R. J. Oosterbaan
The use of composite probability distributions (i.e. integrated from two different distributions, including inverted/mirrored distributions) is useful in case the data sample is drawn under changing external conditions, which frequently occurs. The use of composite distributions is not widespread, with the exception of the Laplace distribution. Also, software for the fitting of such distributions to data series, with the aim to obtain an impression of the frequency of occurrence under changing external conditions, is scarce. This article uses the software package CumFreqA, designed for that purpose, and explains how various well known distributions can be used to obtain composite ones. Simultaneously the software finds the optimal value of the separation point Q of the different distributions left and right of it. Another technique of CumFreqA is to raise the data to a power P, whose value can be optimized numerically using iterative procedures, to reach the condition of minimum sum of squares of deviations of the theoretical from the observed values. The transformation of data to obtain a better fit is not often done with the exception of the log-normal distribution which uses a logarithmic transformation of the data instead of an exponential, which offers more flexibility. Thus the distribution is generalized and made composite to enhance the goodness of fit. Further, the parameters of the distributions are found from transformations of the cumulative distribution function leading to linear equations where after a linear regression is applied, which simplifies the algorithm. The confidence belts of the cumulative distribution functions in CumFreqA, are constructed with the help of the binomial distribution. This leads to the possibility to construct confidence intervals of the return period as well. Various examples of distributions and confidence belts are given. CumFreqA offers the possibility to create histograms with intervals by choice and constructs the corresponding probability density functions, of which examples are given.
Probability distribution, composite, generalization, transformation, linear regression, confidence belt
Cite this paper
R. J. Oosterbaan. (2019) Software for Generalized and Composite Probability Distributions. International Journal of Mathematical and Computational Methods, 4, 1-9