Quasi-arithmetic mean
In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean[1] is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function [math]\displaystyle{ 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]\displaystyle{ I }[/math] of the real line to the real numbers, and is both continuous and injective, the f-mean of [math]\displaystyle{ n }[/math] numbers [math]\displaystyle{ x_1, \dots, x_n \in I }[/math] is defined as [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ f^{-1} }[/math] to exist. Since [math]\displaystyle{ f }[/math] is defined over an interval, [math]\displaystyle{ \frac{f(x_1)+ \cdots + f(x_n)}n }[/math] lies within the domain of [math]\displaystyle{ 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]\displaystyle{ x }[/math] nor smaller than the smallest number in [math]\displaystyle{ x }[/math].
Examples
- If [math]\displaystyle{ I }[/math] = ℝ, the real line, and [math]\displaystyle{ f(x) = x }[/math], (or indeed any linear function [math]\displaystyle{ x\mapsto a\cdot x + b }[/math], [math]\displaystyle{ a }[/math] not equal to 0) then the f-mean corresponds to the arithmetic mean.
- If [math]\displaystyle{ I }[/math] = ℝ+, the positive real numbers and [math]\displaystyle{ 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]\displaystyle{ I }[/math] = ℝ+ and [math]\displaystyle{ f(x) = \frac{1}{x} }[/math], then the f-mean corresponds to the harmonic mean.
- If [math]\displaystyle{ I }[/math] = ℝ+ and [math]\displaystyle{ f(x) = x^p }[/math], then the f-mean corresponds to the power mean with exponent [math]\displaystyle{ p }[/math].
- If [math]\displaystyle{ I }[/math] = ℝ and [math]\displaystyle{ 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]\displaystyle{ M_f(x_1, \dots, x_n) = \mathrm{LSE}(x_1, \dots, x_n)-\log(n) }[/math]. The [math]\displaystyle{ -\log(n) }[/math] 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 [math]\displaystyle{ M_f }[/math] for any single function [math]\displaystyle{ f }[/math]:
Symmetry: The value of [math]\displaystyle{ M_f }[/math]is unchanged if its arguments are permuted.
Idempotency: for all x, [math]\displaystyle{ M_f(x,\dots,x) = x }[/math].
Monotonicity: [math]\displaystyle{ M_f }[/math] is monotonic in each of its arguments (since [math]\displaystyle{ f }[/math] is monotonic).
Continuity: [math]\displaystyle{ M_f }[/math] is continuous in each of its arguments (since [math]\displaystyle{ 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]\displaystyle{ m=M_f(x_1,\dots,x_k) }[/math] it holds:
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ M }[/math] of two variables: [math]\displaystyle{ M(x,M(y,z))=M(M(x,y),M(x,z)) }[/math].
Mediality: For any quasi-arithmetic mean [math]\displaystyle{ M }[/math] of two variables:[math]\displaystyle{ M(M(x,y),M(z,w))=M(M(x,z),M(y,w)) }[/math].
Balancing: For any quasi-arithmetic mean [math]\displaystyle{ M }[/math] of two variables:[math]\displaystyle{ 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]\displaystyle{ \sqrt{n}\{M_f(X_1, \dots, X_n) - f^{-1}(E_f(X_1, \dots, X_n))\} }[/math] is approximately normal.[2] A similar result is available for Bajraktarević means, which are generalizations of quasi-arithmetic means.[3]
Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of [math]\displaystyle{ f }[/math]: [math]\displaystyle{ \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.[4]:chapter 17
- Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.[4]:chapter 17
- Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.[5]
- 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,[6] but that if one additionally assumes [math]\displaystyle{ M }[/math] to be an analytic function then the answer is positive.[7]
Homogeneity
Means are usually homogeneous, but for most functions [math]\displaystyle{ 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]\displaystyle{ C }[/math].
- [math]\displaystyle{ 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]\displaystyle{ F }[/math]. Then the gradient map [math]\displaystyle{ \nabla F }[/math] is globally invertible and the weighted multivariate quasi-arithmetic mean[8] is defined by [math]\displaystyle{ 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]\displaystyle{ w }[/math] is a normalized weight vector ([math]\displaystyle{ w_i=\frac{1}{n} }[/math] by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean [math]\displaystyle{ M_{\nabla F^*} }[/math] associated to the quasi-arithmetic mean [math]\displaystyle{ M_{\nabla F} }[/math]. For example, take [math]\displaystyle{ F(X)=-\log\det(X) }[/math] for [math]\displaystyle{ X }[/math] a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: [math]\displaystyle{ M_{\nabla F}(\theta_1,\theta_2)=2(\theta_1^{-1}+\theta_2^{-1})^{-1}. }[/math]
See also
References
- ↑ Nielsen, Frank; Nock, Richard (June 2017). "Generalizing skew Jensen divergences and Bregman divergences with comparative convexity". IEEE Signal Processing Letters 24 (8): 2. doi:10.1109/LSP.2017.2712195.
- ↑ de Carvalho, Miguel (2016). "Mean, what do you Mean?". The American Statistician 70 (3): 764‒776. doi:10.1080/00031305.2016.1148632. https://zenodo.org/record/895400.
- ↑ Barczy, M. & Burai, P. (2019). "Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables". arXiv:1909.02968 [math.PR].
- ↑ 4.0 4.1 Aczél, J.; Dhombres, J. G. (1989). 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.. Cambridge: Cambridge Univ. Press.
- ↑ Grudkin, Anton (2019). "Characterization of the quasi-arithmetic mean". https://math.stackexchange.com/a/3261514/29780.
- ↑ Aumann, Georg (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". Journal für die reine und angewandte Mathematik 1937 (176): 49–55. doi:10.1515/crll.1937.176.49.
- ↑ Aumann, Georg (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften: 45–81.
- ↑ Nielsen, Frank (2023). "Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry". arXiv:2301.10980 [cs.IT].
- 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.
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