Editing Multimodal distribution (section)

==Summary statistics==

Bimodal distributions are a commonly used example of how summary statistics such as the [[mean]], [[median]], and [[standard deviation]] can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value.  The standard deviation is also larger than deviation of each normal distribution.

Although several have been suggested, there is no presently generally agreed summary statistic (or set of statistics) to quantify the parameters of a general bimodal distribution. For a mixture of two normal distributions the means and standard deviations along with the mixing parameter (the weight for the combination) are usually used – a total of five parameters.

===Ashman's D===

A statistic that may be useful is Ashman's D:<ref name=Ashman1994>{{cite journal |author1=Ashman KM |author2=Bird CM |author3=Zepf SE |date=1994 |title=Detecting bimodality in astronomical datasets |journal=The Astronomical Journal |volume=108 |pages=2348–2361 |arxiv=astro-ph/9408030 |doi=10.1086/117248 |bibcode=1994AJ....108.2348A|s2cid=13464256 }}</ref>

<math display="block"> D = \frac{ \left| \mu_1 - \mu_2 \right| }{ \sqrt{ 2 \left( \sigma_1^2 + \sigma_2^2 \right) } } </math>

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

For a mixture of two normal distributions ''D'' > 2 is required for a clean separation of the distributions.

===van der Eijk's A===

This measure is a weighted average of the degree of agreement the frequency distribution.<ref name=Van_der_Eijk2001>{{cite journal | last1 = Van der Eijk | first1 = C | year = 2001 | title = Measuring agreement in ordered rating scales | journal = Quality & Quantity | volume = 35 | issue = 3| pages = 325–341 | doi=10.1023/a:1010374114305| s2cid = 189822180 }}</ref> ''A'' ranges from -1 (perfect [[bimodal]]ity) to +1 (perfect [[unimodal]]ity). It is defined as

<math display="block"> A = U \left( 1 - \frac{ S - 1 }{ K - 1 } \right) </math>

where ''U'' is the unimodality of the distribution, ''S'' the number of categories that have nonzero frequencies and ''K'' the total number of categories.

The value of U is 1 if the distribution has any of the three following characteristics:

* all responses are in a single category
* the responses are evenly distributed among all the categories
* the responses are evenly distributed among two or more contiguous categories, with the other categories with zero responses

With distributions other than these the data must be divided into 'layers'. Within a layer the responses are either equal or zero. The categories do not have to be contiguous. A value for ''A'' for each layer (''A''<sub>i</sub>) is calculated and a weighted average for the distribution is determined. The weights (''w''<sub>i</sub>) for each layer are the number of responses in that layer. In symbols

<math display="block"> A_\text{overall} = \sum_i w_i A_i </math>

A [[Uniform distribution (discrete)|uniform distribution]] has ''A'' = 0: when all the responses fall into one category ''A'' = +1.

One theoretical problem with this index is that it assumes that the intervals are equally spaced. This may limit its applicability.

===Bimodal separation===

This index assumes that the distribution is a mixture of two normal distributions with means (''μ''<sub>1</sub> and ''μ''<sub>2</sub>) and standard deviations (''σ''<sub>1</sub> and ''σ''<sub>2</sub>):<ref name=Zhang2003>{{cite journal | last1 = Zhang | first1 = C | last2 = Mapes | first2 = BE | last3 = Soden | first3 = BJ | year = 2003 | title = Bimodality in tropical water vapour | journal = Quarterly Journal of the Royal Meteorological Society | volume = 129 | issue = 594| pages = 2847–2866 | doi =  10.1256/qj.02.166| bibcode = 2003QJRMS.129.2847Z | s2cid = 17153773 }}</ref>

<math display="block"> S = \frac{ \mu_1 - \mu_2 }{ 2( \sigma_1 +\sigma_2 ) } </math>

===Bimodality coefficient===

Sarle's bimodality coefficient ''b'' is<ref name=Ellison1987>{{cite journal | last1 = Ellison | first1 = AM | year = 1987 | title = Effect of seed dimorphism on the density-dependent dynamics of experimental populations of ''Atriplex triangularis'' (Chenopodiaceae) | journal = American Journal of Botany | volume = 74 | issue = 8| pages = 1280–1288 | doi=10.2307/2444163| jstor = 2444163 }}</ref>

