basis function

The term basis function is used for functions selected to help describe some function or relation of interest, to specify it exactly or to approximate it. One might (often roughly) specify the relation-of-interest by a parameterized combination of such basis functions. Modeling, e.g., through numerical approximations of partial differential equations (PDEs), can be organized through the use of basis functions, parameterized and combined (e.g., summed) to approximate a function of interest. An example is the sine function, which can serve a basis function for a Fourier series. The concept is also used in the finite element method and similar methods of solving PDEs.

The term basis function is also used regarding models, e.g., statistical analysis or machine learning. There are well-known techniques to find a good fit when a relationship is known to be linear. However, for data without such a simple relation, some particular function (e.g., square, polynomial of some degree, exponential) may fit; for example, by guessing that the data fits some square relation, you can confirm and parameterize that fit. The term basis function is used for such functions you presume might fit. In some cases, it is possible to reduce trials by trying a linear combination of such functions simultaneously, revealing one or which combination of the basis functions that fit the data well. Some phenomena you model will have an inherent function that will fit if you know to try it, but also there are very flexible types of functions that would not be termed inherent, but can be the basis functions for useful heuristic models.