General Purpose Convolution Algorithm in S4 Classes by Means of FFT

Peter Ruckdeschel, Matthias Kohl

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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 fast Fourier transform. 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 for approximations of non-central χ2 distributions. By means of object orientation this default algorithm is overloaded by more specific algorithms where possible, in particular where explicit convolution formulae are available. Our focus is on R package distr which implements this approach, overloading operator + for convolution; based on this convolution, we define a whole arithmetics of mathematical operations acting on distribution objects, comprising operators +, -, *, /, and ^.

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