Generalized mean

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In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder)[1] are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

Definition[edit]

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:[2]

(See p-norm). For p = 0 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 wi with sum we define the weighted power mean as:

The unweighted means correspond to setting all wi = 1/n.

Special cases[edit]

A visual depiction of some of the specified cases for n = 2 with a = x1 = M and b = x2 = M−∞:
  harmonic mean, H = M−1(a, b),
  geometric mean, G = M0(a, b)
  arithmetic mean, A = M1(a, b)
  quadratic mean, Q = M2(a, b)

A few particular values of yield special cases with their own names:[3]

minimum
harmonic mean
geometric mean
arithmetic mean
root mean square
or quadratic mean[4][5]
cubic mean
maximum

Properties[edit]

Let be a sequence of positive real numbers, then the following properties hold:[1]

  1. .
    Each generalized mean always lies between the smallest and largest of the x values.
  2. , where is a permutation operator.
    Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.
  3. .
    Like most means, the generalized mean is a homogeneous function of its arguments x1, ..., xn. That is, if 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 x1, …, xn.
  4. .
    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.

Generalized mean inequality[edit]

Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two positive numbers a and b [6]

In general,

if p < q, then

and the two means are equal if and only if x1 = x2 = ... = xn.

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.

Proof of power means inequality[edit]

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 wi = 1/n.

Equivalence of inequalities between means of opposite signs[edit]

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.

Geometric mean[edit]

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 qth powers of the xi, we are done for the inequality with positive q; the case for negatives is identical.

Inequality between any two power means[edit]

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 replacing them with −q and −p, respectively.

Generalized f-mean[edit]

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) = xp.

Applications[edit]

Signal processing[edit]

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)

See also[edit]

Notes[edit]

  1. ^ a b Sýkora, Stanislav (2009). Mathematical means and averages: basic properties. 3. Stan’s Library: Castano Primo, Italy. doi:10.3247/SL3Math09.001.
  2. ^ a b P. S. Bullen: Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177
  3. ^ Weisstein, Eric W. "Power Mean". MathWorld. (retrieved 2019-08-17)
  4. ^ Thompson, Sylvanus P. (1965). Calculus Made Easy. Macmillan International Higher Education. p. 185. ISBN 9781349004874. Retrieved 5 July 2020.
  5. ^ Jones, Alan R. (2018). Probability, Statistics and Other Frightening Stuff. Routledge. p. 48. ISBN 9781351661386. Retrieved 5 July 2020.
  6. ^ If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
    Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
    Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
    Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM.

References and further reading[edit]

  • P. S. Bullen: Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003, chapter III (The Power Means), pp. 175-265

External links[edit]