Function approximation

In general, a function approximation problem asks us to select a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> among a well-defined class that closely matches ("approximates") a target function in a task-specific way.[better source needed] The need for function approximations arises in many branches of <a href="/facts/Applied_mathematics/qyqklXXm">applied mathematics</a>, and <a href="/facts/Computer_science/lN3pEyPz">computer science</a> in particular [why?], such as predicting the growth of microbes in <a href="/facts/Microbiology/k7udsq81">microbiology</a>. 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 <a href="/facts/Approximation_theory/87AaPXLp">approximation theory</a> is the branch of <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a> that investigates how certain known functions (for example, <a href="/facts/Special_function/JETSeucA">special functions</a>) can be approximated by a specific class of functions (for example, <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> or <a href="/facts/Rational_function/2nJGu0Ko">rational functions</a>) 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 <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> and <a href="/facts/Codomain/MGc43nwQ">codomain</a> of g, several techniques for approximating g may be applicable. For example, if g is an operation on the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, techniques of <a href="/facts/Interpolation/8PyetX4f">interpolation</a>, <a href="/facts/Extrapolation/wggUrqjF">extrapolation</a>, <a href="/facts/Regression_analysis/n6z5Tf7K">regression analysis</a>, and <a href="/facts/Curve_fitting/zrLKw6Xj">curve fitting</a> can be used. If the <a href="/facts/Codomain/MGc43nwQ">codomain</a> (range or target set) of g is a finite set, one is dealing with a <a href="/facts/Statistical_classification/jXXHRkXR">classification</a> problem instead.
To some extent, the different problems (regression, classification, <a href="/facts/Fitness_approximation/PpSnne4V">fitness approximation</a>) have received a unified treatment in <a href="/facts/Statistical_learning_theory/eK1V4mxN">statistical learning theory</a>, where they are viewed as <a href="/facts/Supervised_learning/JlNFNPFt">supervised learning</a> problems.

Function approximation open-in-new

Function approximation