Editing Multimodal distribution (section)

==Statistical tests==

A number of tests are available to determine if a data set is distributed in a bimodal (or multimodal) fashion.

===Graphical methods===

In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log(&nbsp;size&nbsp;) of the particles.<ref name=Folk1957>{{cite journal | last1 = Folk | first1 = RL | last2 = Ward | first2 = WC | year = 1957 | title = Brazos River bar: a study in the significance of grain size parameters | url = https://doi.pangaea.de/10.1594/PANGAEA.896129| journal = Journal of Sedimentary Research | volume = 27 | issue = 1| pages = 3–26 | doi=10.1306/74d70646-2b21-11d7-8648000102c1865d|bibcode = 1957JSedR..27....3F }}</ref><ref name=Dyer1970>{{cite journal | last1 = Dyer | first1 = KR | year = 1970 | title = Grain-size parameters for sandy gravels | journal = Journal of Sedimentary Research | volume = 40 | issue = 2| pages = 616–620 |doi=10.1306/74D71FE6-2B21-11D7-8648000102C1865D}}</ref> This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the [[logarithm]] is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the [[Grain size|Krumbein]] (or phi) scale.

An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode.

;Statistics

Approximate values for several statistics can be derived from the graphic plots.<ref name=Folk1957/>

<math display="block">\begin{align}
\text{mean} &= \frac{ \phi_{16} + \phi_{50} + \phi_{84} }{ 3 } \\[1ex]
\text{std. dev.} &= \frac{ \phi_{84} - \phi_{16} }{ 4 } + \frac{ \phi_{95} - \phi_5 }{ 6.6 } \\[1ex]
\text{skewness} &= \frac{ \phi_{84} +  \phi_{16} - 2  \phi_{50} }{ 2 ( \phi_{84} -  \phi_{16} ) } + \frac{ \phi_{95} +  \phi_{ 5 } -  2 \phi_{50} }{ 2( \phi_{95} - \phi_5 ) } \\[1ex]
\text{kurtosis} &= \frac{ \phi_{95} - \phi_5 }{ 2.44 ( \phi_{75} - \phi_{25} ) }
\end{align}</math>

where ''φ''<sub>x</sub> is the value of the variate ''φ'' at the ''x''<sup>th</sup> percentage of the distribution.

===Unimodal vs. bimodal distribution===

Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.<ref name=Pearson1894>{{cite journal | last1 = Pearson | first1 = K | year = 1894 | title = Contributions to the mathematical theory of evolution: On the dissection of asymmetrical frequency-curves | journal = Philosophical Transactions of the Royal Society A | volume = 185 | pages = 71–90 | doi=10.1098/rsta.1894.0003| bibcode = 1894RSPTA.185...71P| doi-access = free }}</ref> This method required the solution of a ninth order [[polynomial]]. In a subsequent paper Pearson reported that for any distribution skewness<sup>2</sup> + 1 < kurtosis.<ref name=Pearson1916/> Later Pearson showed that<ref name=Pearson1929>{{cite journal | last1 = Pearson | first1 = K | year = 1929 | title = Editorial note | journal = Biometrika | volume = 21 | pages = 370–375 }}</ref>

<math display="block"> b_2 - b_1 \ge 1 </math>

where ''b''<sub>2</sub> is the kurtosis and ''b''<sub>1</sub> is the square of the skewness. Equality holds only for the two point [[Bernoulli distribution]] or the sum of two different [[Dirac delta function]]s. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.

Baker proposed a transformation to convert a bimodal to a unimodal distribution.<ref name=Baker1930>{{cite journal | last1 = Baker | first1 = GA | year = 1930 | title = Transformations of bimodal distributions | journal = Annals of Mathematical Statistics | volume = 1 | issue = 4| pages = 334–344 | doi=10.1214/aoms/1177733063| doi-access = free }}</ref>

Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences.<ref name=Haldane1951>{{cite journal | last1 = Haldane | first1 = JBS | year = 1951 | title = Simple tests for bimodality and bitangentiality | journal = Annals of Eugenics  | volume = 16 | issue = 1| pages = 359–364 | doi = 10.1111/j.1469-1809.1951.tb02488.x | pmid = 14953132 }}</ref> Larkin later introduced a test based on the F test;<ref name=Larkin1979>{{cite journal | last1 = Larkin | first1 = RP | year = 1979 | title = An algorithm for assessing bimodality vs. unimodality in a univariate distribution | journal = Behavior Research Methods & Instrumentation | volume = 11 | issue = 4| pages = 467–468 | doi = 10.3758/BF03205709 | doi-access = free }}</ref> Benett created one based on [[G-test|Fisher's G test]].<ref name=Bennett1992>{{cite journal | last1 = Bennett | first1 = SC | year = 1992 | title = Sexual dimorphism of ''Pteranodon'' and other pterosaurs, with comments on cranial crests | journal = Journal of Vertebrate Paleontology | volume = 12 | issue = 4| pages = 422–434 | doi=10.1080/02724634.1992.10011472}}</ref> Tokeshi has proposed a fourth test.<ref name=Tokeshi1992>{{cite journal | last1 = Tokeshi | first1 = M | year = 1992 | title = Dynamics and distribution in animal communities; theory and analysis | journal = Researches on Population Ecology | volume = 34 | issue = 2| pages = 249–273 | doi=10.1007/bf02514796| s2cid = 22912914 }}</ref><ref name=Barreto2003>{{cite journal | last1 = Barreto | first1 = S | last2 = Borges | first2 = PAV | last3 = Guo | first3 = Q | year = 2003 | title = A typing error in Tokeshi's test of bimodality | journal = Global Ecology and Biogeography | volume = 12 | issue = 2| pages = 173–174 | doi=10.1046/j.1466-822x.2003.00018.x| hdl = 10400.3/1408 | hdl-access = free }}</ref> A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.<ref name=Holzmann2008/>

A method based on the score and Wald tests has been proposed.<ref name=Carolan2001>{{cite journal | last1 = Carolan | first1 = AM | last2 = Rayner | first2 = JCW | year = 2001 | title = One sample tests for the location of modes of nonnormal data | journal =  Journal of Applied Mathematics and Decision Sciences| volume = 5 | issue = 1| pages = 1–19 | doi=10.1155/s1173912601000013| citeseerx = 10.1.1.504.4999 | doi-access = free }}</ref> This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.

===Antimode tests===

Statistical tests for the antimode are known.<ref name=Hartigan2000>{{cite book |last=Hartigan |first=J. A. |date=2000 |chapter=Testing for Antimodes |editor1=Gaul W |editor2=Opitz O |editor3=Schader M |title=Data Analysis |series=Studies in Classification, Data Analysis, and Knowledge Organization |publisher=Springer |pages=169–181 |isbn=3-540-67731-3 |chapter-url=https://books.google.com/books?id=WVDmCAAAQBAJ&pg=PA169 }}</ref>

;Otsu's method

[[Otsu's method]] is commonly employed in computer graphics to determine the optimal separation between two distributions.

===General tests===

To test if a distribution is other than unimodal, several additional tests have been devised: the [[bandwidth test (multimodal)|bandwidth test]],<ref name=Silverman1981/> the [[dip test]],<ref name=Hartigan1985>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Hartigan | first2 = PM | year = 1985 | title = The dip test of unimodality | journal = Annals of Statistics | volume = 13 | issue = 1| pages = 70–84 | doi=10.1214/aos/1176346577| doi-access = free }}</ref> the [[excess mass test]],<ref name=Mueller1991>{{cite journal | last1 = Mueller | first1 = DW | last2 = Sawitzki | first2 = G | year = 1991 | title = Excess mass estimates and tests for multimodality | journal = Journal of the American Statistical Association | volume = 86 | issue = 415| pages = 738–746 |jstor=2290406 | doi=10.1080/01621459.1991.10475103}}</ref> the MAP test,<ref name="Rozál1994">{{cite journal | last1 = Rozál | first1 = GPM Hartigan JA | year = 1994 | title = The MAP test for multimodality | journal = Journal of Classification | volume = 11 | issue = 1| pages = 5–36 | doi = 10.1007/BF01201021 | s2cid = 118500771 }}</ref> the [[mode existence test]],<ref name=Minnotte1997>{{cite journal | last1 = Minnotte | first1 = MC | year = 1997 | title = Nonparametric testing of the existence of modes | journal = Annals of Statistics | volume = 25 | issue = 4| pages = 1646–1660 | doi=10.1214/aos/1031594735| doi-access = free }}</ref> the [[runt test]],<ref name=Hartigan1992>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Mohanty | first2 = S | year = 1992 | title = The RUNT test for multimodality | journal = Journal of Classification | volume = 9 | pages = 63–70 | doi=10.1007/bf02618468| s2cid = 121960832 }}</ref><ref name=Andrushkiw2008>{{cite journal |author1=Andrushkiw RI |author2=Klyushin DD |author3=Petunin YI |date=2008 |title=A new test for unimodality |journal=Theory of Stochastic Processes |volume=14 |issue=1 |pages=1–6}}</ref> the [[span test]],<ref name=Hartigan1988>{{cite book |last=Hartigan |first=J. A. |year=1988 |chapter=The Span Test of Multimodality |title=Classification and Related Methods of Data Analysis |editor-first=H. H. |editor-last=Bock |publisher=North-Holland |location=Amsterdam |pages=229–236 |isbn=0-444-70404-3 }}</ref> and the [[saddle test]].

