Quasi-arithmetic mean

Definition

If \ f\ is a function that maps some continuous interval \ I\ of the real line to some other continuous subset \ J \equiv f(I)\ of the real numbers, and \ f\ is both continuous, and injective (one-to-one). : (We require \ f\ to be injective on \ I\ in order for an inverse function \ f^{-1}\ to exist. We require \ I\ and \ J\ to both be continuous intervals in order to ensure that an average of any finite (or infinite) subset of values within \ J\ will always correspond to a value in \ I\ .) Subject to those requirements, the ** of \ n\ numbers** \ x_1, \ldots, x_n \in I\ is defined to be : \ M_f(x_1, \dots, x_n); \equiv; f^{-1}!\left(\ \frac{1}{n}\Bigl(\ f(x_1) + \cdots + f(x_n)\ \Bigr)\ \right)\ , or equivalently : \ M_f(\vec x); =; f^{-1}!!\left(\ \frac{1}{n} \sum_{k=1}^{n}f(x_k)\ \right) ~.

A consequence of \ f\ being defined over some selected interval, \ I\ , mapping to yet another interval, \ J\ , is that \ \frac{1}{n} \left(\ f(x_1) + \cdots + f(x_n)\ \right)\ must also lie within \ J\ ~. And because \ J\ is the domain of \ f^{-1}\ , so in turn \ f^{-1}\ must produce a value inside the same domain the values originally came from, \ I ~.

Because \ f\ is injective and continuous, it necessarily follows that \ f\ is a strictly monotonic function, and therefore that the **** is neither larger than the largest number of the tuple \ x_1, \ldots\ , x_n \equiv X\ nor smaller than the smallest number contained in \ X\ , hence contained somewhere among the values of the original sample.

Examples

• If I = \mathbb{R}\ , the real line, and \ f(x) = x\ , (or indeed any linear function \ x \mapsto a\cdot x + b\ , for \ a \ne 0\ , otherwise any \ a\ and any \ b\ ) then the corresponds to the arithmetic mean.

• If \ I = \mathbb{R}^+\ , the strictly positive real numbers, and \ f(x)\ =\ \log(x)\ , then the corresponds to the geometric mean. (The result is the same for any logarithm; it does not depend on the base of the logarithm, as long as that base is strictly positive but not 1.)

• If \ I = \mathbb{R}^+\ and \ f(x)\ =\ \frac{\ 1\ }{ x }\ , then the corresponds to the harmonic mean.

• If \ I = \mathbb{R}^+\ and \ f(x)\ =\ x^{\ !p}\ , then the corresponds to the power mean with exponent \ p\ (e.g., for \ p = 2\ one gets the root mean square

• If \ I = \mathbb{R}\ and \ f(x)\ =\ \exp(x)\ , then the 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,\ \ldots,\ x_n\ )\ =\ \operatorname\mathsf{LSE}\left(\ x_1,\ \ldots,\ x_n\ \right) - \log(n) ~. (The \ -\log(n)\ in the expression corresponds to dividing by n, since logarithmic division is linear subtraction.) The LogSumExp function is a smooth maximum: It is 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\ , the repeated average \ M_f(\ x,\ \dots,\ 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\ \equiv\ M_f!\left(\ x_1,\ \ldots\ ,\ x_k\ \right)\ it holds: : \ M_f!\left(\ x_1,\ \dots,\ x_k,\ x_{k+1},\ \ldots\ ,\ x\ n\ \right)\ =\ M_f!\left(; \underbrace{m,,\ \ldots\ ,\ m}{\ k \text{ times}\ }\ ,; x_{k+1}\ ,\ \ldots\ ,\ x_n; \right) ~.

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks: : M_f!\left(\ x_1,\ \dots,\ x_{n\cdot k}\ \right); =; M_f!\Bigl(; M_f\left(\ x_1,\ \ldots\ ,\ x_{k}\ \right),; M_f!\left(\ x_{k+1},\ \ldots\ ,\ x_{2\cdot k}\ \right),; \dots,; M_f!\left(\ x_{(n-1)\cdot k + 1},\ \ldots\ ,\ x_{n\cdot k}\ \right); \Bigr) ~.

