Fast Bayesian Functional Data Analysis: Application to basal body temperature data.
James Mbugua Ciera, David B. Dunson, Bruno Scarpa · Padua@research (University of Padova) · 2009
In many clinical studies, data is collected repeatedly from many subjects over a period of time. Using massive datasets, physicians require fast automated tools to estimate data trajectories and predict clinically important events for a current patient. For example, in reproductive studies, trajectories of hormonal level or daily basal body temperature (bbt) among women can help to identify or predict early pregnancy loss and occurrence of the ovulation day (Bigelow and Dunson, 2008). Our research is motivated by the bbt data from European fecundability study (Colombo and Masarotto, 2000). The study consists of daily bbt measurements from women that contributed temperature measurements from at least one menstrual cycle. A standard bbt curve from a healthy ovulating female has a biphasic pattern. The data is characterized with unequal cycle lengths and unequally-spaced measurements causing problems in estimating bbt curves. Thus, estimation of accurate and smooth curves is based on borrowing information flexibly across cycles. Functional data analysis (FDA) can be used to estimate trajectories but relies on large number of basis functions (Ramsay and Silverman, 1997). Bayesian methods can be implemented to estimate basis coefficients but the posterior sampling is based on slow Markov Chain Monte Carlo (MCMC) algorithms. This raises a practical motivation for fast approximate Bayes approaches that bypass MCMC while maintaining some of the benefits of a Bayesian analysis. In this article, we approximate the bbt projectiles using Multi-Task Relevant Vector machine (MT-RVM) method an extension of Relevant Vector machine (RVM) method (Ji, et al, 2008). RVM is a fast Bayesian method based on Empirical Bayes methodology and penalizes the basis coefficients through a scale mixture of normals prior, which is carefully-chosen so that maximum a posteriori (MAP) estimates of many of the coefficients are zero. This provides a natural mechanism in variable selection leading to a sparse models that is fast to compute. We present an application of the MT-RVM method to the bbt data. We also evaluate the performance of the MT-RVM method relative to classical Bayesian method as the number of observations increases. References: Bigelow, J.L., and Dunson, D.B. (2008). Bayesian adaptive regression splines for hierarchical data. Biometrics, 63,724-732. Colombo, B. and Masarotto, G. (2000). Daily fecundability: First results from a new data base. Demographic Research, 3, 5. Ji, S., Dunson, D.B. and Carin, L. (2008). Multi-task compressive sensing. IEEE Transactions on Signal Processing, to appear. Ramsay, J. O. and Silverman, B. W. (1997). Functional data analysis. New York: Springer Verlag.