Weight function

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' and 'meta-calculus'. In the discrete setting, a weight function w : A → R + {displaystyle scriptstyle wcolon A o {mathbb {R} }^{+}} is a positive function defined on a discrete set A {displaystyle A} , which is typically finite or countable. The weight function w ( a ) := 1 {displaystyle 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 f : A → R {displaystyle scriptstyle fcolon A o {mathbb {R} }} is a real-valued function, then the unweighted sum of f {displaystyle f} on A {displaystyle A} is defined as but given a weight function w : A → R + {displaystyle scriptstyle wcolon A o {mathbb {R} }^{+}} , the weighted sum or conical combination is defined as 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 If A is a finite non-empty set, one can replace the unweighted mean or average by the weighted mean or weighted average