Lehmer Mean

In mathematics, the Lehmer mean of a tuple $x$ of positive real numbers, named after Derrick Henry Lehmer, is defined as:
:$L_p(\mathbf) = \frac.$

The weighted Lehmer mean with respect to a tuple $w$ of positive weights is defined as:
:$L_(\mathbf) = \frac.$

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 $p \mapsto L_p(\mathbf)$ is non-negative
:$\frac L_p(\mathbf) = \frac ,$

thus this function is monotonic and the inequality
:$p \le q \Longrightarrow L_p(\mathbf) \le L_q(\mathbf)$

holds.

The derivative of the weighted Lehmer mean is:
:$\frac = \frac$

Special cases

*$\lim_ L_p(\mathbf)$ is the minimum of the elements of $\mathbf$.
*$L_0(\mathbf)$ is the harmonic mean.
*$L_\frac\left((x_1, x_2)\right)$ is the geometric mean of the two values $x_1$ and $x_2$.
*$L_1(\mathbf)$ is the arithmetic mean.
*$L_2(\mathbf)$ is the contraharmonic mean.
*$\lim_ L_p(\mathbf)$ is the maximum of the elements of $\mathbf$. Sketch of a proof: Without loss of generality let $x_1,\dots,x_k$ be the values which equal the maximum. Then $L_p(\mathbf) = x_1\cdot\frac$

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 you can implement a moving Lehmer mean according to the following Haskell code.

lehmerSmooth :: Floating 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'':

:$f(x) = \frac.$

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

See also

*Mean
*Power mean

Notes

External links

Lehmer Mean at MathWorld

{{Statistics, descriptive

Means
Articles with example Haskell code