Generalized Mean

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In mathematics, **generalized means** are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). The generalized mean is also known as **power mean** or **Hölder mean** (named after Otto Hölder).

If *p* is a non-zero real number, and are positive real numbers, then the **generalized mean** or **power mean** with exponent *p* of these positive real numbers is:

Note the relationship to the *p*-norm. For we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

Furthermore, for a sequence of positive weights *w _{i}* with sum we define the

The unweighted means correspond to setting all .

- Each generalized mean always lies between the smallest and largest of the
*x*values. - Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.
- Like most means, the generalized mean is a homogeneous function of its arguments
*x*_{1}, ...,*x*. That is, if_{n}*b*is a positive real number, then the generalized mean with exponent*p*of the numbers is equal to*b*times the generalized mean of the numbers*x*_{1}, ...,*x*._{n} - Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks. This enables use of a divide and conquer algorithm to calculate the means, when desirable.

In general,

- if
*p*<*q*, then

and the two means are equal if and only if *x*_{1} = *x*_{2} = ... = *x _{n}*.

The inequality is true for real values of *p* and *q*, as well as positive and negative infinity values.

It follows from the fact that, for all real *p*,

which 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.

We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality:

Proof for unweighted power means is easily obtained by substituting *w _{i}* = 1/

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

applying this, 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.

For any *q* > 0 and non-negative weights summing to 1, the following inequality holds:

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:

By applying the exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get

Taking *q*th powers of the *x*_{i}, we are done for the inequality with positive *q*; the case for negatives is identical.

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* : **R**_{+} -> **R**_{+}. *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.

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

This covers the geometric mean without using a limit with *f*(*x*) = *log*(*x*). The power mean is obtained for *f*(*x*) = *x ^{p}*.

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`

one 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.

- Arithmetic mean
- Arithmetic-geometric mean
- Average
- Geometric mean
- Harmonic mean
- Heronian mean
- Inequality of arithmetic and geometric means
- Lehmer mean – also a mean related to powers
- Pythagorean means
- Root mean square
- Minkowski distance

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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