Weight function

Encyclopedia : W : WE : WEI : Weight function

A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be constructed in both discrete and continuous settings.

Discrete weights

In the discrete setting, a weight function [w: A \to ^+] is a positive function defined on a discrete set A, which is typically finite or countable. 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.

[f: A \to ]

is a real-valued function, then the unweighted sum of f on A is

[\sum_ f(a)];

but for a weight function

[w: A \to ^+],

the weighted sum is

[\sum_ f(a) w(a)].

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

[\sum_ w(a).]

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

[\frac

> \sum_ f(a)]

by the weighted mean or weighted average

[ \frac f(a) w(a)} w(a)}.].

In this case only the relative weights are relevant. Weighted means are commonly used in statistics to compensate for the presence of bias.

The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights

[w_1, \ldots, w_n]

(where weight is now interpreted in the physical sense) and locations

[x_1,\ldots,x_n],

then the lever will be in balance if the fulcrum of the lever is at the center of mass

[\frac^n w_i x_i}^n w_i}],

which is also the weighted average of the positions [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 subset of an Euclidean space [^n], for instance [\Omega] could be an interval [[a,b]]. Here dx is Lebesgue measure and [w: \Omega \to \R^+] is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

If [f: \Omega \to ] is a real-valued function, then the unweighted integral [\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.
If E is a subset of [\Omega], then the volume vol(E) of E can be generalized to the weighted volume [ \int_E w(x)\ dx].
If [\Omega] has finite non-zero weighted volume, then we can replace the unweighted average [\frac \int_\Omega f(x)\ dx] by the weighted average [ \frac.]
If [f: \Omega \to ] and [g: \Omega \to ] are two functions, one can generalize the unweighted inner product [\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx] to a weighted inner product [\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x) dx]. See the entry on Orthogonality for more details.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.