May 2020 Archives

Moments Of Pathological Behaviour

Last time we took a look at basis function interpolation with which we approximate functions from their values at given sets of arguments, known as nodes, using weighted sums of distinct functions, known as basis functions. We began by constructing approximations using polynomials before moving on to using bell shaped curves, such as the normal probability density function, centred at the nodes. The latter are particularly useful for approximating multi-dimensional functions, as we saw by using multivariate normal PDFs.
An easy way to create rotationally symmetric functions, known as radial basis functions, is to apply univariate functions that are symmetric about zero to the distance between the interpolation's argument and their associated nodes. PDFs are a rich source of such functions and, in fact, the second bell shaped curve that we considered is related to that of the Cauchy distribution, which has some rather interesting properties.

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