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 ${\displaystyle f}$. It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

Definition

If ${\displaystyle \ f\ }$ is a function that maps some continuous interval ${\displaystyle \ I\ }$ of the real line to some other continuous subset ${\displaystyle \ J\equiv f(I)\ }$ of the real numbers, and ${\displaystyle \ f\ }$ is both continuous, and injective (one-to-one).

(We require ${\displaystyle \ f\ }$ to be injective on ${\displaystyle \ I\ }$ in order for an inverse function ${\displaystyle \ f^{-1}\ }$ to exist. We require ${\displaystyle \ I\ }$ and ${\displaystyle \ J\ }$ to both be continuous intervals in order to ensure that an average of any finite (or infinite) subset of values within ${\displaystyle \ J\ }$ will always correspond to a value in ${\displaystyle \ I\ }$.)

Subject to those requirements, the f mean of ${\displaystyle \ n\ }$ numbers ${\displaystyle \ x_{1},\ldots ,x_{n}\in I\ }$ is defined to be

${\displaystyle \ M_{f}(x_{1},\dots ,x_{n})\;\equiv \;f^{-1}\!\left(\ {\frac {1}{n}}{\Bigl (}\ f(x_{1})+\cdots +f(x_{n})\ {\Bigr )}\ \right)\ ,}$

or equivalently

${\displaystyle \ M_{f}({\vec {x}})\;=\;f^{-1}\!\!\left(\ {\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\ \right)~.}$

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

Because ${\displaystyle \ f\ }$ is injective and continuous, it necessarily follows that ${\displaystyle \ f\ }$ is a strictly monotonic function, and therefore that the f mean is neither larger than the largest number of the tuple ${\displaystyle \ x_{1},\ldots \ ,x_{n}\equiv X\ }$ nor smaller than the smallest number contained in ${\displaystyle \ X\ ,}$ hence contained somewhere among the values of the original sample.

Examples

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

• If ${\displaystyle \ I=\mathbb {R} ^{+}\ ,}$ the strictly positive real numbers, and ${\displaystyle \ f(x)\ =\ \log(x)\ ,}$ then the f mean 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 ${\displaystyle \ I=\mathbb {R} ^{+}\ }$ and ${\displaystyle \ f(x)\ =\ {\frac {\ 1\ }{x}}\ ,}$ then the f mean corresponds to the harmonic mean.

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

• If ${\displaystyle \ I=\mathbb {R} \ }$ and ${\displaystyle \ f(x)\ =\ \exp(x)\ ,}$ 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), ${\displaystyle \ M_{f}(\ x_{1},\ \ldots ,\ x_{n}\ )\ =\ \operatorname {\mathsf {LSE}} \left(\ x_{1},\ \ldots ,\ x_{n}\ \right)-\log(n)~.}$ (The ${\displaystyle \ -\log(n)\ }$ 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 ${\displaystyle \ M_{f}\ }$ for any single function ${\displaystyle \ f\ }$:

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

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

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

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

Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With ${\displaystyle \ m\ \equiv \ M_{f}\!\left(\ x_{1},\ \ldots \ ,\ x_{k}\ \right)\ }$ it holds:

${\displaystyle \ M_{f}\!\left(\ x_{1},\ \dots ,\ x_{k},\ x_{k+1},\ \ldots \ ,\ x\ _{n}\ \right)\ =\ M_{f}\!\left(\;\underbrace {m,\,\ \ldots \ ,\ m} _{\ k{\text{ times}}\ }\ ,\;x_{k+1}\ ,\ \ldots \ ,\ x_{n}\;\right)~.}$

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

${\displaystyle M_{f}\!\left(\ x_{1},\ \dots ,\ x_{n\cdot k}\ \right)\;=\;M_{f}\!{\Bigl (}\;M_{f}\left(\ x_{1},\ \ldots \ ,\ x_{k}\ \right),\;M_{f}\!\left(\ x_{k+1},\ \ldots \ ,\ x_{2\cdot k}\ \right),\;\dots ,\;M_{f}\!\left(\ x_{(n-1)\cdot k+1},\ \ldots \ ,\ x_{n\cdot k}\ \right)\;{\Bigr )}~.}$

Self-distributivity: For any quasi-arithmetic (q.a.) mean ${\displaystyle \ M_{\mathsf {q\ \!a}}\ }$ of two variables:

${\displaystyle \ M{\mathsf {q\ \!a\ \!}}\!{\Bigl (}\;x,\ M{\mathsf {q\ \!a\ \!}}\!\left(\ y,\ z\ \right)\;{\Bigr )}=M{\mathsf {q\ \!a\ \!}}\!{\Bigl (}\;M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ y\ \right),\;M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ z\ \right)\;{\Bigr )}~.}$

Mediality: For any quasi-arithmetic mean ${\displaystyle \ M{\mathsf {q\ \!a}}\ }$ of two variables:

${\displaystyle \ M{\mathsf {q\ \!a\ \!}}\!{\Bigl (}\;M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ y\ \right),\;M{\mathsf {q\ \!a\ \!}}\!\left(\ z,\ w\ \right)\;{\Bigr )}=M{\mathsf {q\ \!a\ \!}}\!{\Bigl (}\;M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ z\ \right),\;M{\mathsf {q\ \!a\ \!}}\!\left(\ y,\ w\ \right)\;{\Bigr )}~.}$

Balancing: For any quasi-arithmetic mean ${\displaystyle \ M{\mathsf {q\ \!a}}\ }$ of two variables:

${\displaystyle \ M{\mathsf {q\ \!a\ \!}}\!{\biggl (}\;\ M{\mathsf {q\ \!a\ \!}}\!{\Bigl (}\;x,\;M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ y\ \right)\;{\Bigr )},\;\ M{\mathsf {q\ a\ \!}}\!{\Bigl (}\;y,\ M{\mathsf {q\ \!a\ \!}}\!\left(\ x,\ y\ \right)\;{\Bigr )}\;\ {\biggr )}~=~M{\mathsf {q\ \!a\ \!}}\!{\bigl (}\ x,\ y\ {\bigr )}~.}$

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and non-trivial scaling of quasi-arithmetic ${\displaystyle \ f\$ :} For any ${\displaystyle \ p(t)\ \equiv \ a+b\cdot q(t)\ ,}$ with ${\displaystyle \ a\ }$ and ${\displaystyle \ b\neq 0\ }$ constants, and ${\displaystyle \ q\ }$ a quasi-aritmetic function, ${\displaystyle \ M_{p}(\ x\ )\ }$ and ${\displaystyle M_{q}(\ x\ )\ }$ are always the same. In mathematical notation:

Given ${\displaystyle \ q\ }$ quasi-aritmetic, and ${\displaystyle \ p\$ :\ {\bigl (}\ p(t)=a+b\cdot q(t)\;\ \forall \ t\ {\bigr )}\;\forall \ a\;\forall \ b\neq 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,

${\displaystyle \ z~\equiv ~{\sqrt {n\ }}\ {\biggl [}\;M_{f}(\ X_{1},\ \ldots \ ,\ X_{n}\ )\;-\;\operatorname {\mathbb {E} } _{X}\!{\Bigl (}\ M_{f}(\ X_{1},\ \ldots \ ,\ X_{n}\ )\ {\Bigr )}\;{\biggr ]}\ }$

is approximately normally distributed.^[2] A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.^[3][4]

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.^{[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 ${\displaystyle M}$ to be an analytic function then the answer is positive.^[8]

Homogeneity

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

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

However this modification may violate monotonicity and the partitioning property of the mean.

Generalizations

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

See also

• Generalized mean
• Jensen's inequality