# 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
class sage.stats.hmm.distributions.DiscreteDistribution

x.__init__(...) initializes x; see help(type(x)) for signature

class sage.stats.hmm.distributions.Distribution

Bases: object

A distribution.

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

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

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

class sage.stats.hmm.distributions.GaussianDistribution

x.__init__(...) initializes x; see help(type(x)) for signature

class sage.stats.hmm.distributions.GaussianMixtureDistribution

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

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

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

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

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

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

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

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


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