Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised "f"-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function $f$. It is also called Kolmogorov mean after Russian scientist Andrey Kolmogorov.

Definition

If "f" is a function which maps a connected subset $S$ of the real line to the real numbers, and is both continuous and injective then we can define the "f"-mean of two numbers :$\{x\_1,\; x\_2\}\; subset\; S$as:$M\_f(x\_1,x\_2)\; =\; f^\{-1\}left(\; frac\{f(x\_1)+f(x\_2)\}2\; ight).$

For $n$ numbers :$\{x\_1,\; dots,\; x\_n\}\; subset\; S$,the f-mean is:$M\_f\; x\; =\; f^\{-1\}left(\; frac\{f(x\_1)+\; cdots\; +\; f(x\_n)\}n\; ight).$

We require "f" to be injective in order for the inverse function $f^\{-1\}$ to exist. Continuity is required to ensure :$frac\{fleft(x\_1\; ight)\; +\; fleft(x\_2\; ight)\}2$ lies within the domain of $f^\{-1\}$.

Since "f" is injective and continuous, it follows that "f" is a strictly monotonic function, and therefore that the "f"-mean is neither larger than the largest number of the tuple $x$ nor smaller than the smallest number in $x$.

Properties

* Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks.:$M\_f(x\_1,dots,x\_\{ncdot\; k\})\; =\; M\_f(M\_f(x\_1,dots,x\_\{k\}),\; M\_f(x\_\{k+1\},dots,x\_\{2cdot\; k\}),\; dots,\; M\_f(x\_\{(n-1)cdot\; k\; +\; 1\},dots,x\_\{ncdot\; k\}))$
* Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.:With $m=M\_f(x\_1,dots,x\_\{k\})$ it holds:$M\_f(x\_1,dots,x\_\{k\},x\_\{k+1\},dots,x\_\{n\})\; =\; M\_f(underbrace\{m,dots,m\}\_\{k\; mbox\{\; times,x\_\{k+1\},dots,x\_\{n\})$
* The quasi-arithmetic mean is invariant with respect to offsets and scaling of $f$::$forall\; a\; forall\; b\; e0\; ((forall\; t\; g(t)=a+bcdot\; f(t))\; Rightarrow\; forall\; x\; M\_f\; x\; =\; M\_g\; x)$.
* If $f$ is monotonic, then $M\_f$ is monotonic.

Examples

* If we take $S$ to be the real line and $f\; =\; mathrm\{id\}$, (or indeed any linear function $xmapsto\; acdot\; x\; +\; b$, $a$ not equal to 0) then the "f"-mean corresponds to the arithmetic mean.

* If we take $S$ to be the set of positive real numbers and $f(x)\; =\; ln(x)$, then the "f"-mean corresponds to the geometric mean. According to the "f"-mean properties, the result does not depend on the base of the logarithm as long as it is positive and not 1.

* If we take $S$ to be the set of positive real numbers and $f(x)\; =\; frac\{1\}\{x\}$, then the "f"-mean corresponds to the harmonic mean.

* If we take $S$ to be the set of positive real numbers and $f(x)\; =\; x^p$, then the "f"-mean corresponds to the power mean with exponent $p$.

Homogenity

Means are usually homogenous,but for most functions $f$, the "f"-mean is not.You can achieve that property by normalizingthe input values by some (homogenous) mean $C$.:$M\_\{f,C\}\; x\; =\; C\; x\; cdot\; f^\{-1\}left(\; frac\{fleft(frac\{x\_1\}\{C\; x\}\; ight)\; +\; dots\; +\; fleft(frac\{x\_n\}\{C\; x\}\; ight)\}\{n\}\; ight)$However this modification may violate monotonicity and the partitioning property of the mean.

Literature

* Andrey Kolmogorov (1930) “Mathematics and mechanics”, Moscow — pp.136-138. (In Russian)
* John Bibby (1974) “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.

See also

* Generalized mean
* Jensen's inequality