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 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

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

For $n$ numbers

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

the f-mean is

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

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

Examples

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

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

$M_f(x_1,\dots,x_k,x_{k+1},\dots,x_n) = M_f(\underbrace{m,\dots,m}_{k \text{ times}},x_{k+1},\dots,x_n)$

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

$\forall a\ \forall b\ne0 ((\forall t\ g(t)=a+b\cdot f(t)) \Rightarrow \forall x\ M_f (x) = M_g (x)$.

• If $f$ is monotonic, then $M_f$ is monotonic.
• Any quasi-arithmetic mean $M$ of two variables has the mediality property $M(M(x,y),M(z,w))=M(M(x,z),M(y,w))$ and the self-distributivity property $M(x,M(y,z))=M(M(x,y),M(x,z))$. 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 $M\big((x, M(x, y)), M(y, M(x, y))\big)=M(x, y)$. 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, but that if one additionally assumes $M$ to be an analytic function then the answer is positive.

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 $C$.

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

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.

See also

• Generalized mean
• Jensen's inequality