Generalized mean

File:Generalized means of 1, x.svg

Plot of several generalized means $M_p(1,x)$

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

Definition

If Template:Mvar is a non-zero real number, and $x_1,\dots ,x_n$ are positive real numbers, then the generalized mean or power mean with exponent Template:Mvar of these positive real numbers is

$$M_p(x_1,\dots ,x_n)={\left(\frac{1}{n}\sum _{i=1}^n{x}_i^p\right)}^{1/p}.$$

(See [[Norm (mathematics)#p-norm|Template:Mvar-norm]]). For Template:Math we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

$$M_0(x_1,\dots ,x_n)={\left(\prod _{i=1}^n{x}_i\right)}^{1/n}.$$

Furthermore, for a sequence of positive weights Template:Mvar we define the weighted power mean as $$M_p(x_1,\dots ,x_n)={\left(\frac{\sum _{i=1}^n{w}_i{x}_i^p}{\sum _{i=1}^n{w}_i}\right)}^{1/p}$$ and when Template:Math, it is equal to the weighted geometric mean:

$$M_0(x_1,\dots ,x_n)={\left(\prod _{i=1}^n{x}_i^{w_i}\right)}^{1/\sum _{i=1}^n{w}_i}.$$

The unweighted means correspond to setting all Template:Math.

Special cases

For some values of $p$, the mean $M_p(x_1,\dots ,x_n)$ corresponds to a well known mean.

File:Generalized Means.svg

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

  Harmonic mean: $M_{-1}(a,b)$.

  Geometric mean: $M_0(a,b)$.

  Arithmetic mean: $M_1(a,b)$.

  Quadratic mean: $M_2(a,b)$.

Name	Exponent	Value
Minimum	$p=-\mathrm{\infty }$	$\min \{x_1,\dots ,x_n\}$
Harmonic mean	$p=-1$	$\frac{n}{\frac{1}{x_1}+\dots +\frac{1}{x_n}}$
Geometric mean	$p=0$	$\sqrt[n]{x_1\dots x_n}$
Arithmetic mean	$p=1$	$\frac{x_1+\dots +x_n}{n}$
Root mean square	$p=2$	$\sqrt{\frac{x_1^2+\dots +x_n^2}{n}}$
Cubic mean	$p=3$	$\sqrt[3]{\frac{x_1^3+\dots +x_n^3}{n}}$
Maximum	$p=+\mathrm{\infty }$	$\max \{x_1,\dots ,x_n\}$

Properties

Let $x_1,\dots ,x_n$ be a sequence of positive real numbers, then the following properties hold:

1. $\min (x_1,\dots ,x_n)\le M_p(x_1,\dots ,x_n)\le \max (x_1,\dots ,x_n)$.Template:Block indent
2. $M_p(x_1,\dots ,x_n)=M_p(P(x_1,\dots ,x_n))$, where $P$ is a permutation operator.Template:Block indent
3. $M_p(b{x}_1,\dots ,b{x}_n)=b\cdot M_p(x_1,\dots ,x_n)$.Template:Block indent
4. $M_p(x_1,\dots ,x_{n\cdot k})=M_p[M_p(x_1,\dots ,x_k),M_p(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_p(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k})]$.Template:Block indent

Generalized mean inequality

Template:QM AM GM HM inequality visual proof.svg In general, if Template:Math, 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 Template:Math.

The inequality is true for real values of Template:Mvar and Template:Mvar, as well as positive and negative infinity values.

It follows from the fact that, for all real Template:Mvar, $$\frac{\partial }{\partial p}{M}_p(x_1,\dots ,x_n)\ge 0$$ which can be proved using Jensen's inequality.

In particular, for Template:Mvar in Template:Math, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

Proof of the weighted inequality

We will prove the weighted power mean inequality. For the purpose of the proof we will assume the following without loss of generality: $$\begin{array}{r}{w}_i\in [0,1]\\ \sum _{i=1}^n{w}_i=1\end{array}$$

The proof for unweighted power means can be easily obtained by substituting Template:Math.

