Weight function - Wikipedia, the free encyclopedia
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Weight function
From Wikipedia, the free encyclopedia
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 "meta-calculus"[2].
Contents
1 Discrete weights
1.1 General definition
1.2 Statistics
1.3 Mechanics
2 Continuous weights
2.1 General definition
2.2 Weighted volume
2.3 Weighted average
2.4 Inner product
3 See also
4 References
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.
is a real-valued function, then the unweighted sum of f on A is defined as
If the function
;
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
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Weight function - Wikipedia, the free encyclopedia
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If A is a finite non-empty 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 fi with variance
averaging all the measurements with weight
independent measurements
, the best estimate of the signal is obtained by
, and the resulting variance is smaller than each of the
. The Maximum likelihood method weights the difference between
fit and data using the same weights wi .
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 an Euclidean space
, for instance Ω could be an interval [a,b]. Here dx is Lebesgue
measure and
is a non-negative measurable function. In this context, the weight function w(x) is
sometimes referred to as a density.
General definition
If
is a real-valued function, then the unweighted integral
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Weight function - Wikipedia, the free encyclopedia
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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 non-zero 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
References
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1. ^ Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and
Integral Calculus (http://books.google.com/books?as_brr=0&
q=%22The+First+Systems+of+Weighted+Differential+and+Integral+Calculus%E2%80%8E%22&
btnG=Search+Books,) , ISBN 0977117014, 1980.
2. ^ Jane Grossman.Meta-Calculus: Differential and Integral (http://books.google.com
/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0,) , ISBN 0977117022,
1981.
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Categories: Mathematical analysis | Measure theory | Combinatorial optimization | Functional analysis
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