Weight function

From Exampleproblems

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 use 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 $Ω$ , which is typically a subset of an Euclidean space ${\Bbb R}^n$ , for instance $Ω$ 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 $f: \Omega \to {\Bbb R}$ 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 $Ω$ , then the volume vol(E) of E can be generalized to the weighted volume $\int_E w(x)\ dx$ .
If $Ω$ 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.