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[source]

Bases: Distribution

class sage.stats.hmm.distributions.Distribution[source]

Bases: object

A distribution.

plot(*args, **kwds)[source]

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)                                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.plot(-Integer(10),Integer(30))                                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive
prob(x)[source]

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

>>> from sage.all import *
>>> import sage.stats.hmm.distributions
>>> sage.stats.hmm.distributions.Distribution().prob(Integer(0))
Traceback (most recent call last):
...
NotImplementedError
sample(n=None)[source]

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

>>> from sage.all import *
>>> import sage.stats.hmm.distributions
>>> sage.stats.hmm.distributions.Distribution().sample()
Traceback (most recent call last):
...
NotImplementedError
class sage.stats.hmm.distributions.GaussianDistribution[source]

Bases: Distribution

class sage.stats.hmm.distributions.GaussianMixtureDistribution[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.3'),Integer(1),Integer(2)),(RealNumber('.7'),-Integer(1),Integer(1))]); P
0.3*N(1.0,2.0) + 0.7*N(-1.0,1.0)
>>> P[Integer(0)]
(0.3, 1.0, 2.0)
>>> P.is_fixed()
False
>>> P.fix(Integer(1))
>>> P.is_fixed(Integer(0))
False
>>> P.is_fixed(Integer(1))
True
>>> P.unfix(Integer(1))
>>> P.is_fixed(Integer(1))
False
fix(i=None)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.fix(Integer(1)); P.is_fixed()
False
>>> P.is_fixed(Integer(1))
True
>>> P.fix(); P.is_fixed()
True
is_fixed(i=None)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.is_fixed()
False
>>> P.is_fixed(Integer(0))
False
>>> P.fix(Integer(0)); P.is_fixed()
False
>>> P.is_fixed(Integer(0))
True
>>> P.fix(); P.is_fixed()
True
prob(x)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.prob(RealNumber('.5'))
0.21123919605857971
>>> P.prob(-Integer(100))
0.0
>>> P.prob(Integer(20))
0.1595769121605731
prob_m(x, m)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.prob_m(RealNumber('.5'), Integer(0))
2.7608117680508...e-97
>>> P.prob_m(RealNumber('.5'), Integer(1))
0.21123919605857971
>>> P.prob_m(RealNumber('.5'), Integer(2))
0.0
sample(n=None)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> type(P.sample())
<class 'float'>
>>> l = P.sample(Integer(1))
>>> len(l)
1
>>> type(l)
<class 'sage.stats.time_series.TimeSeries'>
>>> l = P.sample(Integer(5))
>>> len(l)
5
>>> type(l)
<class 'sage.stats.time_series.TimeSeries'>
>>> l = P.sample(Integer(0))
>>> len(l)
0
>>> type(l)
<class 'sage.stats.time_series.TimeSeries'>
>>> P.sample(-Integer(3))
Traceback (most recent call last):
...
ValueError: n must be nonnegative
unfix(i=None)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> P.fix(Integer(1)); P.is_fixed(Integer(1))
True
>>> P.unfix(Integer(1)); P.is_fixed(Integer(1))
False
>>> P.fix(); P.is_fixed()
True
>>> P.unfix(); P.is_fixed()
False
sage.stats.hmm.distributions.unpickle_gaussian_mixture_distribution_v1(c0, c1, param, fixed)[source]

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

>>> from sage.all import *
>>> P = hmm.GaussianMixtureDistribution([(RealNumber('.2'),-Integer(10),RealNumber('.5')),(RealNumber('.6'),Integer(1),Integer(1)),(RealNumber('.2'),Integer(20),RealNumber('.5'))])
>>> loads(dumps(P)) == P          # indirect doctest
True