Self-distributivity: For any quasi-arithmetic (q.a.) mean \ M_\mathsf{q\ !a}\ of two variables: : \ M\mathsf{q\ !a\ !}!\Bigl(; x,\ M\mathsf{q\ !a\ !}!\left(\ y,\ z\ \right); \Bigr) = M\mathsf{q\ !a\ !}!\Bigl(; M\mathsf{q\ !a\ !}!\left(\ x,\ y\ \right),; M\mathsf{q\ !a\ !}!\left(\ x,\ z\ \right); \Bigr) ~.

Mediality: For any quasi-arithmetic mean \ M\mathsf{q\ !a}\ of two variables: : \ M\mathsf{q\ !a\ !}!\Bigl(; M\mathsf{q\ !a\ !}!\left(\ x,\ y\ \right),; M\mathsf{q\ !a\ !}!\left(\ z,\ w\ \right); \Bigr) = M\mathsf{q\ !a\ !}!\Bigl(; M\mathsf{q\ !a\ !}!\left(\ x,\ z\ \right),; M\mathsf{q\ !a\ !}!\left(\ y,\ w\ \right); \Bigr) ~.

Balancing: For any quasi-arithmetic mean \ M\mathsf{q\ !a}\ of two variables: : \ M\mathsf{q\ !a\ !}!\biggl(;\ M\mathsf{q\ !a\ !}!\Bigl(; x,; M\mathsf{q\ !a\ !}!\left(\ x,\ y\ \right); \Bigr),;\ M\mathsf{q\ a\ !}!\Bigl(; y,\ M\mathsf{q\ !a\ !}!\left(\ x,\ y\ \right); \Bigr);\ \biggr) = M\mathsf{q\ !a\ !}!\bigl(\ x,\ y\ \bigr) ~.

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and non-trivial scaling of quasi-arithmetic \ f\ : For any \ p(t)\ \equiv\ a + b \cdot q(t)\ , with \ a\ and \ b \ne 0\ constants, and \ q\ a quasi-aritmetic function, \ M_p(\ x\ )\ and M_q(\ x\ )\ are always the same. In mathematical notation: : Given \ q\ quasi-aritmetic, and \ p\ :\ \bigl(\ p(t) = a + b \cdot q(t);\ \forall\ t\ \bigr); \forall\ a; \forall\ b \ne 0 \quad \Rightarrow \quad M_p(\ x\ ) = M_q(\ x\ ); \forall\ x ~.

Central limit theorem : Under certain regularity conditions, and for a sufficiently large sample, : \ z \equiv \sqrt{n\ }\ \biggl[; M_f(\ X_1,\ \ldots\ ,\ X_n\ ); -; \operatorname\mathbb{E}_X! \Bigl(\ M_f(\ X_1,\ \ldots\ ,\ X_n\ )\ \Bigr); \biggr]\ is approximately normally distributed. A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means. |url-access=subscription |via=Springer.com |url-access=subscription |via=Springer.com

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.
• 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.
• 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, but that if one additionally assumes M to be an analytic function then the answer is positive.

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 \cdot f^{-1}\left( \frac{f\left(\frac{x_1}{C x}\right) + \cdots + f\left(\frac{x_n}{C x}\right)}{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 is defined by 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 is a normalized weight vector (w_i=\frac{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 F}. 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 F}(\theta_1,\theta_2)=2(\theta_1^{-1}+\theta_2^{-1})^{-1}.

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.

References

1. (June 2017). "Generalizing skew Jensen divergences and Bregman divergences with comparative convexity". IEEE Signal Processing Letters.
2. Aczél, J.. (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 Univ. Press.
3. Grudkin, Anton. (2019). "Characterization of the quasi-arithmetic mean".
4. Aumann, Georg. (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". [[Journal für die reine und angewandte Mathematik]].
5. Aumann, Georg. (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften.
6. Nielsen, Frank. (2023). "Beyond scalar quasi-arithmetic means: Quasi-arithmetic averages and quasi-arithmetic mixtures in information geometry".