Statistics 695U

Multivariate Function Estimation using Splines

Spring 2006

Course Outline

This course presents a systematic treatment of multivariate function estimation via the penalized likelihood method. Emphasis will be placed on the structural model construction, the selection of smoothing parameters, and the use of software tools. A tentative outline of coverage follows.

• Introduction: Cubic smoothing spline; functional ANOVA decomposition; additive models.
• Model construction: Reproducing kernel Hilbert space; shrinkage estimates; univariate splines; tensor product splines; empirical Bayes model.
• Gaussian regression: Smoothing parameter selection and cross validation; Bayesian confidence intervals; cosine diagnostics; software tools.
• More splines: Partial splines; splines on the circle; L-splines; thin-plate splines.
• Non-Gaussian regression: Smoothing parameter selection; Bayesian confidence intervals; software tools.

If time permits, some of the following topics may also be covered.

• Probability density estimation: Logistic density transform; biased sampling and random truncation; conditional density estimation; smoothing parameter selection.
• Hazard rate estimation: Proportional hazard model and beyond; accelerated life models; smoothing parameter selection.

Some of the latest results and on-going research projects by your instructor and his students may also be discussed.

Prerequisite

Working knowledge of statistical inference, linear models, generalized linear models, and matrix algebra. Prior knowledge of Hilbert space is helpful but not required.

Textbook

Smoothing Spline ANOVA Models, by your instructor.

References

Spline Models for Observational Data, by Wahba.

Software

Software tools implementing some of the techniques discussed in this course have been developed under R, an open-source clone of S/Splus. R resources are to be found at The Comprehensive R Archive Network.

Course Work

There will be homework assignments every 2 to 3 weeks. In lieu of exams, a project will be required of each registered student. The project can be comprehensive data analysis, literature reading/presentation, or programming exercises. In case substantial effort is needed, especially for programming exercises, team project can be arranged.

Lecture Notes