Lehmer mean

In mathematics, the Lehmer mean of a tuple ${\displaystyle x}$ of positive

• ${\displaystyle \lim _{p\to -\infty }L_{p}(\mathbf {x} )}$ is the
  minimum
  of the elements of ${\displaystyle \mathbf {x} }$.
• ${\displaystyle L_{0}(\mathbf {x} )}$ is the harmonic mean.
• ${\displaystyle L_{\frac {1}{2}}\left((x_{1},x_{2})\right)}$ is the geometric mean of the two values ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$.
• ${\displaystyle L_{1}(\mathbf {x} )}$ is the arithmetic mean.
• ${\displaystyle L_{2}(\mathbf {x} )}$ is the contraharmonic mean.
• ${\displaystyle \lim _{p\to \infty }L_{p}(\mathbf {x} )}$ is the
  maximum
  of the elements of ${\displaystyle \mathbf {x} }$.
  Sketch of a proof: Without loss of generality let ${\displaystyle x_{1},\dots ,x_{k}}$ be the values which equal the maximum. Then ${\displaystyle L_{p}(\mathbf {x} )=x_{1}\cdot {\frac {k+\left({\frac {x_{k+1}}{x_{1}}}\right)^{p}+\cdots +\left({\frac {x_{n}}{x_{1}}}\right)^{p}}{k+\left({\frac {x_{k+1}}{x_{1}}}\right)^{p-1}+\cdots +\left({\frac {x_{n}}{x_{1}}}\right)^{p-1}}}}$

Applications

Signal processing

Like a

which is shifted towards small signal values for small ${\displaystyle p}$ and emphasizes big signal values for big ${\displaystyle p}$. Given an efficient implementation of a

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 ${\displaystyle p}$ it can serve an envelope detector on a rectified signal.
• For small ${\displaystyle p}$ 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 ${\displaystyle p=2}$). Their convention is to substitute p with the order of the filter Q:

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

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

See also

• Mean
• Power mean

Notes