Lehmer mean

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::$L\_\{p,w\}(x)\; =\; frac\{sum\_\{k=1\}^\{n\}\; w\_kcdot\; x\_k^p\}\{sum\_\{k=1\}^\{n\}\; w\_kcdot\; x\_k^\{p-1$.

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

Properties

The derivative of $p\; mapsto\; L\_p(x)$ is non-negative:$frac\{partial\}\{partial\; p\}\; L\_p(x)\; =frac\; \{sum\_\{j=1\}^\{n\}sum\_\{k=j+1\}^\{n\}\; (x\_j-x\_k)cdot(ln\; x\_j\; -\; ln\; x\_k)cdot(x\_jcdot\; x\_k)^\{p-1\; \{left(sum\_\{k=1\}^\{n\}\; x\_k^\{p-1\}\; ight)^2\},$thus this function is monotonic and the inequality:$ple\; q\; Rightarrow\; L\_p(x)\; le\; L\_q(x)$holds.

pecial cases

*$lim\_\{p\; o-infty\}\; L\_p(x)$ is the minimum of the elements of $x$.
*$L\_0(x)$ is the harmonic mean.
*$L\_frac\{1\}\{2\}left((x\_0,x\_1)\; ight)$ is the geometric mean of the two values $x\_0$ and $x\_1$.
*$L\_1(x)$ is the arithmetic mean.
*$L\_2(x)$ is the contraharmonic mean.
*$lim\_\{p\; oinfty\}\; L\_p(x)$ is the maximum of the elements of $x$.:Sketch of a proof: Without loss of generality let $x\_1,dots,x\_k$ be the values which equal the maximum. Then $L\_p(x)=x\_1cdotfrac\{k+left(frac\{x\_\{k+1\{x\_1\}\; ight)^p+dots+left(frac\{x\_\{n\{x\_1\}\; ight)^p\}\{k+left(frac\{x\_\{k+1\{x\_1\}\; ight)^\{p-1\}+dots+left(frac\{x\_\{n\{x\_1\}\; ight)^\{p-1$

Applications

Signal processing

Like a power mean,a Lehmer mean serves a non-linear moving averagewhich 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 meancalled smooth you can implement a moving Lehmer meanaccording 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))

* 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.

ee also

*mean

External links

* [http://mathworld.wolfram.com/LehmerMean.html Lehmer Mean at MathWorld]