# Seeking assistance creating a frequency matrix

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John on 3 Jan 2013
Hello,
I was hoping that somebody might be able to help me out in creating a frequency matrix?
I normally use excel for statistical modelling but this particular problem takes hours to execute using spreadsheets.
I have created a variables called 'data' and it contains numbers ranging from 0-828.
I was hoping to create a frequency matrix of the number of times one number follows another number. Or put a different way, the number of times a number transitions to another number.
Here is a simple example with numbers ranging from 1-5.
For example, the cell in red, the number 3 is followed by 2 twice, or transitions to 2 twice.
I'd appreciate any help that is offered.
Kind Regards
John

Walter Roberson on 3 Jan 2013
dcol = data(:); %need column vector form
freqmatrix = accumarray( [dcol(1:end-1) + 1, dcol(2:end) + 1], 1 );
This would create a matrix up to 829 x 829. Any one position, (R,C) in it, would indicate a transition from (R-1) to (C-1). The offset of 1 is needed because you have 0 values and subscripts cannot be 0.
If the number of unique values is considerably smaller than 829, then it would be possible to make the matrix much smaller, at the expense of making it less natural to read.
Roger Stafford on 3 Jan 2013
If your data values are not positive integers or are not closely-packed near zero, you can do a 'unique' call first. To use Walter's nomenclature,
[ud,~,nd] = unique(data(:));
N = length(nd);
freqmatrix = accumarray([nd(1:N-1),nd(2:N)],1,[N,N]);
The rows and columns in N x N 'freqmatrix' would then correspond to values in 'ud'.

Image Analyst on 3 Jan 2013
John, though you don't know it (yet), what you are describing is called the "gray level co-occurence matrix". This is done by graycomatrix() in the Image Processing Toolbox.
Roger Stafford on 3 Jan 2013
Yes, T^20. Let's suppose you have the transition matrix
T = [ .3 , .7 ;
.6 , .4 ]
Then T^2 = T*T would be
T^2 = [ .3*.3+.7*.6 , .3*.7+.7*.4 ;
.6*.3+.4*.6 , .6*.7+.4*.4 ]
= [ .51 , .49 ;
.42 , .58 ]
In other words, for example, given that you have started with 1 the probability that after two steps you will end with 2 (.49) is the probability that after one step you will remain at 1 (.3) times the conditional probability that on the next step you will go to 2 (.7) plus the probability that on the first step you will go to 2 (.7) times the conditional probability that on the next step you will remain at 2 (.4) giving a total of .3*.7+.7*.4 = .49 . This is just what matrix multiplication accomplishes.