# Function approximation

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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]}

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[edit]

**^**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.- ^
^{a}^{b}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. **^**Mhaskar, Hrushikesh Narhar; Pai, Devidas V. (2000).*Fundamentals of Approximation Theory*. CRC Press. ISBN 978-0-8493-0939-7.**^**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[edit]

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