A class that represents a hidden markov model with an arbitrary type of emission distribution.
HMM (const size_t states, const Distribution emissions, const double tolerance=1e-5)
Create the Hidden Markov Model with the given number of hidden states and the given default distribution for emissions. HMM (const arma::vec &initial, const arma::mat &transition, const std::vector< Distribution > &emission, const double tolerance=1e-5)
Create the Hidden Markov Model with the given initial probability vector, the given transition matrix, and the given emission distributions. size_t Dimensionality () const
Get the dimensionality of observations. size_t & Dimensionality ()
Set the dimensionality of observations. const std::vector< Distribution > & Emission () const
Return the emission distributions. std::vector< Distribution > & Emission ()
Return a modifiable emission probability matrix reference. double Estimate (const arma::mat &dataSeq, arma::mat &stateProb, arma::mat &forwardProb, arma::mat &backwardProb, arma::vec &scales) const
Estimate the probabilities of each hidden state at each time step for each given data observation, using the Forward-Backward algorithm. double Estimate (const arma::mat &dataSeq, arma::mat &stateProb) const
Estimate the probabilities of each hidden state at each time step of each given data observation, using the Forward-Backward algorithm. void Generate (const size_t length, arma::mat &dataSequence, arma::Col< size_t > &stateSequence, const size_t startState=0) const
Generate a random data sequence of the given length. const arma::vec & Initial () const
Return the vector of initial state probabilities. arma::vec & Initial ()
Modify the vector of initial state probabilities. double LogLikelihood (const arma::mat &dataSeq) const
Compute the log-likelihood of the given data sequence. double Predict (const arma::mat &dataSeq, arma::Col< size_t > &stateSeq) const
Compute the most probable hidden state sequence for the given data sequence, using the Viterbi algorithm, returning the log-likelihood of the most likely state sequence. double Tolerance () const
Get the tolerance of the Baum-Welch algorithm. double & Tolerance ()
Modify the tolerance of the Baum-Welch algorithm. std::string ToString () const
Returns a string representation of this object. void Train (const std::vector< arma::mat > &dataSeq)
Train the model using the Baum-Welch algorithm, with only the given unlabeled observations. void Train (const std::vector< arma::mat > &dataSeq, const std::vector< arma::Col< size_t > > &stateSeq)
Train the model using the given labeled observations; the transition and emission matrices are directly estimated. const arma::mat & Transition () const
Return the transition matrix. arma::mat & Transition ()
Return a modifiable transition matrix reference.
void Backward (const arma::mat &dataSeq, const arma::vec &scales, arma::mat &backwardProb) const
The Backward algorithm (part of the Forward-Backward algorithm). void Forward (const arma::mat &dataSeq, arma::vec &scales, arma::mat &forwardProb) const
The Forward algorithm (part of the Forward-Backward algorithm).
size_t dimensionality
Dimensionality of observations. std::vector< Distribution > emission
Set of emission probability distributions; one for each state. arma::vec initial
Initial state probability vector. double tolerance
Tolerance of Baum-Welch algorithm. arma::mat transition
Transition probability matrix.
A class that represents a Hidden Markov Model with an arbitrary type of emission distribution.
This HMM class supports training (supervised and unsupervised), prediction of state sequences via the Viterbi algorithm, estimation of state probabilities, generation of random sequences, and calculation of the log-likelihood of a given sequence.
The template parameter, Distribution, specifies the distribution which the emissions follow. The class should implement the following functions:
class Distribution
{
public:
// The type of observation used by this distribution.
typedef something DataType;
// Return the probability of the given observation.
double Probability(const DataType& observation) const;
// Estimate the distribution based on the given observations.
void Estimate(const std::vector<DataType>& observations);
// Estimate the distribution based on the given observations, given also
// the probability of each observation coming from this distribution.
void Estimate(const std::vector<DataType>& observations,
const std::vector<double>& probabilities);
};
See the mlpack::distribution::DiscreteDistribution class for an example. One would use the DiscreteDistribution class when the observations are non-negative integers. Other distributions could be Gaussians, a mixture of Gaussians (GMM), or any other probability distribution implementing the four Distribution functions.
