Function Approximation

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, polynomials or rational functions) 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 numbers, 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.

Other articles related to "function approximation, functions, approximation, function":

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... In order to address the last issue mentioned in the previous section, function approximation methods are used ... In linear function approximation one starts with a mapping that assigns a finite dimensional vector to each state-action pair ... However, linear function approximation is not the only choice ...
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... Again, we choose the number of basis functions as five and the size of the training set to be 100 exemplars generated by the chaotic time series ... Typically accuracy with normalized basis functions increases even more over unnormalized functions as input dimensionality increases ...
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... Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that ... is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough.) A “calculus book” approach divides the ... Spaced function values x −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 f(x) 2.22800 2.45663 2.67200 2.32475 0.64400 −0.92575 −0.94000 −0.16963 0.83600 x −1.75 −1.25 −0.75 −0.25 0.25 0.75 1.25 1 ...

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