Generalized mean

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(Redirected from Power 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.

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

$M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \cdot \sum_{i=1}^n x_{i}^p \right)^{1/p}.$

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 $b\cdot x_1,\dots, b\cdot x_n$ 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.

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

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.

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.

Proof of power means inequality

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents p and q holds:

$\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

then:

$\sqrt[p]{\sum_{i=1}^n\frac{w_i}{x_i^p}}\leq \sqrt[q]{\sum_{i=1}^n\frac{w_i}{x_i^q}}$

We raise both sides to the power of -1 (strictly decreasing function in positive reals):

$\sqrt[-p]{\sum_{i=1}^nw_ix_i^{-p}}=\sqrt[p]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}}\geq \sqrt[q]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}}=\sqrt[-q]{\sum_{i=1}^nw_ix_i^{-q}}$

We get the inequality for means with exponents -p and -q, and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.

Geometric mean

For any q the inequality between mean with exponent q and geometric mean can be transformed in the following way:

$\prod_{i=1}^nx_i^{w_i} \leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

$\sqrt[q]{\sum_{i=1}^nw_ix_i^q}\leq \prod_{i=1}^nx_i^{w_i}$

(the first inequality is to be proven for positive q, and the latter otherwise)

We raise both sides to the power of q:

$\prod_{i=1}^nx_i^{w_i\cdot q} \leq \sum_{i=1}^nw_ix_i^q$

in both cases we get the inequality between weighted arithmetic and geometric means for the sequence $x_i^q$ , which can be proved by Jensen's inequality, making use of the fact the logarithmic function is concave:

$\sum_{i=1}^nw_i\log(x_i) \leq \log(\sum_{i=1}^nw_ix_i)$

$log(\prod_{i=1}^nx_i^{w_i}) \leq log(\sum_{i=1}^nw_ix_i)$

By applying (strictly increasing) exp function to both sides we get the inequality:

$\prod_{i=1}^nx_i^{w_i} \leq \sum_{i=1}^nw_ix_i$

Thus for any postive q it is true that:

$\sqrt[-q]{\sum_{i=1}^nw_ix_i^{-q}}\leq \prod_{i=1}^nx_i^{w_i} \leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

since the inequality holds for any q, however small, and, as will be shown later, the expressions on the left and right approximate the geometric mean better as q approaches 0, the limit of the power mean for q approaching 0 is the geometric mean:

$\lim_{q\rightarrow 0}\sqrt[q]{\sum_{i=1}^nw_ix_i^{q}}=\prod_{i=1}^nx_i^{w_i}$

Inequality between any two power means

We are to prove that for any p<q the following inequality holds:

$\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

if p is negative, and q is positive, the inequality is equivalent to the one proved above:

$\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \prod_{i=1}^nx_i^{w_i} \leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

The proof for positive p and q is as follows: Define the following function: $f:{\mathbb R_+}\rightarrow{\mathbb R_+},$ $f(x)=x^{\frac{q}{p}}$ . f is a power function, so it does have a second derivative: $f''(x)=(\frac{q}{p})(\frac{q}{p}-1)x^{\frac{q}{p}-2},$ which is strictly positive within the domain of f, since q > p, so we know f is convex.

Using this, and the Jensen's inequality we get:

$f(\sum_{i=1}^nw_ix_i^p)\leq\sum_{i=1}^nw_if(x_i^p)$

$\sqrt[\frac{p}{q}]{\sum_{i=1}^nw_ix_i^p}\leq\sum_{i=1}^nw_ix_i^q$

after raising both side to the power of 1/q (an increasing function, since 1/q is positive) we get the inequality which was to be proven:

$\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}$

Using the previously shown equivalence we can prove the inequality for negative p and q by substituting them with, respectively, -q and -p, QED.

Minimum and maximum

Minimum and maximum are assumed to be the power means with exponents of $-/+\infty$ . Thus for any q:

$\min (x_1,x_2,\ldots ,x_n)\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}\leq \max (x_1,x_2,\ldots ,x_n)$

For maximum the proof is as follows: Assume WLoG that the sequence x_i is nonincreasing and no weight is zero.

Then the inequality is equivalent to:

$\sqrt[q]{\sum_{i=1}^nw_ix_i^q}\leq x_1$

After raising both sides to the power of q we get (depending on the sign of q) one of the inequalities:

$\sum_{i=1}^nw_ix_i^q\leq {\color{red} \geq} x_1^q$

≤ for q>0, ≥ for q<0.

After substracting $w 1 x 1$ from the both sides we get:

$\sum_{i=2}^nw_ix_i^q\leq {\color{red} \geq} (1-w_1)x_1^q$

After dividing by $(1 - w 1)$ :

$\sum_{i=2}^n\frac{w_i}{(1-w_1)}x_i^q\leq {\color{red} \geq} x_1^q$

1 - w₁ is nonzero, thus:

$\sum_{i=2}^n\frac{w_i}{(1-w_1)}=1$

Substacting x₁^q leaves:

$\sum_{i=2}^n\frac{w_i}{(1-w_1)}(x_i^q-x_1^q)\leq {\color{red} \geq} 0$

which is obvious, since x₁ is greater or equal to any x_i, and thus:

$x_i^q-x_1^q\leq {\color{red} \geq} 0$

For minimum the proof is almost the same, only instead of x₁, w₁ we use x_n, w_n, QED.

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. The power mean is obtained for $f\left(x\right)=x^p$ .

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.

 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.

External links

nl:Wortelgemiddelde su:Generalized mean sv:Generaliserat medelvärde

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

Generalized mean

Definition