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Concept# Cumulative frequency analysis

Summary

Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called frequency of non-exceedance.
Cumulative frequency analysis is performed to obtain insight into how often a certain phenomenon (feature) is below a certain value. This may help in describing or explaining a situation in which the phenomenon is involved, or in planning interventions, for example in flood protection.
This statistical technique can be used to see how likely an event like a flood is going to happen again in the future, based on how often it happened in the past. It can be adapted to bring in things like climate change causing wetter winters and drier summers.
Frequency analysis is the analysis of how often, or how frequently, an observed phenomenon occurs in a certain range.
Frequency analysis applies to a record of length N of observed data X1, X2, X3 . . . XN on a variable phenomenon X. The record may be time-dependent (e.g. rainfall measured in one spot) or space-dependent (e.g. crop yields in an area) or otherwise.
The cumulative frequency MXr of a reference value Xr is the frequency by which the observed values X are less than or equal to Xr.
The relative cumulative frequency Fc can be calculated from:
where N is the number of data
Briefly this expression can be noted as:
When Xr = Xmin, where Xmin is the unique minimum value observed, it is found that Fc = 1/N, because M = 1. On the other hand, when Xr = Xmax, where Xmax is the unique maximum value observed, it is found that Fc = 1, because M = N. Hence, when Fc = 1 this signifies that Xr is a value whereby all data are less than or equal to Xr.
In percentage the equation reads:
The cumulative probability Pc of X to be smaller than or equal to Xr can be estimated in several ways on the basis of the cumulative frequency M.
One way is to use the relative cumulative frequency Fc as an estimate.

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