Function approximation

Not to be confused with Curve fitting.

In general, a function approximation problem asks us to select a function among a well-defined class^{[citation needed][clarification needed]} that closely matches ("approximates") a target function^{[citation needed]} in a task-specific way.^{[1][better source needed]} The need for function approximations arises in many branches of applied mathematics, and computer science in particular ^[why?],^{[citation needed]} such as predicting the growth of microbes in microbiology.^[2] Function approximations are used where theoretical models are unavailable or hard to compute.^[2]

Several progressively more accurate approximations of the step function.

An asymmetrical Gaussian function fit to a noisy curve using regression.

One can distinguish^{[citation needed]} 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.).^[3]

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.^{[citation needed]} 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.^[4]

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.^{[citation needed]}

References

1. Lakemeyer, Gerhard; Sklar, Elizabeth; Sorrenti, Domenico G.; Takahashi, Tomoichi (2007-09-04). RoboCup 2006: Robot Soccer World Cup X. Springer. ISBN 978-3-540-74024-7.
2. Basheer, I.A.; Hajmeer, M. (2000). "Artificial neural networks: fundamentals, computing, design, and application" (PDF). Journal of Microbiological Methods. 43 (1): 3–31. doi:10.1016/S0167-7012(00)00201-3. PMID 11084225. S2CID 18267806.
3. Mhaskar, Hrushikesh Narhar; Pai, Devidas V. (2000). Fundamentals of Approximation Theory. CRC Press. ISBN 978-0-8493-0939-7.
4. Charte, David; Charte, Francisco; García, Salvador; Herrera, Francisco (2019-04-01). "A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations". Progress in Artificial Intelligence. 8 (1): 1–14. arXiv:1811.12044. doi:10.1007/s13748-018-00167-7. ISSN 2192-6360. S2CID 53715158.

See also

• Approximation theory
• Fitness approximation
• Kriging
• Least squares (function approximation)
• Radial basis function network