Generalized mean
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A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means.
Contents |
[edit] Definition
If 
is a non-zero real number, we can define the generalized mean with exponent of the positive real numbers 
as
[edit] Properties
- Like most means, the generalized mean is a homogeneous function of its arguments . That is, if
is a positive real number, then the generalized mean with exponent of the numbers 
is equal to times the generalized mean of the numbers 
.
- Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.
[edit] Generalized mean inequality
In general, if 
, then 
and the two means are equal if and only if 
. This follows from the fact that 
, which can be proved using Jensen's inequality.
In particular, for 
, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.
[edit] Special cases
-
- minimum,
-
- harmonic mean,
-
- geometric mean,
-
- arithmetic mean,
-
- quadratic mean,
-
- maximum.
[edit] Proof of power means inequality
[edit] Equivalence of inequalities between means of opposite signs
Suppose an average between power means with exponents p and q holds:
then:
We raise both sides to the power of -1 (strictly decreasing function in positive reals):
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.
[edit] Geometric mean
For any q the inequality between mean with exponent q and geometric mean can be transformed in the following way:
(the first inequality is to be proven for positive q, and the latter otherwise)
We raise both sides to the power of q:
in both cases we get the inequality between weighted arithmetic and geometric means for the sequence 
, which can be proved by Jensen's inequality, making use of the fact the logarithmic function is concave:
By applying (strictly increasing) exp function to both sides we get the inequality:
Thus for any positive q it is true that:
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:
[edit] Inequality between any two power means
We are to prove that for any p<q the following inequality holds:
if p is negative, and q is positive, the inequality is equivalent to the one proved above:
The proof for positive p and q is as follows:
Define the following function: 
. f is a power function, so it does have a second derivative: 
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:
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:
Using the previously shown equivalence we can prove the inequality for negative p and q by substituting them with, respectively, -q and -p, QED.
[edit] Minimum and maximum
Minimum and maximum are assumed to be the power means with exponents of 
. Thus for any q:
For maximum the proof is as follows: Assume WLoG that the sequence xi is nonincreasing and no weight is zero.
Then the inequality is equivalent to:
After raising both sides to the power of q we get (depending on the sign of q) one of the inequalities:
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entering extended mode
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LaTeX2e <2003/12/01>
Babel
{red} \geq} x_1^q\end{equ...
[1] (./691c0898b941ac711e6efc1ba0498.aux) ) (see the transcript file for additional information) Output written on 691c0898b941ac711e6efc1ba0498.dvi (1 page, 520 bytes).
Transcript written on 691c0898b941ac711e6efc1ba0498.log.≤ for q>0, ≥ for q<0.
After subtracting 
from the both sides we get:
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entering extended mode
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Babel
{red} \geq} (1-w_1)x_1^q\e...
[1] (./3a16fe491c658c3b25cda902e82de.aux) ) (see the transcript file for additional information) Output written on 3a16fe491c658c3b25cda902e82de.dvi (1 page, 588 bytes).
Transcript written on 3a16fe491c658c3b25cda902e82de.log.After dividing by 
:
- WikiTeX: latex reported a failure, namely:
This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./5705ea034427fa067d56a3daa2368
LaTeX2e <2003/12/01>
Babel
{red} \geq} x_1^q\end{equa...
[1] (./5705ea034427fa067d56a3daa2368.aux) ) (see the transcript file for additional information) Output written on 5705ea034427fa067d56a3daa2368.dvi (1 page, 624 bytes).
Transcript written on 5705ea034427fa067d56a3daa2368.log.1 - w1 is nonzero, thus:
Substacting x1q leaves:
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entering extended mode
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LaTeX2e <2003/12/01>
Babel
{red} \geq} 0\end{equation*}
[1] (./b5f917efb4620cc4b70c09ff6504d.aux) ) (see the transcript file for additional information) Output written on b5f917efb4620cc4b70c09ff6504d.dvi (1 page, 640 bytes).
Transcript written on b5f917efb4620cc4b70c09ff6504d.log.which is obvious, since x1 is greater or equal to any xi, and thus:
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This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
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Babel
{red} \geq} 0\end{equation*}
[1] (./d437c960a633cad4ac47e495d5e21.aux) ) (see the transcript file for additional information) Output written on d437c960a633cad4ac47e495d5e21.dvi (1 page, 464 bytes).
Transcript written on d437c960a633cad4ac47e495d5e21.log.For minimum the proof is almost the same, only instead of x1, w1 we use xn, wn, QED.
[edit] Generalized 
-mean
The power mean could be generalized further to the generalized f-mean:
which covers e.g. the geometric mean without using a limit. The power mean is obtained for 
.
[edit] Applications
[edit] Signal processing
A power mean serves a non-linear moving average
which is shifted towards small signal values for small
and emphasizes big signal values for big .
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 it can serve an envelope detector on a rectified signal.
- For small 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
- Power mean at MathWorld
- Examples of Generalized Mean
- A proof of the Generalized Mean on PlanetMath
- Rational Mean
| This page uses content from the English-language version of Wikipedia. The original article was at Generalized mean. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
