Lehmer mean

Properties

The derivative of p \mapsto L_p(\mathbf{x}) is non-negative : \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 :p \le q \Longrightarrow 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 wx^p\ln{x}) - (\sum wx^p)(\sum wx^{p-1}\ln{x})}{(\sum wx^{p-1})^2}

Special cases

• \lim_{p \to -\infty} 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}\left((x_1, x_2)\right) 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.
• \lim_{p \to \infty} 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 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.

Notes

References

1. P. S. Bullen. ''Handbook of means and their inequalities''. Springer, 1987.
2. (2008). "Digital Image Processing". Prentice Hall.