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

$M_{f}(x_{1},x_{2})=f^{-1}\left({\frac {f(x_{1})+f(x_{2})}{2}}\right).$

For $n$ numbers

$\{x_{1},\dots ,x_{n}\}\subset S$,

the f-mean is

$M_{f}x=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right).$

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

${\frac {f\left(x_{1}\right)+f\left(x_{2}\right)}{2}}$

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.

$M_{f}(x_{1},\dots ,x_{n\cdot k})=M_{f}(M_{f}(x_{1},\dots ,x_{k}),M_{f}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{f}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))$

• Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.

With $m=M_{f}(x_{1},\dots ,x_{k})$ 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 $f$:

$\forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}x=M_{g}x)$.

• 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 $x\mapsto a\cdot x+b$, $a$ 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 normalizing the input values by some (homogenous) mean $C$.

$M_{f,C}x=Cx\cdot f^{-1}\left({\frac {f\left({\frac {x_{1}}{Cx}}\right)+\dots +f\left({\frac {x_{n}}{Cx}}\right)}{n}}\right)$

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.

See also

• Generalized mean
• Jensen's inequality