Generalized mean

File:Generalized means of 1, x.svg

Plot of several generalized means <math>M_p(1, x)</math>.

In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder)<ref name=sykora/> 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 <math>x_1, \dots, x_n</math> are positive real numbers, then the generalized mean or power mean with exponent Template:Mvar of these positive real numbers is<ref name="Bullen1"/><ref name = "dC2016">Template:Cite journal</ref>

<math display=block>M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} .</math>

(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):

<math display="block">M_0(x_1, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} .</math>

Furthermore, for a sequence of positive weights Template:Mvar we define the weighted power mean as<ref name="Bullen1"/> <math display=block>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}}</math> and when Template:Math, it is equal to the weighted geometric mean:

<math display=block>M_0(x_1,\dots,x_n) = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} .</math>

The unweighted means correspond to setting all Template:Math.

Special cases

A few particular values of Template:Mvar yield special cases with their own names:<ref name="mw">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PowerMean%7CPowerMean.html}} |title = Power Mean |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }} (retrieved 2019-08-17)</ref>

minimum

<math>M_{-\infty}(x_1,\dots,x_n) = \lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\}</math>

File:MathematicalMeans.svg

A visual depiction of some of the specified cases for Template:Math with Template:Math and Template:Math: Template:Legend Template:Legend Template:Legend Template:Legend

harmonic mean

<math>M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}</math>

geometric mean <math>M_0(x_1,\dots,x_n) = \lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n}</math>

arithmetic mean

<math>M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n}</math>

root mean squareTemplate:Anchor
or quadratic mean<ref>Template:Cite bookTemplate:Dead link</ref><ref>Template:Cite book</ref>

<math>M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}</math>

cubic mean

<math>M_3(x_1,\dots,x_n) = \sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n}}</math>

maximum

<math>M_{+\infty}(x_1,\dots,x_n) = \lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\}</math>

Template:Math proof{p} \right) }</math>

In the limit Template:Math, we can apply L'Hôpital's rule to the argument of the exponential function. We assume that <math>p \isin \mathbb{R}</math> but Template:Math, and that the sum of Template:Mvar is equal to 1 (without loss in generality);<ref>Template:Cite book</ref> Differentiating the numerator and denominator with respect to Template:Mvar, we have <math display=block>\begin{align}

\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} &= \lim_{p \to 0} \frac{\frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{j=1}^n w_j x_j^p}}{1} \\
&= \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{j=1}^n w_j x_j^p} \\
&= \frac{\sum_{i=1}^n w_i \ln{x_i}}{\sum_{j=1}^n w_j} \\
&= \sum_{i=1}^n w_i \ln{x_i} \\
&= \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}

\end{align}</math>

By the continuity of the exponential function, we can substitute back into the above relation to obtain <math display=block>\lim_{p \to 0} M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \right)} = \prod_{i=1}^n x_i^{w_i} = M_0(x_1,\dots,x_n)</math> as desired.<ref name="Bullen1">P. S. Bullen: Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177</ref>}}

Properties

Let <math>x_1, \dots, x_n</math> be a sequence of positive real numbers, then the following properties hold:<ref name=sykora>Template:Cite journal</ref>

1. <math>\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n)</math>.Template:Block indent
2. <math>M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n))</math>, where <math>P</math> is a permutation operator.Template:Block indent
3. <math>M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n)</math>.Template:Block indent
4. <math>M_p(x_1, \dots, x_{n \cdot k}) = M_p\left[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})\right]</math>.Template:Block indent

Generalized mean inequality

Template:QM AM GM HM inequality visual proof.svg In general, if Template:Math, then <math display=block>M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n)</math> 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, <math display=block>\frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0</math> 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: <math display=block>\begin{align}

 w_i \in [0, 1] \\
 \sum_{i=1}^nw_i = 1

\end{align}</math>

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: <math display="block">\left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \geq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}</math> applying this, then: <math display="block">\left(\sum_{i=1}^n\frac{w_i}{x_i^p}\right)^{1/p} \geq \left(\sum_{i=1}^n\frac{w_i}{x_i^q}\right)^{1/q}</math>

We raise both sides to the power of −1 (strictly decreasing function in positive reals): <math display="block">\left(\sum_{i=1}^nw_ix_i^{-p}\right)^{-1/p} = \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}\right)^{1/p} \leq \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}\right)^{1/q} = \left(\sum_{i=1}^nw_ix_i^{-q}\right)^{-1/q}</math>

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: <math display="block">\left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.</math>

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: <math display=block>\log \prod_{i=1}^n x_i^{w_i} = \sum_{i=1}^n w_i\log x_i \leq \log \sum_{i=1}^n w_i x_i.</math>

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 <math display=block>\prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i.</math>

Taking Template:Mvar-th powers of the Template:Mvar yields <math display=block>\begin{align} &\prod_{i=1}^n x_i^{q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^q \\ &\prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.\end{align}</math>

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:

<math display=block>\prod_{i=1}^n x_i^{-q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^{-q}.</math>

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

<math display=block>\prod_{i=1}^n x_i^{w_i} \geq \left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q}.</math>

Inequality between any two power means

We are to prove that for any Template:Math the following inequality holds: <math display="block">\left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^nw_ix_i^q\right)^{1/q}</math> if Template:Mvar is negative, and Template:Mvar is positive, the inequality is equivalent to the one proved above: <math display="block">\left(\sum_{i=1}^nw_i x_i^p\right)^{1/p} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}</math>

The proof for positive Template:Mvar and Template:Mvar is as follows: Define the following function: Template:Math <math>f(x)=x^{\frac{q}{p}}</math>. Template:Mvar is a power function, so it does have a second derivative: <math display="block">f(x) = \left(\frac{q}{p} \right) \left( \frac{q}{p}-1 \right)x^{\frac{q}{p}-2}</math> 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: <math display="block">\begin{align}

    f \left( \sum_{i=1}^nw_ix_i^p \right) &\leq \sum_{i=1}^nw_if(x_i^p) \\[3pt]
 \left(\sum_{i=1}^n w_i x_i^p\right)^{q/p} &\leq \sum_{i=1}^nw_ix_i^q

\end{align}</math> 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:

<math display="block">\left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}</math>

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

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

<math display=block> M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math>

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).<ref name = "dC2016"/>

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.

<syntaxhighlight lang="haskell"> powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a] powerSmooth smooth p = map (** recip p) . smooth . map (**p) </syntaxhighlight>

• 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

Further reading

• Template:Cite book

External links

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