# Generalized mean

Short description: N-th root of the arithmetic mean of the given numbers raised to the power n

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

If p is a non-zero real number, and $\displaystyle{ x_1, \dots, x_n }$ are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is:[2]

$\displaystyle{ M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} . }$

(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):

$\displaystyle{ M_0(x_1, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} . }$

Furthermore, for a sequence of positive weights wi we define the weighted power mean as:[2] $\displaystyle{ M_p(x_1,\dots,x_n) = \left(\frac{\sum_{i=1}^n w_i x_i^p}{\sum_{i=1}^n w_i} \right)^{{1}/{p}} }$ and when p = 0, it is equal to the weighted geometric mean:

$\displaystyle{ M_0(x_1,\dots,x_n) = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} . }$

The unweighted means correspond to setting all wi = 1.

## Special cases

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 p yield special cases with their own names:[3]

minimum
$\displaystyle{ M_{-\infty}(x_1,\dots,x_n) = \lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\} }$
harmonic mean
$\displaystyle{ M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}} }$
geometric mean
$\displaystyle{ M_0(x_1,\dots,x_n) = \lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n} }$
arithmetic mean
$\displaystyle{ M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n} }$
root mean square
$\displaystyle{ M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}} }$
cubic mean
$\displaystyle{ M_3(x_1,\dots,x_n) = \sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n}} }$
maximum
$\displaystyle{ M_{+\infty}(x_1,\dots,x_n) = \lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\} }$

Proof of $\displaystyle{ \lim_{p \to 0} M_p = M_0 }$ (geometric mean) We can rewrite the definition of Mp using the exponential function

$\displaystyle{ M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left[\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}\right]} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) } }$

In the limit p → 0, we can apply L'Hôpital's rule to the argument of the exponential function. Differentiating the numerator and denominator with respect to p, we have \displaystyle{ \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} \\ &= \sum_{i=1}^n \frac{ w_i \ln{x_i}}{ \lim_{p \to 0} \sum_{j=1}^n w_j \left( \frac{x_j}{x_i} \right)^p} \\ &= \sum_{i=1}^n w_i \ln{x_i} \\ &= \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \end{align} }

By the continuity of the exponential function, we can substitute back into the above relation to obtain $\displaystyle{ \lim_{p \to 0} M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \right)} = \prod_{i=1}^n x_i^{w_i} = M_0(x_1,\dots,x_n) }$ as desired.[2]

## Properties

Let $\displaystyle{ x_1, \dots, x_n }$ be a sequence of positive real numbers, then the following properties hold:[1]

1. $\displaystyle{ \min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n) }$.
Each generalized mean always lies between the smallest and largest of the x values.
2. $\displaystyle{ M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n)) }$, where $\displaystyle{ P }$ 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. $\displaystyle{ M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n) }$.
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 $\displaystyle{ b\cdot x_1,\dots, b\cdot x_n }$ is equal to b times the generalized mean of the numbers x1, ..., xn.
4. $\displaystyle{ M_p(x_1, \dots, x_{n \cdot k}) = M_p\left[M_p(x_1, \dots, x_{k}), M_p(x_{k + 1}, \dots, x_{2 \cdot k}), \dots, M_p(x_{(n - 1) \cdot k + 1}, \dots, x_{n \cdot k})\right] }$.
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

In general, if p < q, then $\displaystyle{ M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n) }$ and the two means are equal if and only if 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, $\displaystyle{ \frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0 }$ 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

We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality: \displaystyle{ \begin{align} w_i \in [0, 1] \\ \sum_{i=1}^nw_i = 1 \end{align} }

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: $\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \geq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }$ applying this, then: $\displaystyle{ \left(\sum_{i=1}^n\frac{w_i}{x_i^p}\right)^{1/p} \geq \left(\sum_{i=1}^n\frac{w_i}{x_i^q}\right)^{1/q} }$

We raise both sides to the power of −1 (strictly decreasing function in positive reals): $\displaystyle{ \left(\sum_{i=1}^nw_ix_i^{-p}\right)^{-1/p} = \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}\right)^{1/p} \leq \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}\right)^{1/q} = \left(\sum_{i=1}^nw_ix_i^{-q}\right)^{-1/q} }$

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 > 0 and non-negative weights summing to 1, the following inequality holds: $\displaystyle{ \left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}. }$

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: $\displaystyle{ \log \prod_{i=1}^n x_i^{w_i} = \sum_{i=1}^n w_i\log x_i \leq \log \sum_{i=1}^n w_i x_i. }$

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 $\displaystyle{ \prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i. }$

Taking q-th 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

We are to prove that for any p < q the following inequality holds: $\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^nw_ix_i^q\right)^{1/q} }$ if p is negative, and q is positive, the inequality is equivalent to the one proved above: $\displaystyle{ \left(\sum_{i=1}^nw_i x_i^p\right)^{1/p} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }$

The proof for positive p and q is as follows: Define the following function: f : R+R+ $\displaystyle{ f(x)=x^{\frac{q}{p}} }$. f is a power function, so it does have a second derivative: $\displaystyle{ f''(x) = \left(\frac{q}{p} \right) \left( \frac{q}{p}-1 \right)x^{\frac{q}{p}-2} }$ 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: \displaystyle{ \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} } 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:

$\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }$

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

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

$\displaystyle{ M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) }$

This covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) = xp.

## 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 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)