strictly decreasing function

Generalized mean

A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

Definition

If p is a non-zero real number, we can define the generalized mean with exponent p of the positive real numbers x_1,dots,x_n as

M_p(x_1,dots,x_n) = left(frac{1}{n} cdot sum_{i=1}^n x_{i}^p right)^{1/p}.

Properties

  • Like most means, the generalized mean is a homogeneous function of its arguments x_1,dots,x_n. That is, if b is a positive real number, then the generalized mean with exponent p of the numbers bcdot x_1,dots, bcdot x_n is equal to b times the generalized mean of the numbers x_1,dots, x_n.
  • Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.

M_p(x_1,dots,x_{ncdot k}) = M_p(M_p(x_1,dots,x_{k}), M_p(x_{k+1},dots,x_{2cdot k}),
     dots,
M_p(x_{(n-1)cdot k + 1},dots,x_{ncdot k}))

Generalized mean inequality

In general, if p < q, then M_p(x_1,dots,x_n) le M_q(x_1,dots,x_n) and the two means are equal if and only if x_1 = x_2 = cdots = x_n. This follows from the fact that

forall pinmathbb{R} frac{partial M_p(x_1,dots,x_n)}{partial p}geq 0,

which can be proved using Jensen's inequality.

In particular, for pin{-1, 0, 1}, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

Special cases

  • lim_{pto-infty} M_p(x_1,dots,x_n) = min {x_1,dots,x_n} - minimum,
  • M_{-1}(x_1,dots,x_n) = frac{n}{frac{1}{x_1}+dots+frac{1}{x_n}} - harmonic mean,
  • lim_{pto0} M_p(x_1,dots,x_n) = sqrt[n]{x_1cdotdotscdot x_n} - geometric mean,
  • M_1(x_1,dots,x_n) = frac{x_1 + dots + x_n}{n} - arithmetic mean,
  • M_2(x_1,dots,x_n) = sqrt{frac{x_1^2 + dots + x_n^2}{n}} - quadratic mean,
  • lim_{ptoinfty} M_p(x_1,dots,x_n) = max {x_1,dots,x_n} - maximum.

Proof of power means inequality

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents p and q holds:
sqrt[p]{sum_{i=1}^nw_ix_i^p}leq sqrt[q]{sum_{i=1}^nw_ix_i^q}
then:
sqrt[p]{sum_{i=1}^nfrac{w_i}{x_i^p}}leq sqrt[q]{sum_{i=1}^nfrac{w_i}{x_i^q}}
We raise both sides to the power of -1 (strictly decreasing function in positive reals):
sqrt[-p]{sum_{i=1}^nw_ix_i^{-p}}=sqrt[p]{frac{1}{sum_{i=1}^nw_ifrac{1}{x_i^p}}}geq sqrt[q]{frac{1}{sum_{i=1}^nw_ifrac{1}{x_i^q}}}=sqrt[-q]{sum_{i=1}^nw_ix_i^{-q}}
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

For any q the inequality between mean with exponent q and geometric mean can be transformed in the following way:
prod_{i=1}^nx_i^{w_i} leq sqrt[q]{sum_{i=1}^nw_ix_i^q}
sqrt[q]{sum_{i=1}^nw_ix_i^q}leq prod_{i=1}^nx_i^{w_i}
(the first inequality is to be proven for positive q, and the latter otherwise)

We raise both sides to the power of q:

prod_{i=1}^nx_i^{w_icdot q} leq sum_{i=1}^nw_ix_i^q
in both cases we get the inequality between weighted arithmetic and geometric means for the sequence x_i^q, which can be proved by Jensen's inequality, making use of the fact the logarithmic function is concave:
sum_{i=1}^nw_ilog(x_i) leq log(sum_{i=1}^nw_ix_i)
log(prod_{i=1}^nx_i^{w_i}) leq log(sum_{i=1}^nw_ix_i)
By applying (strictly increasing) exp function to both sides we get the inequality:
prod_{i=1}^nx_i^{w_i} leq sum_{i=1}^nw_ix_i

