Quasi-arithmetic mean

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean{{cite journal |last1=Nielsen |first1=Frank |last2=Nock |first2=Richard |title=Generalizing skew Jensen divergences and Bregman divergences with comparative convexity |journal=IEEE Signal Processing Letters |date=June 2017 |volume=24 |issue=8 |page=2 |doi=10.1109/LSP.2017.2712195 |arxiv=1702.04877 |bibcode=2017ISPL...24.1123N |s2cid=31899023 }} 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 {{mvar|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.{{cite journal|last=de Carvalho|first=Miguel|title=Mean, what do you Mean?|journal=The American Statistician|year=2016|volume=70|issue=3|pages=764‒776|doi=10.1080/00031305.2016.1148632|url=https://zenodo.org/record/895400|hdl=20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c|s2cid=219595024 |hdl-access=free}}

A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.{{Cite journal |last1=Barczy |first1=Mátyás |last2=Burai |first2=Pál |date=2022-04-01 |title=Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables |url=https://link.springer.com/article/10.1007/s00010-021-00813-x |journal=Aequationes Mathematicae |language=en |volume=96 |issue=2 |pages=279–305 |doi=10.1007/s00010-021-00813-x |issn=1420-8903}}{{Cite journal |last1=Barczy |first1=Mátyás |last2=Páles |first2=Zsolt |date=2023-09-01 |title=Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables |url=https://link.springer.com/article/10.1007/s10959-022-01225-6 |journal=Journal of Theoretical Probability |language=en |volume=36 |issue=3 |pages=1626–1666 |doi=10.1007/s10959-022-01225-6 |issn=1572-9230|arxiv=2112.05183 }}

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.{{Cite book|title=Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31.|author=Aczél, J.|author2=Dhombres, J. G.|publisher=Cambridge Univ. Press|year=1989|location=Cambridge}}{{Rp|chapter 17}}
Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.{{Rp|chapter 17}}
Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.{{Cite web|url=https://math.stackexchange.com/a/3261514/29780|title=Characterization of the quasi-arithmetic mean|last=Grudkin|first=Anton|date=2019|website=Math stackexchange}}
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,{{cite journal|last=Aumann|first=Georg|year=1937|title=Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften|journal=Journal für die reine und angewandte Mathematik|volume=1937|issue=176|pages=49–55|doi=10.1515/crll.1937.176.49|s2cid=115392661}} but that if one additionally assumes $M$ to be an analytic function then the answer is positive.{{cite journal|last=Aumann|first=Georg|year=1934|title=Grundlegung der Theorie der analytischen Analytische Mittelwerte|journal=Sitzungsberichte der Bayerischen Akademie der Wissenschaften|pages=45–81}}

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{{cite arXiv|last=Nielsen|first=Frank|year=2023|title=Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry|eprint= 2301.10980| class = cs.IT}} 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:

$M_{\nabla F}(\theta_1,\theta_2)=2(\theta_1^{-1}+\theta_2^{-1})^{-1}.$

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, [http://www.brunodefinetti.it/Opere/concettodiMedia.pdf "Sul concetto di media"], vol. 3, p. 36996, 1931, istituto italiano degli attuari.

Category:Means

[10] [https://rdcu.be/dVZA1 MR4355191 - Characterization of quasi-arithmetic means without regularity condition]

[https://rdcu.be/dVZA1 Burai, P.; Kiss, G.; Szokol, P.]

[https://rdcu.be/dVZA1 Acta Math. Hungar. 165 (2021), no. 2, 474–485.]

[11]

MR4574540 - A dichotomy result for strictly increasing bisymmetric maps

Burai, Pál; Kiss, Gergely; Szokol, Patricia

J. Math. Anal. Appl. 526 (2023), no. 2, Paper No. 127269, 9 pp.