 Weight function

A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more "weight" or influence on the result than other elements in the same set. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus"^{[1]} and "metacalculus".^{[2]}
Contents
Discrete weights
General definition
In the discrete setting, a weight function is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a): = 1 corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.
If the function is a realvalued function, then the unweighted sum of f on A is defined as
 ;
but given a weight function , the weighted sum is defined as
 .
One common application of weighted sums arises in numerical integration.
If B is a finite subset of A, one can replace the unweighted cardinality B of B by the weighted cardinality
If A is a finite nonempty set, one can replace the unweighted mean or average
by the weighted mean or weighted average
In this case only the relative weights are relevant.
Statistics
Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity f measured multiple independent times f_{i} with variance , the best estimate of the signal is obtained by averaging all the measurements with weight , and the resulting variance is smaller than each of the independent measurements . The Maximum likelihood method weights the difference between fit and data using the same weights w_{i} .
Mechanics
The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations :, then the lever will be in balance if the fulcrum of the lever is at the center of mass
 ,
which is also the weighted average of the positions .
Continuous weights
In the continuous setting, a weight is a positive measure such as w(x)dx on some domain Ω,which is typically a subset of a Euclidean space , for instance Ω could be an interval [a,b]. Here dx is Lebesgue measure and is a nonnegative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.
General definition
If is a realvalued function, then the unweighted integral
can be generalized to the weighted integral
Note that one may need to require f to be absolutely integrable with respect to the weight w(x)dx in order for this integral to be finite.
Weighted volume
If E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume
 .
Weighted average
If Ω has finite nonzero weighted volume, then we can replace the unweighted average
by the weighted average
Inner product
If and are two functions, one can generalize the unweighted inner product
to a weighted inner product
See the entry on Orthogonality for more details.
See also
 Center of mass
 numerical integration
 Orthogonality
 Weighted average
 Weighted mean
 Kernel (statistics)
References
 ^ Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980.
 ^ Jane Grossman.MetaCalculus: Differential and Integral, ISBN 0977117022, 1981.
Categories: Mathematical analysis
 Measure theory
 Combinatorial optimization
 Functional analysis
 Types of functions
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