129.55.200.20: Undid revision 1253037847 by 223.123.19.139 (talk)

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Undid revision 1253037847 by 223.123.19.139 (talk)

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{{Use American English|date = March 2019}}
{{Short description|Construct related to weighted sums and averages}}
A '''weight function''' is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a '''weighted sum''' or [[weighted average]]. Weight functions occur frequently in [[statistics]] and [[mathematical analysis|analysis]], and are closely related to the concept of a [[measure (mathematics)|measure]]. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus"<ref>Jane Grossman, Michael Grossman, Robert Katz. [https://books.google.com/books?as_brr=0&q=%22The+First+Systems+of+Weighted+Differential+and+Integral+Calculus%E2%80%8E%22&btnG=Search+Books, ''The First Systems of Weighted Differential and Integral Calculus''], {{isbn|0-9771170-1-4}}, 1980.</ref> and "meta-calculus".<ref>Jane Grossman.[https://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0, ''Meta-Calculus: Differential and Integral''], {{isbn|0-9771170-2-2}}, 1981.</ref>

== Discrete weights ==
=== General definition ===
In the discrete setting, a weight function <math>w \colon A \to \R^+</math> is a positive function defined on a [[discrete mathematics|discrete]] [[Set (mathematics)|set]] <math>A</math>, which is typically [[finite set|finite]] or [[countable]]. The weight function <math>w(a) := 1</math> corresponds to the ''unweighted'' situation in which all elements have equal weight. One can then apply this weight to various concepts.

If the function <math>f\colon A \to \R</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted [[summation|sum]] of <math>f</math> on <math>A</math>'' is defined as

:<math>\sum_{a \in A} f(a);</math>

but given a ''weight function'' <math>w\colon A \to \R^+</math>, the '''weighted sum''' or [[conical combination]] is defined as

:<math>\sum_{a \in A} f(a) w(a).</math>

One common application of weighted sums arises in [[numerical integration]].

If ''B'' is a [[finite set|finite]] subset of ''A'', one can replace the unweighted [[cardinality]] |''B''| of ''B'' by the ''weighted cardinality''

:<math>\sum_{a \in B} w(a).</math>

If ''A'' is a [[finite set|finite]] non-empty set, one can replace the unweighted [[mean]] or [[average]]

:<math>\frac{1}{|A|} \sum_{a \in A} f(a)</math>

by the [[weighted mean]] or [[weighted average]]

:<math> \frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.</math>

In this case only the ''relative'' weights are relevant.

=== Statistics ===
Weighted means are commonly used in [[statistics]] to compensate for the presence of [[Bias_(statistics)|bias]]. For a quantity <math>f</math> measured multiple independent times <math>f_i</math> with [[variance]] <math>\sigma^2_i</math>, the best estimate of the signal is obtained by averaging all the measurements with weight {{nowrap|<math display="inline">w_i = 1 / {\sigma_i^2}</math>,}} and the resulting variance is smaller than each of the independent measurements {{nowrap|<math display="inline"> \sigma^2 = 1 / \sum_i w_i</math>.}} The [[maximum likelihood]] method weights the difference between fit and data using the same weights {{nowrap|<math>w_i</math>.}}

The [[expected value]] of a random variable is the weighted average of the possible values it might take on, with the weights being the respective [[probability|probabilities]]. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.

In [[linear regression|regressions]] in which the [[dependent variable]] is assumed to be affected by both current and lagged (past) values of the [[independent variable]], a [[distributed lag]] function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a [[moving average model]] specifies an evolving variable as a weighted average of current and various lagged values of a random variable.

=== Mechanics ===
The terminology ''weight function'' arises from [[mechanics]]: if one has a collection of <math>n</math> objects on a [[lever]], with weights <math>w_1, \ldots, w_n</math> (where [[weight]] is now interpreted in the physical sense) and locations {{nowrap|<math>\boldsymbol{x}_1,\dotsc,\boldsymbol{x}_n</math>,}} then the lever will be in balance if the [[Lever|fulcrum]] of the lever is at the [[center of mass]]

:<math>\frac{\sum_{i=1}^n w_i \boldsymbol{x}_i}{\sum_{i=1}^n w_i},</math>

which is also the weighted average of the positions {{nowrap|<math>\boldsymbol{x}_i</math>.}}

== Continuous weights ==
In the continuous setting, a weight is a positive [[measure (mathematics)|measure]] such as <math>w(x) \, dx</math> on some [[domain (mathematical analysis)|domain]] <math>\Omega</math>, which is typically a [[subset]] of a [[Euclidean space]] <math>\R^n</math>, for instance <math>\Omega</math> could be an [[Interval (mathematics)|interval]] <math>[a,b]</math>. Here <math>dx</math> is [[Lebesgue measure]] and <math>w\colon \Omega \to \R^+</math> is a non-negative [[measurable]] [[mathematical function|function]]. In this context, the weight function <math>w(x)</math> is sometimes referred to as a [[density]].

=== General definition ===
If <math>f\colon \Omega \to \R</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted'' [[integral]]

:<math>\int_\Omega f(x)\ dx</math>

can be generalized to the ''weighted integral''

:<math>\int_\Omega f(x) w(x)\, dx</math>

Note that one may need to require <math>f</math> to be [[absolutely integrable function|absolutely integrable]] with respect to the weight <math>w(x) \, dx</math> in order for this integral to be finite.

=== Weighted volume ===
If ''E'' is a subset of <math>\Omega</math>, then the [[volume]] vol(''E'') of ''E'' can be generalized to the ''weighted volume''
:<math> \int_E w(x)\ dx,</math>

=== Weighted average ===
If <math>\Omega</math> has finite non-zero weighted volume, then we can replace the unweighted [[average]]

:<math>\frac{1}{\mathrm{vol}(\Omega)} \int_\Omega f(x)\ dx</math>

by the '''weighted average'''

:<math> \frac{\displaystyle\int_\Omega f(x)\, w(x) \, dx}{\displaystyle\int_\Omega w(x) \, dx}</math>

=== Bilinear form ===
If <math> f\colon \Omega \to {\mathbb R}</math> and <math> g\colon \Omega \to {\mathbb R}</math> are two functions, one can generalize the unweighted [[bilinear form]]

:<math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx</math>

to a weighted bilinear form

:<math>{\langle f, g \rangle}_w := \int_\Omega f(x) g(x)\ w(x)\ dx.</math>

See the entry on [[orthogonal polynomials]] for examples of weighted [[orthogonal functions]].

== See also ==
* [[Center of mass]]
* [[Numerical integration]]
* [[Orthogonality]]
* [[Weighted mean]]
* [[Linear combination]]
* [[Kernel (statistics)]]
* [[Measure (mathematics)]]
* [[Riemann–Stieltjes integral]]
* [[Weighting]]
* [[Window function]]

==References==
{{Reflist}}

{{DEFAULTSORT:Weight Function}}
[[Category:Mathematical analysis]]
[[Category:Measure theory]]
[[Category:Combinatorial optimization]]
[[Category:Functional analysis]]
[[Category:Types of functions]]

Weight function - Revision history

129.55.200.20: Undid revision 1253037847 by 223.123.19.139 (talk)