A method for choosing the smoothing parameter in a semi-parametric model for detecting change-points in blood flow

Abstract

In a smoothing spline model with unknown change-points, the choice of the smoothing parameter strongly influences the estimation of the change-point locations and the function at the change-points. In a tumor biology example, where change-points in blood flow in response to treatment were of interest, choosing the smoothing parameter based on minimizing generalized cross-validation (GCV) gave unsatisfactory estimates of the change-points. We propose a new method, aGCV, that re-weights the residual sum of squares and generalized degrees of freedom terms from GCV. The weight is chosen to maximize the decrease in the generalized degrees of freedom as a function of the weight value, while simultaneously minimizing aGCV as a function of the smoothing parameter and the change-points. Compared with GCV, simulation studies suggest that the aGCV method yields improved estimates of the change-point and the value of the function at the change-point.

Keywords:

• change-points
• smoothing spline
• partial spline
• generalized cross-validation
• generalized degrees of freedom

Acknowledgements

We are grateful to two anonymous reviewers for their helpful, thought-provoking comments on our original manuscript. We thank Professor Wensheng Guo for initially suggesting the smoothing spline approach, Professor Eli Glatstein for sharing his wealth of knowledge regarding PDT with us, and Professor Arjun Yodh for his insights into DCS. Dr Andrew Pole contributed valuable insights into the original analyses of these data. We thank Joann Miller and Steven S. Schenkel for help with data collection. Research supported by NIH-NCI 5-P01-CA-087971, R01 CA85831, and CA087971-S1 and CA129554. Code for implementing the programs is available from the first author upon request.