[R] nonlinear modeling with rational functions?

[Martin Maechler]

>>>>> "Tamas" == Tamas Papp <tpapp at axelero.hu>
>>>>>     on Thu, 17 Jun 2004 08:40:13 +0200 writes:

    Tamas> On Wed, Jun 16, 2004 at 10:28:39AM -0500, Spencer Graves wrote:
    >> Rational functions (ratios of polynomials) often provide good 
    >> approximations to many functions.  Does anyone know of any literature on 
    >> nonlinear modeling with rational functions, sequential estimation, 
    >> diagnostics, etc.?  I know I can do it with "nls" and other nonlinear 
    >> regression functions, but I'm wondering what literature might exist 
    >> discussing how a search for an appropriate rational approximation? 

    Tamas> The book

    Tamas> @Book{judd98,
    Tamas>  author =       {Judd, Kenneth L},
    Tamas>  title =        {Numerical methods in economics},
    Tamas>  publisher =    {MIT Press},
    Tamas>  year =         1998
    Tamas> }

    Tamas> provides some information about constructing rational approximations
    Tamas> (aka Padé approximations) for functions with known algebraic forms,
    Tamas> but not for the estimation from data.

    Tamas> However, the text has some references, in particular

    Tamas> Cuyt, A and L Wuytack. 1986. Nonlinear Numerical Methods: Theory and
    Tamas> Practice. Amsterdam: North-Holland

    Tamas> which you might find helpful.

But be aware:  
In these contexts, one is interested in  L_inf() norm
approximations, not in L_2 or even L_1 as we are in statistics.

L_inf aka "sup-norm" aka "minimax" or "worst case" approximation
makes much sense in the context of numerical approximation:
e.g. you might want a Padé approximation of the sin() function with
a maximal (absolute) error of 1e-12 to be implemented for
your pocket calculator.

OTOH, the sup-norm is quite a bad idea for data fitting, since
there, errors are typically either normal or more heavy tailed,
i.e. if using an L_p norm, it should be p <= 2.

Hoping this helps even more,
Martin Maechler