Hidden Markov Models¶
This is a complete pureCython optimized implementation of Hidden Markov Models. It fully supports Discrete, Gaussian, and Mixed Gaussian emissions.
The best references for the basic HMM algorithms implemented here are:
Tapas Kanungo’s “Hidden Markov Models”
 Jackson’s HMM tutorial:
http://personal.ee.surrey.ac.uk/Personal/P.Jackson/tutorial/
LICENSE: Some of the code in this file is based on reading Kanungo’s GPLv2+ implementation of discrete HMM’s, hence the present code must be licensed with a GPLv2+ compatible license.
AUTHOR:
William Stein, 201003

class
sage.stats.hmm.hmm.
DiscreteHiddenMarkovModel
¶ Bases:
sage.stats.hmm.hmm.HiddenMarkovModel
A discrete Hidden Markov model implemented using double precision floating point arithmetic.
INPUT:
A
– a list of lists or a square N x N matrix, whose (i,j) entry gives the probability of transitioning from state i to state j.B
– a list of N lists or a matrix with N rows, such that B[i,k] gives the probability of emitting symbol k while in state i.pi
– the probabilities of starting in each initial state, i.e,. pi[i] is the probability of starting in state i.emission_symbols
– None or list (default: None); if None, the emission_symbols are the ints [0..N1], where N is the number of states. Otherwise, they are the entries of the list emissions_symbols, which must all be hashable.normalize
–bool (default: True); if given, input is normalized to define valid probability distributions, e.g., the entries of A are made nonnegative and the rows sum to 1, and the probabilities in pi are normalized.
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.5,.5]); m Discrete Hidden Markov Model with 2 States and 2 Emissions Transition matrix: [0.4 0.6] [0.1 0.9] Emission matrix: [0.1 0.9] [0.5 0.5] Initial probabilities: [0.5000, 0.5000] sage: m.log_likelihood([0,1,0,1,0,1]) 4.66693474691329... sage: m.viterbi([0,1,0,1,0,1]) ([1, 1, 1, 1, 1, 1], 5.378832842208748) sage: m.baum_welch([0,1,0,1,0,1]) (0.0, 22) sage: m # rel tol 1e10 Discrete Hidden Markov Model with 2 States and 2 Emissions Transition matrix: [1.0134345614745788e70 1.0] [ 1.0 3.9974352713558623e19] Emission matrix: [ 7.380221566254936e54 1.0] [ 1.0 3.9974352626002193e19] Initial probabilities: [0.0000, 1.0000] sage: m.sample(10) [0, 1, 0, 1, 0, 1, 0, 1, 0, 1] sage: m.graph().plot() Graphics object consisting of 6 graphics primitives
A 3state model that happens to always outputs ‘b’:
sage: m = hmm.DiscreteHiddenMarkovModel([[1/3]*3]*3, [[0,1,0]]*3, [1/3]*3, ['a','b','c']) sage: m.sample(10) ['b', 'b', 'b', 'b', 'b', 'b', 'b', 'b', 'b', 'b']

baum_welch
(obs, max_iter=100, log_likelihood_cutoff=0.0001, fix_emissions=False)¶ Given an observation sequence obs, improve this HMM using the BaumWelch algorithm to increase the probability of observing obs.
INPUT:
obs
– list of emissionsmax_iter
– integer (default: 100) maximum number of BaumWelch steps to takelog_likelihood_cutoff
– positive float (default: 1e4); the minimal improvement in likelihood with respect to the last iteration required to continue. Relative value to log likelihood.fix_emissions
– bool (default: False); if True, do not change emissions when updating
OUTPUT:
changes the model in places, and returns the log likelihood and number of iterations.
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.9,0.1]], [[.5,.5],[0,1]], [.2,.8]) sage: m.baum_welch([1,0]*20, log_likelihood_cutoff=0) (0.0, 4) sage: m # rel tol 1e14 Discrete Hidden Markov Model with 2 States and 2 Emissions Transition matrix: [1.3515269707707603e51 1.0] [ 1.0 0.0] Emission matrix: [ 1.0 6.462537138850569e52] [ 0.0 1.0] Initial probabilities: [0.0000, 1.0000]
The following illustrates how BaumWelch is only a local optimizer, i.e., the above model is far more likely to produce the sequence [1,0]*20 than the one we get below:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.5,0.5],[0.5,0.5]], [[.5,.5],[.5,.5]], [.5,.5]) sage: m.baum_welch([1,0]*20, log_likelihood_cutoff=0) (27.725887222397784, 1) sage: m Discrete Hidden Markov Model with 2 States and 2 Emissions Transition matrix: [0.5 0.5] [0.5 0.5] Emission matrix: [0.5 0.5] [0.5 0.5] Initial probabilities: [0.5000, 0.5000]
We illustrate fixing emissions:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.9,0.1]], [[.5,.5],[.2,.8]], [.2,.8]) sage: set_random_seed(0); v = m.sample(100) sage: m.baum_welch(v,fix_emissions=True) (66.98630856918774, 100) sage: m.emission_matrix() [0.5 0.5] [0.2 0.8] sage: m = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.9,0.1]], [[.5,.5],[.2,.8]], [.2,.8]) sage: m.baum_welch(v) (66.782360659293..., 100) sage: m.emission_matrix() # rel tol 1e14 [ 0.5303085748626447 0.46969142513735535] [ 0.2909775550173978 0.7090224449826023]

