Sparsity Inducing Spline Penalties

In my final project for PUBH 8492 - Richly Parameterized Models, I explored via a Monte Carlo simulation study the performance of three different methods for estimating penalized spline knot coefficients- lasso, fused lasso, and the mixed linear model (MLM). Comparisons were made for fitting splines to two "difficult" underlying functions - a bathtub function with abruptly changing smoothness, and a doppler function with differing degrees of curvature over time. Lasso- and fusion-penalized splines were superior when fitting doppler-generated data, but the MLM spline estimated using the restricted likelihood was superior when the true underlying function was a bathtub.

The plot shows a lasso- and fusion-penalized spline with identical degrees of freedom fit to the same data. Fits are nearly identical (top panels), but the knot coefficient patterns (bottom panels) are very different. Lasso penalization uses the minimal number of non-zero coefficients possible, while the fusion penalty encourages the minimal number of fused groups of coefficients.

Code for estimating splines using these penalties, as well as for implementing the simulation study, can be found on my Github page, and the final paper is below.