Generalized mean

From Wikipedia, the free encyclopedia

A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means.

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

[edit] 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

$M_p(x_1,\dots,x_n) = \sqrt[p]{\frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p}$

[edit] 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 $b\cdot x_1,\dots, b\cdot x_n$ 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.

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

[edit] Generalized mean inequality

In general, if $p < q$ , then $M_p(x_1,\dots,x_n) \le M_q(x_1,\dots,x_n)$ and the two means are equal if and only if $x_1 = x_2 = \dots = x_n$ . That follows from the fact that $\forall p\in\mathbb{R}\ \frac{\partial M_p(x_1,\dots,x_n)}{\partial p}\geq 0$ , that can be proved using Jensen's inequality.

In particular, for $p\in\{-1, 0, 1\}$ , the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

[edit] Special cases

$\lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\}$ - minimum,
$M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}$ - harmonic mean,
$\lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n}$ - geometric mean,
$M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n}$ - arithmetic mean,
$M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}$ - quadratic mean,
$\lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\}$ - maximum.

[edit] Generalized $f$ -mean

The power mean could be generalized further to the generalized f-mean:

$M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right)$

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

[edit] Applications

[edit] 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.

 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.

[edit] See also

Inequality of arithmetic and geometric means
arithmetic mean
geometric mean
harmonic mean
Heronian mean
Lehmer mean - also a mean related to powers
average
root mean square

[edit] External links

Retrieved from "http://en.wikipedia.org../../../g/e/n/Generalized_mean.html"

Categories: Means | Inequalities | Articles with example Haskell code

Generalized mean

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Properties

[edit] Generalized mean inequality

[edit] Special cases

[edit] Generalized $f$ -mean

[edit] Applications

[edit] Signal processing

[edit] See also

[edit] External links

Views

Navigation

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In other languages

Generalized mean

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Properties

[edit] Generalized mean inequality

[edit] Special cases

[edit] Generalized f-mean

[edit] Applications

[edit] Signal processing

[edit] See also

[edit] External links

Views

Navigation

interaction

Search

In other languages

[edit] Generalized $f$ -mean