Lehmer mean

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The Lehmer mean of a tuple $x$ of positive real numbers is defined as:

$L_p(x) = \frac{\sum_{k=1}^{n} x_k^p}{\sum_{k=1}^{n} x_k^{p-1}}$ .

The Weighted Lehmer mean with respect to a tuple $w$ of positive weights is defined as:

$L_{p,w}(x) = \frac{\sum_{k=1}^{n} w_k\cdot x_k^p}{\sum_{k=1}^{n} w_k\cdot x_k^{p-1}}$ .

The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.

1 Properties
2 Special cases
3 Applications
- 3.1 Signal processing
4 See also
5 External links

[edit] Properties

The derivative of $p \mapsto L_p(x)$ is non-negative

$\frac{\partial}{\partial p} L_p(x) = \frac {\sum_{j=1}^{n}\sum_{k=j+1}^{n} (x_j-x_k)\cdot(\ln x_j - \ln x_k)\cdot(x_j\cdot x_k)^{p-1}} {\left(\sum_{k=1}^{n} x_k^{p-1}\right)^2},$

thus this function is monotonic and the inequality

$p\le q \Rightarrow L_p(x) \le L_q(x)$

holds.

[edit] Special cases

$\lim_{p\to-\infty} L_p(x)$ is the minimum of the elements of $x$ .
$L 0 (x)$ is the harmonic mean.
$L_\frac{1}{2}(x)$ is the geometric mean.
$L 1 (x)$ is the arithmetic mean.
$L 2 (x)$ is the contraharmonic mean.
$\lim_{p\to\infty} L_p(x)$ is the maximum of the elements of $x$ .

Sketch of a proof: Let $x_1,\dots,x_k$ be the values which equal the maximum. Then $L_p(x)=x_0\cdot\frac{k+\left(\frac{x_{k+1}}{x_0}\right)^p+\dots+\left(\frac{x_{n}}{x_0}\right)^p}{k+\left(\frac{x_{k+1}}{x_0}\right)^{p-1}+\dots+\left(\frac{x_{n}}{x_0}\right)^{p-1}}$

[edit] Applications

[edit] Signal processing

Like a power mean, a Lehmer 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 Lehmer mean according to the following Haskell code.

 lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
 lehmerSmooth smooth p xs =
    zipWith (/)
       (smooth (map (**p) xs))
       (smooth (map (**(p-1)) xs))