Lehmer mean

In mathematics, the Lehmer mean of a tuple ${\displaystyle x}$ of positive real numbers, named after Derrick Henry Lehmer,^[1] is defined as:

${\displaystyle L_{p}(\mathbf {x} )={\frac {\sum _{k=1}^{n}x_{k}^{p}}{\sum _{k=1}^{n}x_{k}^{p-1}}}.}$

The weighted Lehmer mean with respect to a tuple ${\displaystyle w}$ of positive weights is defined as:

${\displaystyle L_{p,w}(\mathbf {x} )={\frac {\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p}}{\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p-1}}}.}$

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 ${\displaystyle p\mapsto L_{p}(\mathbf {x} )}$ is non-negative

${\displaystyle {\frac {\partial }{\partial p}}L_{p}(\mathbf {x} )={\frac {\left(\sum _{j=1}^{n}\sum _{k=j+1}^{n}\left[x_{j}-x_{k}\right]\cdot \left[\ln(x_{j})-\ln(x_{k})\right]\cdot \left[x_{j}\cdot x_{k}\right]^{p-1}\right)}{\left(\sum _{k=1}^{n}x_{k}^{p-1}\right)^{2}}},}$

thus this function is monotonic and the inequality

${\displaystyle p\leq q\Longrightarrow L_{p}(\mathbf {x} )\leq L_{q}(\mathbf {x} )}$

holds.

The derivative of the weighted Lehmer mean is:

${\displaystyle {\frac {\partial L_{p,w}(\mathbf {x} )}{\partial p}}={\frac {(\sum wx^{p-1})(\sum wx^{p}\ln {x})-(\sum wx^{p})(\sum wx^{p-1}\ln {x})}{(\sum wx^{p-1})^{2}}}}$

Special cases

• ${\displaystyle \lim _{p\to -\infty }L_{p}(\mathbf {x} )}$ 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 power mean, a Lehmer mean serves a non-linear moving average 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 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 ${\displaystyle p}$ 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