-Welch to update the corresponding HMM.
INPUT:
i – None (default) or integer; if given, only return whether the i-th component is fixed
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.is_fixed() False sage: P.is_fixed(0) False sage: P.fix(0); P.is_fixed() False sage: P.is_fixed(0) True sage: P.fix(); P.is_fixed() True
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prob
(x)¶ Return the probability of x.
Since this is a continuous distribution, this is defined to be the limit of the p’s such that the probability of [x,x+h] is p*h.
INPUT:
x – float
OUTPUT:
float
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.prob(.5) 0.21123919605857971 sage: P.prob(-100) 0.0 sage: P.prob(20) 0.1595769121605731
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prob_m
(x, m)¶ Return the probability of x using just the m-th summand.
INPUT:
x – float
m – integer
OUTPUT:
float
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.prob_m(.5, 0) 2.7608117680508...e-97 sage: P.prob_m(.5, 1) 0.21123919605857971 sage: P.prob_m(.5, 2) 0.0
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sample
(n=None)¶ Return a single sample from this distribution (by default), or if n>1, return a TimeSeries of samples.
INPUT:
n – integer or None (default: None)
OUTPUT:
float if n is None (default); otherwise a TimeSeries
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.sample() 19.65824361087513 sage: P.sample(1) [-10.4683] sage: P.sample(5) [-0.1688, -10.3479, 1.6812, 20.1083, -9.9801] sage: P.sample(0) [] sage: P.sample(-3) Traceback (most recent call last): ... ValueError: n must be nonnegative
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unfix
(i=None)¶ Set that this GaussianMixtureDistribution (or its ith component) is not fixed when using Baum-Welch to update the corresponding HMM.
INPUT:
i – None (default) or integer; if given, only fix the i-th component
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: P.fix(1); P.is_fixed(1) True sage: P.unfix(1); P.is_fixed(1) False sage: P.fix(); P.is_fixed() True sage: P.unfix(); P.is_fixed() False
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sage.stats.hmm.distributions.
unpickle_gaussian_mixture_distribution_v1
(c0, c1, param, fixed)¶ Used in unpickling GaussianMixtureDistribution’s.
EXAMPLES:
sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)]) sage: loads(dumps(P)) == P # indirect doctest True