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}(\mathbf {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}(\mathbf {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.

Properties

The derivative of $p\mapsto L_{p}(\mathbf {x} )$ is non-negative

{\frac {\partial }{\partial p}}L_{p}(\mathbf {x} )={\frac {\left(\sum _{j=1}^{n}\sum _{k=j+1}^{n}\left[x_{j}-x_{k}\right]\cdot \left[\ln(x_{j})-\ln(x_{k})\right]\cdot \left[x_{j}\cdot x_{k}\right]^{p-1}\right)}{\left(\sum _{k=1}^{n}x_{k}^{p-1}\right)^{2}}},

thus this function is monotonic and the inequality

p\leq q\Longrightarrow L_{p}(\mathbf {x} )\leq L_{q}(\mathbf {x} )

holds.

The derivative of the weighted Lehmer mean is:

{\frac {\partial L_{p,w}(\mathbf {x} )}{\partial p}}={\frac {(\sum wx^{p-1})(\sum wx^{p}\ln {x})-(\sum wx^{p})(\sum wx^{p-1}\ln {x})}{(\sum wx^{p-1})^{2}}}

Special cases

$\lim _{p\to -\infty }L_{p}(\mathbf {x} )$ is the minimum of the elements of $\mathbf {x}$ .
$L_{0}(\mathbf {x} )$ is the harmonic mean.
$L_{\frac {1}{2}}\left((x_{0},x_{1})\right)$ is the geometric mean of the two values $x_{0}$ and $x_{1}$ .
$L_{1}(\mathbf {x} )$ is the arithmetic mean.
$L_{2}(\mathbf {x} )$ is the contraharmonic mean.
$\lim _{p\to \infty }L_{p}(\mathbf {x} )$ is the maximum of the elements of $\mathbf {x}$ .

Sketch of a proof: Without loss of generality let

x_{1},\dots ,x_{k}

be the values which equal the maximum. Then

L_{p}(\mathbf {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))

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.

Gonzalez and Woods call this a "contraharmonic mean filter" described for varying values of p (however, as above, the contraharmonic mean can refer to the specific case $p=2$ ). Their convention is to substitute p with the order of the filter Q:

f(x)={\frac {\sum _{k=1}^{n}x_{k}^{Q+1}}{\sum _{k=1}^{n}x_{k}^{Q}}}.

Q=0 is the arithmetic mean. Positive Q can reduce pepper noise and negative Q can reduce salt noise.[2]

Notes

P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.
Gonzalez, Rafael C.; Woods, Richard E. (2008). "Chapter 5 Image Restoration and Reconstruction". Digital Image Processing (3 ed.). Prentice Hall. ISBN 9780131687288.

External links

Lehmer Mean at MathWorld

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

[2] Gonzalez, Rafael C.; Woods, Richard E. (2008). "Chapter 5 Image Restoration and Reconstruction". Digital Image Processing (3 ed.). Prentice Hall. ISBN 9780131687288.