Polynomial and rational function modeling
polynomial and rational functions modeling change, polynomial and rational function modeling changeIn statistical modeling especially process modeling, polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting
Contents
 1 Polynomial function models
 11 Advantages
 12 Disadvantages
 2 Rational function models
 21 Advantages
 22 Disadvantages
 3 See also
 4 Bibliography
 41 Historical
 5 External links
Polynomial function models
Main articles: polynomial interpolation and polynomial regressionA 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 x^+a_x^+\cdots +a_x^+a_x+a_}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
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 Highdegree polynomials are notorious for oscillations between exactfit 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 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
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 x^+a_x^+\ldots +a_x^+a_x+a_}x^+b_x^+\ldots +b_x^+b_x+b_}}}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 ie, the case when the denominator is a constant
Advantages
Rational function models have the following advantages:
 Rational function models have a moderately simple form
 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 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 better interpolatory properties than polynomial models Rational functions are typically smoother and less oscillatory than polynomial models
 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 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 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
 Rational function models are moderately easy to handle computationally Although they are nonlinear models, rational function models are particularly easy nonlinear models to fit
 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
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
See also
 Response surface methodology
Bibliography
 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 , Wiley
 Kiefer, Jack Carl 1985 L D Brown; et al, eds Collected Papers III Design of Experiments SpringerVerlag ISBN 038796004X
 R H Hardin and N J A Sloane, "A New Approach to the Construction of Optimal Designs", Journal of Statistical Planning and Inference, vol 37, 1993, pp 339369
 R H Hardin and N J A Sloane, "ComputerGenerated Minimal and Larger Response Surface Designs: I The Sphere"
 R H Hardin and N J A Sloane, "ComputerGenerated Minimal and Larger Response Surface Designs: II The Cube"
 Ghosh, S; Rao, C R, eds 1996 Design and Analysis of Experiments Handbook of Statistics 13 NorthHolland ISBN 0444820612
 Draper, Norman & Lin, Dennis K J "Response Surface Designs" pp 343–375 Missing or empty title= help
 Gaffke, N & Heiligers, B "Polynomial Regression" pp 1149–1199 Missing or empty title= help
 Melas, Viatcheslav B 2006 Functional Approach to Optimal Experimental Design Lecture Notes in Statistics 184 SpringerVerlag ISBN 038798741X Modeling with rational functions
Historical
 Gergonne, J D 1815 "Application de la méthode des moindre quarrés a l'interpolation des suites" Annales de mathématiques pures et appliquées 6: 242–252
 Gergonne, J D 1974 "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:101016/0315086074900342
 Stigler, Stephen M 1974 "Gergonne's 1815 paper on the design and analysis of polynomial regression experiments" Historia Mathematica 1 4: 431–439 doi:101016/0315086074900330
 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:101093/biomet/12121 JSTOR 2331929
External links
 Rational Function Models
 

Computational statistics 
 
Correlation and dependence 
 
Regression analysis 
 
Regression as a statistical model 
 
Decomposition of variance 
 
Model exploration 
 
Background 
 
Design of experiments 
 
Numerical approximation 
 
Applications 
 

 

 
 
 
 
 
 
 

This article incorporates public domain material from the National Institute of Standards and Technology website http://wwwnistgov
polynomial and rational function modeling change, polynomial and rational functions modeling change
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29.10.2014
Polynomial and rational function modeling
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