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.

Contents

Discrete weights

Continuous weights

Discrete weights

In the discrete setting, a weight
function $w:\; A\; o\; \{Bbb\; R\}^+$ 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
:$f:\; A\; o\; \{Bbb\; R\}$
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''
:$w:\; A\; o\; \{Bbb\; R\}^+$,
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
: sum_{a in A} f(a)
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 $f$ measured multiple independent times $f\_i$ with variance $sigma^2\_i$, the best estimate of the signal is obtained
by averaging all the measurements with weigth , and
the resulting variance is smaller than each of the independent measurements
$sigma^2=1/sum\; w\_i$. The Maximum likelihood method weights the
difference between fit and data using the same weights $w\_i$ .
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
:,
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 $[a,b]$. 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.
# If $f:\; Omega\; o\; \{Bbb\; R\}$ 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 by the 'weighted average'
# If $f:\; Omega\; o\; \{Bbb\; R\}$ and $g:\; Omega\; o\; \{Bbb\; R\}$ 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.