Browsing by Subject "Hierarchical model"
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Item Copula Based Hierarchical Bayesian Models(2010-10-12) Ghosh, SouparnoThe main objective of our study is to employ copula methodology to develop Bayesian hierarchical models to study the dependencies exhibited by temporal, spatial and spatio-temporal processes. We develop hierarchical models for both discrete and continuous outcomes. In doing so we expect to address the dearth of copula based Bayesian hierarchical models to study hydro-meteorological events and other physical processes yielding discrete responses. First, we present Bayesian methods of analysis for longitudinal binary outcomes using Generalized Linear Mixed models (GLMM). We allow flexible marginal association among the repeated outcomes from different time-points. An unique property of this copula-based GLMM is that if the marginal link function is integrated over the distribution of the random effects, its form remains same as that of the conditional link function. This unique property enables us to retain the physical interpretation of the fixed effects under conditional and marginal model and yield proper posterior distribution. We illustrate the performance of the posited model using real life AIDS data and demonstrate its superiority over the traditional Gaussian random effects model. We develop a semiparametric extension of our GLMM and re-analyze the data from the AIDS study. Next, we propose a general class of models to handle non-Gaussian spatial data. The proposed model can deal with geostatistical data that can accommodate skewness, tail-heaviness, multimodality. We fix the distribution of the marginal processes and induce dependence via copulas. We illustrate the superior predictive performance of our approach in modeling precipitation data as compared to other kriging variants. Thereafter, we employ mixture kernels as the copula function to accommodate non-stationary data. We demonstrate the adequacy of this non-stationary model by analyzing permeability data. In both cases we perform extensive simulation studies to investigate the performances of the posited models under misspecification. Finally, we take up the important problem of modeling multivariate extreme values with copulas. We describe, in detail, how dependences can be induced in the block maxima approach and peak over threshold approach by an extreme value copula. We prove the ability of the posited model to handle both strong and weak extremal dependence and derive the conditions for posterior propriety. We analyze the extreme precipitation events in the continental United States for the past 98 years and come up with a suite of predictive maps.Item Discriminative object categorization with external semantic knowledge(2013-08) Hwang, Sung Ju; Grauman, Kristen Lorraine, 1979-Visual object category recognition is one of the most challenging problems in computer vision. Even assuming that we can obtain a near-perfect instance level representation with the advances in visual input devices and low-level vision techniques, object categorization still remains as a difficult problem because it requires drawing boundaries between instances in a continuous world, where the boundaries are solely defined by human conceptualization. Object categorization is essentially a perceptual process that takes place in a human-defined semantic space. In this semantic space, the categories reside not in isolation, but in relation to others. Some categories are similar, grouped, or co-occur, and some are not. However, despite this semantic nature of object categorization, most of the today's automatic visual category recognition systems rely only on the category labels for training discriminative recognition with statistical machine learning techniques. In many cases, this could result in the recognition model being misled into learning incorrect associations between visual features and the semantic labels, from essentially overfitting to training set biases. This limits the model's prediction power when new test instances are given. Using semantic knowledge has great potential to benefit object category recognition. First, semantic knowledge could guide the training model to learn a correct association between visual features and the categories. Second, semantics provide much richer information beyond the membership information given by the labels, in the form of inter-category and category-attribute distances, relations, and structures. Finally, the semantic knowledge scales well as the relations between categories become larger with an increasing number of categories. My goal in this thesis is to learn discriminative models for categorization that leverage semantic knowledge for object recognition, with a special focus on the semantic relationships among different categories and concepts. To this end, I explore three semantic sources, namely attributes, taxonomies, and analogies, and I show how to incorporate them into the original discriminative model as a form of structural regularization. In particular, for each form of semantic knowledge I present a feature learning approach that defines a semantic embedding to support the object categorization task. The regularization penalizes the models that deviate from the known structures according to the semantic knowledge provided. The first semantic source I explore is attributes, which are human-describable semantic characteristics of an instance. While the existing work treated them as mid-level features which did not introduce new information, I focus on their potential as a means to better guide the learning of object categories, by enforcing the object category classifiers to share features with attribute classifiers, in a multitask feature learning framework. This approach essentially discovers the common low-dimensional features that support predictions in both semantic spaces. Then, I move on to the semantic taxonomy, which is another valuable source of semantic knowledge. The merging and splitting criteria for the categories on a taxonomy are human-defined, and I aim to exploit this implicit semantic knowledge. Specifically, I propose a tree of metrics (ToM) that learns metrics that capture granularity-specific similarities at different nodes of a given semantic taxonomy, and uses a regularizer to isolate granularity-specific disjoint features. This approach captures the intuition that the features used for the discrimination of the parent class should be different from the features used for the children classes. Such learned metrics can be used for hierarchical classification. The use of a single taxonomy can be limited in that its structure is not optimal for hierarchical classification, and there may exist no single optimal semantic taxonomy that perfectly aligns with visual distributions. Thus, I next propose a way to overcome this limitation by leveraging multiple taxonomies as semantic sources to exploit, and combine the acquired complementary information across multiple semantic views and granularities. This allows us, for example, to synthesize semantics from both 'Biological', and 'Appearance'-based taxonomies when learning the visual features. Finally, as a further exploration of more complex semantic relations different from the previous two pairwise similarity-based models, I exploit analogies, which encode the relational similarities between two related pairs of categories. Specifically, I use analogies to regularize a discriminatively learned semantic embedding space for categorization, such that the displacements between the two category embeddings in both category pairs of the analogy are enforced to be the same. Such a constraint allows for a more confusing pair of categories to benefit from a clear separation in the matched pair of categories that share the same relation. All of these methods are evaluated on challenging public datasets, and are shown to effectively improve the recognition accuracy over purely discriminative models, while also guiding the recognition to be more semantic to human perception. Further, the applications of the proposed methods are not limited to visual object categorization in computer vision, but they can be applied to any classification problems where there exists some domain knowledge about the relationships or structures between the classes. Possible applications of my methods outside the visual recognition domain include document classification in natural language processing, and gene-based animal or protein classification in computational biology.Item The effects of three different priors for variance parameters in the normal-mean hierarchical model(2010-05) Chen, Zhu, 1985-; Greenberg, Betsy S.; Sager, Thomas W.Many prior distributions are suggested for variance parameters in the hierarchical model. The “Non-informative” interval of the conjugate inverse-gamma prior might cause problems. I consider three priors – conjugate inverse-gamma, log-normal and truncated normal for the variance parameters and do the numerical analysis on Gelman’s 8-schools data. Then with the posterior draws, I compare the Bayesian credible intervals of parameters using the three priors. I use predictive distributions to do predictions and then discuss the differences of the three priors suggested.Item Handling complex multilevel data structures(2013-05) Li, Yuanhan; Beretvas, Susan NatashaThis report focuses on introducing two statistical models for dealing with data involving complex social structures. Appropriate handling of data structures is a concern in the context of educational settings. From base single-level data to complex hierarchical with cross-classifications and multiple-memberships, we explain and demonstrate their distinction and establish appropriate regression models. Real data from the National Center for Education Statistics (NECS) is used to demonstrate different way of handling a cross-classified data structure as well as appropriate models. Results will be presented and compared to examine the practical operation for each model.