### Generalized f-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 an interval $I$ of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers

$x_1, x_2 \in I$

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

$x_1, \dots, x_n \in I$,

the f-mean is

$M_f\left(x_1, \dots, x_n\right) = f^\left\{-1\right\}\left\left( \frac\left\{f\left(x_1\right)+ \cdots + f\left(x_n\right)\right\}n \right\right).$

We require f to be injective in order for the inverse function $f^\left\{-1\right\}$ to exist. Since $f$ is defined over an interval, $\frac\left\{f\left\left(x_1\right\right) + f\left\left(x_2\right\right)\right\}2$ 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 $x$.

## Examples

• If we take $I$ to be the real line and $f = \mathrm\left\{id\right\}$, (or indeed any linear function $x\mapsto a\cdot x + b$, $a$ not equal to 0) then the f-mean corresponds to the arithmetic mean.
• If we take $I$ to be the set of positive real numbers and $f\left(x\right) = \log\left(x\right)$, 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 $I$ 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 $I$ 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 $p$.

## Properties

• Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks.


M_f(x_1,\dots,x_{n\cdot k}) =

 M_f(M_f(x_1,\dots,x_{k}),
M_f(x_{k+1},\dots,x_{2\cdot k}),
\dots,
M_f(x_{(n-1)\cdot k + 1},\dots,x_{n\cdot k}))


• Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.
With $m=M_f\left(x_1,\dots,x_k\right)$ it holds
$M_f\left(x_1,\dots,x_k,x_\left\{k+1\right\},\dots,x_n\right) = M_f\left(\underbrace\left\{m,\dots,m\right\}_\left\{k \text\left\{ times\right\}\right\},x_\left\{k+1\right\},\dots,x_n\right)$
• The quasi-arithmetic mean is invariant with respect to offsets and scaling of $f$:
$\forall a\ \forall b\ne0 \left(\left(\forall t\ g\left(t\right)=a+b\cdot f\left(t\right)\right) \Rightarrow \forall x\ M_f \left(x\right) = M_g \left(x\right)$.
• If $f$ is monotonic, then $M_f$ is monotonic.
• Any quasi-arithmetic mean $M$ of two variables has the mediality property $M\left(M\left(x,y\right),M\left(z,w\right)\right)=M\left(M\left(x,z\right),M\left(y,w\right)\right)$ and the self-distributivity property $M\left(x,M\left(y,z\right)\right)=M\left(M\left(x,y\right),M\left(x,z\right)\right)$. Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.
• Any quasi-arithmetic mean $M$ of two variables has the balancing property $M\big\left(M\left(x, M\left(x, y\right)\right), M\left(y, M\left(x, y\right)\right)\big\right)=M\left(x, y\right)$. An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arthmetic. Georg Aumann showed in the 1930s that the answer is no in general,[1] but that if one additionally assumes $M$ to be an analytic function then the answer is positive.[2]

## Homogeneity

Means are usually homogeneous, but for most functions $f$, the f-mean is not. Indeed, the only homogeneous quasi-arithmetic means are the power means and the geometric mean; see Hardy–Littlewood–Pólya, page 68.

The homogeneity property can be achieved by normalizing the input values by some (homogeneous) mean $C$.

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

However this modification may violate monotonicity and the partitioning property of the mean.

## References

• Aczél, J.; Dhombres, J. G. (1989) Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31. Cambridge Univ. Press, Cambridge, 1989.
• Andrey Kolmogorov (1930) “On the Notion of Mean”, in “Mathematics and Mechanics” (Kluwer 1991) — pp. 144–146.
• Andrey Kolmogorov (1930) Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, pp. 388–391.
• John Bibby (1974) “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.
• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) Inequalities. 2nd ed. Cambridge Univ. Press, Cambridge, 1952.