Quasi-arithmetic mean

Short description: Generalization of means

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 that maps some continuous interval $I$ of the real line to some other continuous subset $J\equiv f(I)$ of the real numbers, and $f$ is both continuous, and injective (one-to-one).

(We require $f$ to be injective on $I$ in order for an inverse function $f^{-1}$ to exist. We require $I$ and $J$ to both be continuous intervals in order to ensure that an average of any finite (or infinite) subset of values within $J$ will always correspond to a value in $I$.)

Subject to those requirements, the f mean of $n$ numbers $x_1,\dots ,x_n\in I$ is defined to be

$M_f(x_1,\dots ,x_n)\equiv f^{-1}\left(\frac{1}{n}\right(f(x_1)+\cdots +f(x_n)\left)\right),$

or equivalently

$M_f(\overrightarrow{x})=f^{-1}(\frac{1}{n}\sum _{k=1}^{n}f(x_k)).$

A consequence of $f$ being defined over some selected interval, $I,$ mapping to yet another interval, $J,$ is that $\frac{1}{n}(f(x_1)+\cdots +f(x_n))$ must also lie within $J.$ And because $J$ is the domain of $f^{-1},$ so in turn $f^{-1}$ must produce a value inside the same domain the values originally came from, $I.$

Because $f$ is injective and continuous, it necessarily follows that $f$ is a strictly monotonic function, and therefore that the f mean is neither larger than the largest number of the tuple $x_1,\dots ,x_n\equiv X$ nor smaller than the smallest number contained in $X,$ hence contained somewhere among the values of the original sample.

Examples

• If $I=\mathbb{ℝ},$ the real line, and $f(x)=x,$ (or indeed any linear function $x\mapsto a\cdot x+b,$ for $a\ne 0,$ otherwise any $a$ and any $b$) then the f mean corresponds to the arithmetic mean.

• If $I={\mathbb{ℝ}}^{+},$ the strictly positive real numbers, and $f(x)=\log (x),$ then the f mean corresponds to the geometric mean. (The result is the same for any logarithm; it does not depend on the base of the logarithm, as long as that base is strictly positive but 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 $p$ (e.g., for $p=2$ one gets the root mean square (RMS).)

• 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), $M_f(x_1,\dots ,x_n)=\mathsf{L}\mathsf{S}\mathsf{E}(x_1,\dots ,x_n)-\log (n).$ (The $-\log (n)$ in the expression corresponds to dividing by n, since logarithmic division is linear subtraction.) The LogSumExp function is a smooth maximum: It is 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 $x,$ the repeated average $M_f(x,\dots ,x)=x.$

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 $m\equiv M_f(x_1,\dots ,x_k)$ it holds:

$M_f(x_1,\dots ,x_k,x_{k+1},\dots ,x{}_n)=M_f(\underset{k\text{ times}}{\underbrace{m,\dots ,m}},x_{k+1},\dots ,x_n).$

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 (q.a.) mean $M_{\mathsf{q}\mathsf{a}}$ of two variables:

$M\mathsf{q}\mathsf{a}(x,M\mathsf{q}\mathsf{a}(y,z))=M\mathsf{q}\mathsf{a}(M\mathsf{q}\mathsf{a}(x,y),M\mathsf{q}\mathsf{a}(x,z)).$

Mediality: For any quasi-arithmetic mean $M\mathsf{q}\mathsf{a}$ of two variables:

$M\mathsf{q}\mathsf{a}(M\mathsf{q}\mathsf{a}(x,y),M\mathsf{q}\mathsf{a}(z,w))=M\mathsf{q}\mathsf{a}(M\mathsf{q}\mathsf{a}(x,z),M\mathsf{q}\mathsf{a}(y,w)).$

Balancing: For any quasi-arithmetic mean $M\mathsf{q}\mathsf{a}$ of two variables:

$M\mathsf{q}\mathsf{a}\left(M\mathsf{q}\mathsf{a}\right(x,M\mathsf{q}\mathsf{a}(x,y)),M\mathsf{q}\mathsf{a}(y,M\mathsf{q}\mathsf{a}(x,y)\left)\right)=M\mathsf{q}\mathsf{a}(x,y).$

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and non-trivial scaling of quasi-arithmetic $f:$ For any $p(t)\equiv a+b\cdot q(t),$ with $a$ and $b\ne 0$ constants, and $q$ a quasi-aritmetic function, $M_p(x)$ and $M_q(x)$ are always the same. In mathematical notation:

Given $q$ quasi-aritmetic, and $p:(p(t)=a+b\cdot q(t)\mathrm{\forall }t)\mathrm{\forall }a\mathrm{\forall }b\ne 0\Rightarrow M_p(x)=M_q(x)\mathrm{\forall }x.$

Central limit theorem : Under certain regularity conditions, and for a sufficiently large sample,

$z\equiv \sqrt{n}[M_f(X_1,\dots ,X_n)-{\mathbb{𝔼}}_X(M_f(X_1,\dots ,X_n)\left)\right]$

is approximately normally distributed. A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.

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.
• Continuity is superfluous in the characterization of two variables quasi-arithmetic means. See [10] for the details.
• 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 $C$.

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

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 $\mathrm{\nabla }F$ is globally invertible and the weighted multivariate quasi-arithmetic mean is defined by $M_{\mathrm{\nabla }F}({\theta }_1,\dots ,{\theta }_n;w)={\mathrm{\nabla }F}^{-1}(\sum _{i=1}^n{w}_i\mathrm{\nabla }F({\theta }_i))$, 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 $M_{\mathrm{\nabla }{F}^{*}}$ associated to the quasi-arithmetic mean $M_{\mathrm{\nabla }F}$. 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: $M_{\mathrm{\nabla }F}({\theta }_1,{\theta }_2)=2({\theta }_1^{-1}+{\theta }_2^{-1}{)}^{-1}.$

See also

• Generalized mean
• Jensen's inequality

References

• Andrey Kolmogorov (1930) "On the Notion of Mean", in "Mathematics and Mechanics" (Kluwer 1991) — pp. 144–146.
• Andrey Kolmogorov (1930) Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, pp. 388–391.
• John Bibby (1974) "Axiomatisations of the average and a further generalisation of monotonic sequences," Glasgow Mathematical Journal, vol. 15, pp. 63–65.
• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) Inequalities. 2nd ed. Cambridge Univ. Press, Cambridge, 1952.
• B. De Finetti, "Sul concetto di media", vol. 3, p. 36996, 1931, istituto italiano degli attuari.