Peskin–Takeuchi parameter

In mathematics, the Lehmer mean of a tuple $x$ of positive real numbers, named after Derrick Henry Lehmer, 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

$p\leq q\Rightarrow L_{p}(x)\leq L_{q}(x)$

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.

See also

• mean
• power mean

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Lehmer Mean at MathWorld