Usage of the HMM class generally involves either training an HMM or loading an already-known HMM and taking probability measurements of sequences. Example code for supervised training of a Gaussian HMM (that is, where the emission output distribution is a single Gaussian for each hidden state) is given below.
extern arma::mat observations; // Each column is an observation. extern arma::Col<size_t> states; // Hidden states for each observation. // Create an untrained HMM with 5 hidden states and default (N(0, 1)) // Gaussian distributions with the dimensionality of the dataset. HMM<GaussianDistribution> hmm(5, GaussianDistribution(observations.n_rows)); // Train the HMM (the labels could be omitted to perform unsupervised // training). hmm.Train(observations, states);
Once initialized, the HMM can evaluate the probability of a certain sequence (with LogLikelihood()), predict the most likely sequence of hidden states (with Predict()), generate a sequence (with Generate()), or estimate the probabilities of each state for a sequence of observations (with Estimate()).
Template Parameters:
Distribution Type of emission distribution for this HMM.
Definition at line 93 of file hmm.hpp.
Create the Hidden Markov Model with the given number of hidden states and the given default distribution for emissions. The dimensionality of the observations is taken from the emissions variable, so it is important that the given default emission distribution is set with the correct dimensionality. Alternately, set the dimensionality with Dimensionality(). Optionally, the tolerance for convergence of the Baum-Welch algorithm can be set.
By default, the transition matrix and initial probability vector are set to contain equal probability for each state.
Parameters:
states Number of states.
emissions Default distribution for emissions.
tolerance Tolerance for convergence of training algorithm (Baum-Welch).
Create the Hidden Markov Model with the given initial probability vector, the given transition matrix, and the given emission distributions. The dimensionality of the observations of the HMM are taken from the given emission distributions. Alternately, the dimensionality can be set with Dimensionality().
The initial state probability vector should have length equal to the number of states, and each entry represents the probability of being in the given state at time T = 0 (the beginning of a sequence).
The transition matrix should be such that T(i, j) is the probability of transition to state i from state j. The columns of the matrix should sum to 1.
The emission matrix should be such that E(i, j) is the probability of emission i while in state j. The columns of the matrix should sum to 1.
Optionally, the tolerance for convergence of the Baum-Welch algorithm can be set.
Parameters:
initial Initial state probabilities.
transition Transition matrix.
emission Emission distributions.
tolerance Tolerance for convergence of training algorithm (Baum-Welch).
The Backward algorithm (part of the Forward-Backward algorithm). Computes backward probabilities for each state for each observation in the given data sequence, using the scaling factors found (presumably) by Forward(). The returned matrix has rows equal to the number of hidden states and columns equal to the number of observations.
Parameters:
dataSeq Data sequence to compute probabilities for.
scales Vector of scaling factors.
backwardProb Matrix in which backward probabilities will be saved.
Get the dimensionality of observations.
Definition at line 294 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::dimensionality.
Set the dimensionality of observations.
Definition at line 296 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::dimensionality.
Return the emission distributions.
Definition at line 289 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::emission.
Return a modifiable emission probability matrix reference.
Definition at line 291 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::emission.
Estimate the probabilities of each hidden state at each time step for each given data observation, using the Forward-Backward algorithm. Each matrix which is returned has columns equal to the number of data observations, and rows equal to the number of hidden states in the model. The log-likelihood of the most probable sequence is returned.
Parameters:
dataSeq Sequence of observations.
stateProb Matrix in which the probabilities of each state at each time interval will be stored.
forwardProb Matrix in which the forward probabilities of each state at each time interval will be stored.
backwardProb Matrix in which the backward probabilities of each state at each time interval will be stored.
scales Vector in which the scaling factors at each time interval will be stored.
Returns:
Log-likelihood of most likely state sequence.
Estimate the probabilities of each hidden state at each time step of each given data observation, using the Forward-Backward algorithm. The returned matrix of state probabilities has columns equal to the number of data observations, and rows equal to the number of hidden states in the model. The log-likelihood of the most probable sequence is returned.
Parameters:
dataSeq Sequence of observations.
stateProb Probabilities of each state at each time interval.
Returns:
Log-likelihood of most likely state sequence.
