Bimodal distribution



In statistics, a bimodal distribution is a continuous probability distribution with two different modes. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figure 1.

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.

Examples
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.

Important bimodal distributions include the arcsine distribution and the beta distribution.

Mixture distributions
A bimodal distribution most 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 $$ Y $$ 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. For example, the bimodal distribution of sizes of weaver ant workers shown in Figure 2 arises due to existence of two distinct classes of workers, namely major workers and minor workers. In this case, Y would be the size of a random major worker, Z the size of a random minor worker, and α the proportion of worker weaver ants that are major workers.

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).

Mixtures of other distributions require additional parameters to be estimated.

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 gi is a probability distribution and p is the mixing parameter.

The moments of f(x) are


 * $$ \mu = p \mu_1 + ( 1 - p ) \mu_2 $$


 * $$ \nu_2 = p[ \sigma_1^2 + \delta_1^2 ] + ( 1 - p )[ \sigma_2^2 + \delta_2^2 ]$$


 * $$ \nu_3 = p [ S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 ] + ( 1 - p )[ S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 ] $$


 * $$ \nu_4 = p[ K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 ] + ( 1 - p )[ K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 ]$$

where
 * $$ \mu = \int{ x f( x ) dx }$$


 * $$ \delta_i = \mu_i - \mu $$


 * $$ \nu_r = \int{ ( x - \mu )^r f( x ) dx } $$

and Si and Ki are the skewness and kurtosis of the ith distribution.

Multimodality
More generally, a multimodal distribution is a continuous probability distribution with two or more modes, as illustrated in Figure 3.

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.

Ashman's D
A statistic that may be useful is Ashman's D:


 * $$ D = 2^\frac{ 1 }{ 2 } \frac{ | \mu_1 - \mu_2 | }{ \sqrt{ ( \sigma_1^2 + \sigma_2^2 ) } } $$

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.

Bimodality index
The bimodality index assumes that the distribution is a sum of two normal distributions with equal variances but differing means. It is defined as follow:


 * $$ \delta = \frac{ \mu_1 - \mu_2 }{ \sigma } $$

where μ1, μ2 are the means and σ is the common standard deviation.


 * $$ BI = \delta \sqrt{ p( 1 - p ) } $$

where p is the mixing parameter.

Bimodality coefficient
Sarle's bimodality coefficient b is


 * $$ \beta = \frac{ \gamma^2 + 1 }{ \kappa } $$

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 formula for a finite sample is


 * $$ b = \frac{ g^2 + 1 }{ k + 3 ( 1 - \frac{ ( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } ) } $$

where n is the number of items in the sample, g is the sample skewness and k is the sample 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. 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.

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).

Unimodal vs bimodal distribution
A necessary but not sufficient condition for a symmetrical distribution to be bimodal is that the kurtosis be less than three. Here the kurtosis is defined to be the standardised fourth moment around the mean. The reference given prefers to use the excess kurtosis - the kurtosis less 3.

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 a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis. Later Pearson showed that


 * $$ b_2 - b_1 \ge 1 $$

where b2 is the kurtosis and b1 is the square of the skewness. Equality holds only for the two point Bernoulli distribution 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.

Haldane suggested a test based on second central differences.

To test whether a univariant distribution is unimodal or bimodal, Larkin introduced a test based on the F test. Later Benett instead used a G test.

Tokeshi proposed another test for bimodality.

General tests
To test if a distribution is other than unimodal, several additional tests have been devised: the bandwidth test, the dip test, the excess mass test, the MAP test, the mode-existence test, the runt test, the span test, and the saddle test.