Weight function

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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 <math>w: A \to {\Bbb R}^+</math> is a positive function defined on a discrete set A, which is typically 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.

If

<math>f: A \to {\Bbb R}</math>

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

<math>\sum_{a \in A} f(a)</math>;

but for a weight function

<math>w: A \to {\Bbb R}^+</math>,

the weighted sum is

<math>\sum_{a \in A} f(a) w(a)</math>.

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

<math>\sum_{a \in B} w(a).</math>

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

<math>\frac{1}{|A|} \sum_{a \in A} f(a)</math>

by the weighted mean or weighted average

<math> \frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in A} w(a)}.</math>.

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

<math>w_1, \ldots, w_n</math>

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

<math>x_1,\ldots,x_n</math>,

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

<math>\frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}</math>,

which is also the weighted average of the positions <math>x_i</math>.

Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain <math>\Omega</math>, which is typically a subset of an Euclidean space <math>{\Bbb R}^n</math>, for instance <math>\Omega</math> could be an interval <math>[a,b]</math>. Here dx is Lebesgue measure and <math>w: \Omega \to \R^+</math> is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

1. If <math>f: \Omega \to {\Bbb R}</math> is a real-valued function, then the unweighted integral <math>\int_\Omega f(x)\ dx</math> can be generalized to the weighted integral <math>\int_\Omega f(x)\ w(x) dx</math>. 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.
2. If E is a subset of <math>\Omega</math>, then the volume vol(E) of E can be generalized to the weighted volume <math> \int_E w(x)\ dx</math>.
3. If <math>\Omega</math> has finite non-zero weighted volume, then we can replace the unweighted average <math>\frac{1}{vol(\Omega)} \int_\Omega f(x)\ dx</math> by the weighted average <math> \frac{\int_\Omega f(x)\ w(x) dx}{\int_\Omega w(x)\ dx}.</math>
4. If <math>f: \Omega \to {\Bbb R}</math> and <math>g: \Omega \to {\Bbb R}</math> are two functions, one can generalize the unweighted inner product <math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx</math> to a weighted inner product <math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ w(x) dx</math>. See the entry on Orthogonality for more details.