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 . 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 of the real line to the
real numbers, and is both
continuous and
injective, the f-mean of numbers
is defined as , which can also be written
We require f to be injective in order for the
inverse function to exist. Since is defined over an interval, lies within the domain of .
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 nor smaller than the smallest number in .
Examples
If , the
real line, and , (or indeed any linear function , not equal to 0) then the f-mean corresponds to the
arithmetic mean.
If , the
positive real numbers and , 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 and , then the f-mean corresponds to the
harmonic mean.
If and , then the f-mean corresponds to the
power mean with exponent .
If and , 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), . The 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 for any single function :
Symmetry: The value of is unchanged if its arguments are permuted.
Idempotency: for all x, .
Monotonicity: is monotonic in each of its arguments (since is
monotonic).
Continuity: is continuous in each of its arguments (since is continuous).
Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With it holds:
Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:
Self-distributivity: For any quasi-arithmetic mean of two variables: .
Mediality: For any quasi-arithmetic mean of two variables:.
Balancing: For any quasi-arithmetic mean of two variables:.
Central limit theorem : Under regularity conditions, for a sufficiently large sample, 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 : .
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 to be an
analytic function then the answer is positive.[7]
Homogeneity
Means are usually
homogeneous, but for most functions , 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 .
However this modification may violate
monotonicity and the partitioning property of the mean.
Generalizations
Consider a Legendre-type strictly convex function . Then the gradient map is globally invertible and the weighted multivariate quasi-arithmetic mean[8] is defined by
, where is a normalized weight vector ( by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean associated to the quasi-arithmetic mean .
For example, take for a symmetric positive-definite matrix.
The pair of matrix quasi-arithmetic means yields the matrix harmonic mean:
^Barczy, M. & Burai, P. (2019). "Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables".
arXiv:1909.02968 [
math.PR].
^
abAczél, 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.
^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.