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I'm working on a project to calculate the Area of Mandelbrot Set by Monte Carlo method. I implemented my code by generating standard random numbers and throwing them into the area. I used simple sampling, and I also implemented Latin Hypercube Sampling.

I should also implement my project with Pure Sampling and Orthogonal Sampling method. I read Wikipedia but it is not clear for me what is exactly Orthogonal Sampling and how I should implement it. I implement Latin Hypercube Sampling by dividing the area and generating random arrays for each sub-area and counting.

What about Orthogonal? That would be great if someone help me to understand the concept at least.

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Latin Hypercube (LHC) sampling is a sampling method than ensures that each sampling space dimension is roughly evenly sampled. Orthogonal sampling is basically an extension to latin hypercube sampling that ensures that each subspace is evenly sampled. This also ensures that correlation between sampling dimensions is minimised.

In the following image, from wikipedia, you can see that in the LHC example (II), the two dimension have a strong (negative) correlation, and also the combinations of (low A, low B) and (high A, high B) (the bottom left and top right corners) are not sampled at all. This may confound any results, since it will be difficult to tell whether the A or B variable is causing the effects you're seeing.

enter image description here

In contrast, in the Orthogonal sampling example (III), there is very little correlation between A and B, and each of the 4 subspaces are evenly sampled. Note that the subspaces are somewhat arbitrary, but larger than a single sample, and smaller than the entire sample space.

Sounds to me like you actually implemented orthogonal sampling, rather than plain LHC, but depending on the project, there is no disadvantage to doing that. The usual process for generating a plain LHC is to divide the sample space up evenly into n divisions in each dimension, and then randomly allocate n samples to those divisions, ensuring that no two samples share the same division in a given dimension (which also ensures, by the pigeon hole principle, that all the divisions are sampled).

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  • $\begingroup$ Thanks, it helps me a little bit but it is not totally clear yet. I'm going to read and ask again in case of problem :) $\endgroup$
    – Amir
    Nov 3, 2012 at 18:22

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