Semiparametric regression in the presence of complex variance structures arising from small angle x-ray scattering data
Date
2014
Authors
Bugbee, Bruce D., author
Breidt, F. Jay, advisor
Estep, Don, advisor
Meyer, Mary, committee member
Hoeting, Jennifer, committee member
Luger, Karolin, committee member
Journal Title
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Abstract
An ongoing problem in structural biology is how best to infer structural information for complex, biological macromolecules from indirect observational data. Molecular shape dictates functionality but is not always directly observable. There exists a wide class of experimental methods whose data can be used for indirectly inferring molecular shape features with varying degrees of resolution. Of these methods, small angle X-ray scattering (SAXS) is desirable due to low requirements on the sample of interest. However, SAXS data suffers numerous statistical problems that require the development of novel methodologies. A primary concern is the impact of radially reducing two-dimensional sensor data to a series of smooth mean and variance curves. Additionally, pronounced heteroskedasticity is often observed near sensor boundaries. The work presented here focuses on developing general model frameworks and implementation methods appropriate for SAXS data. Semiparametric regression refers to models that combine known parametric structures with flexible nonparametric components. Three semiparametric regression model frameworks that are well-suited for handling smooth data are presented. The first model introduced is the standard semiparametric regression model, described as a mixed model with low rank penalized splines as random effects. The second model extends the first to the case of heteroskedastic errors, which violate standard model assumptions. The latent variance function in the model is estimated through an additional semiparametric regression, allowing for appropriate uncertainty estimation at the mean level. The final model considers a data structure unique to SAXS experiments. This model incorporates both radial mean and radial variance data in hopes to better infer three-dimensional shape properties and understand experimental effects by including all available data. Each of the three model frameworks is structured hierarchically. Bayesian inference is appealing in this context, as it provides efficient and generalized modeling frameworks in a unified way. The main statistical contributions of this thesis are from the specific methods developed to address the computational challenges of Bayesian inference for these models. The contributions include new Markov Chain Monte Carlo (MCMC) procedures for numerical approximation of posterior distributions and novel variational approximations that are extremely fast and accurate. For the heteroskedastic semiparametric case, known form posterior conditionals are available for all model parameters save for the regression coefficients controlling the latent model variance function. A novel implementation of a multivariate delayed rejection adaptive Metropolis (DRAM) procedure is used to sample from this posterior conditional distribution. The joint model for radial mean and radial variance data is shown to be of comparable structure to the heteroskedastic case and the new DRAM methodology is extended to handle this case. Simulation studies of all three methods are provided, showing that these models provide accurate fits of observed data and latent variance functions. The demands of scientific data processing in the context of SAXS, where large data sets are rapidly attained, lead to consideration of fast approximations as alternatives to MCMC. {Variational approximations} or {Variational Bayes} describes a class of approximation methods where the posterior distribution of the parameters is approximated by minimizing the Kullback-Leibler divergence between the true posterior and a class of distributions under mild structural constraints. Variational approximations have been shown to be good approximations of true posteriors in many cases. A novel variational approximation for the general heteroskedastic semiparametric regression model is derived here. Simulation studies are provided demonstrating fit and coverage properties comparable to the DRAM results at a fraction of the computational cost. A variational approximation for the joint model of radial mean and variance data is also provided but is shown to suffer from poor performance due to high correlation across a subset of regression parameters. The heteroskedastic semiparametric regression framework has some strong structural relationships with a distinct, important problem: spatially adaptive smoothing. A noisy function with different amounts of smoothness over its domain may be systematically under-smoothed or over-smoothed if the smoothing is not spatially adaptive. A novel variational approximation is derived for the problem of spatially adaptive penalized spline regression, and shown to have excellent performance. This approximation method is shown to be able to fit highly oscillatory data while not requiring the traditional tuning and computational resources of standard MCMC implementations. Potential scientific contribution of the statistical methodology developed here are illuminated with SAXS data examples. Analysis of SAXS data typically has two primary concerns: description of experimental effects and estimation of physical shape parameters. Formal statistical procedures for testing the effect of sample concentration and exposure time are presented as alternatives to current methods, in which data sets are evaluated subjectively and often combined in ad hoc ways. Additionally, estimation procedures for the scattering intensity at zero angle, known to be proportional to molecular weight, and the radius of gyration are described along with appropriate measures of uncertainty. Finally, a brief example of the joint radial mean and variance method is provided. Guidelines for extending the models presented here to more complex SAXS problems are also given.
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Subject
Heteroskedasticiy
variational Bayes
penalized splines