In statistical modeling (especially process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.
A polynomial function is one that has the form
where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; with a degree of 1 is a line; with a degree of 2 is a quadratic; with a degree of 3 is a cubic, and so on.
Historically, polynomial models are among the most frequently used empirical models for curve fitting.
These models are popular for the following reasons.
However, polynomial models also have the following limitations.
When modeling via polynomial functions is inadequate due to any of the limitations above, the use of rational functions for modeling may give a better fit.
A rational function is simply the ratio of two polynomial functions.
with n denoting a non-negative integer that defines the degree of the numerator and m denoting a non-negative integer that defines the degree of the denominator. For fitting rational function models, the constant term in the denominator is usually set to 1. Rational functions are typically identified by the degrees of the numerator and denominator. For example, a quadratic for the numerator and a cubic for the denominator is identified as a quadratic/cubic rational function. The rational function model is a generalization of the polynomial model: rational function models contain polynomial models as a subset (i.e., the case when the denominator is a constant).
Rational function models have the following advantages:
Rational function models have the following disadvantages:
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