Smoothing splines approximation using Hilbert curve basis selection.

Area of Science:

• Statistics
• Nonparametric Regression

Background:

• Smoothing splines are widely used in nonparametric regression but face computational challenges with large sample sizes (n) and multiple predictors (d).
• Standard smoothing spline computation is O(n^3), and existing basis selection methods, while faster (O(nq^2)), often assume uniform predictor distributions, limiting their applicability.

Purpose of the Study:

• To develop an efficient and adaptive algorithm for smoothing splines that overcomes the limitations of existing basis selection methods.
• To improve computational efficiency and maintain accuracy for nonparametric regression with large datasets and non-uniformly distributed predictors.

Main Methods:

• Developed a novel algorithm for smoothing splines that is adaptive to the unknown probability density function of predictors.
• Utilized basis function approximation to reduce computational complexity from O(n^3) to O(nq^2).

Main Results:

• The proposed estimator achieves convergence rates comparable to full-basis estimators when the number of basis functions (q) is O(n^2).
• Numerical studies demonstrated superior performance of the new estimator against mainstream competitors on various synthetic datasets.

Conclusions:

• The developed algorithm offers an efficient and robust solution for nonparametric regression with smoothing splines, particularly for large and non-uniformly distributed datasets.
• This adaptive approach enhances the practical applicability of smoothing splines in statistical modeling.