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#

Bases: Distribution

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:

EXAMPLES:

sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)])
sage: P.plot(-10,30)                                                        # needs sage.plot
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#

Bases: Distribution

class sage.stats.hmm.distributions.GaussianMixtureDistribution#

Bases: 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
fix(i=None)#

Set that this GaussianMixtureDistribution (or its \(i\)-th 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: type(P.sample())
<class 'float'>
sage: l = P.sample(1)
sage: len(l)
1
sage: type(l)
<class 'sage.stats.time_series.TimeSeries'>
sage: l = P.sample(5)
sage: len(l)
5
sage: type(l)
<class 'sage.stats.time_series.TimeSeries'>
sage: l = P.sample(0)
sage: len(l)
0
sage: type(l)
<class 'sage.stats.time_series.TimeSeries'>
sage: P.sample(-3)
Traceback (most recent call last):
...
ValueError: n must be nonnegative
unfix(i=None)#

Set that this GaussianMixtureDistribution (or its \(i\)-th 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 objects.

EXAMPLES:

sage: P = hmm.GaussianMixtureDistribution([(.2,-10,.5),(.6,1,1),(.2,20,.5)])
sage: loads(dumps(P)) == P          # indirect doctest
True