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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 mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...

, and

computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

in particular , such as predicting the growth of microbes in

microbiology Microbiology () is the branches of science, scientific study of microorganisms, those being of unicellular organism, unicellular (single-celled), multicellular organism, multicellular (consisting of complex cells), or non-cellular life, acellula ...

. 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 function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...

is the branch of

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

that investigates how certain known functions (for example,

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...

s) can be approximated by a specific class of functions (for example,

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

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 and

codomain In mathematics, a codomain, counter-domain, or set of destination of a function is a set into which all of the output of the function is constrained to fall. It is the set in the notation . The term '' range'' is sometimes ambiguously used to ...

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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, techniques of

interpolation In the mathematics, 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 ...

extrapolation In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...

, regression analysis, and

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 ...

can be used. If the

(range or target set) of ''g'' is a finite set, one is dealing with a

classification Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identif ...

problem instead. To some extent, the different problems (regression, classification,

fitness approximation Fitness approximationY. JinA comprehensive survey of fitness approximation in evolutionary computation ''Soft Computing'', 9:3–12, 2005 aims to approximate the objective or fitness functions in evolutionary optimization by building up machine l ...

) have received a unified treatment in

statistical learning theory Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on da ...

, where they are viewed as

supervised learning In machine learning, supervised learning (SL) is a paradigm where a Statistical model, model is trained using input objects (e.g. a vector of predictor variables) and desired output values (also known as a ''supervisory signal''), which are often ...

problems.

References

See also