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 ${\displaystyle f}${\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 ${\displaystyle I}${\displaystyle I} of the real line to the real numbers, and is both continuous and injective, the f-mean of ${\displaystyle n}${\displaystyle n} numbers ${\displaystyle x_{1},\dots ,x_{n}\in I}${\displaystyle x_{1},\dots ,x_{n}\in I} is defined as ${\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right)}${\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

${\displaystyle M_{f}({\vec {x}})=f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)}${\displaystyle 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 ${\displaystyle f^{-1}}${\displaystyle f^{-1}} to exist. Since ${\displaystyle f}${\displaystyle f} is defined over an interval, ${\displaystyle {\frac {f(x_{1})+\cdots +f(x_{n})}{n}}}${\displaystyle {\frac {f(x_{1})+\cdots +f(x_{n})}{n}}} lies within the domain of ${\displaystyle f^{-1}}${\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 ${\displaystyle x}${\displaystyle x} nor smaller than the smallest number in ${\displaystyle x}${\displaystyle x}.

• If ${\displaystyle I=\mathbb {R} }${\displaystyle I=\mathbb {R} }, the real line, and ${\displaystyle f(x)=x}${\displaystyle f(x)=x}, (or indeed any linear function ${\displaystyle x\mapsto a\cdot x+b}${\displaystyle x\mapsto a\cdot x+b}, ${\displaystyle a}${\displaystyle a} not equal to 0) then the f-mean corresponds to the arithmetic mean.
• If ${\displaystyle I=\mathbb {R} ^{+}}${\displaystyle I=\mathbb {R} ^{+}}, the positive real numbers and ${\displaystyle f(x)=\log(x)}${\displaystyle 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 ${\displaystyle I=\mathbb {R} ^{+}}${\displaystyle I=\mathbb {R} ^{+}} and ${\displaystyle f(x)={\frac {1}{x}}}${\displaystyle f(x)={\frac {1}{x}}}, then the f-mean corresponds to the harmonic mean.
• If ${\displaystyle I=\mathbb {R} ^{+}}${\displaystyle I=\mathbb {R} ^{+}} and ${\displaystyle f(x)=x^{p}}${\displaystyle f(x)=x^{p}}, then the f-mean corresponds to the power mean with exponent ${\displaystyle p}${\displaystyle p}.
• If ${\displaystyle I=\mathbb {R} }${\displaystyle I=\mathbb {R} } and ${\displaystyle f(x)=\exp(x)}${\displaystyle 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), ${\displaystyle M_{f}(x_{1},\dots ,x_{n})=\mathrm {LSE} (x_{1},\dots ,x_{n})-\log(n)}${\displaystyle M_{f}(x_{1},\dots ,x_{n})=\mathrm {LSE} (x_{1},\dots ,x_{n})-\log(n)}. The ${\displaystyle -\log(n)}${\displaystyle -\log(n)} 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.

The following properties hold for ${\displaystyle M_{f}}${\displaystyle M_{f}} for any single function ${\displaystyle f}${\displaystyle f}:

Symmetry: The value of ${\displaystyle M_{f}}${\displaystyle M_{f}} is unchanged if its arguments are permuted.

Idempotency: for all x, ${\displaystyle M_{f}(x,\dots ,x)=x}${\displaystyle M_{f}(x,\dots ,x)=x}.

Monotonicity: ${\displaystyle M_{f}}${\displaystyle M_{f}} is monotonic in each of its arguments (since ${\displaystyle f}${\displaystyle f} is monotonic).

Continuity: ${\displaystyle M_{f}}${\displaystyle M_{f}} is continuous in each of its arguments (since ${\displaystyle f}${\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 ${\displaystyle m=M_{f}(x_{1},\dots ,x_{k})}${\displaystyle m=M_{f}(x_{1},\dots ,x_{k})} it holds:

${\displaystyle M_{f}(x_{1},\dots ,x_{k},x_{k+1},\dots ,x_{n})=M_{f}(\underbrace {m,\dots ,m} _{k{\text{ times}}},x_{k+1},\dots ,x_{n})}${\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})}

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:${\displaystyle M_{f}(x_{1},\dots ,x_{n\cdot k})=M_{f}(M_{f}(x_{1},\dots ,x_{k}),M_{f}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{f}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))}${\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 ${\displaystyle M}${\displaystyle M} of two variables: ${\displaystyle M(x,M(y,z))=M(M(x,y),M(x,z))}${\displaystyle M(x,M(y,z))=M(M(x,y),M(x,z))}.

Mediality: For any quasi-arithmetic mean ${\displaystyle M}${\displaystyle M} of two variables:${\displaystyle M(M(x,y),M(z,w))=M(M(x,z),M(y,w))}${\displaystyle M(M(x,y),M(z,w))=M(M(x,z),M(y,w))}.

Balancing: For any quasi-arithmetic mean ${\displaystyle M}${\displaystyle M} of two variables:${\displaystyle M{\big (}M(x,M(x,y)),M(y,M(x,y)){\big )}=M(x,y)}${\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, ${\displaystyle {\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}}${\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 ${\displaystyle f}${\displaystyle f}: ${\displaystyle \forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}(x)=M_{g}(x)}${\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).

• Mediality is essentially sufficient to characterize quasi-arithmetic means.^{[5] : chapter 17}
• Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.^{[5] : chapter 17}
• Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.^[6]
• 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,^[7] but that if one additionally assumes ${\displaystyle M}${\displaystyle M} to be an analytic function then the answer is positive.^[8]

Means are usually homogeneous, but for most functions ${\displaystyle f}${\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 ${\displaystyle C}${\displaystyle C}.

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

Consider a Legendre-type strictly convex function ${\displaystyle F}${\displaystyle F}. Then the gradient map ${\displaystyle \nabla F}${\displaystyle \nabla F} is globally invertible and the weighted multivariate quasi-arithmetic mean^[9] is defined by ${\displaystyle M_{\nabla F}(\theta _{1},\ldots ,\theta _{n};w)={\nabla F}^{-1}\left(\sum _{i=1}^{n}w_{i}\nabla F(\theta _{i})\right)}${\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 ${\displaystyle w}${\displaystyle w} is a normalized weight vector (${\displaystyle w_{i}={\frac {1}{n}}}${\displaystyle w_{i}={\frac {1}{n}}} by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean ${\displaystyle M_{\nabla F^{*}}}${\displaystyle M_{\nabla F^{*}}} associated to the quasi-arithmetic mean ${\displaystyle M_{\nabla F}}${\displaystyle M_{\nabla F}}. For example, take ${\displaystyle F(X)=-\log \det(X)}${\displaystyle F(X)=-\log \det(X)} for ${\displaystyle X}${\displaystyle X} a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: ${\displaystyle M_{\nabla F}(\theta _{1},\theta _{2})=2(\theta _{1}^{-1}+\theta _{2}^{-1})^{-1}.}${\displaystyle M_{\nabla F}(\theta _{1},\theta _{2})=2(\theta _{1}^{-1}+\theta _{2}^{-1})^{-1}.}

• Generalized mean
• Jensen's inequality

• 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.

• Means

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