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Rational Functions
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A rational function is simply the ratio of two polynomial
functions,
$$ y = \frac{a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{2}x^{2} + a_{1}x + a_{0}} {b_{m}x^{m} + b_{m-1}x^{m-1}
+ ... + b_{2}x^{2} + b_{1}x + b_{0}} \, , $$
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. When 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.
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Rational Function Models
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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.
Note that fitting rational function models is also referred
to as the Pade approximation.
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Advantages
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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.
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Disadvantages
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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.
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General Properties of Rational Functions
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The following are general properties of rational functions.
- If the numerator and denominator
are of the same degree (\(n=m\)),
then
\(y = a_n / b_m\)
is a horizontal asymptote of the function.
- If the degree of the denominator is greater
than the degree of the numerator, then \(y=0\)
is a horizontal asymptote.
- If the degree of the denominator is less
than the degree of the numerator, then
there are no horizontal asymptotes.
- When \(x\)
is equal to a root of the denominator
polynomial, the denominator is zero and there
is a vertical asymptote. The exception is the case
when the root of the denominator is also a
root of the numerator. However, for this
case we can cancel a factor from both the
numerator and denominator (and we effectively have
a lower-degree rational function).
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Starting Values for Rational Function Models
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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,
$$ y = A_0 + A_1x + ... + A_{p_n}x^{p_n} - B_1xy - ... - B_{p_d}x^{p_d}y \, .$$
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 fit made using
the linear least squares algorithm 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
models 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.
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Example
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The thermal expansion of
copper case study contains an example of fitting a
rational function model.
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Specific Rational Functions
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- Constant / Linear Rational
Function
- Linear / Linear Rational
Function
- Linear / Quadratic Rational
Function
- Quadratic / Linear Rational
Function
- Quadratic / Quadratic Rational
Function
- Cubic / Linear Rational
Function
- Cubic / Quadratic Rational
Function
- Linear / Cubic Rational
Function
- Quadratic / Cubic Rational
Function
- Cubic / Cubic Rational
Function
- Determining m and
n for Rational Function Models
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