Quasi-arithmetic mean

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 .) • 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 , 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-arithmetic function, \ M_p(\ x\ )\ and M_q(\ x\ )\ are always the same. In mathematical notation: : Given \ q\ quasi-arithmetic, 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. == 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. • 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}. == See also ==