Min and max of deviations from mean in Matlab

Question

Intro: I need to find the greatest/smallest possible deviations from the average in some given Matlab vectors. The issue is that, due to the large dimensions of the arrays involved, I cannot explicitly construct such vectors. Therefore, I wonder whether there is a smarter way to proceed. I summarise below the procedure that I would implement if there were no memory constraints and then present my question.

(1) Consider 28 matrices in Matlab A{1},..., A{28} . For each j=1,...,28 , A{j} has 4 columns. The number of rows of A{j} can be different across j=1,..., 28 and is stored in the vector r .

clear
rng default
A=cell(28,1);
r=randi(28,10000,1)+10000; %28x1
for j=1:28
A{j}=randn(r(j),4); %r(j)x4
end

(2) For each j=1,...,28 , let b{j} be the r(j) x 1 vector that is obtained by summing the first two rows of A{j} and subtracting the last two rows of A{j} :

b=cell(28,1);
for j=1:28
    b{j}=  A{j}(:,1)+A{j}(:,2)-A{j}(:,3)-A{j}(:,4); %r(j)x1
end

(3) Let B be the R x 28 matrix that is obtained as a 28-D grid from the vectors b{1},...,b{28} :

[ca, cb, cc, cd, ce, ...] = ndgrid(b{1}, b{2}, b{3}  , ..., b{28});
B(:,1)=ca(:);
B(:,2)=cb(:);
...
B(:,28)=...;

Apologies for the incomplete code. As you can imagine, this is the step where I have issues as explained below.

(4) For each row of B , I compute the difference between such row and its mean value and store the results in a R x 28 matrix D :

R=size(B,1);
D=zeros(R,28);
for t=1:R
    D(t,:)=B(t,:)-mean(B(t,:)); 
end

(5) For each j=1,...,28 , I compute the min and max value of D(:,j) and store it in a matrix F

F=zeros(2,28);
for j=1:28
F(1,j)=min(D(:,j));
F(2,j)=max(D(:,j));
end

Question: F is the matrix that I would like to construct. However, step (3) is unfeasible due to the large r(1),..., r(28) . Even a loop would take forever. Hence, I'm asking: is there a smarter way to construct F that saves me from explicitly obtaining B ?

Answer 1

Here is a fast solution (without using ndgrid ):

F = zeros(2,28);
mx = cellfun(@max,b)
mn = cellfun(@min,b)
for k = 1: 28
    F(1,k) = min(b{k}-(b{k}+sum(mx)-mx(k))/28);
    F(2,k) = max(b{k}-(b{k}+sum(mn)-mn(k))/28);
end