Generalized mean

Template:Short description Template:More citations needed

Plot of several generalized means

M_{p} (1, x)

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 Template:Mvar is a non-zero real number, and $x_{1}, \dots, x_{n}$ are positive real numbers, then the generalized mean or power mean with exponent Template:Mvar of these positive real numbers is^[2]^[3]

$M_{p} (x_{1}, \dots, x_{n}) = {(\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{p})}^{1 / p} .$

(See [[Norm (mathematics)#p-norm|Template:Mvar-norm]]). For Template:Math we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

$M_{0} (x_{1}, \dots, x_{n}) = {(\prod_{i = 1}^{n} x_{i})}^{1 / n} .$

Furthermore, for a sequence of positive weights Template:Mvar we define the weighted power mean as^[2] $M_{p} (x_{1}, \dots, x_{n}) = {(\frac{\sum_{i = 1}^{n} w_{i} x_{i}^{p}}{\sum_{i = 1}^{n} w_{i}})}^{1 / p}$ and when Template:Math, it is equal to the weighted geometric mean:

$M_{0} (x_{1}, \dots, x_{n}) = {(\prod_{i = 1}^{n} x_{i}^{w_{i}})}^{1 / \sum_{i = 1}^{n} w_{i}} .$

The unweighted means correspond to setting all Template:Math.

Special cases

For some values of $p$ , the mean $M_{p} (x_{1}, \dots, x_{n})$ corresponds to a well known mean.

File:Generalized Means.svg

A visual depiction of some of the specified cases for

n = 2

. <templatestyles src="Legend/styles.css" />

Harmonic mean:

M_{- 1} (a, b)

.

Geometric mean:

M_{0} (a, b)

.

Arithmetic mean:

M_{1} (a, b)

.

Quadratic mean:

M_{2} (a, b)

.

Name	Exponent	Value
Minimum	$p = - \infty$	$\min {x_{1}, \dots, x_{n}}$
Harmonic mean	$p = - 1$	$\frac{n}{\frac{1}{x_{1}} + \dots + \frac{1}{x_{n}}}$
Geometric mean	$p = 0$	$\sqrt[n]{x_{1} \dots x_{n}}$
Arithmetic mean	$p = 1$	$\frac{x_{1} + \dots + x_{n}}{n}$
Root mean square	$p = 2$	$\sqrt{\frac{x_{1}^{2} + \dots + x_{n}^{2}}{n}}$
Cubic mean	$p = 3$	$\sqrt[3]{\frac{x_{1}^{3} + \dots + x_{n}^{3}}{n}}$
Maximum	$p = + \infty$	$\max {x_{1}, \dots, x_{n}}$

Template:Math proof

Template:Proof

Properties

Let $x_{1}, \dots, x_{n}$ be a sequence of positive real numbers, then the following properties hold:^[1]

$\min (x_{1}, \dots, x_{n}) \leq M_{p} (x_{1}, \dots, x_{n}) \leq \max (x_{1}, \dots, x_{n})$ .Template:Block indent
$M_{p} (x_{1}, \dots, x_{n}) = M_{p} (P (x_{1}, \dots, x_{n}))$ , where $P$ is a permutation operator.Template:Block indent
$M_{p} (b x_{1}, \dots, b x_{n}) = b \cdot M_{p} (x_{1}, \dots, x_{n})$ .Template:Block indent
$M_{p} (x_{1}, \dots, x_{n \cdot k}) = M_{p} [M_{p} (x_{1}, \dots, x_{k}), M_{p} (x_{k + 1}, \dots, x_{2 \cdot k}), \dots, M_{p} (x_{(n - 1) \cdot k + 1}, \dots, x_{n \cdot k})]$ .Template:Block indent

Generalized mean inequality

Template:QM AM GM HM inequality visual proof.svg In general, if Template:Math, then $M_{p} (x_{1}, \dots, x_{n}) \leq M_{q} (x_{1}, \dots, x_{n})$ and the two means are equal if and only if Template:Math.

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

It follows from the fact that, for all real Template:Mvar, $\frac{\partial}{\partial p} M_{p} (x_{1}, \dots, x_{n}) \geq 0$ which can be proved using Jensen's inequality.