emission_matrix
()¶ Return the matrix whose ith row specifies the emission probability distribution for the ith state.
More precisely, the i,j entry of the matrix is the probability of the Markov model outputting the jth symbol when it is in the ith state.
OUTPUT:
a Sage matrix with real double precision (RDF) entries.
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.5,.5]) sage: E = m.emission_matrix(); E [0.1 0.9] [0.5 0.5]
The returned matrix is mutable, but changing it does not change the transition matrix for the model:
sage: E[0,0] = 0; E[0,1] = 1 sage: m.emission_matrix() [0.1 0.9] [0.5 0.5]

generate_sequence
(length, starting_state=None)¶ Return a sample of the given length from this HMM.
INPUT:
length
– positive integerstarting_state
– int (or None); if specified then generate a sequence using this model starting with the given state instead of the initial probabilities to determine the starting state.
OUTPUT:
an IntList or list of emission symbols
IntList of the actual states the model was in when emitting the corresponding symbols
EXAMPLES:
In this example, the emission symbols are not set:
sage: set_random_seed(0) sage: a = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.1,0.9]], [[1,0],[0,1]], [0,1]) sage: a.generate_sequence(5) ([1, 0, 1, 1, 1], [1, 0, 1, 1, 1]) sage: list(a.generate_sequence(1000)[0]).count(0) 90
Here the emission symbols are set:
sage: set_random_seed(0) sage: a = hmm.DiscreteHiddenMarkovModel([[0.5,0.5],[0.1,0.9]], [[1,0],[0,1]], [0,1], ['up', 'down']) sage: a.generate_sequence(5) (['down', 'up', 'down', 'down', 'down'], [1, 0, 1, 1, 1])
Specify the starting state:
sage: set_random_seed(0); a.generate_sequence(5, starting_state=0) (['up', 'up', 'down', 'down', 'down'], [0, 0, 1, 1, 1])

log_likelihood
(obs, scale=True)¶ Return the logarithm of the probability that this model produced the given observation sequence. Thus the output is a nonpositive number.
INPUT:
obs
– sequence of observationsscale
– boolean (default: True); if True, use rescaling to overoid loss of precision due to the very limited dynamic range of floats. You should leave this as True unless the obs sequence is very small.
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.2,.8]) sage: m.log_likelihood([0, 1, 0, 1, 1, 0, 1, 0, 0, 0]) 7.3301308009370825 sage: m.log_likelihood([0, 1, 0, 1, 1, 0, 1, 0, 0, 0], scale=False) 7.330130800937082 sage: m.log_likelihood([]) 0.0 sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.2,.8], ['happy','sad']) sage: m.log_likelihood(['happy','happy']) 1.6565295199679506 sage: m.log_likelihood(['happy','sad']) 1.4731602941415523
Overflow from not using the scale option:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.2,.8]) sage: m.log_likelihood([0,1]*1000, scale=True) 1433.820666652728 sage: m.log_likelihood([0,1]*1000, scale=False) inf

viterbi
(obs, log_scale=True)¶ Determine “the” hidden sequence of states that is most likely to produce the given sequence seq of observations, along with the probability that this hidden sequence actually produced the observation.
INPUT:
seq
– sequence of emitted ints or symbolslog_scale
– bool (default: True) whether to scale the sequence in order to avoid numerical overflow.
OUTPUT:
list
– “the” most probable sequence of hidden states, i.e., the Viterbi path.float
– log of probability that the observed sequence was produced by the Viterbi sequence of states.
EXAMPLES:
sage: a = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.1,0.9]], [[0.9,0.1],[0.1,0.9]], [0.5,0.5]) sage: a.viterbi([1,0,0,1,0,0,1,1]) ([1, 0, 0, 1, ..., 0, 1, 1], 11.06245322477221...)
We predict the state sequence when the emissions are 3/4 and ‘abc’.:
sage: a = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.1,0.9]], [[0.9,0.1],[0.1,0.9]], [0.5,0.5], [3/4, 'abc'])
Note that state 0 is common below, despite the model trying hard to switch to state 1:
sage: a.viterbi([3/4, 'abc', 'abc'] + [3/4]*10) ([0, 1, 1, 0, 0 ... 0, 0, 0, 0, 0], 25.299405845367794)

class
sage.stats.hmm.hmm.
HiddenMarkovModel
¶ Bases:
object
Abstract base class for all Hidden Markov Models.