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents Template:Mvar and Template:Mvar holds: $${\left(\sum _{i=1}^n{w}_i{x}_i^p\right)}^{1/p}\ge {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}$$ applying this, then: $${\left(\sum _{i=1}^n\frac{w_i}{x_i^p}\right)}^{1/p}\ge {\left(\sum _{i=1}^n\frac{w_i}{x_i^q}\right)}^{1/q}$$

We raise both sides to the power of −1 (strictly decreasing function in positive reals): $${\left(\sum _{i=1}^n{w}_i{x}_i^{-p}\right)}^{-1/p}={\left(\frac{1}{\sum _{i=1}^n{w}_i\frac{1}{x_i^p}}\right)}^{1/p}\le {\left(\frac{1}{\sum _{i=1}^n{w}_i\frac{1}{x_i^q}}\right)}^{1/q}={\left(\sum _{i=1}^n{w}_i{x}_i^{-q}\right)}^{-1/q}$$

We get the inequality for means with exponents Template:Math and Template:Math, 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 Template:Math and non-negative weights summing to 1, the following inequality holds: $${\left(\sum _{i=1}^n{w}_i{x}_i^{-q}\right)}^{-1/q}\le \prod _{i=1}^n{x}_i^{w_i}\le {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}.$$

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: $$\log \prod _{i=1}^n{x}_i^{w_i}=\sum _{i=1}^n{w}_i\log x_i\le \log \sum _{i=1}^n{w}_i{x}_i.$$

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 $$\prod _{i=1}^n{x}_i^{w_i}\le \sum _{i=1}^n{w}_i{x}_i.$$

Taking Template:Mvar-th powers of the Template:Mvar yields $$\begin{array}{rl}& \prod _{i=1}^n{x}_i^{q\cdot w_i}\le \sum _{i=1}^n{w}_i{x}_i^q\\ & \prod _{i=1}^n{x}_i^{w_i}\le {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}.\end{array}$$

Thus, we are done for the inequality with positive Template:Mvar; the case for negatives is identical but for the swapped signs in the last step:

$$\prod _{i=1}^n{x}_i^{-q\cdot w_i}\le \sum _{i=1}^n{w}_i{x}_i^{-q}.$$

Of course, taking each side to the power of a negative number Template:Math swaps the direction of the inequality.

$$\prod _{i=1}^n{x}_i^{w_i}\ge {\left(\sum _{i=1}^n{w}_i{x}_i^{-q}\right)}^{-1/q}.$$

Inequality between any two power means

We are to prove that for any Template:Math the following inequality holds: $${\left(\sum _{i=1}^n{w}_i{x}_i^p\right)}^{1/p}\le {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}$$ if Template:Mvar is negative, and Template:Mvar is positive, the inequality is equivalent to the one proved above: $${\left(\sum _{i=1}^n{w}_i{x}_i^p\right)}^{1/p}\le \prod _{i=1}^n{x}_i^{w_i}\le {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}$$

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

Using this, and the Jensen's inequality we get: $$\begin{array}{rl}f\left(\sum _{i=1}^n{w}_i{x}_i^p\right)& \le \sum _{i=1}^n{w}_{i}f(x_i^p)\\ {\left(\sum _{i=1}^n{w}_i{x}_i^p\right)}^{q/p}& \le \sum _{i=1}^n{w}_i{x}_i^q\end{array}$$ after raising both side to the power of Template:Math (an increasing function, since Template:Math is positive) we get the inequality which was to be proven:

$${\left(\sum _{i=1}^n{w}_i{x}_i^p\right)}^{1/p}\le {\left(\sum _{i=1}^n{w}_i{x}_i^q\right)}^{1/q}$$

Using the previously shown equivalence we can prove the inequality for negative Template:Mvar and Template:Mvar by replacing them with Template:Mvar and Template:Mvar, respectively.

Generalized f-mean

The power mean could be generalized further to the [[generalized f-mean|generalized Template:Mvar-mean]]:

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

This covers the geometric mean without using a limit with Template:Math. The power mean is obtained for Template:Mvar. Properties of these means are studied in de Carvalho (2016).

Applications

Signal processing

A power mean serves a non-linear moving average which is shifted towards small signal values for small Template:Mvar and emphasizes big signal values for big Template:Mvar. 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 Template:Mvar it can serve as an envelope detector on a rectified signal.
• For small Template:Mvar it can serve as a baseline detector on a mass spectrum.

See also

• Arithmetic–geometric mean
• Average
• Heronian mean
• Inequality of arithmetic and geometric means
• Lehmer mean – also a mean related to powers
• Minkowski distance
• Quasi-arithmetic mean – another name for the generalized f-mean mentioned above
• Root mean square

External links

• Power mean at MathWorld
• Examples of Generalized Mean
• A proof of the Generalized Mean on PlanetMath