TPC Numerical Methods

This page is currently under development

Introduction

This page contains a brief description of the numerical methods used in the TPC. A full description will be published in a paper for Journal of Statistical Software. This paper is currently in preparation.

Matlab function TPC_20 redirects the calculation effort according to the characteristic functions present in the first argument of this function, the cell structure cCF.
.

The Cell Structure cCF Contains at Least One Symmetric Normal RV

In this case the characteristic function (CF) of the linear combination of the components of cCF decays to zero at least as

exp(-t^2 * sigma^2 / 2),

where sigma^2 is the combined (and weighted by linear combination coefficients, lambda) standard deviation of all normal distributions in cCF. The internal Matlab function integral is capable to calculate the inversion integrals for TP / CDF / PDF calculation. This is done in function Gil_Pelaez_21_normal that is part of this package.

The Cell Structure cCF Contains at Least One Student's t RV

In this case the dominant term in the decay of the CF cCF to zero is given by the asymptotic behaviour of the modified Bessel function of second kind and it is

exp(-c * t)

where c is a constant given by the weighted contribution of all Student's t distributions in cCF. In this case function TPC_20 calls the function Gil-Pelaez_21_t which, in turn, calls function integrate_oscillatory_real to evaluate the oscillatory integral. The latest uses the method described in Davis and Rabinowitz (1984) page 230 to accelerate the convergence process.

The Cell Structure cCF Contains at Least One Symmetric Uniform or Symmetric Triangular RV

By examination of the asymptotic behaviour of CF cCF for large values of t one can see that it decays to zero with a rate

t^-(n_u + 2 * n_t),

where n_u is the number of uniform RV and n_t is the number of triangular RV in the CF cCF. Compared to the previous two cases this is a much slower convergence. Nevertheless the same method as in the previous case can be used here. This is done in Matlab function Gil_Pelaez_21_oscillatory which is part of this package.

References

1. Davis and Rabinowitz (1984): P J Davis and P Rabinowitz, Methods of Numerical Integration, Academic Press (1984), reprinted by Dover Publications, Inc (2007), ISBN 978-0-486-45339-2.