Generalized mean explained

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 is a non-zero real number, and

x _{1,} ... , x _{n} are positive real numbers, then the generalized mean or power mean with exponent of these positive real numbers is^[1]

$$M\_p(x\_1,\backslash dots,x\_n)\; =\; \backslash left(\backslash frac\; \backslash sum\_^n\; x\_i^p\; \backslash right)^\; .$$

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

$$M\_0(x\_1,\; \backslash dots,\; x\_n)\; =\; \backslash left(\backslash prod\_^n\; x\_i\backslash right)^\; .$$

Furthermore, for a sequence of positive weights we define the weighted power mean as$$M\_p(x\_1,\backslash dots,x\_n)\; =\; \backslash left(\backslash frac\; \backslash right)^$$and when, it is equal to the weighted geometric mean:

$$M\_0(x\_1,\backslash dots,x\_n)\; =\; \backslash left(\backslash prod\_^n\; x\_i^\backslash right)^\; .$$

The unweighted means correspond to setting all .

Special cases

A few particular values of yield special cases with their own names:^[2]

minimum :

M _{-infty} (x _{1,...,x} _{n)} = \lim _{p\to-infty} M _{p(x} _{1,...,x} _{n)} = min \{x _{1,...,x} _{n\}}

harmonic mean :

M _{-1} (x _{1,...,x} _{n)} = (n)/((1)/(+...+ (1)/(x _{n}) x _{1})) M _{0(x} _{1,...,x} _{n)} = \lim _{p\to0} M _{p(x} _{1,...,x} _{n)} = \sqrt[n]{x _{1 ⋅ ... ⋅} x _{n}}

arithmetic mean :

M _{1(x} _{1,...,x} _{n)} = (x _{1} +...+ x _{n})/(n)

root mean square
or quadratic mean^{[3] [4]} :

M _{2(x} _{1,...,x} _{n)} = \sqrt{ (x ^{2} _{1} + ... + x ^{2} _{n})/(n) }

cubic mean :

M _{3(x} _{1,...,x} _{n)} = \sqrt[ 3 ]{ (x ^{3} _{1} + ... + x ^{3} _{n})/(n) }

maximum :

M _{+infty} (x _{1,...,x} _{n)} = \lim _{p\toinfty} M _{p(x} _{1,...,x} _{n)} = max \{x _{1,...,x} _{n\}}

Properties

Let

x _{1,} ... , x _{n} be a sequence of positive real numbers, then the following properties hold:^[5] min (x _{1,} ... , x _{n)} \le M _{p(x} _{1,} ... , x _{n)} \le max (x _{1,} ... , x _{n)} .M _{p(x} _{1,} ... , x _{n)} = M _{p(P(x} _{1,} ... , x _{n))} , where P is a permutation operator.M _{p(b} x _{1,} ... , b x _{n)} = b ⋅ M _{p(x} _{1,} ... , x _{n)} .M _{p(x} _{1,} ... , x _{n} ) = M _{p\left[M} _{p(x} _{1,} ... , x _{k} ), M _{p(x} _{k} , ... , x _{2} ), ... , M _{p(x} _{(n} , ... , x _{n} )\right] .

Generalized mean inequality

In general, if, then$$M\_p(x\_1,\; \backslash dots,\; x\_n)\; \backslash le\; M\_q(x\_1,\; \backslash dots,\; x\_n)$$and the two means are equal if and only if .

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

It follows from the fact that, for all real,$$\backslash fracM\_p(x\_1,\; \backslash dots,\; x\_n)\; \backslash geq\; 0$$which can be proved using Jensen's inequality.