The Forward algorithm (part of the Forward-Backward algorithm). Computes forward probabilities for each state for each observation in the given data sequence. The returned matrix has rows equal to the number of hidden states and columns equal to the number of observations.
Parameters:
dataSeq Data sequence to compute probabilities for.
scales Vector in which scaling factors will be saved.
forwardProb Matrix in which forward probabilities will be saved.
Generate a random data sequence of the given length. The data sequence is stored in the dataSequence parameter, and the state sequence is stored in the stateSequence parameter. Each column of dataSequence represents a random observation.
Parameters:
length Length of random sequence to generate.
dataSequence Vector to store data in.
stateSequence Vector to store states in.
startState Hidden state to start sequence in (default 0).
Return the vector of initial state probabilities.
Definition at line 279 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::initial.
Modify the vector of initial state probabilities.
Definition at line 281 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::initial.
Compute the log-likelihood of the given data sequence.
Parameters:
dataSeq Data sequence to evaluate the likelihood of.
Returns:
Log-likelihood of the given sequence.
Compute the most probable hidden state sequence for the given data sequence, using the Viterbi algorithm, returning the log-likelihood of the most likely state sequence.
Parameters:
dataSeq Sequence of observations.
stateSeq Vector in which the most probable state sequence will be stored.
Returns:
Log-likelihood of most probable state sequence.
Get the tolerance of the Baum-Welch algorithm.
Definition at line 299 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::tolerance.
Modify the tolerance of the Baum-Welch algorithm.
Definition at line 301 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::tolerance.
Returns a string representation of this object.
Train the model using the Baum-Welch algorithm, with only the given unlabeled observations. Instead of giving a guess transition and emission matrix here, do that in the constructor. Each matrix in the vector of data sequences holds an individual data sequence; each point in each individual data sequence should be a column in the matrix. The number of rows in each matrix should be equal to the dimensionality of the HMM (which is set in the constructor).
It is preferable to use the other overload of Train(), with labeled data. That will produce much better results. However, if labeled data is unavailable, this will work. In addition, it is possible to use Train() with labeled data first, and then continue to train the model using this overload of Train() with unlabeled data.
The tolerance of the Baum-Welch algorithm can be set either in the constructor or with the Tolerance() method. When the change in log-likelihood of the model between iterations is less than the tolerance, the Baum-Welch algorithm terminates.
Note:
Train() can be called multiple times with different sequences; each time it is called, it uses the current parameters of the HMM as a starting point for training.
Parameters:
dataSeq Vector of observation sequences.
Train the model using the given labeled observations; the transition and emission matrices are directly estimated. Each matrix in the vector of data sequences corresponds to a vector in the vector of state sequences. Each point in each individual data sequence should be a column in the matrix, and its state should be the corresponding element in the state sequence vector. For instance, dataSeq[0].col(3) corresponds to the fourth observation in the first data sequence, and its state is stateSeq[0][3]. The number of rows in each matrix should be equal to the dimensionality of the HMM (which is set in the constructor).
Note:
Train() can be called multiple times with different sequences; each time it is called, it uses the current parameters of the HMM as a starting point for training.
Parameters:
dataSeq Vector of observation sequences.
stateSeq Vector of state sequences, corresponding to each observation.
Return the transition matrix.
Definition at line 284 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::transition.
Return a modifiable transition matrix reference.
Definition at line 286 of file hmm.hpp.
References mlpack::hmm::HMM< Distribution >::transition.
Dimensionality of observations.
Definition at line 350 of file hmm.hpp.
Referenced by mlpack::hmm::HMM< Distribution >::Dimensionality().
Set of emission probability distributions; one for each state.
Definition at line 347 of file hmm.hpp.
Referenced by mlpack::hmm::HMM< Distribution >::Emission().
Initial state probability vector.
Definition at line 341 of file hmm.hpp.
Referenced by mlpack::hmm::HMM< Distribution >::Initial().
Tolerance of Baum-Welch algorithm.
Definition at line 353 of file hmm.hpp.
Referenced by mlpack::hmm::HMM< Distribution >::Tolerance().
Transition probability matrix.
Definition at line 344 of file hmm.hpp.
Referenced by mlpack::hmm::HMM< Distribution >::Transition().
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