<math display="block"> \beta = \frac{ \gamma^2 + 1 }{ \kappa }  </math>

where ''γ'' is the [[skewness]] and ''κ'' is the [[kurtosis]]. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of ''b'' lies between 0 and 1.<ref name=Pearson1916>{{cite journal | last1 = Pearson | first1 = K | year = 1916 | title = Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation | journal = Philosophical Transactions of the Royal Society A | volume = 216 | issue = 538–548| pages = 429–457 | doi = 10.1098/rsta.1916.0009 | jstor = 91092 | bibcode =   1916RSPTA.216..429P| doi-access = free }}</ref> The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient.

The formula for a finite sample is<ref name=SASInst2012>SAS Institute Inc. (2012). SAS/STAT 12.1 user’s guide. Cary, NC: Author.</ref>

<math display="block"> b = \frac{ g^2 + 1 }{ k + \frac{ 3( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } } </math>

where ''n'' is the number of items in the sample, ''g'' is the [[sample skewness]] and ''k'' is the sample [[excess kurtosis]].

The value of ''b'' for the [[uniform distribution (continuous)|uniform distribution]] is 5/9. This is also its value for the [[exponential distribution]]. Values greater than 5/9 may indicate a bimodal or multimodal distribution, though corresponding values can also result for heavily skewed unimodal distributions.<ref>{{cite journal | last1 = Pfister | first1 = R | last2 = Schwarz | first2 = KA | last3 = Janczyk | first3 = M. | last4 = Dale | first4 = R | last5 = Freeman | first5 = JB | year = 2013 | title = Good things peak in pairs: A note on the bimodality coefficient | journal =  Frontiers in Psychology| volume = 4 | pages = 700 | doi = 10.3389/fpsyg.2013.00700| pmid = 24109465 | pmc = 3791391 | doi-access = free }}</ref> The maximum value (1.0) is reached only by a [[Bernoulli distribution]] with only two distinct values or the sum of two different [[Dirac delta function]]s (a bi-delta distribution).

The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson – the difference between the kurtosis and the square of the skewness (''vide infra'').

===Bimodality amplitude===

This is defined as<ref name=Zhang2003/>

<math display="block"> A_B = \frac{A_1 - A_{ an } }{ A_1 } </math>

where ''A''<sub>1</sub> is the amplitude of the smaller peak and ''A''<sub>an</sub> is the amplitude of the antimode.

''A''<sub>B</sub> is always < 1. Larger values indicate more distinct peaks.

===Bimodal ratio===

This is the ratio of the left and right peaks.<ref name=Zhang2003/> Mathematically

<math display="block"> R = \frac{ A_r }{ A_l } </math>

where ''A''<sub>l</sub> and ''A''<sub>r</sub> are the amplitudes of the left and right peaks respectively.

===Bimodality parameter===

This parameter (''B'') is due to Wilcock.<ref name=Wilcock1993>{{cite journal | last1 = Wilcock | first1 = PR | year = 1993 | title = The critical shear stress of natural sediments | journal = Journal of Hydraulic Engineering | volume = 119 | issue = 4| pages = 491–505 | doi=10.1061/(asce)0733-9429(1993)119:4(491)}}</ref>

<math display="block"> B = \sqrt{ \frac{ A_r }{ A_l } } \sum_i P_i </math>

where ''A''<sub>l</sub> and ''A''<sub>r</sub> are the amplitudes of the left and right peaks respectively and ''P''<sub>''i''</sub> is the logarithm taken to the base 2 of the proportion of the distribution in the i<sup>th</sup> interval. The maximal value of the ''ΣP'' is 1 but the value of ''B'' may be greater than this.

To use this index, the log of the values are taken. The data is then divided into interval of width Φ whose value is log 2. The width of the peaks are taken to be four times 1/4Φ centered on their maximum values.

===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>