Thus for any positive q it is true that:

sqrt[-q]{sum_{i=1}^nw_ix_i^{-q}}leq prod_{i=1}^nx_i^{w_i} leq sqrt[q]{sum_{i=1}^nw_ix_i^q}
since the inequality holds for any q, however small, and, as will be shown later, the expressions on the left and right approximate the geometric mean better as q approaches 0, the limit of the power mean for q approaching 0 is the geometric mean:
lim_{qrightarrow 0}sqrt[q]{sum_{i=1}^nw_ix_i^{q}}=prod_{i=1}^nx_i^{w_i}

Inequality between any two power means

We are to prove that for any p<q the following inequality holds:
sqrt[p]{sum_{i=1}^nw_ix_i^p}leq sqrt[q]{sum_{i=1}^nw_ix_i^q}
if p is negative, and q is positive, the inequality is equivalent to the one proved above:
sqrt[p]{sum_{i=1}^nw_ix_i^p}leq prod_{i=1}^nx_i^{w_i} leqsqrt[q]{sum_{i=1}^nw_ix_i^q}
The proof for positive p and q is as follows: Define the following function: f:{mathbb R_+}rightarrow{mathbb R_+}, f(x)=x^{frac{q}{p}}. f is a power function, so it does have a second derivative: f(x)=(frac{q}{p})(frac{q}{p}-1)x^{frac{q}{p}-2}, 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:

f(sum_{i=1}^nw_ix_i^p)leqsum_{i=1}^nw_if(x_i^p)
sqrt[frac{p}{q}]{sum_{i=1}^nw_ix_i^p}leqsum_{i=1}^nw_ix_i^q
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:
sqrt[p]{sum_{i=1}^nw_ix_i^p}leqsqrt[q]{sum_{i=1}^nw_ix_i^q}

Using the previously shown equivalence we can prove the inequality for negative p and q by substituting them with, respectively, -q and -p, QED.

Minimum and maximum

Minimum and maximum are assumed to be the power means with exponents of -infty and +infty. Thus for any q:
min (x_1,x_2,ldots ,x_n)leq sqrt[q]{sum_{i=1}^nw_ix_i^q}leq max (x_1,x_2,ldots ,x_n)
For maximum the proof is as follows: Assume WLoG that the sequence xi is nonincreasing and no weight is zero.

Then the inequality is equivalent to:

sqrt[q]{sum_{i=1}^nw_ix_i^q}leq x_1
After raising both sides to the power of q we get (depending on the sign of q) one of the inequalities:
sum_{i=1}^nw_ix_i^qleq {color{red} geq} x_1^q
≤ for q>0, ≥ for q<0.

After subtracting w_1x_1 from the both sides we get:

sum_{i=2}^nw_ix_i^qleq {color{red} geq} (1-w_1)x_1^q
After dividing by (1-w_1):
sum_{i=2}^nfrac{w_i}{(1-w_1)}x_i^qleq {color{red} geq} x_1^q
1 - w1 is nonzero, thus:
sum_{i=2}^nfrac{w_i}{(1-w_1)}=1
Substacting x1q leaves:
sum_{i=2}^nfrac{w_i}{(1-w_1)}(x_i^q-x_1^q)leq {color{red} geq} 0
which is obvious, since x1 is greater or equal to any xi, and thus:
x_i^q-x_1^qleq {color{red} geq} 0

For minimum the proof is almost the same, only instead of x1, w1 we use xn, wn, QED.

Generalized f-mean

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

M_f(x_1,dots,x_n) = f^{-1}
left({frac{1}{n}cdotsum_{i=1}^n{f(x_i)}}right)

which covers e.g. the geometric mean without using a limit. The power mean is obtained for fleft(xright)=x^p .

Applications

Signal processing

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)

See also

External links

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