In particular, for Template:Mvar in Template:Math, 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: $\begin{aligned} w_{i} \in [0, 1] \\ \sum_{i = 1}^{n} w_{i} = 1 \end{aligned}$

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

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents Template:Mvar and Template:Mvar holds: ${(\sum_{i = 1}^{n} w_{i} x_{i}^{p})}^{1 / p} \geq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q}$ applying this, then: ${(\sum_{i = 1}^{n} \frac{w_{i}}{x_{i}^{p}})}^{1 / p} \geq {(\sum_{i = 1}^{n} \frac{w_{i}}{x_{i}^{q}})}^{1 / q}$

We raise both sides to the power of −1 (strictly decreasing function in positive reals): ${(\sum_{i = 1}^{n} w_{i} x_{i}^{- p})}^{- 1 / p} = {(\frac{1}{\sum_{i = 1}^{n} w_{i} \frac{1}{x_{i}^{p}}})}^{1 / p} \leq {(\frac{1}{\sum_{i = 1}^{n} w_{i} \frac{1}{x_{i}^{q}}})}^{1 / q} = {(\sum_{i = 1}^{n} w_{i} x_{i}^{- q})}^{- 1 / q}$

We get the inequality for means with exponents Template:Math and Template:Math, 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 Template:Math and non-negative weights summing to 1, the following inequality holds: ${(\sum_{i = 1}^{n} w_{i} x_{i}^{- q})}^{- 1 / q} \leq \prod_{i = 1}^{n} x_{i}^{w_{i}} \leq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q} .$

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

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 $\prod_{i = 1}^{n} x_{i}^{w_{i}} \leq \sum_{i = 1}^{n} w_{i} x_{i} .$

Taking Template:Mvar-th powers of the Template:Mvar yields $\begin{aligned} \prod_{i = 1}^{n} x_{i}^{q \cdot w_{i}} \leq \sum_{i = 1}^{n} w_{i} x_{i}^{q} \\ \prod_{i = 1}^{n} x_{i}^{w_{i}} \leq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q} . \end{aligned}$

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

$\prod_{i = 1}^{n} x_{i}^{- q \cdot w_{i}} \leq \sum_{i = 1}^{n} w_{i} x_{i}^{- q} .$

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

$\prod_{i = 1}^{n} x_{i}^{w_{i}} \geq {(\sum_{i = 1}^{n} w_{i} x_{i}^{- q})}^{- 1 / q} .$

Inequality between any two power means

We are to prove that for any Template:Math the following inequality holds: ${(\sum_{i = 1}^{n} w_{i} x_{i}^{p})}^{1 / p} \leq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q}$ if Template:Mvar is negative, and Template:Mvar is positive, the inequality is equivalent to the one proved above: ${(\sum_{i = 1}^{n} w_{i} x_{i}^{p})}^{1 / p} \leq \prod_{i = 1}^{n} x_{i}^{w_{i}} \leq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q}$

The proof for positive Template:Mvar and Template:Mvar is as follows: Define the following function: Template:Math $f (x) = x^{\frac{q}{p}}$ . Template:Mvar 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 Template:Mvar, since Template:Math, so we know Template:Mvar is convex.

Using this, and the Jensen's inequality we get: $\begin{aligned} f (\sum_{i = 1}^{n} w_{i} x_{i}^{p}) & \leq \sum_{i = 1}^{n} w_{i} f (x_{i}^{p}) \\ {(\sum_{i = 1}^{n} w_{i} x_{i}^{p})}^{q / p} & \leq \sum_{i = 1}^{n} w_{i} x_{i}^{q} \end{aligned}$ after raising both side to the power of Template:Math (an increasing function, since Template:Math is positive) we get the inequality which was to be proven:

${(\sum_{i = 1}^{n} w_{i} x_{i}^{p})}^{1 / p} \leq {(\sum_{i = 1}^{n} w_{i} x_{i}^{q})}^{1 / q}$

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

Generalized f-mean

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The power mean could be generalized further to the [[generalized f-mean|generalized Template:Mvar-mean]]:

$M_{f} (x_{1}, \dots, x_{n}) = f^{- 1} (\frac{1}{n} \cdot \sum_{i = 1}^{n} f (x_{i}))$

This covers the geometric mean without using a limit with Template:Math. The power mean is obtained for Template:Mvar. Properties of these means are studied in de Carvalho (2016).^[3]

Applications

Signal processing

A power mean serves a non-linear moving average which is shifted towards small signal values for small Template:Mvar and emphasizes big signal values for big Template:Mvar. 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 Template:Mvar it can serve as an envelope detector on a rectified signal.
For small Template:Mvar it can serve as a baseline detector on a mass spectrum.

Notes

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References

Template:Reflist

External links

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[1]

[2]

[3]

Generalized mean

Contents

Definition

Special cases

Properties

Generalized mean inequality

Proof of the weighted inequality

Equivalence of inequalities between means of opposite signs

Geometric mean

Inequality between any two power means

Generalized f-mean

Applications

Signal processing

See also

Notes

References

Further reading

External links

Navigation menu

Generalized mean

Definition

Special cases

Properties

Generalized mean inequality

Proof of the weighted inequality

Equivalence of inequalities between means of opposite signs

Geometric mean

Inequality between any two power means

Generalized f-mean

Applications

Signal processing

See also

Notes

References

Further reading

External links

Navigation menu

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