[R] nonlinear modeling with rational functions?
Martin Maechler
maechler at stat.math.ethz.ch
Thu Jun 17 10:04:12 CEST 2004
>>>>> "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 Padé 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 Padé 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
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