Editing Multimodal distribution (section)

===Bimodality indices===

==== Wang's index ====

The bimodality index proposed by Wang ''et al'' assumes that the distribution is a sum of two normal distributions with equal variances but differing means.<ref name=Wang2009>{{cite journal | last1 = Wang | first1 = J | last2 = Wen | first2 = S | last3 = Symmans | first3 = WF | last4 = Pusztai | first4 = L | last5 = Coombes | first5 = KR | year = 2009 | title = The bimodality index: a criterion for discovering and ranking bimodal signatures from cancer gene expression profiling data | journal = Cancer Informatics | volume = 7 | pages = 199–216 |doi=10.4137/CIN.S2846| pmid = 19718451 | pmc = 2730180 }}</ref> It is defined as follows:

<math display="block"> \delta = \frac{ | \mu_1 - \mu_2 |}{ \sigma } </math>

where ''μ''<sub>1</sub>, ''μ''<sub>2</sub> are the means and ''σ'' is the common standard deviation.

<math display="block"> BI = \delta \sqrt{ p( 1 - p ) } </math>

where ''p'' is the mixing parameter.

==== Sturrock's index ====

A different bimodality index has been proposed by Sturrock.<ref name=Sturrock2008>{{cite journal | last1 = Sturrock | first1 = P | year = 2008 | title = Analysis of bimodality in histograms formed from GALLEX and GNO solar neutrino data | journal = Solar Physics | volume = 249 | issue = 1| pages = 1–10 | doi=10.1007/s11207-008-9170-3|arxiv = 0711.0216 |bibcode = 2008SoPh..249....1S | s2cid = 118389173 }}</ref>

This index (''B'') is defined as

<math display="block"> B = \frac{ 1 }{ N } \left[ \left( \sum_1^N \cos ( 2 \pi m \gamma ) \right)^2 +  \left( \sum_1^N \sin ( 2 \pi m \gamma ) \right)^2 \right] </math>

When ''m'' = 2 and ''γ'' is uniformly distributed, ''B'' is exponentially distributed.<ref name=Scargle1082>{{cite journal | last1 = Scargle | first1 = JD | year = 1982 | title = Studies in astronomical time series analysis. II – Statistical aspects of spectral analysis of unevenly spaced data | journal = The Astrophysical Journal | volume = 263 | issue = 1| pages = 835–853 | doi=10.1086/160554 | bibcode=1982ApJ...263..835S}}</ref>

This statistic is a form of [[periodogram]]. It suffers from the usual problems of estimation and spectral leakage common to this form of statistic.

==== de Michele and Accatino's index ====

Another bimodality index has been proposed by de Michele and Accatino.<ref name=deMichele2014>{{cite journal | last1 = De Michele | first1 = C | last2 = Accatino | first2 = F | year = 2014 | title = Tree cover bimodality in savannas and forests emerging from the switching between two fire dynamics | journal = PLoS One | volume =  9| issue = 3| pages =  e91195| doi = 10.1371/journal.pone.0091195 |bibcode = 2014PLoSO...991195D | pmid=24663432 | pmc=3963849| doi-access = free }}</ref> Their index (''B'') is

<math display="block"> B = | \mu - \mu_M | </math>

where ''μ'' is the arithmetic mean of the sample and

<math display="block"> \mu_M = \frac{ \sum_{ i = 1 }^L m_i x_i }{ \sum_{ i = 1 }^L m_i } </math>

where ''m''<sub>''i''</sub> is number of data points in the ''i''<sup>th</sup> bin, ''x''<sub>''i''</sub> is the center of the ''i''<sup>th</sup> bin and ''L'' is the number of bins.

The authors suggested a cut off value of 0.1 for ''B'' to distinguish between a bimodal (''B'' > 0.1)and unimodal (''B'' < 0.1) distribution. No statistical justification was offered for this value.

==== Sambrook Smith's index ====

A further index (''B'') has been proposed by Sambrook Smith ''et al''<ref name=SambrookSmith1997>{{cite journal | last1 = Sambrook Smith | first1 = GH | last2 = Nicholas | first2 = AP | last3 = Ferguson | first3 = RI | year = 1997 | title = Measuring and defining bimodal sediments: Problems and implications | journal = Water Resources Research | volume = 33 | issue = 5| pages = 1179–1185 | doi=10.1029/97wr00365 | bibcode=1997WRR....33.1179S| doi-access = free }}</ref>

<math display="block"> B = | \phi_2 - \phi_1 | \frac{ p_2 }{ p_1 } </math>

where ''p''<sub>1</sub> and ''p''<sub>2</sub> are the proportion contained in the primary (that with the greater amplitude) and secondary (that with the lesser amplitude) mode and ''φ''<sub>1</sub> and ''φ''<sub>2</sub> are the ''φ''-sizes of the primary and secondary mode. The ''φ''-size is defined as minus one times the log of the data size taken to the base 2. This transformation is commonly used in the study of sediments.

The authors recommended a cut off value of 1.5 with B being greater than 1.5 for a bimodal distribution and less than 1.5 for a unimodal distribution. No statistical justification for this value was given.

==== Otsu's method ====

[[Otsu's method]] for finding a threshold for separation between two modes relies on minimizing the quantity
<math display=block> \frac{ n_1 \sigma_1^2 + n_2 \sigma_2^2 }{ m \sigma^2 } </math>
where ''n''<sub>''i''</sub> is the number of data points in the ''i''<sup>th</sup> subpopulation, ''σ''<sub>''i''</sub><sup>2</sup> is the variance of the ''i''<sup>th</sup> subpopulation, ''m'' is the total size of the sample and ''σ''<sup>2</sup> is the sample variance. Some researchers (particularly in the field of [[digital image processing]]) have applied this quantity more broadly as an index for detecting bimodality, with a small value indicating a more bimodal distribution.<ref name=Chaudhuri2010>{{cite journal | last1 = Chaudhuri | first1 = D | last2 = Agrawal | first2 = A | year = 2010 | title = Split-and-merge procedure for image segmentation using bimodality detection approach | journal = Defence Science Journal | volume = 60 | issue = 3| pages = 290–301 | doi=10.14429/dsj.60.356| doi-access =  }}</ref>