In statistical modeling (especially process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.
Polynomial function models
A polynomial function is one that has the form
y = a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 {\displaystyle y=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}}
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.
Advantages These models are popular for the following reasons.
Polynomial models have a simple form. Polynomial models have well known and understood properties. Polynomial models have moderate flexibility of shapes. Polynomial models are a closed family. Changes of location and scale in the raw data result in a polynomial model being mapped to a polynomial model. That is, polynomial models are not dependent on the underlying metric. Polynomial models are computationally easy to use.
Disadvantages However, polynomial models also have the following limitations.
Polynomial models have poor interpolatory properties. High-degree polynomials are notorious for oscillations between exact-fit values. Polynomial models have poor extrapolatory properties. Polynomials may provide good fits within the range of data, but they will frequently deteriorate rapidly outside the range of the data. Polynomial models have poor asymptotic properties. By their nature, polynomials have a finite response for finite x values and have an infinite response if and only if the x value is infinite. Thus polynomials may not model asymptotic phenomena very well. While no procedure is immune to the bias-variance tradeoff, polynomial models exhibit a particularly poor tradeoff between shape and degree. In order to model data with a complicated structure, the degree of the model must be high, indicating that the associated number of parameters to be estimated will also be high. This can result in highly unstable models. 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.
Rational function models A rational function is simply the ratio of two polynomial functions.
y = a n x n + a n − 1 x n − 1 + … + a 2 x 2 + a 1 x + a 0 b m x m + b m − 1 x m − 1 + … + b 2 x 2 + b 1 x + b 0 {\displaystyle y={\frac {a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{2}x^{2}+a_{1}x+a_{0}}{b_{m}x^{m}+b_{m-1}x^{m-1}+\ldots +b_{2}x^{2}+b_{1}x+b_{0}}}}
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).
Advantages Rational function models have the following advantages:
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