NationMaster - Encyclopedia: Generalized mean

A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In mathematics, the harmonic mean is one of several methods of calculating an average. ... In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ...

1 Definition
2 Properties
- 2.1 Generalized Mean Inequality
3 Special cases
4 Generalized f-mean
5 Applications
- 5.1 Signal processing
6 See also
7 External links

Definition

If $p$ is a non-zero real number, we can define the generalized mean with exponent $p$ of the positive real numbers $x_1,dots,x_n$ as In mathematics, the real numbers may be described informally in several different ways. ...

$M_p(x) = sqrt[p]{frac{1}{n} cdot sum_{i=1}^n x_{i}^p}$

Properties

Like most means, the generalized mean is a homogeneous function of its arguments $x_1,dots,x_n$ . That is, if $b$ is a positive real number, then the generalized mean with exponent $p$ of the numbers $bcdot x_1,dots, bcdot x_n$ is equal to $b$ times the generalized mean of the numbers $x_1,dots, x_n$ .
Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.

$M_p(x_1,dots,x_{ncdot k}) = M_p(M_p(x_1,dots,x_{k}), M_p(x_{k+1},dots,x_{2cdot k}), dots, M_p(x_{(n-1)cdot k + 1},dots,x_{ncdot k}))$

In statistics, mean has two related meanings: the average in ordinary English, which is also called the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ... In mathematics and statistics, the quasi-arithmetic mean (or generalised f-mean) is the natural generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f(x). ...

Generalized Mean Inequality

In general, if $p < q$ , then $M_p(x) le M_q(x)$ and the two means are equal if and only if $x_1 = x_2 = dots = x_n$ . That follows from the fact that $forall pinmathbb{R} frac{partial M_p(x)}{partial p}geq 0$ , that can be proved using Jensen's inequality. In mathematics, Jensens inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. ...

In particular, for $pin{-1, 0, 1}$ , the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means. The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ... In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal...

Special cases

$lim_{pto-infty} M_p(x)$ - minimum,
$M - 1 (x)$ - harmonic mean,
$lim_{pto0} M_p(x)$ - geometric mean,
$M 1 (x)$ - arithmetic mean,
$M 2 (x)$ - quadratic mean,
$lim_{ptoinfty} M_p(x)$ - maximum.

The largest and the smallest element of a set are called extreme values, or extreme records. ... In mathematics, the harmonic mean is one of several methods of calculating an average. ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In mathematics, root mean square (abbreviated RMS or rms), also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity: its specially useful when variates are positive and negative, ie the cases of waves. ... The largest and the smallest element of a set are called extreme values, or extreme records. ...

Generalized $f$ -mean

The power mean could be generalized further to the generalized f-mean: In mathematics and statistics, the generalised f-mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). ...

$M_f(x) = f^{-1} left({frac{1}{n}cdotsum_{i=1}^n{f(x_i)}}right)$

which covers e.g. the geometric mean without using a limit.

Applications

Signal processing

A power mean serves a non-linear moving average which is shifted towards small signal values for small $p$ and emphasizes big signal values for big $p$ . Given an efficient implementation of a moving arithmetic mean called smooth you can implement a moving power mean according to the following Haskell code. The term moving average is used in different contexts. ... A low-pass filter passes low frequencies fairly well, but attenuates high frequencies. ... Haskell is a standardized pure functional programming language with non-strict semantics, named after the logician Haskell Curry. ...

 powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a] powerSmooth smooth p = map (** recip p) . smooth . map (**p)

For big $p$ it can serve an envelope detector on a rectified signal.
For small $p$ it can serve an baseline detector on a mass spectrum.

An envelope detector is a device which is used to demodulate AM signals. ... AC, half-wave and full wave rectified signals A rectifier is an electrical device, comprising one or more semiconductive devices (such as diodes) or vacuum tubes arranged for converting alternating current to direct current. ... A mass spectrum is an intensity vs. ...

External links

Power mean at MathWorld
Examples of Generalized Mean
A proof of the Generalized Mean on PlanetMath
Rational Mean

Categories: Means | Inequalities PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Results from FactBites:

Amazon.com: "generalized mean value theorem": Key Phrase page (448 words)
	Use this to derive the mean value theorem and the generalized mean value theorem.
	Applications of Taylor's theorem 623 Example 12.20 Use the generalized mean value theorem in two variables to expand the function J '(x, y) = e,-+2,:,about the point (0, 0).
	Use the generalized mean value theorem to prove that if f is twice differentiable,...

Generalized mean - Biocrawler (251 words)
	A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means.
	The case t = 1 yields the arithmetic mean and the case t = −1 yields the harmonic mean.
	As t approaches 0, the limit of M(t) is the geometric mean of the given numbers, and so it makes sense to define M(0) to be the geometric mean.

More results at FactBites »

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