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 mathematician 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

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 $I$ = ℝ, the real line, and $f(x)=x$ , (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 $I$ = ℝ⁺, the 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 $I$ = ℝ⁺ and $f(x)={\frac {1}{x}}$ , then the f-mean corresponds to the harmonic mean.
If $I$ = ℝ⁺ and $f(x)=x^{p}$ , then the f-mean corresponds to the power mean with exponent $p$ .
If $I$ = ℝ and $f(x)=\exp(x)$ , then the f-mean is a constant shifted version of the LogSumExp (LSE) function, $M_{f}(x_{1},\dots ,x_{n})=LSE(x_{1},\dots ,x_{n})-\log(n)$ . The LogSumExp function is used as a smooth approximation to the maximum function.

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 $M\big(M(x, M(x, y)), M(y, M(x, y))\big)=M(x, y)$ . An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. 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]
Under regularity conditions, a central limit theorem can be derived for the generalised f-mean, thus implying that for a large sample ${\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}$ is approximately normal.^[3]

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

↑ Aumann, Georg (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". Journal für die reine und angewandte Mathematik. 176: 49–55. doi:10.1515/crll.1937.176.49.
↑ Aumann, Georg (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften: 45–81.
↑ de Carvalho, Miguel (2016). "Mean, what do you Mean?". The American Statistician. 70: 764‒776. doi:10.1080/00031305.2016.1148632.

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.