Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti 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 Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

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, the f-mean of $$n$$ numbers $$x_1, \dots, x_n \in I$$ is defined as $$M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{f(x_1)+ \cdots + f(x_n)}n \right)$$, which can also be written
 * $$ M_f(\vec x)= f^{-1}\left(\frac{1}{n} \sum_{k=1}^{n}f(x_k) \right)$$

We require f to be injective in order for the inverse function $$f^{-1}$$ to exist. Since $$f$$ is defined over an interval, $$\frac{f(x_1)+ \cdots + f(x_n)}n$$ 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 $$I = \mathbb{R}$$, 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 $$I = \mathbb{R}^+$$, 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 $$I = \mathbb{R}^+$$ and $$f(x) = \frac{1}{x}$$, then the f-mean corresponds to the harmonic mean.
 * If $$I = \mathbb{R}^+$$ and $$f(x) = x^p$$, then the f-mean corresponds to the power mean with exponent $$p$$.
 * If $$I = \mathbb{R}$$ and $$f(x) = \exp(x)$$, then the f-mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), $$M_f(x_1, \dots, x_n) = \mathrm{LSE}(x_1, \dots, x_n)-\log(n)$$. The $$-\log(n)$$ corresponds to dividing by $n$, since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function.

Properties
The following properties hold for $$M_f$$ for any single function $$f$$:

Symmetry: The value of $$M_f$$is unchanged if its arguments are permuted.

Idempotency: for all x, $$M_f(x,\dots,x) = x$$.

Monotonicity: $$M_f$$ is monotonic in each of its arguments (since $$f$$ is monotonic).

Continuity: $$M_f$$ is continuous in each of its arguments  (since  $$f$$ is continuous).

Replacement: 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:


 * $$M_f(x_1,\dots,x_k,x_{k+1},\dots,x_n) = M_f(\underbrace{m,\dots,m}_{k \text{ times}},x_{k+1},\dots,x_n)$$

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:$$ M_f(x_1,\dots,x_{n\cdot k}) = M_f(M_f(x_1,\dots,x_{k}),     M_f(x_{k+1},\dots,x_{2\cdot k}),      \dots,      M_f(x_{(n-1)\cdot k + 1},\dots,x_{n\cdot k})) $$

Self-distributivity: For any quasi-arithmetic mean $$M$$ of two variables: $$M(x,M(y,z))=M(M(x,y),M(x,z))$$.

Mediality: For any quasi-arithmetic mean $$M$$ of two variables:$$M(M(x,y),M(z,w))=M(M(x,z),M(y,w))$$.

Balancing: For any quasi-arithmetic mean $$M$$ of two variables:$$M\big(M(x, M(x, y)), M(y, M(x, y))\big)=M(x, y)$$.

Central limit theorem : Under regularity conditions, for a sufficiently large sample, $$\sqrt{n}\{M_f(X_1, \dots, X_n) - f^{-1}(E_f(X_1, \dots, X_n))\}$$ is approximately normal. A similar result is available for Bajraktarević means, which are generalizations of quasi-arithmetic means.

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of $$f$$: $$\forall a\ \forall b\ne0 ((\forall t\ g(t)=a+b\cdot f(t)) \Rightarrow \forall x\ M_f (x) = M_g (x)$$.

Characterization
There are several different sets of properties that characterize the quasi-arithmetic mean (i.e., each function that satisfies these properties is an f-mean for some function f).


 * Mediality is essentially sufficient to characterize quasi-arithmetic means.
 * Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.
 * Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.
 * Balancing: An interesting problem is whether this condition (together with symmetry, fixed-point, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, but that if one additionally assumes $$M$$ to be an analytic function then the answer is positive.

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 (including 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.

Generalizations
Consider a Legendre-type strictly convex function $$F$$. Then the gradient map $$\nabla F$$ is globally invertible and the weighted multivariate quasi-arithmetic mean is defined by $$ M_{\nabla F}(\theta_1,\ldots,\theta_n;w) = {\nabla F}^{-1}\left(\sum_{i=1}^n w_i \nabla F(\theta_i)\right) $$, where $$w$$ is a normalized weight vector ($$w_i=\frac{1}{n}$$ by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean $$M_{\nabla F^*}$$ associated to the quasi-arithmetic mean $$M_{\nabla F}$$. For example, take $$F(X)=-\log\det(X)$$ for $$X$$ a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: $$M_{\nabla F}(\theta_1,\theta_2)=2(\theta_1^{-1}+\theta_2^{-1})^{-1}. $$