Garritt L. Page

Spatial Product Partition Models

When modeling geostatistical or areal data, spatial structure is commonly accommodated via a covariance function for the former and a neighborhood structure for the latter. In both cases the resulting spatial structure is a consequence of implicit spatial grouping in that observations near in space are assumed to behave similarly. It would be desirable to develop spatial methods that explicitly model the partitioning of spatial locations providing more control over resulting spatial structures and being able to better balance global vs local spatial dependence. To this end, we extend product partition models to a spatial setting so that the partitioning of locations into spatially dependent clusters is explicitly modeled. We explore the spatial structures that result from employing a spatial product partition model and demonstrate its flexibility in accommodating many types of spatial dependencies. We illustrate the methods utility through simulation studies and an education application.

Bayes Statistical Analyses for Particle Sieving Studies

Particle size is commonly used to determine quality and predict performance of particle systems. We consider particle size distributions inferred from a material sample using a fixed number of sieves with progressively smaller size openings, where the weight of the particles in each size interval is measured. In this article, we propose Bayes analyses for data from particle sieving studies based on parsimoniously parameterized multivariate normal approximate models for vectors of log weight fraction ratios. Additionally, we observe that the basic approach extends directly to modeling mixture contexts, which provides model flexibility and is a very natural extension when physical mixtures of materials with fundamentally different particle sizes are encountered. We also consider hierarchical modeling, where a single process produces lots of particles and the data available are (replicated) weight fraction vectors from different lots. Supplementary materials for this article are available online.

Classification via Bayesian Nonparametric Learning of Affine Subspaces

It has become common for datasets to contain large numbers of variables in studies conducted in areas such as genetics, machine vision, image analysis, and many others. When analyzing such data, parametric models are often too inflexible while nonparametric procedures tend to be nonrobust because of insufficient data on these high-dimensional spaces. This is particularly true when interest lies in building efficient classifiers in the presence of many predictor variables. When dealing with these types of data, it is often the case that most of the variability tends to lie along a few directions, or more generally along a much smaller dimensional submanifold of the data space. In this article, we propose a class of models that flexibly learn about this submanifold while simultaneously performing dimension reduction in classification. This methodology allows the cell probabilities to vary nonparametrically based on a few coordinates expressed as linear combinations of the predictors. Also, as opposed to many black-box methods for dimensionality reduction, the proposed model is appealing in having clearly interpretable and identifiable parameters that provide insight into which predictors are important in determining accurate classification boundaries. Gibbs sampling methods are developed for posterior computation, and the methods are illustrated using simulated and real data applications.

Effect of position, usage rate, and per game minutes played on NBA player production curves

Abstract In this paper, we model a basketball player’s on-court production as a function of the percentiles corresponding to the number of games played. A player’s production curve is flexibly estimated using Gaussian process regression. The hierarchical structure of the model allows us to borrow strength across players who play the same position and have similar usage rates and play a similar number of minutes per game. From the results of the modeling, we discuss questions regarding the relative deterioration of production for each of the different player positions. Learning how minutes played and usage rate affect a player’s career production curve should prove to be useful to NBA decision makers.

Nonparametric Bayesian inference in applications

Nonparametric Bayesian (BNP) inference is concerned with inference for infinite dimensional parameters, including unknown distributions, families of distributions, random mean functions and more. Better computational resources and increased use of massive automated or semi-automated data collection makes BNP models more and more common. We briefly review some of the main classes of models, with an emphasis on how they arise from applied research questions, and focus in more depth only on BNP models for spatial inference as a good example of a class of inference problems where BNP models can successfully address limitations of parametric inference.

Spatial Species Sampling and Product Partition Models

Inference for spatial data arises, for example in medical imaging, epidemiology, ecology, and other areas, and gives rise to specific challenges for nonparametric Bayesian modeling. In this chapter we briefly review the fast growing related literature and discuss two specific models in more detail. The two models are the CAR SSM (species sampling with conditional autoregression) prior of Jo et al. (Dependent species sampling models for spatial density estimation. Technical report, Department of Statistics, Seoul National University, 2015) and the spatial PPM (product partition model) of Page and Quintana (Spatial product partition models. Technical report, Pontificia Universidad Catolica de Chile, 2015).

Predictions Based on the Clustering of Heterogeneous Functions via Shape and Subject-Specific Covariates

We consider a study of players employed by teams who are members of the National Basketball Association where units of observation are functional curves that are realizations of production measurements taken through the course of ones career. The observed functional output displays large amounts of between player heterogeneity in the sense that some individuals produce curves that are fairly smooth while others are (much) more erratic. We argue that this variability in curve shape is a feature that can be exploited to guide decision making, learn about processes under study and improve prediction. In this paper we develop a methodology that takes advantage of this feature when clustering functional curves. Individual curves are flexibly modeled using Bayesian penalized B-splines while a hierarchical structure allows the clustering to be guided by the smoothness of individual curves. In a sense, the hierarchical structure balances the desire to fit individual curves well while still producing meaningful clusters that are used to guide prediction. We seamlessly incorporate available covariate information to guide the clustering of curves non-parametrically through the use of a product partition model prior for a random partition of individuals. Clustering based on curve smoothness and subject-specific covariate information is particularly important in carrying out the two types of predictions that are of interest, those that complete a partially observed curve from an active player, and those that predict the entire career curve for a player yet to play in the National Basketball Association.

Bayesian Local Contamination Models for Multivariate Outliers

In studies where data are generated from multiple locations or sources it is common for there to exist observations that are quite unlike the majority. Motivated by the application of establishing a reference value in an inter-laboratory setting when outlying labs are present, we propose a local contamination model that is able to accommodate unusual multivariate realizations in a flexible way. The proposed method models the process level of a hierarchical model using a mixture with a parametric component and a possibly nonparametric contamination. Much of the flexibility in the methodology is achieved by allowing varying random subsets of the elements in the lab-specific mean vectors to be allocated to the contamination component. Computational methods are developed and the methodology is compared to three other possible approaches using a simulation study. We apply the proposed method to a NIST/NOAA sponsored inter-laboratory study which motivated the methodological development.

A Bayesian Approach to Establishing a Reference Particle Size Distribution in the Presence of Outliers

The presence of observations or measurements that are unlike the majority is fairly common in studies conducted to establish particle size (or weight fraction) distributions. Therefore, there is a need to develop methods that are capable of producing estimates of particle size distributions that are not overly sensitive to the presence of a few observations that might be considered outliers. This article proposes a type of contamination mixture model that probabilistically allocates each observation to either a majority component or a contamination component. Observations that are allocated to a contamination component are down-weighted when estimating the particle size distribution (while the uncertainty of contamination classification is automatically accounted for in estimation). Computational methods are developed and the utility of the proposed methodology is demonstrated via a simulation study. The method is then applied to data produced from an inter-laboratory study conducted to establish a particle size distribution in cement.

Calibrating covariate informed product partition models

Covariate informed product partition models incorporate the intuitively appealing notion that individuals or units with similar covariate values a priori have a higher probability of co-clustering than those with dissimilar covariate values. These methods have been shown to perform well if the number of covariates is relatively small. However, as the number of covariates increase, their influence on partition probabilities overwhelm any information the response may provide in clustering and often encourage partitions with either a large number of singleton clusters or one large cluster resulting in poor model fit and poor out-of-sample prediction. This same phenomenon is observed in Bayesian nonparametric regression methods that induce a conditional distribution for the response given covariates through a joint model. In light of this, we propose two methods that calibrate the covariate-dependent partition model by capping the influence that covariates have on partition probabilities. We demonstrate the new methods’ utility using simulation and two publicly available datasets.