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 $w: A \to {\Bbb R}^+$ 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 {\Bbb R}

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

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

;

but for a weight function

w: A \to {\Bbb R}^+

the weighted sum is

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

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

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

by the weighted mean or weighted average

\frac{\sum_{a \in A} f(a) w(a)}{\sum_{a \in 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{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^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 ${\Bbb R}^n$ , for instance $\Omega$ could be an interval ^*. 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 $f: \Omega \to {\Bbb R}$ 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 $\frac{1}{vol(\Omega)} \int_\Omega f(x)\ dx$ by the weighted average $\frac{\int_\Omega f(x)\ w(x) dx}{\int_\Omega w(x)\ dx}.$
If $f: \Omega \to {\Bbb R}$ and $g: \Omega \to {\Bbb R}$ 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.

Mathematical analysis | Measure theory | Combinatorial optimization | Functional analysis

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Discrete weights

Continuous weights