Weight function

Last updated

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" [1] and "meta-calculus". [2]

Contents

Discrete weights

General definition

In the discrete setting, a weight function is a positive function defined on a discrete set , which is typically finite or countable. 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-valued function, then the unweighted sum 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 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

In this case only the relative weights are relevant.

Statistics

Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity measured multiple independent times with variance , 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 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 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.

Mechanics

The terminology weight function arises from mechanics: if one has a collection of objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations , 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 .

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 , for instance could be an interval . Here is Lebesgue measure and is a non-negative measurable function. In this context, the weight function is sometimes referred to as a density.

General definition

If is a real-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.

Weighted volume

If E is a subset of , then the volume vol(E) of E can be generalized to the weighted volume

Weighted average

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

by the weighted average

Bilinear form

If and 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.

See also

Related Research Articles

In probability theory, the expected value of a random variable , denoted or , is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of . The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.

Integral Operation in calculus

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

In vector calculus and differential geometry, the generalized Stokes theorem, also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

Gaussian quadrature

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826. The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as

Fourier transform Mathematical transform that expresses a function of time as a function of frequency

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s.

Jensens inequality Theorem of convex functions

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

In the calculus of variations, a field of mathematical analysis, the functional derivative relates a change in a Functional to a change in a function on which the functional depends.

In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed x-axis and to the y-axis.

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation xf(x), for x ∈ [a, b].

In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as .

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes and hence also belongs to the class of margin-maximizing supervised classification algorithms. Consider a classification function

In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on Rn is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems.

In mathematics, the method of steepest descent or stationary-phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point, in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

In statistics and physics, multicanonical ensemble is a Markov chain Monte Carlo sampling technique that uses the Metropolis–Hastings algorithm to compute integrals where the integrand has a rough landscape with multiple local minima. It samples states according to the inverse of the density of states, which has to be known a priori or be computed using other techniques like the Wang and Landau algorithm. Multicanonical sampling is an important technique for spin systems like the Ising model or spin glasses.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

References

  1. Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN   0-9771170-1-4, 1980.
  2. Jane Grossman.Meta-Calculus: Differential and Integral, ISBN   0-9771170-2-2, 1981.