.TH "mlpack::hmm::HMM< Distribution >" 3 "Tue Sep 9 2014" "Version 1.0.10" "MLPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
mlpack::hmm::HMM< Distribution > \- 
.PP
A class that represents a Hidden Markov Model with an arbitrary type of emission distribution\&.  

.SH SYNOPSIS
.br
.PP
.SS "Public Member Functions"

.in +1c
.ti -1c
.RI "\fBHMM\fP (const size_t states, const Distribution emissions, const double \fBtolerance\fP=1e-5)"
.br
.RI "\fICreate the Hidden Markov Model with the given number of hidden states and the given default distribution for emissions\&. \fP"
.ti -1c
.RI "\fBHMM\fP (const arma::vec &\fBinitial\fP, const arma::mat &\fBtransition\fP, const std::vector< Distribution > &\fBemission\fP, const double \fBtolerance\fP=1e-5)"
.br
.RI "\fICreate the Hidden Markov Model with the given initial probability vector, the given transition matrix, and the given emission distributions\&. \fP"
.ti -1c
.RI "size_t \fBDimensionality\fP () const "
.br
.RI "\fIGet the dimensionality of observations\&. \fP"
.ti -1c
.RI "size_t & \fBDimensionality\fP ()"
.br
.RI "\fISet the dimensionality of observations\&. \fP"
.ti -1c
.RI "const std::vector< Distribution > & \fBEmission\fP () const "
.br
.RI "\fIReturn the emission distributions\&. \fP"
.ti -1c
.RI "std::vector< Distribution > & \fBEmission\fP ()"
.br
.RI "\fIReturn a modifiable emission probability matrix reference\&. \fP"
.ti -1c
.RI "double \fBEstimate\fP (const arma::mat &dataSeq, arma::mat &stateProb, arma::mat &forwardProb, arma::mat &backwardProb, arma::vec &scales) const "
.br
.RI "\fIEstimate the probabilities of each hidden state at each time step for each given data observation, using the Forward-Backward algorithm\&. \fP"
.ti -1c
.RI "double \fBEstimate\fP (const arma::mat &dataSeq, arma::mat &stateProb) const "
.br
.RI "\fIEstimate the probabilities of each hidden state at each time step of each given data observation, using the Forward-Backward algorithm\&. \fP"
.ti -1c
.RI "void \fBGenerate\fP (const size_t length, arma::mat &dataSequence, arma::Col< size_t > &stateSequence, const size_t startState=0) const "
.br
.RI "\fIGenerate a random data sequence of the given length\&. \fP"
.ti -1c
.RI "const arma::vec & \fBInitial\fP () const "
.br
.RI "\fIReturn the vector of initial state probabilities\&. \fP"
.ti -1c
.RI "arma::vec & \fBInitial\fP ()"
.br
.RI "\fIModify the vector of initial state probabilities\&. \fP"
.ti -1c
.RI "double \fBLogLikelihood\fP (const arma::mat &dataSeq) const "
.br
.RI "\fICompute the log-likelihood of the given data sequence\&. \fP"
.ti -1c
.RI "double \fBPredict\fP (const arma::mat &dataSeq, arma::Col< size_t > &stateSeq) const "
.br
.RI "\fICompute 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\&. \fP"
.ti -1c
.RI "double \fBTolerance\fP () const "
.br
.RI "\fIGet the tolerance of the Baum-Welch algorithm\&. \fP"
.ti -1c
.RI "double & \fBTolerance\fP ()"
.br
.RI "\fIModify the tolerance of the Baum-Welch algorithm\&. \fP"
.ti -1c
.RI "std::string \fBToString\fP () const "
.br
.RI "\fIReturns a string representation of this object\&. \fP"
.ti -1c
.RI "void \fBTrain\fP (const std::vector< arma::mat > &dataSeq)"
.br
.RI "\fITrain the model using the Baum-Welch algorithm, with only the given unlabeled observations\&. \fP"
.ti -1c
.RI "void \fBTrain\fP (const std::vector< arma::mat > &dataSeq, const std::vector< arma::Col< size_t > > &stateSeq)"
.br
.RI "\fITrain the model using the given labeled observations; the transition and emission matrices are directly estimated\&. \fP"
.ti -1c
.RI "const arma::mat & \fBTransition\fP () const "
.br
.RI "\fIReturn the transition matrix\&. \fP"
.ti -1c
.RI "arma::mat & \fBTransition\fP ()"
.br
.RI "\fIReturn a modifiable transition matrix reference\&. \fP"
.in -1c
.SS "Private Member Functions"

