Mode (stochastics)

This article describes the measure of distributions. For the mode of a sample, see Mode (Statistics) , further meanings can be found under Mode .

In stochastics, a mode or modal value is a key figure for the distribution of a random variable or a probability measure . The mode belongs to the position measures and thus, like the expected value and the median, has the task of characterizing the position of a distribution.

The mode is defined via the probability density functions or probability functions of a distribution and must be distinguished from the mode in the sense of descriptive statistics . This is a key figure of a sample (like the arithmetic mean ), the mode in stochastics, however, is a figure of an abstract quantity function (like the expected value).

definition

About probability densities

Is a random variable or a probability distribution with probability density function given is the name of a mode or modal value of or when a local maximum of is. ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle f}$ ${\displaystyle x_{m}}$ ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle f(x_{m})}$ ${\displaystyle f}$

If the random variable is real-valued or if the probability distribution is defined on the real numbers, this is equivalent to that ${\displaystyle X}$

${\displaystyle f(x)\leq f(x_{m})}$ for all ${\displaystyle x\in (x_{m}-\varepsilon ;x_{m}+\varepsilon )}$

for a . ${\displaystyle \varepsilon >0}$

About probability functions

Let there be at most a countable set whose elements are sorted in ascending order, that is . If then a random variable with values ​​in and is a probability function or is a probability function on with a probability function , then a mode or modal value of or , if ${\displaystyle \Omega }$ ${\displaystyle x_{k}}$ ${\displaystyle \dots <x_{k-1}<x_{k}<x_{k+1}<\dots }$ ${\displaystyle X}$ ${\displaystyle \Omega }$ ${\displaystyle f}$ ${\displaystyle P}$ ${\displaystyle \Omega }$ ${\displaystyle f}$ ${\displaystyle x_{k}}$ ${\displaystyle X}$ ${\displaystyle P}$

${\displaystyle f(x_{k-1})\leq f(x_{k})\geq f(x_{k+1})}$

is.

More specifically, if a random variable with values ​​in or a probability distribution is on , a mode is if ${\displaystyle X}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle P}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle k}$

${\displaystyle f(k-1)\leq f(k)\geq f(k+1)}$

is.

weaknesses

The mode is not always unproblematic as a measure of position. For example, it may have little or no meaningfulness. If one considers the probability density function of the exponential distribution with the parameter ${\displaystyle \lambda }$

${\displaystyle f_{\lambda }(x)={\begin{cases}\displaystyle \lambda {\rm {e}}^{-\lambda x}&x\geq 0\\0&x<0\end{cases}}}$

so this has a global maximum. Thus zero is the unique mode of the exponential distribution. However, the probability of getting a value less than zero is zero. This is in clear contradiction to the underlying idea of ​​a measure of position, which is supposed to indicate where “a lot of probability” is. ${\displaystyle x_{m}=0}$

Likewise, the mode does not generally have to be unique (see below). In the extreme case of the constant uniform distribution , which is the probability density function ${\displaystyle f(x)={\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b\\0&{\text{sonst}}\end{cases}}}$

each value in the interval is a mode. ${\displaystyle (a,b)}$

Constructive terms

Distributions that only have one mode are called unimodal distributions .

Distributions with more than one mode are called multimodal distributions and are further differentiated according to the number of their modes. We also speak of bimodal distributions (two modes) or trimodal distributions (three modes).

Demarcation

The mode (in the sense of statistics) can be assigned to any sample that is nominally scaled , the elements of which can therefore be grouped into certain categories. The mode is therefore a key figure of a sample, i.e. an arrangement of results from a random experiment carried out.

The mode (in the sense of probability theory) is a key figure of a probability distribution. This is an illustration that assigns a number to specific quantities and can therefore be distinguished from a sample.

The two mode terms are therefore different, especially since they assign numbers to different mathematical constructs: once the sample, once the probability distribution. Both terms can be linked via the empirical distribution . If a sample is given, the mode of the sample corresponds to the mode (in the sense of probability theory) of the empirical distribution of . ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle x}$ ${\displaystyle x}$

Individual evidence

1. ↑ ^{a b} A.V. Prokhorov: fashion . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
2. ↑ Unimodal distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
3. ^ AV Prokhorov: Multimodal Distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
4. ↑ Eric W. Weisstein : Multimodal . In: MathWorld (English).
5. ↑ Eric W. Weisstein : Trimodal . In: MathWorld (English).
6. ↑ bimodal distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).