6
$\begingroup$

I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there's a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.

Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it's stated that the results regarding the 'optimality' of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.

I would like to know if it's a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $\| \cdot \|_2^2$. In many articles I've found phrases like "we will set the knots at the Chebyshev sites, that are well known to be good..." making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.

If your could put some light on the problem I will be very grateful.

Improve this question
$\endgroup$
3
  • $\begingroup$ @RamiroScorolli: You might want to contact the authors of these articles directly to see if they have derived these results themselves or they reviewed them from somewhere. $\endgroup$
    – usεr11852
    Jan 19, 2019 at 22:51
  • $\begingroup$ I am not sure I 100% understand your question. Are you referring to the Schoenberg and Whitney theorem somehow (the theorem says that splines spaces are weak Chebyshev spaces)? On the other hand I do not know if this could help you in a flexible LASSO-regression settings you mentioned in your question. I would rather go for a penalized spline approach to tackle this kind of problems (but this, of course, is my personal taste) $\endgroup$
    – Gi_F.
    Jan 23, 2019 at 14:13
  • $\begingroup$ They might at least make good initializations for an iterative knot estimation method. $\endgroup$
    – John Madden
    May 19, 2021 at 2:16

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.