Quasi-arithmetic mean

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In mathematics and statistics, the quasi-arithmetic mean (or generalised f-mean) is the natural generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function $f$ .

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

$\{x_1, x_2\} \subset S$

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

For $n$ numbers

$\{x_1, \dots, x_n\} \subset S$ ,

the f-mean is

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

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 $x$ .

1 Properties
2 Examples
3 Homogenity
4 See also

[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 $f$ :

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

If $f$ is monotonic, then $M f$ is monotonic.

[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 $f (x) = x p$ , 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 $C$ .

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