February 2017 Archives

Integration, Quasi Mode

Last time we saw how it was possible to use uniformly distributed random variables to approximate the integrals of univariate and multivariate functions, being those that take numbers and vectors as arguments respectively. Specifically, since the integral of a univariate function is equal to the net area under its graph within the interval of integration it must equal its average height multiplied by the length of that interval and, by definition, the expected value of that function for a uniformly distributed random variable on that interval is the average height and can be approximated by the average of a large number of samples of it. This is trivially generalised to multivariate functions with multidimensional volumes instead of areas and lengths.
We have also seen how quasi random sequences fill areas more evenly than pseudo random sequences and so you might be asking yourself whether we could do better by using the former rather than the latter to approximate integrals.

Clever you!

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