Peskin–Takeuchi parameter

2013-11-21T00:29:25Z

131.203.251.94:

In mathematics, the '''Lehmer mean''' of a [[tuple]] <math>x</math> of positive [[real number]]s, named after [[Derrick Henry Lehmer]],<ref>P. S. Bullen. ''Handbook of means and their inequalities''. Springer, 1987.</ref> is defined as:
:<math>L_p(x) = \frac{\sum_{k=1}^n x_k^p}{\sum_{k=1}^n x_k^{p-1}}.</math>

The '''weighted Lehmer mean''' with respect to a tuple <math>w</math> of positive weights is defined as:
:<math>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}}.</math>

The Lehmer mean is an alternative to [[power mean]]s
for [[Interpolation|interpolating]] between [[minimum]] and [[maximum]] via [[arithmetic mean]] and [[harmonic mean]].

== Properties ==

The derivative of <math>p \mapsto L_p(x)</math> is non-negative
:<math>
\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},
</math>
thus this function is monotonic and the inequality
:<math>p\le q \Rightarrow L_p(x) \le L_q(x)</math>
holds.

==Special cases==

*<math>\lim_{p\to-\infty} L_p(x)</math> is the [[minimum]] of the elements of <math>x</math>.
*<math>L_0(x)</math> is the [[harmonic mean]].
*<math>L_\frac{1}{2}\left((x_0,x_1)\right)</math> is the [[geometric mean]] of the two values <math>x_0</math> and <math>x_1</math>.
*<math>L_1(x)</math> is the [[arithmetic mean]].
*<math>L_2(x)</math> is the [[contraharmonic mean]].
*<math>\lim_{p\to\infty} L_p(x)</math> is the [[maximum]] of the elements of <math>x</math>.
:Sketch of a proof: [[Without loss of generality]] let <math>x_1,\dots,x_k</math> be the values which equal the maximum. Then <math>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}}</math>

== 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 <math>p</math> and emphasizes big signal values for big <math>p</math>. Given an efficient implementation of a [[lowpass|moving arithmetic mean]] called <tt>smooth</tt> you can implement a moving Lehmer mean
according to the following [[Haskell (programming language)|Haskell]] code.

<source lang="haskell">
lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
lehmerSmooth smooth p xs = zipWith (/)
(smooth (map (**p) xs))
(smooth (map (**(p-1)) xs))
</source>

* For big <math>p</math> it can serve an [[envelope detector]] on a [[rectifier|rectified]] signal.
* For small <math>p</math> it can serve an [[Baseline (spectrometry)|baseline detector]] on a [[mass spectrum]].

==See also==
*[[mean]]
*[[power mean]]

==Notes==
{{reflist}}

==External links==
*[http://mathworld.wolfram.com/LehmerMean.html Lehmer Mean at MathWorld]

[[Category:Means]]
[[Category:Articles with example Haskell code]]

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Peskin–Takeuchi parameter