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Lehmer mean - Wikipedia

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Mathematic formula for deriving a mean

In mathematics, the Lehmer mean of a tuple x {\displaystyle x} of positive real numbers, named after Derrick Henry Lehmer,^[1] is defined as:

L p ( x ) = ∑ k = 1 n x k p ∑ k = 1 n x k p − 1 . {\displaystyle 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 {\displaystyle w} of positive weights is defined as:

L p , w ( x ) = ∑ k = 1 n w k ⋅ x k p ∑ k = 1 n w k ⋅ x k p − 1 . {\displaystyle 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.

The derivative of p ↦ L p ( x ) {\displaystyle p\mapsto L_{p}(\mathbf {x} )} is non-negative

∂ ∂ p L p ( x ) = ( ∑ j = 1 n ∑ k = j + 1 n [ x j − x k ] ⋅ [ ln ⁡ ( x j ) − ln ⁡ ( x k ) ] ⋅ [ x j ⋅ x k ] p − 1 ) ( ∑ k = 1 n x k p − 1 ) 2 , {\displaystyle {\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 ≤ q ⟹ L p ( x ) ≤ L q ( x ) {\displaystyle p\leq q\Longrightarrow L_{p}(\mathbf {x} )\leq L_{q}(\mathbf {x} )}

holds.

The derivative of the weighted Lehmer mean is:

∂ L p , w ( x ) ∂ p = ( ∑ w x p − 1 ) ( ∑ w x p ln ⁡ x ) − ( ∑ w x p ) ( ∑ w x p − 1 ln ⁡ x ) ( ∑ w x p − 1 ) 2 {\displaystyle {\frac {\partial L_{p,w}(\mathbf {x} )}{\partial p}}={\frac {(\sum wx^{p-1})(\sum wx^{p}\ln {x})-(\sum wx^{p})(\sum wx^{p-1}\ln {x})}{(\sum wx^{p-1})^{2}}}}

Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small p {\displaystyle p} and emphasizes big signal values for big p {\displaystyle 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))

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 {\displaystyle p=2} ). Their convention is to substitute p with the order of the filter Q:

f ( x ) = ∑ k = 1 n x k Q + 1 ∑ k = 1 n x k Q . {\displaystyle 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.^[2]

• Mean
• Power mean

1. ^ P. S. Bullen. Handbook of means and their inequalities. Springer, 1987.
2. ^ Gonzalez, Rafael C.; Woods, Richard E. (2008). "Chapter 5 Image Restoration and Reconstruction". Digital Image Processing (3 ed.). Prentice Hall. ISBN 9780131687288.

• Lehmer Mean at MathWorld

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