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In statistics, a multimodal distribution is a probability distribution with more than one
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
. These appear as distinct peaks (local maxima) in the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
, as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.


Terminology

When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the antimode. The difference between the major and minor modes is known as the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
. In time series the major mode is called the acrophase and the antimode the batiphase.


Galtung's classification

Galtung introduced a classification system (AJUS) for distributions: *A: unimodal distribution – peak in the middle *J: unimodal – peak at either end *U: bimodal – peaks at both ends *S: bimodal or multimodal – multiple peaks This classification has since been modified slightly: *J: (modified) – peak on right *L: unimodal – peak on left *F: no peak (flat) Under this classification bimodal distributions are classified as type S or U.


Examples

Bimodal distributions occur both in mathematics and in the natural sciences.


Probability distributions

Important bimodal distributions include the arcsine distribution and the
beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
(iff both parameters are less than 1). Others include the
U-quadratic distribution In probability theory and statistics, the U-quadratic distribution is a continuous probability distribution defined by a unique convex quadratic function with lower limit ''a'' and upper limit ''b''. : f(x, a,b,\alpha, \beta)=\alpha \left ( x - ...
. The ratio of two normal distributions is also bimodally distributed. Let : R = \frac where ''a'' and ''b'' are constant and ''x'' and ''y'' are distributed as normal variables with a mean of 0 and a standard deviation of 1. ''R'' has a known density that can be expressed as a
confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
. The distribution of the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of a ''t'' distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed. A ''t'' statistic generated from data set drawn from a
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
is bimodal.


Occurrences in nature

Examples of variables with bimodal distributions include the time between eruptions of certain geysers, the color of galaxies, the size of worker weaver ants, the age of incidence of
Hodgkin's lymphoma Hodgkin lymphoma (HL) is a type of lymphoma, in which cancer originates from a specific type of white blood cell called lymphocytes, where multinucleated Reed–Sternberg cells (RS cells) are present in the patient's lymph nodes. The condition w ...
, the speed of inactivation of the drug
isoniazid Isoniazid, also known as isonicotinic acid hydrazide (INH), is an antibiotic used for the treatment of tuberculosis. For active tuberculosis it is often used together with rifampicin, pyrazinamide, and either streptomycin or ethambutol. For la ...
in US adults, the absolute magnitude of novae, and the circadian activity patterns of those crepuscular animals that are active both in morning and evening twilight. In fishery science multimodal length distributions reflect the different year classes and can thus be used for age distribution- and growth estimates of the fish population. Sediments are usually distributed in a bimodal fashion. When sampling mining galleries crossing either the host rock and the mineralized veins, the distribution of geochemical variables would be bimodal. Bimodal distributions are also seen in traffic analysis, where traffic peaks in during the AM rush hour and then again in the PM rush hour. This phenomenon is also seen in daily water distribution, as water demand, in the form of showers, cooking, and toilet use, generally peak in the morning and evening periods.


Econometrics

In
econometric Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
models, the parameters may be bimodally distributed.


Origins


Mathematical

A bimodal distribution most commonly arises as a mixture of two different
unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal p ...
distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as Y with probability \alpha or Z with probability (1-\alpha), where ''Y'' and ''Z'' are unimodal random variables and 0 < \alpha < 1 is a mixture coefficient. Mixtures with two distinct components need not be bimodal and two component mixtures of unimodal component densities can have more than two modes. There is no immediate connection between the number of components in a mixture and the number of modes of the resulting density.


Particular distributions

Bimodal distributions, despite their frequent occurrence in data sets, have only rarely been studied. This may be because of the difficulties in estimating their parameters either with frequentist or Bayesian methods. Among those that have been studied are * Bimodal exponential distribution. * Alpha-skew-normal distribution. * Bimodal skew-symmetric normal distribution. * A mixture of Conway-Maxwell-Poisson distributions has been fitted to bimodal count data. Bimodality also naturally arises in the cusp catastrophe distribution.


Biology

In biology five factors are known to contribute to bimodal distributions of population sizes: *the initial distribution of individual sizes *the distribution of growth rates among the individuals *the size and time dependence of the growth rate of each individual * mortality rates that may affect each size class differently * the DNA methylation in human and mouse genome. The bimodal distribution of sizes of weaver ant workers arises due to existence of two distinct classes of workers, namely major workers and minor workers. The
distribution of fitness effects In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mitos ...
of mutations for both whole
genome In the fields of molecular biology and genetics, a genome is all the genetic information of an organism. It consists of nucleotide sequences of DNA (or RNA in RNA viruses). The nuclear genome includes protein-coding genes and non-coding g ...
s and individual
gene In biology, the word gene (from , ; "... Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a b ...
s is also frequently found to be bimodal with most mutations being either neutral or lethal with relatively few having intermediate effect.


