Editing Multimodal distribution (section)

==Parameter estimation and fitting curves==

Assuming that the distribution is known to be bimodal or has been shown to be bimodal by one or more of the tests above, it is frequently desirable to fit a curve to the data. This may be difficult.

Bayesian methods may be useful in difficult cases.

===Software===

;Two normal distributions

A package for [[R (programming language)|R]] is available for testing for bimodality.<ref>{{cite web |url=http://www.uni-marburg.de/fb12/stoch/research/rpackage/manualbimodlilitytest.pdf |title=Archived copy |access-date=2013-11-01 |url-status=dead |archive-url=https://web.archive.org/web/20131103100209/http://www.uni-marburg.de/fb12/stoch/research/rpackage/manualbimodlilitytest.pdf |archive-date=2013-11-03 }}</ref> This package assumes that the data are distributed as a sum of two normal distributions. If this assumption is not correct the results may not be reliable. It also includes functions for fitting a sum of two normal distributions to the data.

Assuming that the distribution is a mixture of two normal distributions then the expectation-maximization algorithm may be used to determine the parameters. Several programmes are available for this including Cluster,<ref>{{cite web|url=https://engineering.purdue.edu/~bouman/software/cluster/|title=Cluster home page|website=engineering.purdue.edu}}</ref> and the R package nor1mix.<ref>{{cite web|url=https://cran.r-project.org/web/packages/nor1mix/index.html|title=nor1mix: Normal (1-d) Mixture Models (S3 Classes and Methods)|first=Martin|last=Mächler|date=25 August 2016|via=R-Packages}}</ref>

;Other distributions

The mixtools package available for R can test for and estimate the parameters of a number of different distributions.<ref>{{cite web|url=https://cran.r-project.org/web/packages/mixtools/index.html|title=mixtools: Tools for Analyzing Finite Mixture Models|first1=Derek|last1=Young|first2=Tatiana|last2=Benaglia|first3=Didier|last3=Chauveau|first4=David|last4=Hunter|first5=Ryan|last5=Elmore|first6=Thomas|last6=Hettmansperger|first7=Hoben|last7=Thomas|first8=Fengjuan|last8=Xuan|date=10 March 2017|via=R-Packages}}</ref> A package for a mixture of two right-tailed gamma distributions is available.<ref>{{cite web|title=discrimARTs|url=https://cran.r-project.org/web/packages/discrimARTs/discrimARTs.pdf|website=cran.r-project.org|access-date=22 March 2018}}</ref>

Several other packages for R are available to fit mixture models; these include flexmix,<ref>{{cite web|url=https://cran.r-project.org/web/packages/flexmix/index.html|title=flexmix: Flexible Mixture Modeling|first1=Bettina|last1=Gruen|first2=Friedrich|last2=Leisch|first3=Deepayan|last3=Sarkar|first4=Frederic|last4=Mortier|first5=Nicolas|last5=Picard|date=28 April 2017|via=R-Packages}}</ref> mcclust,<ref>{{cite web|url=https://cran.r-project.org/web/packages/mclust/index.html|title=mclust: Gaussian Mixture Modelling for Model-Based Clustering, Classification, and Density Estimation|first1=Chris|last1=Fraley|first2=Adrian E.|last2=Raftery|first3=Luca|last3=Scrucca|first4=Thomas Brendan|last4=Murphy|first5=Michael|last5=Fop|date=21 May 2017|via=R-Packages}}</ref> agrmt,<ref>{{cite web|url=https://cran.r-project.org/web/packages/agrmt/index.html|title=agrmt|first1=Didier|last1=Ruedin|date=2 April 2016|publisher=cran.r-project.org}}</ref> and mixdist.<ref>{{cite web|url=https://cran.r-project.org/web/packages/mixdist/index.html|title=mixdist: Finite Mixture Distribution Models|first1=Peter|last1=Macdonald|first2=with contributions from Juan|last2=Du|date=29 October 2012|via=R-Packages}}</ref>

The statistical programming language [[SAS language|SAS]] can also fit a variety of mixed distributions with the PROC FREQ procedure.
[[File:Joggers.png|thumb|Number of joggers in a park by time of the day (X in hours) in a bimodal probability distribution]]

In Python, the package [[Scikit-learn]] contains a tool for mixture modeling<ref>{{cite web|title=Gaussian mixture models|url=https://scikit-learn.org/stable/modules/mixture.html#mixture|website=scikit-learn.org|access-date=30 November 2023}}</ref>

===Example software application===
The CumFreqA <ref>CumFreq, free program for fitting of probability distributions to a data set. On line: [https://www.waterlog.info/cumfreq.htm]</ref> program for the fitting of composite probability distributions to a data set (X) can divide the set into two parts with a different distribution. The figure shows an example of a double generalized mirrored [[Gumbel distribution]] as in [[distribution fitting]] with cumulative distribution function (CDF) equations:

 X < 8.10 : CDF = 1 - exp[-exp{-(0.092X'''^'''0.01+935)}]
 X > 8.10 : CDF = 1 - exp[-exp{-(-0.0039X'''^'''2.79+1.05)}]