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Generalization of means
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 f {\displaystyle f} . It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.
If f is a function which maps an interval I {\displaystyle I} of the real line to the real numbers, and is both continuous and injective, the f-mean of n {\displaystyle n} numbers x 1 , … , x n ∈ I {\displaystyle x_{1},\dots ,x_{n}\in I} is defined as M f ( x 1 , … , x n ) = f − 1 ( f ( x 1 ) + ⋯ + f ( x n ) n ) {\displaystyle 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
We require f to be injective in order for the inverse function f − 1 {\displaystyle f^{-1}} to exist. Since f {\displaystyle f} is defined over an interval, f ( x 1 ) + ⋯ + f ( x n ) n {\displaystyle {\frac {f(x_{1})+\cdots +f(x_{n})}{n}}} lies within the domain of f − 1 {\displaystyle 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 {\displaystyle x} nor smaller than the smallest number in x {\displaystyle x} .
The following properties hold for M f {\displaystyle M_{f}} for any single function f {\displaystyle f} :
Symmetry: The value of M f {\displaystyle M_{f}} is unchanged if its arguments are permuted.
Idempotency: for all x, M f ( x , … , x ) = x {\displaystyle M_{f}(x,\dots ,x)=x} .
Monotonicity: M f {\displaystyle M_{f}} is monotonic in each of its arguments (since f {\displaystyle f} is monotonic).
Continuity: M f {\displaystyle M_{f}} is continuous in each of its arguments (since f {\displaystyle 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 , … , x k ) {\displaystyle m=M_{f}(x_{1},\dots ,x_{k})} it holds:
Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks: M f ( x 1 , … , x n ⋅ k ) = M f ( M f ( x 1 , … , x k ) , M f ( x k + 1 , … , x 2 ⋅ k ) , … , M f ( x ( n − 1 ) ⋅ k + 1 , … , x n ⋅ k ) ) {\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}))}
Self-distributivity: For any quasi-arithmetic mean M {\displaystyle M} of two variables: M ( x , M ( y , z ) ) = M ( M ( x , y ) , M ( x , z ) ) {\displaystyle M(x,M(y,z))=M(M(x,y),M(x,z))} .
Mediality: For any quasi-arithmetic mean M {\displaystyle M} of two variables: M ( M ( x , y ) , M ( z , w ) ) = M ( M ( x , z ) , M ( y , w ) ) {\displaystyle M(M(x,y),M(z,w))=M(M(x,z),M(y,w))} .
Balancing: For any quasi-arithmetic mean M {\displaystyle M} of two variables: M ( M ( x , M ( x , y ) ) , M ( y , M ( x , y ) ) ) = M ( x , y ) {\displaystyle 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, n { M f ( X 1 , … , X n ) − f − 1 ( E f ( X 1 , … , X n ) ) } {\displaystyle {\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}} is approximately normal.[2] A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.[3][4]
Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of f {\displaystyle f} : ∀ a ∀ b ≠ 0 ( ( ∀ t g ( t ) = a + b ⋅ f ( t ) ) ⇒ ∀ x M f ( x ) = M g ( x ) {\displaystyle \forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}(x)=M_{g}(x)} .
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).
Means are usually homogeneous, but for most functions f {\displaystyle 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 {\displaystyle C} .
However this modification may violate monotonicity and the partitioning property of the mean.
Consider a Legendre-type strictly convex function F {\displaystyle F} . Then the gradient map ∇ F {\displaystyle \nabla F} is globally invertible and the weighted multivariate quasi-arithmetic mean[9] is defined by M ∇ F ( θ 1 , … , θ n ; w ) = ∇ F − 1 ( ∑ i = 1 n w i ∇ F ( θ i ) ) {\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)} , where w {\displaystyle w} is a normalized weight vector ( w i = 1 n {\displaystyle w_{i}={\frac {1}{n}}} by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean M ∇ F ∗ {\displaystyle M_{\nabla F^{*}}} associated to the quasi-arithmetic mean M ∇ F {\displaystyle M_{\nabla F}} . For example, take F ( X ) = − log det ( X ) {\displaystyle F(X)=-\log \det(X)} for X {\displaystyle X} a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: M ∇ F ( θ 1 , θ 2 ) = 2 ( θ 1 − 1 + θ 2 − 1 ) − 1 . {\displaystyle M_{\nabla F}(\theta _{1},\theta _{2})=2(\theta _{1}^{-1}+\theta _{2}^{-1})^{-1}.}
[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.
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