General properties

A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality. Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median. This is clearly the case when the distribution is U shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails.


Moments of mixtures

Let : f( x ) = p g_1( x ) + ( 1 - p ) g_2( x ) \, where ''g''''i'' is a probability distribution and ''p'' is the mixing parameter. The moments of ''f''(''x'') are : \mu = p \mu_1 + ( 1 - p ) \mu_2 : \nu_2 = p \sigma_1^2 + \delta_1^2 + ( 1 - p ) \sigma_2^2 + \delta_2^2 /math> : \nu_3 = p S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 + ( 1 - p ) S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 : \nu_4 = p K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 + ( 1 - p ) K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 /math> where : \mu = \int x f( x ) \, dx : \delta_i = \mu_i - \mu : \nu_r = \int ( x - \mu )^r f( x ) \, dx and ''S''''i'' and ''K''''i'' are the
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...
and
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
of the ''i''th distribution.


Mixture of two normal distributions

It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two normal distributions. Because of this, this mixture has been studied in some detail. A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
s with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation. Estimates of the parameters is simplified if the variances can be assumed to be equal (the
homoscedastic In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The s ...
case). If the means of the two normal distributions are equal, then the combined distribution is unimodal. Conditions for
unimodality In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal pr ...
of the combined distribution were derived by Eisenberger. Necessary and sufficient conditions for a mixture of normal distributions to be bimodal have been identified by Ray and Lindsay. A mixture of two approximately equal mass normal distributions has a negative kurtosis since the two modes on either side of the center of mass effectively reduces the tails of the distribution. A mixture of two normal distributions with highly unequal mass has a positive kurtosis since the smaller distribution lengthens the tail of the more dominant normal distribution. Mixtures of other distributions require additional parameters to be estimated.


Tests for unimodality

*When the components of the mixture have equal variances the mixture is unimodal
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
: d \le 1 or : \left\vert \log( 1 - p ) - \log( p ) \right\vert \ge 2 \log( d - \sqrt ) + 2d \sqrt , where ''p'' is the mixing parameter and : d = \frac, and where ''μ''1 and ''μ''2 are the means of the two normal distributions and ''σ'' is their standard deviation. *The following test for the case ''p'' = 1/2 was described by Schilling ''et al''. Let : r = \frac . The separation factor (''S'') is : S = \frac . If the variances are equal then ''S'' = 1. The mixture density is unimodal if and only if : , \mu_1 - \mu_2 , < S , \sigma_1 + \sigma_2 , . *A sufficient condition for unimodality is :, \mu_1-\mu_2, \le2\min (\sigma_1,\sigma_2). *If the two normal distributions have equal standard deviations \sigma, a sufficient condition for unimodality is :, \mu _1-\mu_2, \le 2\sigma \sqrt.


Summary statistics

Bimodal distributions are a commonly used example of how summary statistics such as the
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
, 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: : D = (2^\frac) \frac where ''μ''1, ''μ''2 are the means and ''σ''1 ''σ''2 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. ''A'' ranges from -1 (perfect
bimodal In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and d ...
ity) to +1 (perfect
unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal p ...
ity). It is defined as : A = U ( 1 - \frac ) 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''i) is calculated and a weighted average for the distribution is determined. The weights (''w''i) for each layer are the number of responses in that layer. In symbols : A_ = \sum w_i A_i A 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''1 and ''μ''2) and standard deviations (''σ''1 and ''σ''2): : S = \frac


Bimodality coefficient

Sarle's bimodality coefficient ''b'' is : \beta = \frac where ''γ'' is the
skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...
and ''κ'' is the
kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of ''b'' lies between 0 and 1. 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 isSAS Institute Inc. (2012). SAS/STAT 12.1 user’s guide. Cary, NC: Author. : b = \frac where ''n'' is the number of items in the sample, ''g'' is the sample skewness and ''k'' is the sample
excess kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
. The value of ''b'' for the 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. The maximum value (1.0) is reached only by a
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
with only two distinct values or the sum of two different Dirac delta functions (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 : A_B = \frac where ''A''1 is the amplitude of the smaller peak and ''A''an is the amplitude of the antimode. ''A''B is always < 1. Larger values indicate more distinct peaks.


Bimodal ratio

This is the ratio of the left and right peaks. Mathematically : R = \frac where ''A''l and ''A''r are the amplitudes of the left and right peaks respectively.


