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| 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. They 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. [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,''The First Systems of Weighted Differential and Integral Calculus''], ISBN 0-9771170-1-4, 1980.</ref> and "meta-calculus".<ref>Jane Grossman.[http://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>
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| == Discrete weights ==
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| === General definition ===
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| In the discrete setting, a weight function <math>\scriptstyle w\colon A \to {\Bbb R}^+</math> is a positive function defined on a [[discrete mathematics|discrete]] [[Set (mathematics)|set]] '''<math>A</math>''', which is typically
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| [[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.
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| If the function <math>\scriptstyle f\colon A \to {\Bbb R}</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted [[sum]] of <math>f</math> on '''<math>A</math>''''' is defined as
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| :<math>\sum_{a \in A} f(a);</math>
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| but given a ''weight function'' <math>\scriptstyle w\colon A \to {\Bbb R}^+</math>, the '''weighted sum''' or [[conical combination]] is defined as
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| :<math>\sum_{a \in A} f(a) w(a).</math>
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| One common application of weighted sums arises in [[numerical integration]].
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| If ''B'' is a [[finite set|finite]] subset of ''A'', one can replace the unweighted [[cardinality]] ''|B|'' of ''B'' by the ''weighted cardinality''
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| :<math>\sum_{a \in B} w(a).</math>
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| If ''A'' is a [[finite set|finite]] non-empty set, one can replace the unweighted [[mean]] or [[average]]
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| :<math>\frac{1}{|A|} \sum_{a \in A} f(a)</math>
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| by the [[weighted mean]] or [[weighted average]]
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| :<math> \frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.</math>
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| In this case only the ''relative'' weights are relevant.
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| === Statistics ===
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| Weighted means are commonly used in [[statistics]] to compensate for the presence of [[bias]]. For a quantity <math>f</math> measured multiple independent times <math>f_i</math> with [[variance]] <math>\scriptstyle\sigma^2_i</math>, the best estimate of the signal is obtained
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| by averaging all the measurements with weight <math>\scriptstyle w_i=\frac 1 {\sigma_i^2}</math>, and
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| the resulting variance is smaller than each of the independent measurements
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| <math>\scriptstyle\sigma^2=1/\sum w_i</math>. The [[Maximum likelihood]] method weights the difference between fit and data using the same weights <math>w_i</math>.
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| === Mechanics ===
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| The terminology ''weight function'' arises from [[mechanics]]: if one has a collection of ''<math>n</math>'' objects on a [[lever]], with weights <math>\scriptstyle w_1, \dotsc, w_n</math> (where [[weight]] is now interpreted in the physical sense) and locations :<math>\scriptstyle\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]]
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| :<math>\frac{\sum_{i=1}^n w_i \boldsymbol{x}_i}{\sum_{i=1}^n w_i},</math>
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| which is also the weighted average of the positions <math>\scriptstyle\boldsymbol{x}_i</math>.
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| == Continuous weights ==
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| In the continuous setting, a weight is a positive [[measure (mathematics)|measure]] such as ''<math>w(x) dx</math>'' on some [[domain (mathematics)|domain]] '''<math>\Omega</math>''',which is typically a [[subset]] of a [[Euclidean space]] <math>\scriptstyle{\Bbb 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>\scriptstyle 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]].
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| === General definition ===
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| If <math>f\colon \Omega \to {\Bbb R}</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted'' [[integral]]
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| :<math>\int_\Omega f(x)\ dx</math> | |
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| can be generalized to the ''weighted integral''
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| :<math>\int_\Omega f(x) w(x)\, dx</math>
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| Note that one may need to require ''<math>f</math>'' to be [[absolutely integrable]] with respect to the weight ''<math>w(x) dx</math>'' in order for this integral to be finite.
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| === Weighted volume ===
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| If ''E'' is a subset of <math>\Omega</math>, then the [[volume]] vol(''E'') of ''E'' can be generalized to the ''weighted volume''
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| :<math> \int_E w(x)\ dx,</math>
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| === Weighted average ===
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| If <math>\Omega</math> has finite non-zero weighted volume, then we can replace the unweighted [[average]]
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| :<math>\frac{1}{\mathrm{vol}(\Omega)} \int_\Omega f(x)\ dx</math>
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| by the '''weighted average'''
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| :<math> \frac{\int_\Omega f(x)\ w(x) dx}{\int_\Omega w(x)\ dx}</math>
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| === Inner product ===
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| If <math>\scriptstyle f\colon \Omega \to {\Bbb R}</math> and <math>\scriptstyle g\colon \Omega \to {\Bbb R}</math> are two functions, one can generalize the unweighted [[inner product]]
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| :<math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx</math>
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| to a weighted inner product
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| :<math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x)\ dx.</math>
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| See the entry on [[Orthogonality]] for more details.
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| == See also ==
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| * [[Center of mass]]
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| * [[Numerical integration]]
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| * [[Orthogonality]]
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| * [[Weighted mean]]
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| * [[Kernel (statistics)]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Weight Function}}
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| [[Category:Mathematical analysis]]
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| [[Category:Measure theory]]
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| [[Category:Combinatorial optimization]]
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| [[Category:Functional analysis]]
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| [[Category:Types of functions]]
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