|This article does not cite any references or sources. (November 2009)|
In mathematics, generalised means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and harmonic means. The generalised mean is also known as power mean or Hölder mean (named after Otto Hölder).
- 1 Definition
- 2 Properties
- 3 Special cases
- 4 Proof of power means inequality
- 5 Generalized f-mean
- 6 Applications
- 7 See also
- 8 External links
If p is a non-zero real number, we can define the generalised mean or power mean with exponent p of the positive real numbers as:
Note the relationship to the p-norm. For p = 0 we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case):
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. For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below):
Proof of (geometric mean)
We can rewrite the definition of Mp using the exponential function
In the limit p → 0, we can apply L'Hôpital's rule to the exponential component,
By the continuity of the exponential function, we can substitute back into the above relation to obtain
Proof of and
Assume (possibly after relabeling and combining terms together) that . Then
The formula for follows from
- 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.
- Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.
Generalized mean inequality
- 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.
|quadratic mean, a.k.a. root mean square|
Proof of power means inequality
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
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.
For any q > 0, and non-negative weights summing to 1, the following inequality holds
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
and taking qth powers of the xi, we are done for the inequality with positive q, and the case for negatives is identical.
Inequality between any two power means
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 substituting them with, respectively, −q and −p, QED.
While the proof given above holds for all powers, we may give a proof relying on induction when the power is an integer without aid from the tools of calculus. We prove the power means inequality for powers which are positive integers (the same inequality follows for negative integers by applying the inequality to reciprocals of the original numbers). Let be positive real numbers, be non-negative weights that add up to one, and let k be a positive integer. First note that by the Cauchy-Schwarz inequality,
with equality if and only if all the 's are equal. The expression on the left hand side is known as the weighted Lehmer mean and is denoted by . Thus the above inequality can be written as .
Our aim is to show
Note that for k=1 this inequality follows easily from Cauchy-Schwarz since
Now suppose the inequality is true for a particular k. Thus
Raising both sides to the k(k+1)th power and simplifying results in
Multiplying both sides by we get
Now using the fact that
which can be simplified to get the power means inequality for k+1, i.e.,
The power mean could be generalized further to the generalized f-mean:
Which covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) = xp.
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 you 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 p it can serve an envelope detector on a rectified signal.
- For small p it can serve an baseline detector on a mass spectrum.
- Arithmetic mean
- Arithmetic-geometric mean
- Geometric mean
- Harmonic mean
- Heronian mean
- Inequality of arithmetic and geometric means
- Lehmer mean – also a mean related to powers
- Root mean square