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 $||.||_2^2$. 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 you could put some light on the problem I will be very grateful.

  • $\begingroup$ Either let the knots be parameters or choose them according to the density of abscissas of your data. $\endgroup$ – Oppenede Jan 18 '19 at 13:56
  • $\begingroup$ I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede $\endgroup$ – RScrlli Jan 18 '19 at 16:31

The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.

To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).

Orthogonal Regression Polynomials family

The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.

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