Discretization of 3dimensional signals

Question

Symbolic Aggregate Approximation talks about converting a time series (x,t) into symbols.Based on this,I have few basic queries. What if the time signal is a composite of (x,y,z,t) representing position coordinates or simply (x,y,t) for 2 dimensional image with a time stamp t. Then how do I use this tool to assign symbols/discretize.Please help.

Answer 1

You could apply the SAX transformation to each dimension separately, then combine the symbols/letters for each time stamp.

Take the (x,y,z,t) for example, you would get the combination b,a,c for t=1, then a,a,c for t=2, etc..

You could then combine the symbols to form "mega-symbols" if you want. Say the set of symbols was Symbols={a,b,c}. Then the new set of letters would simply be the cartesian product SxSxS (one for each dimension).

In other words aaa becomes the new letter A, aab as B, then aac, aba, abb, etc..

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EDIT:

Here is some code to show what I had in mind. Since I don't have an implementation of SAX algorithm, I will be using the following function as placeholder (it returns rubbish):

%# use your actual SAX function instead of this one
my_sax_function = @(x,n,a) randi(a, [n 1]);

Here is the code:

%# time series of length=100, with (x,y,z) at each timestamp
data = cumsum(randn(100,3));

%# apply your SAX function to each dimension independently
N = 20;     %# number of segments to divide the signal into
A = 3;      %# size of alphabet (a,b,c)
dataSAX = zeros(N,3);
for i=1:3
    dataSAX(:,i) = my_sax_function(data(:,i), N, A);
end

%# we assume the above function returns integers denoting the symbols
%# therefore row i corresponds to A=3 symbols for each of the 3 x/y/z dimensions
dataSAX(1,:)

%# build cartesian product of all combinations of the A=3 symbols
[x y z] = ndgrid(1:A,1:A,1:A);
cartProd = [x(:) y(:) z(:)];

%# map to the new alphabet with 3*3*3 = 27 symbols
[~,V] = ismember(dataSAX, cartProd, 'rows')

%# A to Z with $ character to make up 27 symbols
oldSymbols = {'a';'b';'c'};             %# 1: a, 2: b, 3: c
newSymbols = cellstr(['A':'Z' '$']');   %# 1: A, ..., 26: Z, 27: $

%# SAX representation of the entire time series as a string
mappedV = char(newSymbols(V))'