.in +1c
.ti -1c
.RI "void \fBBackward\fP (const arma::mat &dataSeq, const arma::vec &scales, arma::mat &backwardProb) const "
.br
.RI "\fIThe Backward algorithm (part of the Forward-Backward algorithm)\&. \fP"
.ti -1c
.RI "void \fBForward\fP (const arma::mat &dataSeq, arma::vec &scales, arma::mat &forwardProb) const "
.br
.RI "\fIThe Forward algorithm (part of the Forward-Backward algorithm)\&. \fP"
.in -1c
.SS "Private Attributes"

.in +1c
.ti -1c
.RI "size_t \fBdimensionality\fP"
.br
.RI "\fIDimensionality of observations\&. \fP"
.ti -1c
.RI "std::vector< Distribution > \fBemission\fP"
.br
.RI "\fISet of emission probability distributions; one for each state\&. \fP"
.ti -1c
.RI "arma::vec \fBinitial\fP"
.br
.RI "\fIInitial state probability vector\&. \fP"
.ti -1c
.RI "double \fBtolerance\fP"
.br
.RI "\fITolerance of Baum-Welch algorithm\&. \fP"
.ti -1c
.RI "arma::mat \fBtransition\fP"
.br
.RI "\fITransition probability matrix\&. \fP"
.in -1c
.SH "Detailed Description"
.PP 

.SS "template<typename Distribution = distribution::DiscreteDistribution>class mlpack::hmm::HMM< Distribution >"
A class that represents a Hidden Markov Model with an arbitrary type of emission distribution\&. 

This \fBHMM\fP 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\&.
.PP
The template parameter, Distribution, specifies the distribution which the emissions follow\&. The class should implement the following functions:
.PP
.PP
.nf
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);
};
.fi
.PP
.PP
See the \fBmlpack::distribution::DiscreteDistribution\fP 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\&.
.PP
Usage of the \fBHMM\fP class generally involves either training an \fBHMM\fP or loading an already-known \fBHMM\fP and taking probability measurements of sequences\&. Example code for supervised training of a Gaussian \fBHMM\fP (that is, where the emission output distribution is a single Gaussian for each hidden state) is given below\&.
.PP
.PP
.nf
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);
.fi
.PP
.PP
Once initialized, the \fBHMM\fP can evaluate the probability of a certain sequence (with \fBLogLikelihood()\fP), predict the most likely sequence of hidden states (with \fBPredict()\fP), generate a sequence (with \fBGenerate()\fP), or estimate the probabilities of each state for a sequence of observations (with \fBEstimate()\fP)\&.
.PP
\fBTemplate Parameters:\fP
.RS 4
\fIDistribution\fP Type of emission distribution for this \fBHMM\fP\&. 
.RE
.PP

.PP
Definition at line 93 of file hmm\&.hpp\&.
.SH "Constructor & Destructor Documentation"
.PP 
.SS "template<typename Distribution  = distribution::DiscreteDistribution> \fBmlpack::hmm::HMM\fP< Distribution >::\fBHMM\fP (const size_tstates, const Distributionemissions, const doubletolerance = \fC1e-5\fP)"

.PP
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 \fBDimensionality()\fP\&. Optionally, the tolerance for convergence of the Baum-Welch algorithm can be set\&.
.PP
By default, the transition matrix and initial probability vector are set to contain equal probability for each state\&.
.PP
\fBParameters:\fP
.RS 4
\fIstates\fP Number of states\&. 
.br
\fIemissions\fP Default distribution for emissions\&. 
.br
\fItolerance\fP Tolerance for convergence of training algorithm (Baum-Welch)\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> \fBmlpack::hmm::HMM\fP< Distribution >::\fBHMM\fP (const arma::vec &initial, const arma::mat &transition, const std::vector< Distribution > &emission, const doubletolerance = \fC1e-5\fP)"

.PP
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 \fBHMM\fP are taken from the given emission distributions\&. Alternately, the dimensionality can be set with \fBDimensionality()\fP\&.
.PP
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)\&.
.PP
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\&.
.PP
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\&.
.PP
Optionally, the tolerance for convergence of the Baum-Welch algorithm can be set\&.
.PP
\fBParameters:\fP
.RS 4
\fIinitial\fP Initial state probabilities\&. 
.br
\fItransition\fP Transition matrix\&. 
.br
\fIemission\fP Emission distributions\&. 
.br
\fItolerance\fP Tolerance for convergence of training algorithm (Baum-Welch)\&. 
.RE
.PP

