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
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
. Weight functions occur frequently in
statistics and
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, 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".
[Jane Grossma]
''Meta-Calculus: Differential and Integral''
, 1981.
Discrete weights
General definition
In the discrete setting, a weight function
is a positive function defined on a
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
set , which is typically
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
or
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
. 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, then the ''unweighted
sum of
on
'' is defined as
:
but given a ''weight function''
, the weighted sum or
conical combination Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ ...
is defined as
:
One common application of weighted sums arises in
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
.
If ''B'' is a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
subset of ''A'', one can replace the unweighted
cardinality , ''B'', of ''B'' by the ''weighted cardinality''
:
If ''A'' is a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
non-empty set, one can replace the unweighted
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
or
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
:
by the
weighted mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
or
weighted average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
:
In this case only the ''relative'' weights are relevant.
Statistics
Weighted means are commonly used in
statistics to compensate for the presence of
bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
. For a quantity
measured multiple independent times
with
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
, the best estimate of the signal is obtained by averaging all the measurements with weight and the resulting variance is smaller than each of the independent measurements The
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...
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
probabilities. 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 In statistics and econometrics, a distributed lag model is a model for time series data in which a regression equation is used to predict current values of a dependent variable based on both the current values of an explanatory variable and the ...
function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a
moving average model
In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. The moving-average model specifies that the output variable is cross-correlated with a ...
specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
Mechanics
The terminology ''weight function'' arises from
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
: if one has a collection of
objects on a
lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...
, with weights
(where
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
is now interpreted in the physical sense) and locations then the lever will be in balance if the
fulcrum
A fulcrum is the support about which a lever pivots.
Fulcrum may also refer to:
Companies and organizations
* Fulcrum (Anglican think tank), a Church of England think tank
* Fulcrum Press, a British publisher of poetry
* Fulcrum Wheels, a bicy ...
of the lever is at the
center of mass
:
which is also the weighted average of the positions
Continuous weights
In the continuous setting, a weight is a positive
measure such as
on some
domain , which is typically a
subset of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, for instance
could be an
interval