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In general, a function approximation problem asks us to select a function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
in particular , such as predicting the growth of microbes in
microbiology Microbiology () is the scientific study of microorganisms, those being unicellular (single cell), multicellular (cell colony), or acellular (lacking cells). Microbiology encompasses numerous sub-disciplines including virology, bacteriology, ...
. Function approximations are used where theoretical models are unavailable or hard to compute. One can distinguish two major classes of function approximation problems: First, for known target functions
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. Note that what is meant by ''best'' and ''simpler'' wil ...
is the branch of
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example,
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s or
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
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 may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and codomain of ''g'', several techniques for approximating ''g'' may be applicable. For example, if ''g'' is an operation on the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, techniques of
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...
, extrapolation,
regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
, 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 Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...
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.


References


See also

*
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. Note that what is meant by ''best'' and ''simpler'' wil ...
* Fitness approximation * Kriging *
Least squares (function approximation) In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation can be defined as that which minimizes the difference bet ...
*
Radial basis function network In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inp ...
Regression analysis Statistical approximations {{statistics-stub