tangent tangent correlation calculation in matrix form MATLAB

Question

Consider the following calculation of the tangent tangent correlation which is performed in a for loop

v1=rand(25,1);
v2=rand(25,1);
n=25;
nSteps=10;
mean_theta = zeros(nSteps,1);
for j=1:nSteps
    theta=[];
    for i=1:(n-j)
        d = dot([v1(i) v2(i)],[v1(i+j) v2(i+j)]);
        n1 = norm([v1(i) v2(i)]);
        n2 = norm([v1(i+j) v2(i+j)]);
        theta = [theta acosd(d/n1/n2)];  

    end
    mean_theta(j)=mean(theta);
end
plot(mean_theta)

How can matlab matrix calculations be utilized to make this performance better?

Answer 1

Here is a full vectorized solution:

i = 1:n-1;
j = (1:nSteps).';
ij= min(i+j,n);
a = cat(3, v1(i).', v2(i).');
b = cat(3, v1(ij), v2(ij));

d = sum(a .* b, 3);
n1 = sum(a .^ 2, 3);
n2 = sum(b .^ 2, 3);
theta = acosd(d./sqrt(n1.*n2));
idx = (1:nSteps).' <= (n-1:-1:1);

mean_theta = sum(theta .* idx ,2) ./ sum(idx,2);

Result of Octave timings for my method,method4 from the answer provided by @CrisLuengo and the original method (n=250):

Full vectorized    : 0.000864983 seconds
Method4(Vectorize) : 0.002774 seconds
Original(loop)     : 0.340693 seconds