graph
(eps=0.001)¶ Create a weighted directed graph from the transition matrix, not including any edge with a probability less than eps.
INPUT:
eps – nonnegative real number
OUTPUT:
a digraph
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[.3,0,.7],[0,0,1],[.5,.5,0]], [[.5,.5,.2]]*3, [1/3]*3) sage: G = m.graph(); G Looped digraph on 3 vertices sage: G.edges() [(0, 0, 0.3), (0, 2, 0.7), (1, 2, 1.0), (2, 0, 0.5), (2, 1, 0.5)] sage: G.plot() Graphics object consisting of 11 graphics primitives

initial_probabilities
()¶ Return the initial probabilities, which as a TimeSeries of length N, where N is the number of states of the Markov model.
EXAMPLES:
sage: m = hmm.DiscreteHiddenMarkovModel([[0.4,0.6],[0.1,0.9]], [[0.1,0.9],[0.5,0.5]], [.2,.8]) sage: pi = m.initial_probabilities(); pi [0.2000, 0.8000] sage: type(pi) <... 'sage.finance.time_series.TimeSeries'>
The returned time series is a copy, so changing it does not change the model.
sage: pi[0] = .1; pi[1] = .9 sage: m.initial_probabilities() [0.2000, 0.8000]
Some other models:
sage: hmm.GaussianHiddenMarkovModel([[.1,.9],[.5,.5]], [(1,1), (1,1)], [.1,.9]).initial_probabilities() [0.1000, 0.9000] sage: hmm.GaussianMixtureHiddenMarkovModel([[.9,.1],[.4,.6]], [[(.4,(0,1)), (.6,(1,0.1))],[(1,(0,1))]], [.7,.3]).initial_probabilities() [0.7000, 0.3000]

sample
(length, number=None, starting_state=None)¶ Return number samples from this HMM of given length.
INPUT:
length
– positive integernumber
– (default: None) if given, compute list of this many sample sequencesstarting_state
– int (or None); if specified then generate a sequence using this model starting with the given state instead of the initial probabilities to determine the starting state.
OUTPUT:
if number is not given, return a single TimeSeries.
if number is given, return a list of TimeSeries.
EXAMPLES:
sage: set_random_seed(0) sage: a = hmm.DiscreteHiddenMarkovModel([[0.1,0.9],[0.1,0.9]], [[1,0],[0,1]], [0,1]) sage: print(a.sample(10, 3)) [[1, 0, 1, 1, 1, 1, 0, 1, 1, 1], [1, 1, 0, 0, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 0, 1, 0, 1, 1, 1]] sage: a.sample(15) [1, 1, 1, 1, 0 ... 1, 1, 1, 1, 1] sage: a.sample(3, 1) [[1, 1, 1]] sage: list(a.sample(1000)).count(0) 88
If the emission symbols are set:
sage: set_random_seed(0) sage: a = hmm.DiscreteHiddenMarkovModel([[0.5,0.5],[0.1,0.9]], [[1,0],[0,1]], [0,1], ['up', 'down']) sage: a.sample(10) ['down', 'up', 'down', 'down', 'down', 'down', 'up', 'up', 'up', 'up']
Force a starting state:
sage: set_random_seed(0); a.sample(10, starting_state=0) ['up', 'up', 'down', 'down', 'down', 'down', 'up', 'up', 'up', 'up']

transition_matrix
()¶ Return the state transition matrix.
OUTPUT:
a Sage matrix with real double precision (RDF) entries.
EXAMPLES:
sage: M = hmm.DiscreteHiddenMarkovModel([[0.7,0.3],[0.9,0.1]], [[0.5,.5],[.1,.9]], [0.3,0.7]) sage: T = M.transition_matrix(); T [0.7 0.3] [0.9 0.1]
The returned matrix is mutable, but changing it does not change the transition matrix for the model:
sage: T[0,0] = .1; T[0,1] = .9 sage: M.transition_matrix() [0.7 0.3] [0.9 0.1]
Transition matrices for other types of models:
sage: hmm.GaussianHiddenMarkovModel([[.1,.9],[.5,.5]], [(1,1), (1,1)], [.5,.5]).transition_matrix() [0.1 0.9] [0.5 0.5] sage: hmm.GaussianMixtureHiddenMarkovModel([[.9,.1],[.4,.6]], [[(.4,(0,1)), (.6,(1,0.1))],[(1,(0,1))]], [.7,.3]).transition_matrix() [0.9 0.1] [0.4 0.6]


sage.stats.hmm.hmm.
unpickle_discrete_hmm_v0
(A, B, pi, emission_symbols, name)¶

sage.stats.hmm.hmm.
unpickle_discrete_hmm_v1
(A, B, pi, n_out, emission_symbols, emission_symbols_dict)¶ Return a
DiscreteHiddenMarkovModel
, restored from the arguments.This function is used internally for unpickling.