Quasi-arithmetic mean explained

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 . 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 numbersx _{1,} ... , x _{n} \in I is defined as M _{f(x} _{1,} ... , x _{n)} = f ^{-1} \left( (f(x _{1)+} … + f(x _{n)})/(n) \right) , which can also be written M _{f(\vec} x) = f ^{-1} \left( (1)/(n) \sum ^{n} _{k =1} f(x _{k)} \right) f ^{-1} to exist. Since f is defined over an interval, (f(x _{1)+} … + 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 = R , the real line, and f(x) = x , (or indeed any linear function x\mapsto a ⋅ x + b , a not equal to 0) then the f-mean corresponds to the arithmetic mean.

• If

I = 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 = R ^{+} and f(x) = (1)/(x) , then the f-mean corresponds to the harmonic mean.

• If

I = R ^{+} and f(x) = x ^{p} , then the f-mean corresponds to the power mean with exponent p .

• If

I = 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,} ... , x _{n)} = LSE (x _{1,} ... , x _{n)-log(n)} . The -log (n) corresponds to dividing by, 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,...,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,...,x} _{k)} it holds:M _{f(x} _{1,...,x} _{k,x} _{k+1} , ... ,x _{n)} = M _{f(\underbrace{m,...,m}} _{k} ,x _{k+1} , ... ,x _{n)}

Partitioning

The computation of the mean can be split into computations of equal sized sub-blocks:

M _{f(x} _{1,...,x} _{n ⋅} ) = M _{f(M} _{f(x} _{1,...,x} _{k} ), M _{f(x} _{k+1} , ... ,x _{2 ⋅} ), ... , M _{f(x} _{(n-1) ⋅} , ... ,x _{n ⋅} ))

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 ( M(x, M(x, y)), M(y, M(x, y)) ) = M(x, y) .

Central limit theorem : Under regularity conditions, for a sufficiently large sample,

\sqrt{n}\{M _{f(X} _{1,} ... , X _{n)} - f ^{-1} (E _{f(X} _{1,} ... , X _{n))\}} 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

f : \forall a \forall b\ne 0 ((\forall t g(t) = a + b ⋅ f(t)) ⇒ \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.^[4]
• Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.
• 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

M to be an analytic function then the answer is positive.^[7]

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 ⋅ f ^{-1} \left( (f\left( (x _{1})/(\right) + … + f\left( (x _{n})/(C x) \right) C x))/(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^[8] is defined byM _{\nabla} (\theta _{1,\ldots,\theta} _{n;w)} = {\nabla F} ^{-1} \left(\sum ^{n} _{i =1} w _{i} \nabla F(\theta _{i)\right)} , where w is a normalized weight vector (w _{i = (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} .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} (\theta _{1,\theta} _{2)=2(\theta} ^{-1} ^{1} + \theta ^{-1} _{2} ) ^{-1} .

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.

Notes and References

1. Nielsen . Frank . Nock . Richard . Generalizing skew Jensen divergences and Bregman divergences with comparative convexity . IEEE Signal Processing Letters . June 2017 . 24 . 8 . 2 . 10.1109/LSP.2017.2712195 . 1702.04877 . 2017ISPL...24.1123N . 31899023 .
2. de Carvalho. Miguel. Mean, what do you Mean?. The American Statistician. 2016. 70. 3. 764‒776. 10.1080/00031305.2016.1148632. 20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c. 219595024 . free.
3. 1909.02968. Barczy, M.. Burai, P.. amp. Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables. 2019. math.PR.
4. Book: 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.. Aczél, J.. Dhombres, J. G.. Cambridge Univ. Press. 1989. Cambridge.
5. Web site: Characterization of the quasi-arithmetic mean. Grudkin. Anton. 2019. Math stackexchange.
6. Aumann. Georg. 1937. Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften. Journal für die reine und angewandte Mathematik. 1937. 176. 49–55. 10.1515/crll.1937.176.49. 115392661.
7. Aumann. Georg. 1934. Grundlegung der Theorie der analytischen Analytische Mittelwerte. Sitzungsberichte der Bayerischen Akademie der Wissenschaften. 45–81.
8. Nielsen. Frank. 2023. Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry. 2301.10980. cs.IT.