Bimodality parameter

This parameter (''B'') is due to Wilcock. : B = \sqrt \sum P_i where ''A''l and ''A''r are the amplitudes of the left and right peaks respectively and ''P''''i'' is the logarithm taken to the base 2 of the proportion of the distribution in the ith 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. It is defined as follows: : \delta = \frac where ''μ''1, ''μ''2 are the means and ''σ'' is the common standard deviation. : BI = \delta \sqrt where ''p'' is the mixing parameter. ;Sturrock's index A different bimodality index has been proposed by Sturrock. This index (''B'') is defined as : B = \frac \left \left( \sum_1^N \cos ( 2 \pi m \gamma ) \right)^2 + \left( \sum_1^N \sin ( 2 \pi m \gamma ) \right)^2 \right When ''m'' = 2 and ''γ'' is uniformly distributed, ''B'' is exponentially distributed. This statistic is a form of
periodogram In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most ...
. 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. Their index (''B'') is : B = , \mu - \mu_M , where ''μ'' is the arithmetic mean of the sample and : \mu_M = \frac where ''m''''i'' is number of data points in the ''i''th bin, ''x''''i'' is the center of the ''i''th 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'' B = , \phi_2 - \phi_1 , \frac where ''p''1 and ''p''2 are the proportion contained in the primary (that with the greater amplitude) and secondary (that with the lesser amplitude) mode and ''φ''1 and ''φ''2 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 In computer vision and image processing, Otsu's method, named after , is used to perform automatic image thresholding. In the simplest form, the algorithm returns a single intensity threshold that separate pixels into two classes, foreground a ...
for finding a threshold for separation between two modes relies on minimizing the quantity \frac where ''n''''i'' is the number of data points in the ''i''th subpopulation, ''σ''''i''2 is the variance of the ''i''th subpopulation, ''m'' is the total size of the sample and ''σ''2 is the sample variance. Some researchers (particularly in the field of
digital image processing Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
) have applied this quantity more broadly as an index for detecting bimodality, with a small value indicating a more bimodal distribution.


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( size ) of the particles. This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the 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. : \mathit = \frac : \mathit = \frac + \frac : \mathit = \frac + \frac : \mathit = \frac where ''Mean'' is the mean, ''StdDev'' is the standard deviation, ''Skew'' is the skewness, ''Kurt'' is the kurtosis and ''φ''x is the value of the variate ''φ'' at the ''x''th 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. This method required the solution of a ninth order
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
. In a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis. Later Pearson showed that : b_2 - b_1 \ge 1 where ''b''2 is the kurtosis and ''b''1 is the square of the skewness. Equality holds only for the two point
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
or the sum of two different Dirac delta functions. 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. Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences. Larkin later introduced a test based on the F test; Benett created one based on Fisher's G test. Tokeshi has proposed a fourth test. A test based on a likelihood ratio has been proposed by Holzmann and Vollmer. A method based on the score and Wald tests has been proposed. This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.


Antimode tests

Statistical tests for the antimode are known. ;Otsu's method
Otsu's method In computer vision and image processing, Otsu's method, named after , is used to perform automatic image thresholding. In the simplest form, the algorithm returns a single intensity threshold that separate pixels into two classes, foreground a ...
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 Throughput of a network can be measured using various tools available on different platforms. This page explains the theory behind what these tools set out to measure and the issues regarding these measurements. Reasons for measuring throughput i ...
, the dip test, the excess mass test, the MAP test, the mode existence test, the
runt test In a group of animals (usually a Litter (animal), litter of animals born in multiple births), a runt is a member which is significantly smaller or weaker than the others. Owing to its small size, a runt in a litter faces obvious disadvantage, inc ...
, the span test, and the saddle test. An implementation of the dip test is available for the
R programming language R is a programming language for statistical computing and graphics supported by the R Core Team and the R Foundation for Statistical Computing. Created by statisticians Ross Ihaka and Robert Gentleman, R is used among data miners, bioinforma ...
. 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.


Silverman's test

Silverman introduced a bootstrap method for the number of modes. 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.


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. In
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
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.


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 is available for testing for bimodality. 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, and the R package nor1mix. ;Other distributions The mixtools package available for R can test for and estimate the parameters of a number of different distributions. A package for a mixture of two right-tailed gamma distributions is available. Several other packages for R are available to fit mixture models; these include flexmix, mcclust, agrmt, and mixdist. The statistical programming language SAS can also fit a variety of mixed distributions with the PROC FREQ procedure.


Example software application

The CumFreqA CumFreq, free program for fitting of probability distributions to a data set. On line

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 In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Th ...
as in
distribution fitting Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon. The aim of distribution fitting is to predict the probab ...
with cumulative distribution function (CDF) equations: X < 8.10 : CDF = 1 - exp
exp Exp may stand for: * Exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the pos ...
X > 8.10 : CDF = 1 - exp
exp Exp may stand for: * Exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the pos ...


See also

*
Overdispersion In statistics, overdispersion is the presence of greater variability (statistical dispersion) in a data set than would be expected based on a given statistical model. A common task in applied statistics is choosing a parametric model to fit a ...


References

{{ProbDistributions Continuous distributions