Generalized mean

{{Short description|N-th root of the arithmetic mean of the given numbers raised to the power n}} {{More citations needed|date=June 2020}}

[[File:Generalized means of 1, x.svg|400px|thumb|right|Plot of several generalized means <math>M_p(1, x)</math>]]

In [[mathematics]], '''generalized 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 mean|arithmetic]], [[geometric mean|geometric]], and [[harmonic mean|harmonic]] [[mean]]s).

==Definition== If {{mvar|p}} 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 {{mvar|p}} of these positive real numbers is<ref name="Bullen1"/><ref name = "dC2016">{{cite journal|last=de Carvalho|first=Miguel|title=Mean, what do you Mean?|journal=[[The American Statistician]]|year=2016|volume=70|issue=3|pages=764‒776|doi=10.1080/00031305.2016.1148632|url=https://zenodo.org/record/895400|hdl=20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c|hdl-access=free}}</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|{{mvar|p}}-norm]]). For {{math|1=''p'' = 0}} 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 {{mvar|wi}} 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 {{math|1=''p'' = 0}}, 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 {{math|1=''wi'' = 1}}.

== Special cases ==

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

[[File:Generalized Means.svg|thumb|A visual depiction of some of the specified cases for <math>n = 2</math>. {{legend|magenta|Harmonic mean: <math>M_{-1}(a, b)</math>.}} {{legend|blue|Geometric mean: <math>M_0(a, b)</math>.}} {{legend|red|Arithmetic mean: <math>M_1(a, b)</math>.}} {{legend|lime|Quadratic mean: <math>M_2(a, b)</math>.}}]]

{| class="wikitable" |- ! Name ! Exponent ! Value |- | [[Minimum]] | <math>p = -\infty</math> | <math>\min \{x_1, \dots, x_n\}</math> |- | [[Harmonic mean]] | <math>p = -1</math> | <math>\frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}</math> |- | [[Geometric mean]] | <math>p = 0</math> | <math>\sqrt[n]{x_1\dots x_n}</math> |- | [[Arithmetic mean]] | <math>p = 1</math> | <math>\frac{x_1 + \dots + x_n}{n}</math> |- | [[Root mean square]] | <math>p = 2</math> | <math>\sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}</math> |- | [[Cubic mean]] | <math>p = 3</math> | <math>\sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n}}</math> |- | [[Maximum]] | <math>p = +\infty</math> | <math>\max\{x_1, \dots, x_n\}</math> |}

{{Math proof|title=Proof of <math display="inline"> \lim_{p \to 0} M_p = M_0 </math> (geometric mean)|proof=For the purpose of the proof, we will assume without loss of generality that <math display="block"> w_i \in [0,1] </math> and <math display="block"> \sum_{i=1}^n w_i = 1. </math>

We can rewrite the definition of <math>M_p</math> using the exponential function as

<math display=block>M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left[\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}\right]} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) }</math>

In the limit {{math|''p'' → 0}}, we can apply [[L'Hôpital's rule]] to the argument of the exponential function. We assume that <math>p \isin \mathbb{R}</math> but {{math|''p'' ≠ 0}}, and that the sum of {{mvar|wi}} is equal to 1 (without loss in generality);<ref>{{Cite book |title=Handbook of Means and Their Inequalities (Mathematics and Its Applications)}}</ref> differentiating the numerator and denominator with respect to {{mvar|p}}, 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>}}

{{Proof|title= Proof of <math display="inline">\lim_{p \to \infty} M_p = M_\infty</math> and <math display="inline">\lim_{p \to -\infty} M_p = M_{-\infty}</math> |proof= Assume (possibly after relabeling and combining terms together) that <math>x_1 \geq \dots \geq x_n</math>. Then

<math display=block>\begin{align} \lim_{p \to \infty} M_p(x_1,\dots,x_n) &= \lim_{p \to \infty} \left( \sum_{i=1}^n w_i x_i^p \right)^{1/p} \\ &= x_1 \lim_{p \to \infty} \left( \sum_{i=1}^n w_i \left( \frac{x_i}{x_1} \right)^p \right)^{1/p} \\ &= x_1 = M_\infty (x_1,\dots,x_n). \end{align}</math>

The formula for <math>M_{-\infty}</math> follows from <math display="block">M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)} = x_n.</math> }}

==Properties==

Let <math>x_1, \dots, x_n</math> be a sequence of positive real numbers, then the following properties hold:<ref name=sykora>{{cite journal|last=Sýkora|first=Stanislav|year=2009|title=Mathematical means and averages: basic properties|journal=Stan's Library |location=Castano Primo, Italy|volume=III |doi=10.3247/SL3Math09.001 }}</ref>

