Definitions

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

$\left\{x_1, x_2\right\} subset S$
as
$M_f\left(x_1,x_2\right) = f^\left\{-1\right\}left\left(frac\left\{f\left(x_1\right)+f\left(x_2\right)\right\}2 right\right).$

For $n$ numbers

$\left\{x_1, dots, x_n\right\} subset S$,
the f-mean is
$M_f x = f^\left\{-1\right\}left\left(frac\left\{f\left(x_1\right)+ cdots + f\left(x_n\right)\right\}n right\right).$

We require f to be injective in order for the inverse function $f^\left\{-1\right\}$ to exist. Continuity is required to ensure

$frac\left\{fleft\left(x_1right\right) + fleft\left(x_2right\right)\right\}2$
lies within the domain of $f^\left\{-1\right\}$.

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 $m=M_f\left(x_1,dots,x_\left\{k\right\}\right)$ it holds
$M_f\left(x_1,dots,x_\left\{k\right\},x_\left\{k+1\right\},dots,x_\left\{n\right\}\right) = M_f\left(underbrace\left\{m,dots,m\right\}_\left\{k mbox\left\{ times\right\}\right\},x_\left\{k+1\right\},dots,x_\left\{n\right\}\right)$

• The quasi-arithmetic mean is invariant with respect to offsets and scaling of $f$:

$forall a forall bne0 \left(\left(forall t g\left(t\right)=a+bcdot f\left(t\right)\right) Rightarrow forall x M_f x = M_g x\right)$.

• If $f$ is monotonic, then $M_f$ is monotonic.

Examples

• If we take $S$ to be the real line and $f = mathrm\left\{id\right\}$, (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 $f\left(x\right) = ln\left(x\right)$, 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 $f\left(x\right) = frac\left\{1\right\}\left\{x\right\}$, then the f-mean corresponds to the harmonic mean.
• If we take $S$ to be the set of positive real numbers and $f\left(x\right) = x^p$, 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 normalizing the input values by some (homogenous) mean $C$.

$M_\left\{f,C\right\} x = C x cdot f^\left\{-1\right\}left\left(frac\left\{fleft\left(frac\left\{x_1\right\}\left\{C x\right\}right\right) + dots + fleft\left(frac\left\{x_n\right\}\left\{C x\right\}right\right)\right\}\left\{n\right\} right\right)$
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.