Multidimensional arrays in Matlab
Xanadu Halkias
Overview
• Creating/accessing/indexing multidimensional
arrays
• Using and computing with multidimensional
arrays
• Multidimensional cell arrays
• Multidimensional structure arrays
Corpus hypercubus
%Creating multidimensional
arrays
a = [1 2 3;0 6 4;2 5 4];
a(:,:,2) = [4 2 1;0 5 1;6 3 2];
a(:,:,3) = 3;
%turning a into a 3x3x3x2
a(:,:,:,2) = ones(3,3,3);
a(:,:,1,2) = 4;
More on creating
%different way
b = repmat(2,[3 3 3 2]);
c = repmat(rand(2),[3 1 2]);
%one more
d = cat(4,[1 4; 2 3],[5 8;2 3]);
Memory: try to pre-allocate eg. zeros or use repmat for best
performance
A few good functions…
%reshape and permute
%a bunch of data
x = rand(3,3,2);
new_x = reshape(x,[6 3]);
tranx = permute(x,[2 1 3]);
Ind = sub2ind(size(x),1,3,2);
Working with them
%computing with multidimensional
arrays
x = rand(3,3,3);
s = sum(x,3);
d = sin(x);
e = eig(x(:,:,2));
ee = eig(squeeze(x(2,:,:)));
Multidimensional cell arrays
%multidimensional cells
a = {[1 2 5 3 77 2 4 5], 1:10; rand(1,3), 2};
b = {1:2:15, rand(1,5); [1 0 0 1], 5;}
c = cat(3,a,b);
[D{1}] = conv(c{1:2});
[D{2}] = conv(c{1,1,:});
d = cell2mat(D);
for i=1:length(D{1}),
m1(i) = D{1}(i);
end
for i=1:length(D{2}),
m2(i) = D{2}(i);
end
m = [m1, m2];
Multidimensional structure arrays
%multidimensional structures
dolphin(1,1,1).species = 'bottlenose';
dolphin(1,1,1).sampfreq = 44100;
dolphin(1,1,1).signal = rand(1,100);
dolphin(1,2,1).species = 'atlantic';
dolphin(1,2,1).sampfreq = 44100;
dolphin(1,2,1).signal = rand(1,100);
dolphin(1,1,2).species = 'pink';
dolphin(1,1,2).sampfreq = 44100;
dolphin(1,1,2).signal = rand(1,100);
dolphin(1,2,2).species = 'riso';
dolphin(1,2,2).sampfreq = 44100;
dolphin(1,2,2).signal = rand(1,100);
Multidimensional structures more…
%computing some stuff
s = sum([dolphin.sampfreq]);
[b{1}] = fft(dolphin(1,1,1).signal);
dolphin(1,1,1).fourier = b;
Murphy’s mhmm_em.m
%each page gets its own cell
if ~iscell(data)
data = num2cell(data, [1 2]);
end
numex = length(data); %num of pages is number
mixtures
O = size(data{1},1); %getting the dimensionality
Q = length(prior); %number of states
if isempty(mixmat)%number of mixtures
mixmat = ones(Q,1);
end
M = size(mixmat,2);
if M == 1
adj_mix = 0;
end
Murphy’s more-EM
while (num_iter <= max_iter) & ~converged
% E step
[loglik, exp_num_trans, exp_num_visits1,
postmix, m, ip, op] = ...
ess_mhmm(prior, transmat, mixmat, mu,
Sigma, data);
% M step
if verbose, fprintf(1, 'iteration %d, loglik =
%f\n', num_iter, loglik); end
num_iter = num_iter + 1;
converged =
em_converged(loglik,previous_loglik, thresh);
previous_loglik = loglik;
LL = [LL loglik];
end
The E of EM
function [loglik, exp_num_trans, exp_num_visits1,
postmix, m, ip, op] = ... ess_mhmm(prior,
transmat, mixmat, mu, Sigma, data)
Loglik: likelihood of the model
Exp_num_trans: expected number of transitions is
the transition matrix
Exp_num_visits1: expected number of visits to
each state is our priors
Postmix: is the mixing matrix
M: is the weighted observations
Ip: is the weighted inner product of the
observations
Op: is the weighted outer product of the
observations
More on E
%initialization
verbose = 0;
numex = length(data);
O = size(data{1},1);
Q = length(prior);
M = size(mixmat,2);
exp_num_trans = zeros(Q,Q);
exp_num_visits1 = zeros(Q,1);
postmix = zeros(Q,M);
m = zeros(O,Q,M);
op = zeros(O,O,Q,M);
ip = zeros(Q,M);
mix = (M>1);
One more on expectation
loglik = 0;
if verbose, fprintf(1, 'forwards-backwards example #
'); end
for ex=1:numex
if verbose, fprintf(1, '%d ', ex); end
obs = data{ex};
T = size(obs,2);
if mix %if you have mixtures
[B, B2] = mixgauss_prob(obs, mu, Sigma, mixmat);
[alpha, beta, gamma, current_loglik, xi, gamma2] =
...
fwdback(prior, transmat, B, 'obslik2', B2, 'mixmat',
mixmat);
else %one single gaussian as an emission
B = mixgauss_prob(obs, mu, Sigma);
[alpha, beta, gamma, current_loglik, xi] =
fwdback(prior, transmat, B);
end
Almost done
loglik = loglik + current_loglik;
if verbose, fprintf(1, 'll at ex %d = %f\n',
ex, loglik); end
exp_num_trans = exp_num_trans + sum(xi,3);
exp_num_visits1 = exp_num_visits1 + gamma(:,1);
if mix
postmix = postmix + sum(gamma2,3);
else
postmix = postmix + sum(gamma,2);
gamma2 = reshape(gamma, [Q 1 T]);
end
Last one
for i=1:Q
for k=1:M
w = reshape(gamma2(i,k,:), [1 T]); % w(t) =
w(i,k,t,l)
wobs = obs .* repmat(w, [O 1]); % wobs(:,t)
= w(t) * obs(:,t)
m(:,i,k) = m(:,i,k) + sum(wobs, 2); % m(:)
= sum_t w(t) obs(:,t)
op(:,:,i,k) = op(:,:,i,k) + wobs * obs'; %
op(:,:) = sum_t w(t) * obs(:,t) * obs(:,t)'
ip(i,k) = ip(i,k) + sum(sum(wobs .* obs,
2)); % ip = sum_t w(t) * obs(:,t)' * obs(:,t)
end
end
end
if verbose, fprintf(1, '\n'); end