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.

1 Discrete weights
2 Continuous weights
3 See also

[edit] Discrete weights

[edit] General definition

In the discrete setting, a weight function $\scriptstyle w: A \to {\Bbb R}^+$ 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.

If the function $\scriptstyle f: A \to {\Bbb R}$ is a real-valued function, then the unweighted sum of $f$ on $A$ is defined as

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

but given a weight function $\scriptstyle w: A \to {\Bbb R}^+$ , the weighted sum is defined as

$\sum_{a \in A} f(a) w(a)$ .

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.

[edit] Statistics

Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity $f$ measured multiple independent times $f i$ with variance $\scriptstyle\sigma^2_i$ , the best estimate of the signal is obtained by averaging all the measurements with weight $\scriptstyle w_i=\frac 1 {\sigma_i^2}$ , and the resulting variance is smaller than each of the independent measurements $\scriptstyle\sigma^2=1/\sum w_i$ . The Maximum likelihood method weights the difference between fit and data using the same weights $w i$ .

[edit] Mechanics

The terminology weight function arises from mechanics: if one has a collection of $n$ objects on a lever, with weights $\scriptstyle w_1, \ldots, w_n$ (where weight is now interpreted in the physical sense) and locations : $\scriptstyle\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n$ , then the lever will be in balance if the fulcrum of the lever is at the center of mass

$\frac{\sum_{i=1}^n w_i \boldsymbol{x}_i}{\sum_{i=1}^n w_i}$ ,

which is also the weighted average of the positions $\scriptstyle\boldsymbol{x}_i$ .

[edit] Continuous weights

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