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 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" and "meta-calculus".
Contents
General definition
In the discrete setting, a weight function
If the function
but given a weight function
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 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
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 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 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
which is also the weighted average of the positions
Continuous weights
In the continuous setting, a weight is a positive measure such as
General definition
If
can be generalized to the weighted integral
Note that one may need to require
Weighted volume
If E is a subset of
Weighted average
If
by the weighted average
Bilinear form
If
to a weighted bilinear form
See the entry on orthogonal polynomials for examples of weighted orthogonal functions.