.SH "Member Function Documentation"
.PP 
.SS "template<typename Distribution  = distribution::DiscreteDistribution> void \fBmlpack::hmm::HMM\fP< Distribution >::Backward (const arma::mat &dataSeq, const arma::vec &scales, arma::mat &backwardProb) const\fC [private]\fP"

.PP
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 \fBForward()\fP\&. The returned matrix has rows equal to the number of hidden states and columns equal to the number of observations\&.
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Data sequence to compute probabilities for\&. 
.br
\fIscales\fP Vector of scaling factors\&. 
.br
\fIbackwardProb\fP Matrix in which backward probabilities will be saved\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> size_t \fBmlpack::hmm::HMM\fP< Distribution >::Dimensionality () const\fC [inline]\fP"

.PP
Get the dimensionality of observations\&. 
.PP
Definition at line 294 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::dimensionality\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> size_t& \fBmlpack::hmm::HMM\fP< Distribution >::Dimensionality ()\fC [inline]\fP"

.PP
Set the dimensionality of observations\&. 
.PP
Definition at line 296 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::dimensionality\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> const std::vector<Distribution>& \fBmlpack::hmm::HMM\fP< Distribution >::Emission () const\fC [inline]\fP"

.PP
Return the emission distributions\&. 
.PP
Definition at line 289 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::emission\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> std::vector<Distribution>& \fBmlpack::hmm::HMM\fP< Distribution >::Emission ()\fC [inline]\fP"

.PP
Return a modifiable emission probability matrix reference\&. 
.PP
Definition at line 291 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::emission\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::Estimate (const arma::mat &dataSeq, arma::mat &stateProb, arma::mat &forwardProb, arma::mat &backwardProb, arma::vec &scales) const"

.PP
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\&.
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Sequence of observations\&. 
.br
\fIstateProb\fP Matrix in which the probabilities of each state at each time interval will be stored\&. 
.br
\fIforwardProb\fP Matrix in which the forward probabilities of each state at each time interval will be stored\&. 
.br
\fIbackwardProb\fP Matrix in which the backward probabilities of each state at each time interval will be stored\&. 
.br
\fIscales\fP Vector in which the scaling factors at each time interval will be stored\&. 
.RE
.PP
\fBReturns:\fP
.RS 4
Log-likelihood of most likely state sequence\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::Estimate (const arma::mat &dataSeq, arma::mat &stateProb) const"

.PP
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\&.
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Sequence of observations\&. 
.br
\fIstateProb\fP Probabilities of each state at each time interval\&. 
.RE
.PP
\fBReturns:\fP
.RS 4
Log-likelihood of most likely state sequence\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> void \fBmlpack::hmm::HMM\fP< Distribution >::Forward (const arma::mat &dataSeq, arma::vec &scales, arma::mat &forwardProb) const\fC [private]\fP"

.PP
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\&.
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Data sequence to compute probabilities for\&. 
.br
\fIscales\fP Vector in which scaling factors will be saved\&. 
.br
\fIforwardProb\fP Matrix in which forward probabilities will be saved\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> void \fBmlpack::hmm::HMM\fP< Distribution >::Generate (const size_tlength, arma::mat &dataSequence, arma::Col< size_t > &stateSequence, const size_tstartState = \fC0\fP) const"

.PP
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\&.
.PP
\fBParameters:\fP
.RS 4
\fIlength\fP Length of random sequence to generate\&. 
.br
\fIdataSequence\fP Vector to store data in\&. 
.br
\fIstateSequence\fP Vector to store states in\&. 
.br
\fIstartState\fP Hidden state to start sequence in (default 0)\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> const arma::vec& \fBmlpack::hmm::HMM\fP< Distribution >::Initial () const\fC [inline]\fP"

.PP
Return the vector of initial state probabilities\&. 
.PP
Definition at line 279 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::initial\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> arma::vec& \fBmlpack::hmm::HMM\fP< Distribution >::Initial ()\fC [inline]\fP"

.PP
Modify the vector of initial state probabilities\&. 
.PP
Definition at line 281 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::initial\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::LogLikelihood (const arma::mat &dataSeq) const"

.PP
Compute the log-likelihood of the given data sequence\&. 
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Data sequence to evaluate the likelihood of\&. 
.RE
.PP
\fBReturns:\fP
.RS 4
Log-likelihood of the given sequence\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::Predict (const arma::mat &dataSeq, arma::Col< size_t > &stateSeq) const"

.PP
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\&. 
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Sequence of observations\&. 
.br
\fIstateSeq\fP Vector in which the most probable state sequence will be stored\&. 
.RE
.PP
\fBReturns:\fP
.RS 4
Log-likelihood of most probable state sequence\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::Tolerance () const\fC [inline]\fP"

