Function approximation
Encyclopedia
The need for function approximations arises in many branches of applied mathematics
, and computer science
in particular. In general, a function approximation problem asks us to select a function
among a well-defined class that closely matches ("approximates") a target function in a task-specific way.
One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory
is the branch of numerical analysis
that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomial
s or rational function
s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).
Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain
and codomain
of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real number
s, techniques of interpolation
, extrapolation
, regression analysis
, and curve fitting
can be used. If the codomain
(range or target set) of g is a finite set, one is dealing with a classification problem instead.
To some extent the different problems (regression, classification, fitness approximation
) have received a unified treatment in statistical learning theory
, where they are viewed as supervised learning
problems.
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
in particular. In general, a function approximation problem asks us to select a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
among a well-defined class that closely matches ("approximates") a target function in a task-specific way.
One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
is the branch of numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s or rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).
Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
and codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, techniques of interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
, extrapolation
Extrapolation
In mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...
, regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
, and curve fitting
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...
can be used. If the codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
(range or target set) of g is a finite set, one is dealing with a classification problem instead.
To some extent the different problems (regression, classification, fitness approximation
Fitness approximation
In function optimization, fitness approximation is a method for decreasing the number of fitness function evaluations to reach a target solution...
) have received a unified treatment in statistical learning theory
Statistical learning theory
Statistical learning theory is an ambiguous term.#It may refer to computational learning theory, which is a sub-field of theoretical computer science that studies how algorithms can learn from data....
, where they are viewed as supervised learning
Supervised learning
Supervised learning is the machine learning task of inferring a function from supervised training data. The training data consist of a set of training examples. In supervised learning, each example is a pair consisting of an input object and a desired output value...
problems.
See also
- Radial basis function networkRadial basis function networkA radial basis function network is an artificial neural network that uses radial basis functions as activation functions. It is a linear combination of radial basis functions...
- Fitness approximationFitness approximationIn function optimization, fitness approximation is a method for decreasing the number of fitness function evaluations to reach a target solution...