yawiki.org entry for 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.

    Weight function
        Discrete weights
        Continuous weights

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Discrete weights

In the discrete setting, a weight
function

w: A 	o ^+

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

$f: A o$

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

$sum_ f(a)$ ;

but for a weight function

$w: A o ^+$ ,

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

$rac sum_ f(a)$

by the weighted mean or weighted average

$rac.$ .

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

$rac$ ,

which is also the weighted average of the positions

x_i

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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 	o R^+

is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

f: Omega 	o

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

Omega

has finite non-zero weighted volume, then we can replace the unweighted average

rac int_Omega f(x) dx

by the weighted average

rac.

f: Omega 	o

and

g: Omega 	o

are two functions, one can generalize the unweighted inner product

langle f, g

angle

= int_Omega f(x) g(x) dx to a weighted inner product $langle f, g$

angle

= int_Omega f(x) g(x) w(x) dx. See the entry on Orthogonality for more details.

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