The weighted Lehmer mean with respect to a tuple of positive weights is defined as:
The derivative of is non-negative
thus this function is monotonic and the inequality
- is the minimum of the elements of .
- is the harmonic mean.
- is the geometric mean of the two values and .
- is the arithmetic mean.
- is the contraharmonic mean.
- is the maximum of the elements of .
- Sketch of a proof: Without loss of generality let be the values which equal the maximum. Then
Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small and emphasizes big signal values for big . 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 it can serve an envelope detector on a rectified signal.
- For small it can serve an baseline detector on a mass spectrum.
- P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.