.PP
Get the tolerance of the Baum-Welch algorithm\&. 
.PP
Definition at line 299 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::tolerance\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> double& \fBmlpack::hmm::HMM\fP< Distribution >::Tolerance ()\fC [inline]\fP"

.PP
Modify the tolerance of the Baum-Welch algorithm\&. 
.PP
Definition at line 301 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::tolerance\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> std::string \fBmlpack::hmm::HMM\fP< Distribution >::ToString () const"

.PP
Returns a string representation of this object\&. 
.SS "template<typename Distribution  = distribution::DiscreteDistribution> void \fBmlpack::hmm::HMM\fP< Distribution >::Train (const std::vector< arma::mat > &dataSeq)"

.PP
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 \fBHMM\fP (which is set in the constructor)\&.
.PP
It is preferable to use the other overload of \fBTrain()\fP, 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 \fBTrain()\fP with labeled data first, and then continue to train the model using this overload of \fBTrain()\fP with unlabeled data\&.
.PP
The tolerance of the Baum-Welch algorithm can be set either in the constructor or with the \fBTolerance()\fP method\&. When the change in log-likelihood of the model between iterations is less than the tolerance, the Baum-Welch algorithm terminates\&.
.PP
\fBNote:\fP
.RS 4
\fBTrain()\fP can be called multiple times with different sequences; each time it is called, it uses the current parameters of the \fBHMM\fP as a starting point for training\&. 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Vector of observation sequences\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> void \fBmlpack::hmm::HMM\fP< Distribution >::Train (const std::vector< arma::mat > &dataSeq, const std::vector< arma::Col< size_t > > &stateSeq)"

.PP
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 \fBHMM\fP (which is set in the constructor)\&.
.PP
\fBNote:\fP
.RS 4
\fBTrain()\fP can be called multiple times with different sequences; each time it is called, it uses the current parameters of the \fBHMM\fP as a starting point for training\&. 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIdataSeq\fP Vector of observation sequences\&. 
.br
\fIstateSeq\fP Vector of state sequences, corresponding to each observation\&. 
.RE
.PP

.SS "template<typename Distribution  = distribution::DiscreteDistribution> const arma::mat& \fBmlpack::hmm::HMM\fP< Distribution >::Transition () const\fC [inline]\fP"

.PP
Return the transition matrix\&. 
.PP
Definition at line 284 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::transition\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> arma::mat& \fBmlpack::hmm::HMM\fP< Distribution >::Transition ()\fC [inline]\fP"

.PP
Return a modifiable transition matrix reference\&. 
.PP
Definition at line 286 of file hmm\&.hpp\&.
.PP
References mlpack::hmm::HMM< Distribution >::transition\&.
.SH "Member Data Documentation"
.PP 
.SS "template<typename Distribution  = distribution::DiscreteDistribution> size_t \fBmlpack::hmm::HMM\fP< Distribution >::dimensionality\fC [private]\fP"

.PP
Dimensionality of observations\&. 
.PP
Definition at line 350 of file hmm\&.hpp\&.
.PP
Referenced by mlpack::hmm::HMM< Distribution >::Dimensionality()\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> std::vector<Distribution> \fBmlpack::hmm::HMM\fP< Distribution >::emission\fC [private]\fP"

.PP
Set of emission probability distributions; one for each state\&. 
.PP
Definition at line 347 of file hmm\&.hpp\&.
.PP
Referenced by mlpack::hmm::HMM< Distribution >::Emission()\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> arma::vec \fBmlpack::hmm::HMM\fP< Distribution >::initial\fC [private]\fP"

.PP
Initial state probability vector\&. 
.PP
Definition at line 341 of file hmm\&.hpp\&.
.PP
Referenced by mlpack::hmm::HMM< Distribution >::Initial()\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> double \fBmlpack::hmm::HMM\fP< Distribution >::tolerance\fC [private]\fP"

.PP
Tolerance of Baum-Welch algorithm\&. 
.PP
Definition at line 353 of file hmm\&.hpp\&.
.PP
Referenced by mlpack::hmm::HMM< Distribution >::Tolerance()\&.
.SS "template<typename Distribution  = distribution::DiscreteDistribution> arma::mat \fBmlpack::hmm::HMM\fP< Distribution >::transition\fC [private]\fP"

.PP
Transition probability matrix\&. 
.PP
Definition at line 344 of file hmm\&.hpp\&.
.PP
Referenced by mlpack::hmm::HMM< Distribution >::Transition()\&.

.SH "Author"
.PP 
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