Probability distribution fitting: Difference between revisions

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Some probability distributions, like the [[exponential distribution|exponential]], do not support data values (''X'') equal to or less than zero. Yet, when negative data are present, such distributions can still be used replacing ''X'' by ''Y''=''X''-''Xm'', where ''Xm'' is the minimum value of ''X''. This replacement represents a shift of the probability distribution in positive direction, i.e. to the right, because ''Xm'' is negative. After completing the distribution fitting of ''Y'', the corresponding ''X''-values are found from ''X''=''Y''+''Xm'', which represents a back-shift of the distribution in negative direction, i.e. to the left.<br>

The technique of distribution shifting augments the chance to find a properly fitting probability distribution.

==Composite distributions==

[[File:SanLor.jpg|thumb|left|Composite (discontinuous) distribution with confidence belt <ref>Intro to composite probability distributions [https://www.waterlog.info/composite.htm]</ref> ]]

The option exists to use two different probability distributions, one for the lower data range, and one for the higher like for example the [[Laplace distribution]]. The ranges are separated by a break-point. The use of such composite (discontinuous) probability distributions can be opportune when the data of the phenomenon studied were obtained under two sets different conditions.

[[File:SampleFreqCurves.tif|thumb|Variations of nine ''[[return period]]'' curves of 50-year samples from a theoretical 1000 year record (base line), data from Benson <ref>Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (Ed.), Flood frequency analysis. U.S. Geological Survey Water Supply Paper, 1543-A, pp. 51-71.</ref>]]

== Uncertainty of prediction ==

[[File:SampleFreqCurves.tif|thumb|Variations of nine ''[[return period]]'' curves of 50-year samples from a theoretical 1000 year record (base line), data from Benson <ref>Benson, M.A. 1960. Characteristics of frequency curves based on a theoretical 1000 year record. In: T.Dalrymple (Ed.), Flood frequency analysis. U.S. Geological Survey Water Supply Paper, 1543-A, pp. 51-71.</ref>]]