In statistics, a multimodal distribution is a probability distribution with more than one mode (i.e., more than one local peak of the distribution). These appear as distinct peaks (local maxima) in the probability density function, 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. 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 (if both parameters a and b are less than 1). Others include the U-quadratic distribution.
The ratio of two normal distributions is also bimodally distributed. Let
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.
The distribution of the reciprocal 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 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, the speed of inactivation of the drug isoniazid 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.Introduction to tropical fish stock assessment 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. Some genes in bacteria have also exhibited bimodal distributions of gene expression both in normal as well as in stress conditions.
Econometrics
In econometric models, the parameters may be bimodally distributed.
Origins
Mathematical
A bimodal distribution commonly arises as a mixture of two different unimodal distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as with probability or with probability where Y and Z are unimodal random variables and 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, several 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 dynamics of transcription at the promoter region.
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 of mutations for both whole genomes and individual genes 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 when the two distribution curves are combined.
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
where is a probability distribution and is the mixing parameter.
The moments of are
where
and and are the skewness and kurtosis of the -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 distributions 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 case).
If the means of the two normal distributions are equal, then the combined distribution is unimodal. Conditions for unimodality 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 or where p is the mixing parameter and and where μ1 and μ2 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 The separation factor (S) is If the variances are equal then S = 1. The mixture density is unimodal if and only if A sufficient condition for unimodality is If the two normal distributions have equal standard deviations a sufficient condition for unimodality is
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:
where μ1, μ2 are the means and σ1, σ2 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 bimodality) to +1 (perfect unimodality). It is defined as
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 (Ai) is calculated and a weighted average for the distribution is determined. The weights (wi) for each layer are the number of responses in that layer. In symbols
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 (μ1 and μ2) and standard deviations (σ1 and σ2):
Bimodality coefficient
Sarle's bimodality coefficient b is
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. 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.
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 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 with only two distinct values or the sum of two different Dirac delta functions (a bi-delta distribution).