An implementation of the dip test is available for the [[R (programming language)|R programming language]].<ref>{{cite web|url=https://cran.r-project.org/web/packages/diptest/index.html|title=diptest: Hartigan's Dip Test Statistic for Unimodality - Corrected|first1=Martin Maechler (originally from Fortran and S.-plus by Dario|last1=Ringach|last2=NYU.edu)|date=5 December 2016|via=R-Packages}}</ref> The p-values for the dip statistic values range between 0 and 1. P-values less than 0.05 indicate significant multimodality and p-values greater than 0.05 but less than 0.10 suggest multimodality with marginal significance.<ref name=FreemanDale2012>{{cite journal | last1 = Freeman | last2 = Dale | year = 2012 | title = Assessing bimodality to detect the presence of a dual cognitive process | journal = Behavior Research Methods | volume = 45 | issue = 1 | pages = 83–97 | doi = 10.3758/s13428-012-0225-x | pmid = 22806703 | s2cid = 14500508 | url = http://psych.nyu.edu/freemanlab/pubs/2012_BRM.pdf| doi-access = free }}</ref>

===Silverman's test===

Silverman introduced a bootstrap method for the number of modes.<ref name=Silverman1981>{{cite journal | last1 = Silverman | first1 = B. W. | year = 1981 | title = Using kernel density estimates to investigate multimodality | journal = Journal of the Royal Statistical Society, Series B | volume = 43 | issue = 1| pages = 97–99 |jstor=2985156| bibcode = 1981JRSSB..43...97S | doi=10.1111/j.2517-6161.1981.tb01155.x}}</ref> The test uses a fixed bandwidth which reduces the power of the test and its interpretability. Under smoothed densities may have an excessive number of modes whose count during bootstrapping is unstable.

===Bajgier-Aggarwal test===

Bajgier and Aggarwal have proposed a test based on the kurtosis of the distribution.<ref name=Bajgier1991>{{cite journal |author1=Bajgier SM |author2=Aggarwal LK |date=1991 |title=Powers of goodness-of-fit tests in detecting balanced mixed normal distributions |journal=Educational and Psychological Measurement |volume=51 |issue=2 |pages=253–269 |doi=10.1177/0013164491512001|s2cid=121113601 }}</ref>

===Special cases===

Additional tests are available for a number of special cases:

;Mixture of two normal distributions

A study of a mixture density of two normal distributions data found that separation into the two normal distributions was difficult unless the means were separated by 4–6 standard deviations.<ref name=Jackson1898>{{cite journal | last1 = Jackson | first1 = PR | last2 = Tucker | first2 = GT | last3 = Woods | first3 = HF | year = 1989 | title = Testing for bimodality in frequency distributions of data suggesting polymorphisms of drug metabolism--hypothesis testing | journal = British Journal of Clinical Pharmacology | volume = 28 | issue = 6| pages = 655–662 | doi=10.1111/j.1365-2125.1989.tb03558.x| pmid = 2611088 | pmc = 1380036 }}</ref>

In [[astronomy]] the Kernel Mean Matching algorithm is used to decide if a data set belongs to a single normal distribution or to a mixture of two normal distributions.

;Beta-normal distribution

This distribution is bimodal for certain values of is parameters. A test for these values has been described.<ref>{{cite conference|url=http://www.amstat.org/sections/srms/Proceedings/y2002/Files/JSM2002-000150.pdf|archive-url=https://web.archive.org/web/20160304051936/http://www.amstat.org/sections/srms/Proceedings/y2002/Files/JSM2002-000150.pdf|archive-date=2016-03-04|contribution=Beta-normal distribution: Bimodality properties and application|first1=Felix|last1=Famoye|first2=Carl|last2=Lee|first3=Nicholas|last3=Eugene|title=Joint Statistical Meetings - Section on Physical & Engineering Sciences (SPES)|publisher=American Statistical Society|pages=951-956}}</ref>