Content deleted Content added
→Shifting of distributions: Composite distributions added |
m Images shifted |
||
Line 70:
[[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: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:
Line 86 ⟶ 78:
* The occurrence of events in another situation or in the future may deviate from the fitted distribution as this occurrence can also be subject to random error
* 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.
Line 94 ⟶ 88:
[[File:GEVdistrHistogr+Density.png|thumb|220px|Histogram and probability density of a data set fitting the [[GEV distribution]] ]]
==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==
| |||