#<math>\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n)</math>.{{block indent|left=1|text= Each generalized mean always lies between the smallest and largest of the {{mvar|x}} values.}} #<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.{{block indent|left=1|text= Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.}} #<math>M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n)</math>.{{block indent|left=1|text= Like most [[Mean#Properties|mean]]s, the generalized mean is a [[homogeneous function]] of its arguments {{math|''x''1, ..., ''xn''}}. That is, if {{mvar|b}} is a positive real number, then the generalized mean with exponent {{mvar|p}} of the numbers <math>b\cdot x_1,\dots, b\cdot x_n</math> is equal to {{mvar|b}} times the generalized mean of the numbers {{math|''x''1, ..., ''xn''}}.}} #<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>.{{block indent|left=1|text= Like the [[quasi-arithmetic mean]]s, 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.}}

=== Generalized mean inequality === {{QM_AM_GM_HM_inequality_visual_proof.svg}} In general, if {{math|''p'' < ''q''}}, 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 {{math|1= ''x''1 = ''x''2 = ... = ''xn''}}.

The inequality is true for real values of {{mvar|p}} and {{mvar|q}}, as well as positive and negative infinity values.

It follows from the fact that, for all real {{mvar|p}}, <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 {{mvar|p}} in {{math|{−1, 0, 1}<nowiki/>}}, 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 {{math|1= ''wi'' = 1/''n''}}.

===Equivalence of inequalities between means of opposite signs=== Suppose an average between power means with exponents {{mvar|p}} and {{mvar|q}} 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 {{math|−''p''}} and {{math|−''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 {{math|''q'' > 0}} 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 {{mvar|q}}-th powers of the {{mvar|xi}} 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 {{mvar|q}}; the case for negatives is identical but for the swapped signs in the last step:

Of course, taking each side to the power of a negative number {{math|-1/''q''}} 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 {{math|''p'' < ''q''}} 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 {{mvar|p}} is negative, and {{mvar|q}} 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 {{mvar|p}} and {{mvar|q}} is as follows: Define the following function: {{math|''f'' : '''R'''+ → '''R'''+}} <math>f(x)=x^{\frac{q}{p}}</math>. {{mvar|f}} 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 {{mvar|f}}, since {{math|''q'' > ''p''}}, so we know {{mvar|f}} 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 {{math|1/''q''}} (an increasing function, since {{math|1/''q''}} 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 {{mvar|p}} and {{mvar|q}} by replacing them with {{mvar|−q}} and {{mvar|−p}}, respectively.

== Generalized ''f''-mean == {{Main|Generalized f-mean|l1=Generalized {{mvar|f}}-mean}}

The power mean could be generalized further to the [[generalized f-mean|generalized {{mvar|f}}-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 {{math|1= ''f''(''x'') {{=}} log(''x'')}}. The power mean is obtained for {{mvar|1= ''f''(''x'') {{=}} ''xp''}}. 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 {{mvar|p}} and emphasizes big signal values for big {{mvar|p}}. Given an efficient implementation of a [[lowpass|moving arithmetic mean]] called <code>smooth</code> one can implement a moving power mean according to the following [[Haskell (programming language)|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 {{mvar|p}} it can serve as an [[envelope detector]] on a [[rectifier|rectified]] signal. * For small {{mvar|p}} it can serve as a [[Baseline (spectrometry)|baseline detector]] on a [[mass spectrum]].

==See also== {{cols|colwidth=26em}} * [[Arithmetic–geometric mean]] * [[Average]] * [[Heronian mean]] * [[Inequality of arithmetic and geometric means]] * [[Lehmer mean]] – also a mean related to [[Power (mathematics)|powers]] * [[Minkowski distance]] * [[Quasi-arithmetic mean]] – another name for the [[generalized f-mean]] mentioned above * [[Root mean square]] {{colend}}

== Notes == {{notelist}} {{reflist|group=note}}

== References == {{reflist}}

== Further reading == * {{cite book|first1=P. S. |last1=Bullen|title=Handbook of Means and Their Inequalities|location=Dordrecht, Netherlands|publisher=Kluwer|year=2003|chapter=Chapter III - The Power Means|pages=175–265}}

==External links== *[https://mathworld.wolfram.com/PowerMean.html Power mean at MathWorld] *[https://people.revoledu.com/kardi/tutorial/BasicMath/Average/Generalized%20mean.html Examples of Generalized Mean] *A [https://planetmath.org/ProofOfGeneralMeansInequality proof of the Generalized Mean] on [[PlanetMath]]

[[Category:Means]] [[Category:Inequalities (mathematics)]] [[Category:Articles with example Haskell code]]