Quasi-arithmetic mean

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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.

Contents

[edit] 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.

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

[edit] Examples

  • If we take S to be the real line and f = 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) = xp, then the f-mean corresponds to the power mean with exponent p.

[edit] 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 = C x \cdot f^{-1}\left( \frac{f\left(\frac{x_1}{C x}\right) + \dots + f\left(\frac{x_n}{C x}\right)}{n} \right)

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

[edit] 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.

[edit] See also