Lehmer mean

Short description: Mathematic formula for deriving a mean

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

$L_p(\mathbf{𝐱})=\frac{\sum _{k=1}^n{x}_k^p}{\sum _{k=1}^n{x}_k^{p-1}}.$

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

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

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

$\frac{\partial }{\partial p}{L}_p(\mathbf{𝐱})=\frac{(\sum _{j=1}^n\sum _{k=j+1}^n[x_j-x_k]\cdot [\ln (x_j)-\ln (x_k)]\cdot {[x_j\cdot x_k]}^{p-1})}{{\left(\sum _{k=1}^n{x}_k^{p-1}\right)}^2},$

thus this function is monotonic and the inequality

$p\le q⟹L_p(\mathbf{𝐱})\le L_q(\mathbf{𝐱})$

holds.

The derivative of the weighted Lehmer mean is:

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

Special cases

• $\underset{p\to -\mathrm{\infty }}{\lim }{L}_p(\mathbf{𝐱})$ is the minimum of the elements of $\mathbf{𝐱}$.
• $L_0(\mathbf{𝐱})$ is the harmonic mean.
• $L_{\frac{1}{2}}((x_1,x_2))$ is the geometric mean of the two values $x_1$ and $x_2$.
• $L_1(\mathbf{𝐱})$ is the arithmetic mean.
• $L_2(\mathbf{𝐱})$ is the contraharmonic mean.
• $\underset{p\to \mathrm{\infty }}{\lim }{L}_p(\mathbf{𝐱})$ is the maximum of the elements of $\mathbf{𝐱}$.
  Sketch of a proof: Without loss of generality let $x_1,\dots ,x_k$ be the values which equal the maximum. Then $L_p(\mathbf{𝐱})=x_1\cdot \frac{k+{\left(\frac{x_{k+1}}{x_1}\right)}^p+\cdots +{\left(\frac{x_n}{x_1}\right)}^p}{k+{\left(\frac{x_{k+1}}{x_1}\right)}^{p-1}+\cdots +{\left(\frac{x_n}{x_1}\right)}^{p-1}}$

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 $p$ 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.

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:

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

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

See also

• Mean
• Power mean

External links

• Lehmer Mean at MathWorld