MATLAB: What are the formats of the “seq” and “states” vectors used by the HMMESTIMATE function in the Statistics Toolbox 4.0 (R13)

.seq"hiddenhmmestimatehmmgeneratemarkovmodelstatesStatistics and Machine Learning Toolbox

I would like an example and explanation for the structure of the "seq" and "states" vectors used by HMMESTIMATE.

Best Answer

To explain the format of the "seq" and "states" parameters of the HMMESTIMATE function, a hidden Markov model will be used.

First, take a non-fair coin, with a probability "p" of coming up heads. The two states shall be defined as "heads" and "tails", and the transition from one state to the next is defined by the next flip of the coin. Therefore, the transition matrix would be:

 TM = [p 1-p; ...
 p 1-p];

Assume that whenever the coin is in state 1, the value of the model is 2, and whenever the coin is in state 2, the value of the model is 1 or 3, with equal probability. Therefore, the emission matrix will be:

 E = [0 1 0; ...
 1/2 0 1/2];

If you simulate the hidden Markov model states using the TM matrix (choosing a state at random each simulation step based on the probabilities given in TM), and record the state of the model at each simulation step, the "states" parameter for HMMESTIMATE is obtained. Likewise, if you take that "states" parameter and generate a random value for the model at each simulation step (using the probabilities from the E matrix), the vector of sequence values "seq" is obtained.

To clarify, if the coin is twice as likely to come up heads as tails, you would expect "states" to have twice as many 1's as 2's. You would also expect that the "seq" vector will be one-fourth 1, one-half 2, and one-fourth 3:

p = 2/3; % (1-p) = tails probability = 1/3 = p/2 = half the heads probability
TM = [p 1-p; ...
p 1-p];
E = [0 1 0; ...
1/2 0 1/2];
[SEQ, STATES] = hmmgenerate(1000, TM, E); % Generate 1000 simulation steps
[TM_est, E_est] = hmmestimate(SEQ, STATES) % TM_est should be roughly TM and E_est should be roughly E

Related Solutions

MATLAB: Does the “SEQ” output from the HMMGENERATE function from the Statistics Toolbox not return emission probabilty

This bug has been fixed for Release 14 (R14). For previous releases, documentation on the HMMGENERATE function is missing from the Statistics Toolbox.

The output "SEQ" from the HMMGENERATE function should return a random sequence of emission symbols. The documentation should mention "random sequence of emission symbols".

MATLAB: How to use continuous sequence values with HMMESTIMATE in the Statistics Toolbox 7.1 (R2009a)

The ability to do hidden Markov model estimation for continuous-valued emissions is not available in the Statistics Toolbox 7.1 (R2009a).

As a workaround, you can use the following function to see the probability density function estimates of the state's emission values. The function uses quantization to collect values into different bins, and indexes the bins. These indexed bin values are used to create a new sequence array, and HMMESTIMATE is run as usual.

For probability density function estimation, a histogram is obtained and smoothed.

Inputs to the function: Sequence array, States array, and the number of bins to quantize with.

Outputs of the function: Transition matrix, Emission matrix, bin centre array, smoothed estimated PDF matrix.

function [transition_mat, emission_mat, bins, em_pdf] = run_hmm(s_seq, s_mat, x)
% check if number of bins is an integer
if abs(x-round(x))~=0
    disp('Argument must be an integer! Stopping.');
    return;
end
nBins = x;
% obtain histogram for the sequence array
[freq, bins] = hist(s_seq, nBins);
% obtain the bin distance
bin_dist = (bins(2)-bins(1));
fprintf('Bin width: %f\n', bin_dist);
quant_error = 0;
% indexing each float value to the nearest bin
for i = 1:length(s_seq)
    for j = 1:nBins
        dist = abs(bins - s_seq(i));
        [minval, indx] = min(dist);
        quant_error = quant_error + (s_seq(i) - bins(indx));
        s_seq_ind(i) = indx;
    end
end
% total quantization error in indexing
fprintf('Total quantization error in indexing: %f\n', (quant_error));
% obtain HMMESTIMATE
[transition_mat, emission_mat] = hmmestimate(s_seq_ind, s_mat);
% smoothing filter coeffs. to obtain PDF plot
B = (0.3905/4)*[1 2 1];
A = [1 -0.9428 .3333];
% shifting bins to account for filtering delay
bins = bins+(1.1*bin_dist);
% smoothing the histogram
em_pdf = (filter(B, A, emission_mat'))';
figure;
hold on;
legend_str = {};
for i = 1:size(emission_mat,1)
    %normalizing to PDF
    em_pdf(i, :) = em_pdf(i, :)/(bin_dist*sum(em_pdf(i, :)));
    % to get changing color for each State's PDF
    color = (i-1)/size(emission_mat,1);
    % plotting each State's PDF
    plot(bins, em_pdf(i, :), 'Color', [sqrt(color), color, (color)^2]);
    % concatenating to the legend string
    legend_str{i} = strcat('State ', num2str(i));
end
grid;
legend(legend_str);
title('Estimated Prob. Dist. Functions of State Emission values');
xlabel('Emission value');
ylabel('PDF value');
end

This toolbox may also be of interest:

https://www.cs.ubc.ca/~murphyk/Software/HMM/hmm.html

Note that The MathWorks does not guarantee or warrant the use or content of these submissions. Any questions, issues, or complaints should be directed to the contributing author.

Best Answer

Related Solutions

MATLAB: Does the “SEQ” output from the HMMGENERATE function from the Statistics Toolbox not return emission probabilty

MATLAB: How to use continuous sequence values with HMMESTIMATE in the Statistics Toolbox 7.1 (R2009a)

Related Question