ⓘ Polynomial and rational function modeling
In statistical modeling, polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.
1. 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_{n1}x^{n1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}}where n is a nonnegative 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.
1.1. Polynomial function models Advantages
These models are popular for the following reasons.
 Polynomial models have well known and understood properties.
 Polynomial models have a simple form.
 Polynomial models have moderate flexibility of shapes.
 Polynomial models are computationally easy to use.
 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.
1.2. Polynomial function models Disadvantages
However, polynomial models also have the following limitations.
 Polynomial models have poor interpolatory properties. Highdegree polynomials are notorious for oscillations between exactfit values.
 While no procedure is immune to the biasvariance 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.
 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.
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.
2. 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_{n1}x^{n1}+\ldots +a_{2}x^{2}+a_{1}x+a_{0}}{b_{m}x^{m}+b_{m1}x^{m1}+\ldots +b_{2}x^{2}+b_{1}x+b_{0}}}}with n denoting a nonnegative integer that defines the degree of the numerator and m denoting a nonnegative 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.
2.1. Rational function models Advantages
Rational function models have the following advantages:
 Rational function models are moderately easy to handle computationally. Although they are nonlinear models, rational function models are particularly easy nonlinear models to fit.
 Rational functions have excellent extrapolatory powers. Rational functions can typically be tailored to model the function not only within the domain of the data, but also so as to be in agreement with theoretical/asymptotic behavior outside the domain of interest.
 Rational function models are a closed family. As with polynomial models, this means that rational function models are not dependent on the underlying metric.
 Rational function models have better interpolatory properties than polynomial models. Rational functions are typically smoother and less oscillatory than polynomial models.
 One common difficulty in fitting nonlinear models is finding adequate starting values. A major advantage of rational function models is the ability to compute starting values using a linear least squares fit. To do this, p points are chosen from the data set, with p denoting the number of parameters in the rational model. For example, given the linear/quadratic model
 Rational function models have excellent asymptotic properties. Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model.
 Rational function models can take on an extremely wide range of shapes, accommodating a much wider range of shapes than does the polynomial family.
 Rational function models have a moderately simple form.
 Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator. This in turn means that fewer coefficients will be required compared to the polynomial model.
2.2. Rational function models Disadvantages
Rational function models have the following disadvantages:
 The properties of the rational function family are not as well known to engineers and scientists as are those of the polynomial family. The literature on the rational function family is also more limited. Because the properties of the family are often not well understood, it can be difficult to answer the following modeling question: Given that data has a certain shape, what values should be chosen for the degree of the numerator and the degree on the denominator?
 Unconstrained rational function fitting can, at times, result in undesired vertical asymptotes due to roots in the denominator polynomial. The range of x values affected by the function "blowing up" may be quite narrow, but such asymptotes, when they occur, are a nuisance for local interpolation in the neighborhood of the asymptote point. These asymptotes are easy to detect by a simple plot of the fitted function over the range of the data. These nuisance asymptotes occur occasionally and unpredictably, but practitioners argue that the gain in flexibility of shapes is well worth the chance that they may occur, and that such asymptotes should not discourage choosing rational function models for empirical modeling.
3. Bibliography
 Smith, Kirstine 1918. "On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations". Biometrika. 12 1/2: 1–85. doi:10.1093/biomet/12.12.1. JSTOR 2331929.
 Atkinson, A. C. and Donev, A. N. and Tobias, R. D. 2007. Optimum Experimental Designs, with SAS. Oxford University Press. pp. 511+xvi. ISBN 9780199296606. CS1 maint: multiple names: authors list link
 Box, G. E. P. and Draper, Norman. 2007. Response Surfaces, Mixtures, and Ridge Analyses, Second Edition. "The application of the method of least squares to the interpolation of sequences". Historia Mathematica Translated by Ralph St. John and S. M. Stigler from the 1815 French ed. 1 4: 439–447. doi:10.1016/0315086074900342.
 Stigler, Stephen M. 1974. "Gergonnes 1815 paper on the design and analysis of polynomial regression experiments". Historia Mathematica. 1 4: 431–439. doi:10.1016/0315086074900330.
 Curve fitting Line regression Local polynomial regression Polynomial and rational function modeling Polynomial interpolation Response surface methodology
 The Tutte polynomial also called the dichromate or the Tutte Whitney polynomial is a graph polynomial It is a polynomial in two variables which plays
 Quartic function Fourth degree polynomial Quintic function Fifth degree polynomial Sextic function Sixth degree polynomial Rational functions A ratio
 Architecture Polynomial function model Rational function model Scientific modeling Unified Modeling Language View model This article incorporates public domain
 response  surface methodology Optimal designs Polynomial regression Polynomial and rational function modeling Surrogate model Probabilistic design Gradient  enhanced
 Non  uniform rational basis spline NURBS is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces
 used. Thorough composite rational and polynomial approximations have been given by Wichura and Acklam. Non  composite rational approximations have been
 M., Hanzon, B, Rational Approximation of Transfer Functions for Non  Negative EPT Densities Draft paper 2  Exponential  Polynomial  Trigonometric 2  EPT
 of polynomials with rational number coefficients, the subring of integer  valued polynomials is a free abelian group. It has as basis the polynomials Pk t
 ideal Function field of an algebraic variety Function field scheme theory Genus mathematics Polynomial lemniscate Quartic plane curve Rational normal
 solution of differential and integral equations are based on polynomial interpolation. The technique of rational function modeling is a generalization that
Users also searched:
...
