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Lehmer mean

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Title: Lehmer mean  
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Subject: Contraharmonic mean, Derrick Henry Lehmer, Means, Generalized mean, Mean
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Lehmer mean

In mathematics, the Lehmer mean of a tuple x of positive real numbers, named after Derrick Henry Lehmer,[1] 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.

Contents

  • Properties 1
  • Special cases 2
  • Applications 3
    • Signal processing 3.1
  • See also 4
  • Notes 5
  • External links 6

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.

Special cases

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_1\cdot\frac{k+\left(\frac{x_{k+1}}{x_1}\right)^p+\cdots+\left(\frac{x_{n}}{x_1}\right)^p}{k+\left(\frac{x_{k+1}}{x_1}\right)^{p-1}+\cdots+\left(\frac{x_{n}}{x_1}\right)^{p-1}}

Applications

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))

See also

Notes

  1. ^ P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.

External links

  • Lehmer Mean at MathWorld
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