In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
Derrick Henry "Dick" Lehmer was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes...
, 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
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
.
Properties
The derivative of is non-negative
thus this function is monotonic and the inequality
holds.
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L_2.- Definition :...
Without loss of generality is a frequently used expression in mathematics...
let be the values which equal the maximum. Then
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 smooth you can implement a moving Lehmer mean
according to the following Haskell
Haskell (programming language)
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
code.
lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
lehmerSmooth smooth p xs = zipWith (/)
(smooth (map (**p) xs))
(smooth (map (**(p-1)) xs))
An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the "envelope" of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls...
A rectifier is an electrical device that converts alternating current , which periodically reverses direction, to direct current , which flows in only one direction. The process is known as rectification...
signal.
For small it can serve an baseline detector on a mass spectrum
Mass spectrum
A mass spectrum is an intensity vs. m/z plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of ions by mass in a sample. It is a histogram usually acquired using an instrument called a mass spectrometer...
.
The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
of positive real number
of positive weights is defined as:
is non-negative
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
is the minimum of the elements of 
.
is the harmonic meanHarmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
is the geometric meanGeometric meanThe geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
and 
.
is the arithmetic meanArithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
is the contraharmonic meanContraharmonic meanIn mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, L_2.- Definition :...
is the maximum of the elements of 
.
be the values which equal the maximum. Then 
it can serve an envelope detectorEnvelope detectorAn envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the "envelope" of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls...
it can serve an baseline detector on a mass spectrumMass spectrumA mass spectrum is an intensity vs. m/z plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of ions by mass in a sample. It is a histogram usually acquired using an instrument called a mass spectrometer...