Unimodality

Property of having a unique mode or maximum value

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.

In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's t distribution, chi squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability. Figure 2 and Figure 3 illustrate bimodal distributions.

One reason for the importance of distribution unimodality is that it allows for several important results. Several inequalities are given below which are only valid for unimodal distributions. Thus, it is important to assess whether or not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the article on multimodal distribution.

A first important result is Gauss's inequality. Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends on unimodality.

A second is the Vysochanskiï–Petunin inequality, a refinement of the Chebyshev inequality. The Chebyshev inequality guarantees that in any probability distribution, "nearly all" the values are "close to" the mean value. The Vysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is continuous and unimodal. Further results were shown by Sellke and Sellke.

A function f(x) is "S unimodal" (often referred to as "S unimodal map") if its Schwarzian derivative is negative for all , where is the critical point. In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function. A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one to one differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian matrix. Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher dimensional Euclidean spaces.