- Source: Lehmer mean
In mathematics, the Lehmer mean of a tuple
x
{\displaystyle x}
of positive real numbers, named after Derrick Henry Lehmer, 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.
Properties
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}}}}
Special cases
lim
p
→
−
∞
L
p
(
x
)
{\displaystyle \lim _{p\to -\infty }L_{p}(\mathbf {x} )}
is the minimum of the elements of
x
{\displaystyle \mathbf {x} }
.
L
0
(
x
)
{\displaystyle L_{0}(\mathbf {x} )}
is the harmonic mean.
L
1
2
(
(
x
1
,
x
2
)
)
{\displaystyle L_{\frac {1}{2}}\left((x_{1},x_{2})\right)}
is the geometric mean of the two values
x
1
{\displaystyle x_{1}}
and
x
2
{\displaystyle x_{2}}
.
L
1
(
x
)
{\displaystyle L_{1}(\mathbf {x} )}
is the arithmetic mean.
L
2
(
x
)
{\displaystyle L_{2}(\mathbf {x} )}
is the contraharmonic mean.
lim
p
→
∞
L
p
(
x
)
{\displaystyle \lim _{p\to \infty }L_{p}(\mathbf {x} )}
is the maximum of the elements of
x
{\displaystyle \mathbf {x} }
. Sketch of a proof: Without loss of generality let
x
1
,
…
,
x
k
{\displaystyle x_{1},\dots ,x_{k}}
be the values which equal the maximum. Then
L
p
(
x
)
=
x
1
⋅
k
+
(
x
k
+
1
x
1
)
p
+
⋯
+
(
x
n
x
1
)
p
k
+
(
x
k
+
1
x
1
)
p
−
1
+
⋯
+
(
x
n
x
1
)
p
−
1
{\displaystyle 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
{\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.
For big
p
{\displaystyle p}
it can serve an envelope detector on a rectified signal.
For small
p
{\displaystyle 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
{\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.
See also
Mean
Power mean
Notes
External links
Lehmer Mean at MathWorld
Kata Kunci Pencarian:
- Rata-rata
- Hipotesis Riemann
- Daftar masalah matematika yang belum terpecahkan
- Lehmer mean
- Contraharmonic mean
- D. H. Lehmer
- Mean
- Generalized mean
- Lehmer
- Lehmer code
- Average
- Geometric mean
- Arithmetic mean
