Rational functions are typically identified by the orders 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.

If modeling via polynomial models is inadequate, you should consider a rational function model.

Note that fitting rational function models is also referred to Pade approximation.

1. Rational function models have a moderately simple form.

2. rational function models are theoretically exact for high enough degree for the numerator and denominator. An exact fit can be made if the degree of the numerator plus the degree of the denominator is equal to the number of observations minus one.

3. Rational function models are a closed family. As with polynomial models, this means that rational function models are not dependent on the underlying metric.

4. Rational function models can entertain an extemely wide range of shapes and behavior in the data. Specifically, they accomodate a much wider range of shapes than the polynomial family.

5. Rational function models have better interpolatory properties than polynomial models. Rational functions are typically smoother and less oscillatory than polynomial models between exact fit points.

6. 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.

7. 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.

8. Rational function models can often be used to model complicated structure with fairly low degree in both the numerator and denominator. This in turn means that fewer coefficients will be required compared to the polynomial model.

9. Rational function models are moderately easy computationally. Although they are a nonlinear model, rational function models are a particularly easy nonlinear model to fit.

1. The properties of the rational function family are not as well known to engineers and scientists as those of the polynomial family. The literature on rational function family is more limited and the typical analyst's knowledge of the behavior of various members of the family is more limited. From a practical point of view, if the properties of the family are not well understood, then this translates into difficulty in answering the following modeling question:
   Given that data has such and such shape, what value does the analyst choose for the degree of the numerator and for the degree on the denominator?

2. 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.

• If the degree of the numerator and denominator terms is equal (n=m), then y = a_n/b_m is a horizontal asymptote of the function.

• If the degree of the denominator term is greater than the degree of the numerator term, then y = 0 is a horizontal asymptote.

• If the degree of the denominator term is less than the degree of the numerator term, then there are no horizontal asymptotes.

• Where x is equal to a root of the denominator polynomial, the denominator term is zero and there is a vertical asymptote. The exception is the case where the root of the denominator term is also a root of the numerator term. However, for this case, we can cancel a factor from both the numerator and denominator (and we effectively have a lower degree rational function).

To do this, choose p points from the data set, where p is the number of parameters in the rational model. For example, given the linear/quadratic model

We then perform a linear fit on the model

The subset of points should be selected over the range of the data. It is not critical which exact points are selected, although you should avoid points that are obvious outliers.

1. Constant / Linear Rational Function
2. Linear / Linear Rational Function
3. Linear / Quadratic Rational Function
4. Quadratic / Linear Rational Function
5. Quadratic / Quadratic Rational Function
6. Cubic / Linear Rational Function
7. Cubic / Quadratic Rational Function
8. Linear / Cubic Rational Function
9. Quadratic / Cubic Rational Function
10. Cubic / Cubic Rational Function
11. Determining m and n for Rational Function Models