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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. The result of this application of a weight function is a weighted sum or . Weight functions occur frequently in 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"Jane Grossman, Michael Grossman, Robert Katz. First Systems of Weighted Differential and Integral Calculus, , 1980. and "meta-calculus".Jane Grossman. Meta-Calculus: Differential and Integral, , 1981.


Discrete weights

General definition
In the discrete setting, a weight function \scriptstyle w\colon A \to {\Bbb R}^+ is a positive function defined on a discrete set A, which is typically or . 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 \scriptstyle f\colon A \to {\Bbb R} is a -valued function, then the unweighted of f on A is defined as

\sum_{a \in A} f(a);

but given a weight function \scriptstyle w\colon A \to {\Bbb R}^+, the weighted sum or conical combination is defined as

\sum_{a \in A} f(a) w(a).

One common application of weighted sums arises in numerical integration.

If B is a subset of A, one can replace the unweighted |B| of B by the weighted cardinality

\sum_{a \in B} w(a).

If A is a non-empty set, one can replace the unweighted or

\frac{1}
\sum_{a \in A} f(a)

by the or

\frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.

In this case only the relative weights are relevant.


Statistics
Weighted means are commonly used in to compensate for the presence of bias. For a quantity f measured multiple independent times f_i with \scriptstyle\sigma^2_i, the best estimate of the signal is obtained by averaging all the measurements with weight \scriptstyle w_i=\frac 1 {\sigma_i^2}, and the resulting variance is smaller than each of the independent measurements \scriptstyle\sigma^2=1/\sum w_i. The maximum likelihood method weights the difference between fit and data using the same weights w_i.

The of a random variable is the weighted average of the possible values it might take on, with the weights being the respective . 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 regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a 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 : if one has a collection of n objects on a , with weights \scriptstyle w_1, \ldots, w_n (where is now interpreted in the physical sense) and locations :\scriptstyle\boldsymbol{x}_1,\dotsc,\boldsymbol{x}_n, then the lever will be in balance if the of the lever is at the center of mass

\frac{\sum_{i=1}^n w_i \boldsymbol{x}_i}{\sum_{i=1}^n w_i},

which is also the weighted average of the positions \scriptstyle\boldsymbol{x}_i.


Continuous weights
In the continuous setting, a weight is a positive measure such as w(x) \, dx on some domain \Omega,which is typically a of a \scriptstyle{\Bbb R}^n, for instance \Omega could be an interval a,b. Here dx is and \scriptstyle w\colon \Omega \to \R^+ is a non-negative function. In this context, the weight function w(x) is sometimes referred to as a .


General definition
If f\colon \Omega \to {\Bbb R} is a -valued function, then the unweighted

\int_\Omega f(x)\ dx

can be generalized to the weighted integral

\int_\Omega f(x) w(x)\, dx

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 \Omega, then the vol( E) of E can be generalized to the weighted volume
\int_E w(x)\ dx,


Weighted average
If \Omega has finite non-zero weighted volume, then we can replace the unweighted

\frac{1}{\mathrm{vol}(\Omega)} \int_\Omega f(x)\ dx

by the weighted average

\frac{\int_\Omega f(x)\ w(x) dx}{\int_\Omega w(x)\ dx}


Bilinear form
If f\colon \Omega \to {\Bbb R} and g\colon \Omega \to {\Bbb R} are two functions, one can generalize the unweighted

\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx

to a weighted bilinear form

\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x)\ dx.

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


See also
  • Center of mass
  • Numerical integration
  • Kernel (statistics)
  • Measure (mathematics)
  • Riemann–Stieltjes integral

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