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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.

Definition[]

If is a non-zero real number, we can define the generalized mean with exponent of the positive real numbers as

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.

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.

Special cases[]

File:RMS-AM-GM-HM.gif

A visual depiction of some of the specified cases for n=2.

  • - minimum,
  • - harmonic mean,
  • - geometric mean,
  • - arithmetic mean,
  • - quadratic mean,
  • - 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:

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.

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:

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.

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:

≤ for q>0, ≥ for q<0.

After subtracting from the both sides we get:

After dividing by :

1 - w1 is nonzero, thus:

Substacting x1q leaves:

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

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

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 .

Applications[]

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.

See also[]

External links[]


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