Probability distribution fitting: Difference between revisions

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[[File:FitGumbelDistr.tif|thumb|220px|Cumulative Gumbel distribution fitted to maximum one-day October rainfalls in [[Surinam]] by the regression method with added '''[[confidence band]]''' using [[CumFreq|cumfreq]] ]]

*''Regression method'', using a transformation of the [[cumulative distribution function]] so that a [[linear relation]] is found between the [[cumulative probability]] and the values of the data, which may also need to be transformed, depending on the selected probability distribution. In this method the cumulative probability needs to be estimated by the [[plotting position]] <ref name="gen">Software for Generalized and Composite Probability Distributions. International Journal of Mathematical and Computational Methods, 4, 1-9 [https://www.iaras.org/iaras/home/caijmcm/software-for-generalized-and-composite-probability-distributions] or [https://www.waterlog.info/pdf/MathJournal.pdf]</ref>

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More generally one can raise the data to a power ''p'' in order to fit symmetrical distributions to data obeying a distribution of any skewness, whereby ''p'' < 1 when the skewness is positive and ''p'' > 1 when the skewness is negative. The optimal value of ''p'' is to be found by a [[numerical method]]. The numerical method may consist of assuming a range of ''p'' values, then applying the distribution fitting procedure repeatedly for all the assumed ''p'' values, and finally selecting the value of ''p'' for which the sum of squares of deviations of calculated probabilities from measured frequencies ([[Chi-squared test|chi squared]]) is minimum, as is done in [[CumFreq]].

The generalization enhances the flexibility of probability distributions and increases their applicability in distribution fitting. <ref name="gen">Software for Generalized and Composite Probability Distributions. International Journal of Mathematical and Computational Methods, 4, 1-9 [https://www.iaras.org/iaras/home/caijmcm/software-for-generalized-and-composite-probability-distributions] or [https://www.waterlog.info/pdf/MathJournal.pdf]</ref>

The versatility of generalization makes it possible, for example, to fit approximately normally distributed data sets to a large number of different probability distributions, <ref>Example of an approximately normally

distributed data set to which a large number of different probability distributions can be fitted, [https://www.waterlog.info/pdf/Multiple%20fit.pdf] </ref> while negatively skewed distributions can be fitted to

square normal and mirrored Gumbel distributions. <ref>Left (negatively) skewed frequency histograms can be

fitted to square normal or mirrored Gumbel probability functions.

[https://www.waterlog.info/pdf/LeftSkew.pdf]</ref>