4.
Process Modeling
4.6.
Case Studies in Process Modeling
4.6.4.
Thermal Expansion of Copper Case Study
4.6.4.2.

Rational Function Models



Before proceeding with the case study, some explanation
of rational function models is required.

Polynomial Functions

A polynomial function is one that has the form,
$$ f(x) = a_{n}x^{n} + a_{n1}x^{n1} + ... + a_{2}x^{2} + a_{1}x + a_{0} \, , $$
with \(n\)
denoting a nonnegative integer that defines
the degree of the polynomial. A polynomial with a degree of 0
is simply a constant, 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.

Rational Functions

A rational function is simply the ratio of two polynomial
functions,
$$ f(x) = \frac{a_{n}x^{n} + a_{n1}x^{n1} + ... + a_{2}x^{2} + a_{1}x + a_{0}} {b_{m}x^{m} + b_{m1}x^{m1}
+ ... + b_{2}x^{2} + b_{1}x + b_{0}} \, ,$$
with \(n\)
denoting a nonnegative integer that defines
the degree of the numerator and \(m\)
is 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
graphs of some common
rational functions are shown in an appendix.

Polynomial Models

Historically, polynomial models are among the most frequently
used empirical models for fitting functions. 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.
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.
 Polynomial models have a shape/degree tradeoff. In order
to model data with a complicated structure, the degree of
the model must be high, indicating and the associated
number of parameters to be estimated will also be high.
This can result in highly unstable models.

Rational Function Models

A 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).
If modeling via polynomial models is inadequate due to any
of the limitations above, you should consider a rational
function model.

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 a particularly easy
nonlinear models to fit.

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 nusiance asymptotes (vertically)
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. Such asymptotes
should not discourage you from considering rational
function models as a choice for empirical modeling.
These nuisance asymptotes occur occasionally and
unpredictably, but the gain in flexibility of shapes
is well worth the chance that they may occur.

Starting Values for Rational Function 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, choose \(p\)
points from the data set, with \(p\)
denoting the number of parameters in the rational model.
For example, given the linear/quadratic model,
$$ \frac{A_0 + A_1x} {1 + B_1x + B_2x^{2}} \, , $$
we need to select four representative points.
We then perform a linear fit on the model
$$ y = A_0 + A_1x + ... + A_{p_n}x^{p_n}  B_1xy  ...  B_{p_d}x^{p_d}y + \varepsilon \, . $$
Here, \(p_n\)
and \(p_d\)
are the degrees of the numerator and denominator, respectively, and
the \(x\)
and \(y\)
contain the subset of points, not the full data set.
The estimated coefficients from this linear fit are
used as the starting values for fitting the nonlinear
model to the full data set.
Note:This type of fit, with the response
variable appearing on both sides of the function, should
only be used to obtain starting values
for the nonlinear fit. The statistical properties of fits
like this are not well understood.
The subset of points should be selected over the range
of the data. It is not critical which points
are selected, although you should avoid points that
are obvious outliers.
