HMM工具箱命令使用说明

HOW TO USE THE HMM TOOLBOX (MATLAB)

一、离散输出的隐马尔科夫模型（DHMM，HMM with discrete outputs）

最大似然参数估计EM（Baum Welch算法）

The script dhmm_em_demo.m gives an example of how to learn an HMM with discrete outputs. Let there be Q=2 states and O=3 output symbols. We create random stochastic matrices as follows.

O = 3;

Q = 2;

prior0 = normalise(rand(Q,1));

transmat0 = mk_stochastic(rand(Q,Q));

obsmat0 = mk_stochastic(rand(Q,O));

Now we sample nex=20 sequences of length T=10 each from this model, to use as training data.

T=10; %序列长度

nex=20; %样本序列数目

data = dhmm_sample(prior0, transmat0, obsmat0, nex, T);

Here data is 20x10. Now we make a random guess as to what the parameters are, prior1 = normalise(rand(Q,1)); %初始状态概率

transmat1 = mk_stochastic(rand(Q,Q)); %初始状态转移矩阵

obsmat1 = mk_stochastic(rand(Q,O)); %初始观测状态到隐藏状态间的概率转移矩阵

and improve our guess using 5 iterations of EM...

[LL, prior2, transmat2, obsmat2] = dhmm_em(data, prior1, transmat1, obsmat1,

'max_iter', 5);

%prior2, transmat2, obsmat2 为训练好后的初始概率，状态转移矩阵及混合状态概率转移矩阵

LL(t) is the log-likelihood after iteration t, so we can plot the learning curve.

序列分类

To evaluate the log-likelihood of a trained model given test data, proceed as follows: loglik = dhmm_logprob(data, prior2, transmat2, obsmat2) %HMM测试

Note: the discrete alphabet is assumed to be {1, 2, ..., O}, where O = size(obsmat, 2). Hence data cannot contain any 0s.

To classify a sequence into one of k classes, train up k HMMs, one per class, and then compute the log-likelihood that each model gives to the test sequence; if the i'th model is the most likely, then declare the class of the sequence to be class i. Computing the most probable sequence (Viterbi)

First you need to evaluate B(i,t) = P(y_t | Q_t=i) for all t,i:

B = multinomial_prob(data, obsmat);

Then you can use

[path] = viterbi_path(prior, transmat, B)

二、具有高斯混合输出的隐马尔科夫模型（GHMM，HMM with mixture of Gaussians outputs）

Maximum likelihood parameter estimation using EM (Baum Welch)

Let us generate nex=50 vector-valued sequences of length T=50; each vector has size O=2.

O = 2;

T = 50;

nex = 50;

data = randn(O,T,nex);

Now let use fit a mixture of M=2 Gaussians for each of the Q=2 states using

K-means.

M = 2;

Q = 2;

left_right = 0;

prior0 = normalise(rand(Q,1));

transmat0 = mk_stochastic(rand(Q,Q));

[mu0, Sigma0] = mixgauss_init(Q*M, reshape(data, [O T*nex]), cov_type);

mu0 = reshape(mu0, [O Q M]);

Sigma0 = reshape(Sigma0, [O O Q M]);

mixmat0 = mk_stochastic(rand(Q,M));

Finally, let us improve these parameter estimates using EM.

[LL, prior1, transmat1, mu1, Sigma1, mixmat1] = ...

mhmm_em(data, prior0, transmat0, mu0, Sigma0, mixmat0, 'max_iter', 2);

Since EM only finds a local optimum, good initialisation is crucial. The initialisation procedure illustrated above is very crude, and is probably not adequate for real applications... Click here for a real-world example of EM with mixtures of Gaussians using BNT.

What to do if the log-likelihood becomes positive?

It is possible for p(x) > 1 if p(x) is a probability density function, such as a Gaussian. (The requirements for a density are p(x)>0 for all x and int_x p(x) = 1.) In practice this usually means your covariance is shrinking to a point/delta function, so you should increase the width of the prior (see below), or constrain the matrix to be spherical or diagonal, or clamp it to a large fixed constant (not learn it at all). It is also very helpful to ensure the components of the data vectors have small and comparable magnitudes (use e.g., KPMstats/standardize).

This is a well-known pathology of maximum likelihood estimation for Gaussian mixtures: the global optimum may place one mixture component on a single data point, and give it 0 covariance, and hence infinite likelihood. One usually relies on the fact that EM cannot find the global optimum to avoid such pathologies.

What to do if the log-likelihood decreases during EM?

Since I implicitly add a prior to every covariance matrix (see below), what increases is loglik + log(prior), but what I print is just loglik, which may occasionally decrease.