General Purpose Convolution Algorithm in S4-Classes by means of FFT

Details General Purpose Convolution Algorithm in S4-Classes by means of FFT General Purpose Convolution Algorithm in S4-Classes by means of FFT

2010-06-04

arxiv.org/abs/1006.0764v2

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Object orientation provides a flexible framework for the implementation of the convolution of arbitrary distributions of real-valued random variables. We discuss an algorithm which is based on the Discrete Fourier Transformation and its fast computability via the Fast Fourier Transformation. It directly applies to lattice-supported distributions. In the case of continuous distributions an additional discretization to a linear lattice is necessary and the resulting lattice-supported distributions are suitably smoothed after convolution. We compare our algorithm to other approaches aiming at a similar generality as to accuracy and speed. In situations where the exact results are known, several checks confirm a high accuracy of the proposed algorithm which is also illustrated at approximations of non-central $\chi^2$-distributions. By means of object orientation this default algorithm can be overloaded by more specific algorithms where possible, in particular where explicit convolution formulae are available. Our focus is on \proglang{R} package \pkg{distr} which includes an implementation of this approach overloading operator "+" for convolution; based on this convolution, we define a whole arithmetics of mathematical operations acting on distribution objects, comprising, among others, operators \code{+}, \code{-}, \code{*}, \code{/}, and \code{^}.

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