Which function allow me to calculate cumulative variance over a vector? - matlab

Question

I need to calculate the cumulative variance of a vector. I have tried to build and script, but this script takes too much time to calculate the cumulative variance of my vectors of size 1*100000. Do you know if there exists a faster way to find this cumulative variance?
This is the code I am using
%%Creation of the rand vectors. ans calculation of the variances
d=100000; %dimension of the vectors
nv=6 %quantity of vectors
for j=1:nv;
VItimeseries(:,j)=rand(d,1); % Final matrix with vectors
end
%% script to calculate the cumulative variance in the columns of my matrix
VectorVarianza=0;
VectoFinalVar=0;
VectorFinalTotalVAriances=zeros(d,nv);
for k=1:nv %number of columns
for j=1:numel(VItimeseries(:,k)) %size of the rows
Vector=VItimeseries(:,k);
VectorVarianza(1:j)= Vector(1:j); % Vector to calculate the variance...
...Independently
VectorFinalVar(j,k)= var(VectorVarianza);%Calculation of variances
end
VectorFinalTotalVAriances(:,k)=VectorFinalVar(:,k)% construction of the...
...Final Vector with the cumulative variances
end

Answer 1

Looping over the n elements of x, and within the loop computing the variance of all elements up to i using var(x(1:i)) amounts to an algorithm O(n2). This is inherently expensive.
Sample variance (what var computes) is defined as sum((x-mean(x)).^2) / (n-1), with n = length(x). This can be rewritten as (sum(x.^2) - sum(x).^2 / n) / (n-1). This formula allows us to accumulate sum(x) and sum(x.^2) within a single loop, then compute the variance later. It also allows us to compute the cumulative variance in O(n).
For a vector x, we'd have the following loop:
x = randn(100,1); % some data
v = zeros(size(x)); % cumulative variance
s = x(1); % running sum of x
s2 = x(1).^2; % running sum of square of x
for ii = 2:numel(x) % loop starts at 2, for ii=1 we cannot compute variance
s = s + x(ii);
s2 = s2 + x(ii).^2;
v(ii) = (s2 - s.^2 / ii) / (ii-1);
end
We can avoid the explicit loop by using cumsum:
s = cumsum(x);
s2 = cumsum(x.^2);
n = (1:numel(x)).';
v = (s2 - s.^2 ./ n) ./ (n-1); % v(1) will be NaN, rather than 0 as in the first version
v(1) = 0; % so we set it to 0 explicitly here
The code in the OP computes the cumulative variance for each column of a matrix. The code above can be trivially adapted to do the same:
s = cumsum(VItimeseries,1); % cumulative sum explicitly along columns
s2 = cumsum(VItimeseries.^2,1);
n = (1:size(VItimeseries,1)).'; % use number of rows, rather than `numel`.
v = (s2 - s.^2 ./ n) ./ (n-1);
v(1,:) = 0; % fill first row with zeros, not just first element