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.^{[citation needed]}

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.^{[citation needed]}

Galtung's classification

Galtung introduced a classification system (AJUS) for distributions:^[1]

• 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 (iff 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

${\displaystyle R={\frac {a+x}{b+y}}}$

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.^[2]

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.^[3]

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.^[4] 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 models, the parameters may be bimodally distributed.^[5]

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 ${\displaystyle Y}$ with probability ${\displaystyle \alpha }$ or ${\displaystyle Z}$ with probability ${\displaystyle (1-\alpha ),}$ where Y and Z are unimodal random variables and ${\displaystyle 0<\alpha <1}$ 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^{[citation needed]}. 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.^[6]
• Alpha-skew-normal distribution.^[7]
• Bimodal skew-symmetric normal distribution.^[8]
• A mixture of Conway-Maxwell-Poisson distributions has been fitted to bimodal count data.^[9]

Bimodality also naturally arises in the cusp catastrophe distribution.

Biology

In biology five factors are known to contribute to bimodal distributions of population sizes^{[citation needed]}:

• 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.^[10]

The distribution of fitness effects of mutations for both whole genomes^[11][12] and individual genes^[13] 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.^[14]

Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median.^[15] 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

${\displaystyle f(x)=pg_{1}(x)+(1-p)g_{2}(x)\,}$

where g_i is a probability distribution and p is the mixing parameter.

The moments of f(x) are^[16]

${\displaystyle \mu =p\mu _{1}+(1-p)\mu _{2}}$

${\displaystyle \nu _{2}=p+(1-p)}$

${\displaystyle \nu _{3}=pS_{1}\sigma _{1}^{3}+3\delta _{1}\sigma _{1}^{2}+\delta _{1}^{3}+(1-p)S_{2}\sigma _{2}^{3}+3\delta _{2}\sigma _{2}^{2}+\delta _{2}^{3}}$

${\displaystyle {\mathrm{\nu <!--\; \nu \; -->}}_4=p{K}_1{\mathrm{\sigma <!--\; \sigma \; -->}}_1^4+4S_1{\mathrm{\delta <!--\; \delta \; -->}}_1{\mathrm{\sigma <!--\; \sigma \; -->}}_1^3+6{\mathrm{\delta <!--\; \delta \; -->}}_1^2{\mathrm{\sigma <!--\; \sigma \; -->}}_1^2+{\mathrm{\delta <!--\; \delta \; -->}}_1^4+(1-<!--\; -\; -->p)K_2{\mathrm{\sigma <!--\; \sigma \; -->}}_2^{}}$Zdroj:https://en.wikipedia.org?pojem=Bimodal_distribution
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