In mathematics, the Lehmer mean of a tuple of positive real numbers, named after Derrick Henry Lehmer,P. S. Bullen. Handbook of means and their inequalities. Springer, 1987. is defined as:
The weighted Lehmer mean with respect to a tuple of positive weights is defined as:
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 is non-negative
thus this function is monotonic and the inequality
holds.
The derivative of the weighted Lehmer mean is:
Special cases
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
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 and emphasizes big signal values for big . 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 it can serve an envelope detector on a rectified signal. For small 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 ). Their convention is to substitute p with the order of the filter 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
Notes
External links Lehmer Mean at MathWorld