Lehmer mean explained

x of positive real numbers, named after Derrick Henry Lehmer,^[1] is defined as:L _{p(x)} = (\sum ^{n} _{k =1} x ^{p} _{k})/(\sum ^{n} _{k =1} x ^{p -1} _{k}) .

The weighted Lehmer mean with respect to a tuple

w of positive weights is defined as:L _{p,w} ( x ) = (\sum ^{n} _{k =1} w _{k ⋅} x ^{p} _{k})/(\sum ^{n} _{k =1} w _{k ⋅} x ^{p -1} _{k}) .

The Lehmer mean is an alternative to power meansfor interpolating between minimum and maximum via arithmetic mean and harmonic mean.

Properties

The derivative of

p \mapsto L _{p(x)} is non-negative(\partial)/(\partial p) L _{p(x)} = (\left(\sum ^{n} _{j =1} \sum ^{n} _{k = j +1} \left[x _{j} - x _{k\right]} ⋅ \left[ ln (x _{j)} - ln (x _{k)\right]} ⋅ \left[x _{j} ⋅ x ^{p -1} _{k\right]} \right))/(\left(\sum ^{n} _{k =1} x ^{p -1} _{k} \right) ^{2}) ,

thus this function is monotonic and the inequality

p \le q \Longrightarrow L _{p(x)} \le L _{q(x)}

holds.

The derivative of the weighted Lehmer mean is:

(\partial L _{p,w} ( x ))/(\partial p) = ((\sum w x ^{p-1} )(\sum wx ^{pln{x})/()) - (\sum wx ^{p)(\sum} wx ^{p-1} ln {x})}{(\sum wx ^{p-1} ) ^{2}}

Special cases

\lim _{p} L _{p(x)} is the minimum of the elements of x .L _{0(x)} is the harmonic mean.L _{(1)/(2)} \left((x _{1,} x _{2)\right)} is the geometric mean of the two values x _{1} and x _{2} .L _{1(x)} is the arithmetic mean.L _{2(x)} is the contraharmonic mean.\lim _{p} L _{p(x)} is the maximum of the elements of x . Sketch of a proof: Without loss of generality let x _{1,...,x} _{k} be the values which equal the maximum. Then L _{p(x)} = x _{1 ⋅ (k + \left( (x _{k+1})/(x _{1}) \right) ^{p} + … + \left( (x _{n})/(x _{1}) \right) ^{p})/(k + \left( (x _{k+1})/(x _{1}) \right) ^{p-1} + … + \left( (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) = (\sum ^{n} _{k =1} x ^{Q +1} _{k})/(\sum ^{n} _{k =1} x ^{Q} _{k}) .

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

External links

• Lehmer Mean at MathWorld

Notes and References

1. P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.
2. Book: Digital Image Processing . 3 . Gonzalez . Rafael C. . Woods . Richard E. . 2008 . Chapter 5 Image Restoration and Reconstruction . 9780131687288 . Prentice Hall .