Probability distribution fitting: Difference between revisions

Revision as of 20:34, 31 August 2020

Revision as of 11:18, 16 October 2020

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{{Use dmy dates|date=AugustOctober 20122020}}

'''Probability distribution fitting''' or simply '''distribution fitting''' is the fitting of a [[probability distribution]] to a series of data concerning the repeated measurement of a variable phenomenon.

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''Skew distributions to the left''

When the smaller values tend to be farther away from the mean than the larger values, one has a skew distribution to the left (i.e. there is negative skewness), one may for example select the ''square-normal distribution'' (i.e. the normal distribution applied to the square of the data values) ,<ref name="skew"> Left (negatively) skewed frequency histograms can be fitted to square Normal or mirrored Gumbel probability functions. On line: [https://www.researchgate.net/publication/338633570_Left_negatively_skewed_frequency_histograms_can_be_fitted_to_square_Normal_or_mirrored_Gumbel_probability_functions] </ref>, the inverted (mirrored) Gumbel distribution ,<ref name=skew/>, the [[Dagum distribution]] (mirrored Burr distribution), or the [[Gompertz distribution]], which is bounded to the left.

== Techniques of fitting ==

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==Composite distributions==

[[File:SanLor.jpg|thumb|left|Composite (discontinuous) distribution with confidence belt <ref>[https://www.waterlog.info/composite.htm Intro to composite probability distributions]</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.<ref>''Software for Generalized and Composite Probability Distributions''. In: International Journal of Mathematical and Computational Methods, January 2019. On line: [https://www.iaras.org/iaras/filedownloads/ijmcm/2019/001-0001(2019).pdf]</ref>

== Uncertainty of prediction ==

[[File:BinomialConfBelts.jpg|thumb|<small>Uncertainty analysis with confidence belts using the binomial distribution </small> <ref>Frequency predictions and their binomial confidence limits. In: International Commission on Irrigation and Drainage, Special Technical Session: Economic Aspects of Flood Control and non -Structural Measures, Dubrovnik, Yougoslavia, 1988. [http://www.waterlog.info/pdf/binomial.pdf On line]</ref>]]

Predictions of occurrence based on fitted probability distributions are subject to [[uncertainty]], which arises from the following conditions:

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* A change of environmental conditions may cause a change in the probability of occurrence of the phenomenon

[[File:SampleFreqCurves.tif|thumb|left|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>]]

An estimate of the uncertainty in the first and second case can be obtained with the [[Binomial distribution|binomial probability distribution]] using for example the probability of exceedance ''Pe'' (i.e. the chance that the event ''X'' is larger than a reference value ''Xr'' of ''X'') and the probability of non-exceedance ''Pn'' (i.e. the chance that the event ''X'' is smaller than or equal to the reference value ''Xr'', this is also called [[cumulative probability]]). In this case there are only two possibilities: either there is exceedance or there is non-exceedance. This duality is the reason that the binomial distribution is applicable.

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==Goodness of fit==

By ranking the [[goodness of fit]] of various distributions one can get an impression of which distribution is acceptable and which is not.

==Histogram and density function==