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 $$pand emphasizes big signal values for big $$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]