Generating iterative sequence without using for-loop - matlab

Question

I want to generate a random series x with length N through the following rule related to non-central chi-square distribution:
xn+1~χν2(λxn)
where ν is a given constant representing degrees of freedom, λ is also pre-specified and the multiplication of λ and xn is the non-centrality parameter, x1 is supposed to be given.
I wrote the following code to generate such sequence and time the running with x1=0.04, ν=0.005, λ=100 and N=1e5:
tic;
N = 1e5;
x = zeros(1,N);
x(1) = 0.04;
nu = 0.005;
lambda = 100;
for i = 1:N-1
x(i+1) = ncx2rnd(nu,lambda*x(i));
end
toc;
To illustrate my question, I have tested another example, which is different from above. Here I considered generating N=1e5 samples from the distribution χν2(λ) with ν=0.005 and λ=100:
tic;
N = 1e5;
x = zeros(1,N);
nu = 0.005;
lambda = 100;
for i = 1:N
x(i) = ncx2rnd(nu,lambda);
end
toc;
tic;
N = 1e5;
nu = 0.005;
lambda = 100;
x = ncx2rnd(nu,lambda*ones(1,N));
toc;
These two approaches work equivalently. However, it turns out that the second approach which does not use for-loop is much faster than the first one. The difference between both examples is, in the second example, the rule to generate some sample does not require the information of previous samples, which is not the case in the first, therefore all samples can be generated simultaneously without using for-loop. Based on this I wonder whether avoiding for-loop would accelerate the code execution. So would there be any MATLAB built-in function to generate random series shown in the first example without using for-loop when the rule of dependence on previous samples is explicit? If the rule is linear I know the function filter would be a possible choice, what about cases like the first example?

Answer 1

Logically it's impossible to calculate something iterative without doing the iterations. If x(n+1) is dependent on x(n) then you must calculate x(n) first, there is no "clever trick" here.
That just leaves us to optimise the calculation within the loop, specifically ncx2rnd. As with most MATLAB in-built functions, it is already fairly concise and performant, but there are some things to consider. Note that what I'm about to suggest involves using edit ncx2nrd to look inside this in-built function which contains code under MathWorks copyright, I'm simply noting observations about how it works.
There are some input checks to handle incorrectly sized inputs and/or inputs with negative values. If you can take the burden of validation on yourself (i.e. you know your inputs are valid) then you can reduce the function to its single mathematical operation:
% function r = ncx2rnd(v,delta)
r = 2.*randg(poissrnd(delta./2, sizeOut)) + 2.*randg(v./2,sizeOut);
Running this standalone saves around 20% of the processing time, which was for input validation (with a nominal N=1e5).
In the MathWorks syntax, delta is equal to your lambda*x(i), the other term including v is independent of your x, so you could compute it outside of the loop, i.e. vectorising one of the calls to randg. Again using N=1e5 this brings the total time saving to around 25%.
The result would mean this change to your example:
% Common inputs
N = 1e5;
nu = 0.1;
lambda = 0.1;
% Baseline example
x = zeros(1,N);
x(1) = 0.04;
for i = 1:N-1
x(i+1) = ncx2rnd(nu,lambda*x(i));
end
% ~25% faster alternative, with no input validation and partially vectorised
x = zeros(1,N);
x(1) = 0.04;
vTerm = 2.*randg(nu./2, [1,N]);
for i = 1:N-1
x(i+1) = 2.*randg(poissrnd(lambda*x(i)./2, [1,1])) + vTerm(i);
end