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 $$f$$.

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 $$f = \mathrm{id}$$, (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 $$S$$ 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.