Quasi-arithmetic mean - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar mean

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

s such as the arithmetic mean

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

and the geometric mean

Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

, using a function

. It is also called Kolmogorov mean after Russian scientist Andrey Kolmogorov

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

Definition

If f is a function which maps an interval

of the real line to the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, and is both continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

and injective

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

then we can define the f-mean of two numbers

For

numbers

,
the f-mean is

We require f to be injective in order for the inverse function

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

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

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

, 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 we take to be the real line and , (or indeed any linear function , not equal to 0) then the f-mean corresponds to the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

.

If we take to be the set of positive real numbers and , then the f-mean corresponds to the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

. According to the f-mean properties, the result does not depend on the base of the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

as long as it is positive and not 1.

If we take to be the set of positive real numbers and , then the f-mean corresponds to the harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

.

If we take to be the set of positive real numbers and , then the f-mean corresponds to the power mean with exponent .

Properties

Partitioning
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...

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

Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.

With

it holds

The quasi-arithmetic mean is invariant with respect to offsets and scaling of :

If is monotonic
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

, then is monotonic.
Any quasi-arithmetic mean of two variables has the mediality property and the self-distributivity property . Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.
Any quasi-arithmetic mean of two variables has the balancing property . An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arthmetic. Georg Aumann showed in the 1930's that the answer is no in general , but that if one additionally assumes to be an analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

then the answer is positive .

Homogenity

Mean

In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

s are usually homogeneous

Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

,
but for most functions

, the f-mean is not.
Indeed, the only homogeneous quasi-arithmetic means are the power means and the geometric mean

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