A weight function
is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum
or weighted average
. Weight functions occur frequently in statistics
and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus"
[Jane Grossman, Michael Grossman, Robert Katz. First Systems of Weighted Differential and Integral Calculus, , 1980.]
[Jane Grossman. Meta-Calculus: Differential and Integral, , 1981.]
In the discrete setting, a weight function
is a positive function defined on a discrete set
, which is typically
. The weight function
corresponds to the unweighted
situation in which all elements have equal weight. One can then apply this weight to various concepts.
If the function is a real number-valued function, then the unweighted summation of on is defined as
but given a weight function , the weighted sum or conical combination is defined as
One common application of weighted sums arises in numerical integration.
If B is a finite set subset of A, one can replace the unweighted cardinality |B| of B by the weighted cardinality
If A is a finite set non-empty set, one can replace the unweighted mean or average
by the weighted mean or weighted average
In this case only the relative weights are relevant.
Weighted means are commonly used in statistics
to compensate for the presence of bias. For a quantity
measured multiple independent times
, the best estimate of the signal is obtained
by averaging all the measurements with weight
the resulting variance is smaller than each of the independent measurements
. The maximum likelihood method weights the difference between fit and data using the same weights
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probability. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
In regressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
The terminology weight function
arises from mechanics
: if one has a collection of
objects on a lever
, with weights
is now interpreted in the physical sense) and locations :
, then the lever will be in balance if the Lever
of the lever is at the center of mass
which is also the weighted average of the positions .
In the continuous setting, a weight is a positive measure such as
on some domain
,which is typically a subset
of a Euclidean space
, for instance
could be an interval
is Lebesgue measure
is a non-negative measurable
function. In this context, the weight function
is sometimes referred to as a density
is a real number
-valued function, then the unweighted integral
can be generalized to the weighted integral
Note that one may need to require to be absolutely integrable with respect to the weight in order for this integral to be finite.
is a subset of
, then the volume
) of E
can be generalized to the weighted volume
has finite non-zero weighted volume, then we can replace the unweighted average
by the weighted average
are two functions, one can generalize the unweighted bilinear form
to a weighted bilinear form
See the entry on orthogonal polynomials for examples of weighted orthogonal functions.