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 $f$ .

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

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

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 $x$ .

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.

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 $p$ .

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.

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

Quasi-arithmetic mean

Definition

Properties

Examples

Homogenity

See also

Acknowledgement and Attribution Regarding Sources of Content

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