Quasi-arithmetic mean

Template:Short description In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean<ref>Template:Cite journal</ref> is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function <math>f</math>. 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 <math>I</math> of the real line to the real numbers, and is both continuous and injective, the f-mean of <math>n</math> numbers <math>x_1, \dots, x_n \in I</math> is defined as <math>M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{f(x_1)+ \cdots + f(x_n)}n \right)</math>, which can also be written

<math> M_f(\vec x)= f^{-1}\left(\frac{1}{n} \sum_{k=1}^{n}f(x_k) \right)</math>

We require f to be injective in order for the inverse function <math>f^{-1}</math> to exist. Since <math>f</math> is defined over an interval, <math>\frac{f(x_1)+ \cdots + f(x_n)}n</math> lies within the domain of <math>f^{-1}</math>.

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 <math>x</math> nor smaller than the smallest number in <math>x</math>.

Examples

• If <math>I = \mathbb{R}</math>, the real line, and <math>f(x) = x</math>, (or indeed any linear function <math>x\mapsto a\cdot x + b</math>, <math>a</math> not equal to 0) then the f-mean corresponds to the arithmetic mean.
• If <math>I = \mathbb{R}^+</math>, the positive real numbers and <math>f(x) = \log(x)</math>, 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 <math>I = \mathbb{R}^+</math> and <math>f(x) = \frac{1}{x}</math>, then the f-mean corresponds to the harmonic mean.
• If <math>I = \mathbb{R}^+</math> and <math>f(x) = x^p</math>, then the f-mean corresponds to the power mean with exponent <math>p</math>.
• If <math>I = \mathbb{R}</math> and <math>f(x) = \exp(x)</math>, 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), <math>M_f(x_1, \dots, x_n) = \mathrm{LSE}(x_1, \dots, x_n)-\log(n)</math>. The <math>-\log(n)</math> corresponds to dividing by Template:Mvar, 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 <math>M_f</math> for any single function <math>f</math>:

Symmetry: The value of <math>M_f</math> is unchanged if its arguments are permuted.

Idempotency: for all x, <math>M_f(x,\dots,x) = x</math>.

Monotonicity: <math>M_f</math> is monotonic in each of its arguments (since <math>f</math> is monotonic).

Continuity: <math>M_f</math> is continuous in each of its arguments (since <math>f</math> is continuous).

Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With <math>m=M_f(x_1,\dots,x_k)</math> it holds:

<math>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)</math>

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:<math> 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}))

</math>

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

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

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

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

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

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.<ref name=":0">Template:Cite book</ref>Template:Rp
• Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.<ref name=":0" />Template:Rp
• Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

• 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,<ref>Template:Cite journal</ref> but that if one additionally assumes <math>M</math> to be an analytic function then the answer is positive.<ref>Template:Cite journal</ref>

Homogeneity

Means are usually homogeneous, but for most functions <math>f</math>, 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 <math>C</math>.

<math>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)</math>

However this modification may violate monotonicity and the partitioning property of the mean.

Generalizations

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

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.

<references />[10] MR4355191 - Characterization of quasi-arithmetic means without regularity condition Burai, P.; Kiss, G.; Szokol, P. 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.