Weight function

Summary

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"[1] and "meta-calculus".[2]

Discrete weightsEdit

General definitionEdit

In the discrete setting, a weight function   is a positive function defined on a discrete set  , which is typically finite or countable. The weight function   corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.

If the function   is a real-valued function, then the unweighted sum of   on   is defined as

 

but given a weight function  , the weighted sum or conical combination 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

 

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.

StatisticsEdit

Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity   measured multiple independent times   with variance  , the best estimate of the signal is obtained by averaging all the measurements with weight  , and the resulting variance is smaller than each of the independent measurements  . The maximum likelihood method weights the difference between fit and data using the same weights  .

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.

MechanicsEdit

The terminology weight function arises from mechanics: if one has a collection of   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 weightsEdit

In the continuous setting, a weight is a positive measure such as   on some domain  , which is typically a subset of a Euclidean space  , for instance   could be an interval  . Here   is Lebesgue measure and   is a non-negative measurable function. In this context, the weight function   is sometimes referred to as a density.

General definitionEdit

If   is a real-valued function, then the unweighted integral

 

can be generalized to the weighted integral

 

Note that one may need to require   to be absolutely integrable with respect to the weight   in order for this integral to be finite.

Weighted volumeEdit

If E is a subset of  , then the volume vol(E) of E can be generalized to the weighted volume

 

Weighted averageEdit

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

 

by the weighted average

 

Bilinear formEdit

If   and   are two functions, one can generalize the unweighted bilinear form

 

to a weighted bilinear form

 

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

See alsoEdit

ReferencesEdit

  1. ^ Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0-9771170-1-4, 1980.
  2. ^ Jane Grossman.Meta-Calculus: Differential and Integral, ISBN 0-9771170-2-2, 1981.