In particular, for in, 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:$$\backslash begin\; w\_i\; \backslash in\; [0,\; 1]\; \backslash \backslash \; \backslash sum\_^nw\_i\; =\; 1\backslash end$$

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

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents and holds:$$\backslash left(\backslash sum\_^n\; w\_i\; x\_i^p\backslash right)^\; \backslash geq\; \backslash left(\backslash sum\_^n\; w\_i\; x\_i^q\backslash right)^$$applying this, then:$$\backslash left(\backslash sum\_^n\backslash frac\backslash right)^\; \backslash geq\; \backslash left(\backslash sum\_^n\backslash frac\backslash right)^$$

We raise both sides to the power of −1 (strictly decreasing function in positive reals):$$\backslash left(\backslash sum\_^nw\_ix\_i^\backslash right)^=\; \backslash left(\backslash frac\backslash right)^\backslash leq\; \backslash left(\backslash frac\backslash right)^=\; \backslash left(\backslash sum\_^nw\_ix\_i^\backslash right)^$$

We get the inequality for means with exponents and, 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 and non-negative weights summing to 1, the following inequality holds:$$\backslash left(\backslash sum\_^n\; w\_i\; x\_i^\backslash right)^\; \backslash leq\; \backslash prod\_^n\; x\_i^\; \backslash leq\; \backslash left(\backslash sum\_^n\; w\_i\; x\_i^q\backslash right)^.$$

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:$$\backslash log\; \backslash prod\_^n\; x\_i^\; =\; \backslash sum\_^n\; w\_i\backslash log\; x\_i\; \backslash leq\; \backslash log\; \backslash sum\_^n\; w\_i\; x\_i.$$

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$$\backslash prod\_^n\; x\_i^\; \backslash leq\; \backslash sum\_^n\; w\_i\; x\_i.$$

Taking -th powers of the yields

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

$$\backslash prod\_^n\; x\_i^\; \backslash leq\; \backslash sum\_^n\; w\_i\; x\_i^.$$

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

$$\backslash prod\_^n\; x\_i^\; \backslash geq\; \backslash left(\backslash sum\_^n\; w\_i\; x\_i^q\backslash right)^.$$

Inequality between any two power means

We are to prove that for any the following inequality holds:$$\backslash left(\backslash sum\_^n\; w\_i\; x\_i^p\backslash right)^\; \backslash leq\; \backslash left(\backslash sum\_^nw\_ix\_i^q\backslash right)^$$if is negative, and is positive, the inequality is equivalent to the one proved above:$$\backslash left(\backslash sum\_^nw\_i\; x\_i^p\backslash right)^\; \backslash leq\; \backslash prod\_^n\; x\_i^\; \backslash leq\; \backslash left(\backslash sum\_^n\; w\_i\; x\_i^q\backslash right)^$$

The proof for positive and is as follows: Define the following function:

f(x) = x ^{(q)/(p)} . is a power function, so it does have a second derivative:$$f$$(x) = \left(\frac \right) \left(\frac-1 \right)x^which is strictly positive within the domain of, since, so we know is convex.

Using this, and the Jensen's inequality we get:after raising both side to the power of (an increasing function, since is positive) we get the inequality which was to be proven:

$$\backslash left(\backslash sum\_^n\; w\_i\; x\_i^p\backslash right)^\; \backslash leq\; \backslash left(\backslash sum\_^n\; w\_i\; x\_i^q\backslash right)^$$

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

Generalized f-mean

See main article: Generalized {{mvar.

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

$$M\_f(x\_1,\backslash dots,x\_n)\; =\; f^\; \backslash left(\backslash right)$$

This covers the geometric mean without using a limit with . The power mean is obtained for . Properties of these means are studied in de Carvalho (2016).^[1]

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 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 it can serve as an envelope detector on a rectified signal.
• For small 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

• Book: P. S. . Bullen. Handbook of Means and Their Inequalities. Dordrecht, Netherlands. Kluwer. 2003. Chapter III - The Power Means. 175–265.

External links

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

Notes and References

1. de Carvalho. Miguel. Mean, what do you Mean?. The American Statistician. 2016. 70. 3. 764‒776. 10.1080/00031305.2016.1148632. 20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c. free.
2. (retrieved 2019-08-17)
3. Book: Thompson . Sylvanus P. . Calculus Made Easy . 1965 . Macmillan International Higher Education . 9781349004874 . 185 . 5 July 2020 .
4. Book: Jones . Alan R. . Probability, Statistics and Other Frightening Stuff . 2018 . Routledge . 9781351661386 . 48 . 5 July 2020.
5. Sýkora. Stanislav. 2009. Mathematical means and averages: basic properties. Stan's Library . Stan's Library . Castano Primo, Italy. III . 10.3247/SL3Math09.001 .