Basketball shot location data provide valuable summary information regarding players to coaches, sports analysts, fans, statisticians, as well as players themselves. Represented by spatial points, such data are naturally analyzed with spatial point process models. We present a novel nonparametric Bayesian method for learning the underlying intensity surface built upon a combination of Dirichlet process and Markov random field. Our method has the advantage of effectively encouraging local spatial homogeneity when estimating a globally heterogeneous intensity surface. Posterior inferences are performed with an efficient Markov chain Monte Carlo (MCMC) algorithm. Simulation studies show that the inferences are accurate and that the method is superior compared to the competing methods. Application to the shot location data of $20$ representative NBA players in the 2017-2018 regular season offers interesting insights about the shooting patterns of these players. A comparison against the competing method shows that the proposed method can effectively incorporate spatial contiguity into the estimation of intensity surfaces.
We propose a multidimensional tensor clustering approach for studying how professional basketball players' shooting patterns vary over court locations and game time. Unlike most existing methods that only study continuous-valued tensors or have to assume the same cluster structure along different tensor directions, we propose a Bayesian nonparametric model that deals with count-valued tensors and projects the heterogeneity among players onto tensor dimensions while allowing cluster structures to be different over directions. Our method is fully probabilistic; hence allows simultaneous inferenc…
We propose a Bayesian nonparametric matrix clustering approach to analyze the latent heterogeneity structure in the shot selection data collected from professional basketball players in the National Basketball Association (NBA). The proposed method adopts a mixture of finite mixtures framework and fully utilizes the spatial information via a mixture of matrix normal distribution representation. We propose an efficient Markov chain Monte Carlo algorithm for posterior sampling that allows simultaneous inference on both the number of clusters and the cluster configurations. We also establish larg…
Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as $C(α)$ tests, as in Neyman (1959), and shown to be locally, asymptotically optimal. These $C(α)$ tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimat…
Assessing homogeneity of distributions is an old problem that has received considerable attention, especially in the nonparametric Bayesian literature. To this effect, we propose the semi-hierarchical Dirichlet process, a novel hierarchical prior that extends the hierarchical Dirichlet process of Teh et al. (2006) and that avoids the degeneracy issues of nested processes recently described by Camerlenghi et al. (2019a). We go beyond the simple yes/no answer to the homogeneity question and embed the proposed prior in a random partition model; this procedure allows us to give a more comprehensiv…
An overwhelming number of true and false news stories are posted and shared in social networks, and users diffuse the stories based on multiple factors. Diffusion of news stories from one user to another depends not only on the stories' content and the genuineness but also on the alignment of the topical interests between the users. In this paper, we propose a novel Bayesian nonparametric model that incorporates homogeneity of news stories as the key component that regulates the topical similarity between the posting and sharing users' topical interests. Our model extends hierarchical Dirichle…
Regional frequency analysis is an important tool to properly estimate hydrological characteristics at ungauged or partially gauged sites in order to prevent hydrological disasters. The delineation of homogeneous groups of sites is an important first step in order to transfer information and obtain accurate quantile estimates at the target site. The Hosking-Wallis homogeneity test is usually used to test the homogeneity of the selected sites. Despite its usefulness and good power, it presents some drawbacks including the subjective choice of a parametric distribution for the data and a poorly j…
Many panel data have the latent subgroup effect on individuals, and it is important to correctly identify these groups since the efficiency of resulting estimators can be improved significantly by pooling the information of individuals within each group. However, the currently assumed parametric and semiparametric relationship between the response and predictors may be misspecified, which leads to a wrong grouping result, and the nonparametric approach hence can be considered to avoid such mistakes. Moreover, the response may depend on predictors in different ways at various quantile levels, a…
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；School of Computer Science, Fudan University（复旦大学计算机科学技术学院）；School of Computing and Information Systems, University of Melbourne（墨尔本大学计算与信息系统学院）
Bayesian nonparametric space partition (BNSP) models provide a variety of strategies for partitioning a $D$-dimensional space into a set of blocks. In this way, the data points lie in the same block would share certain kinds of homogeneity. BNSP models can be applied to various areas, such as regression/classification trees, random feature construction, relational modeling, etc. In this survey, we investigate the current progress of BNSP research through the following three perspectives: models, which review various strategies for generating the partitions in the space and discuss their theore…
The Bayesian approach to inference stands out for naturally allowing borrowing information across heterogeneous populations, with different samples possibly sharing the same distribution. A popular Bayesian nonparametric model for clustering probability distributions is the nested Dirichlet process, which however has the drawback of grouping distributions in a single cluster when ties are observed across samples. With the goal of achieving a flexible and effective clustering method for both samples and observations, we investigate a nonparametric prior that arises as the composition of two dif…
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observ…
Change point analysis is a statistical tool to identify homogeneity within time series data. We propose a pruning approach for approximate nonparametric estimation of multiple change points. This general purpose change point detection procedure `cp3o' applies a pruning routine within a dynamic program to greatly reduce the search space and computational costs. Existing goodness-of-fit change point objectives can immediately be utilized within the framework. We further propose novel change point algorithms by applying cp3o to two popular nonparametric goodness of fit measures: `e-cp3o' uses E-s…
Bayesian Mixture Models for Histograms: with Applications to Large Datasets
Brigham Young University（杨百翰大学）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Politecnico di Milano（米兰理工大学）；University of Milano-Bicocca（米兰比可卡大学）
This paper considers identification and estimation of ceteris paribus effects of continuous regressors in nonseparable panel models with time homogeneity. The effects of interest are derivatives of the average and quantile structural functions of the model. We find that these derivatives are identified with two time periods for "stayers", i.e. for individuals with the same regressor values in two time periods. We show that the identification results carry over to models that allow location and scale time effects. We propose nonparametric series methods and a weighted bootstrap scheme to estima…
We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known compactly supported density g with a finite number of well separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic minimax lower bounds for the global risk of an estimator of f and construct adaptive wavelet nonlinear th…
