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.

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

$\lim_{p\to-\infty} L_p(x)$ is the minimum of the elements of $x$ .
$L_0(x)$ is the harmonic mean.
$L_\frac{1}{2}\left((x_0,x_1)\right)$ 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.
$\lim_{p\to\infty} L_p(x)$ 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_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.

Notes

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

External links

Lehmer Mean at MathWorld

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