Sequential Logistic Principal Component Analysis (SLPCA): Dimensional Reduction in Streaming Multivariate Binary-State System

Sequential or online dimensional reduction is of interests due to the explosion of streaming data based applications and the requirement of adaptive statistical modeling, in many emerging fields, such as the modeling of energy end-use profile. Principal Component Analysis (PCA), is the classical way of dimensional reduction. However, traditional Singular Value Decomposition (SVD) based PCA fails to model data which largely deviates from Gaussian distribution. The Bregman Divergence was recently introduced to achieve a generalized PCA framework. If the random variable under dimensional reduction follows Bernoulli distribution, which occurs in many emerging fields, the generalized PCA is called Logistic PCA (LPCA). In this paper, we extend the batch LPCA to a sequential version (i.e. SLPCA), based on the sequential convex optimization theory. The convergence property of this algorithm is discussed compared to the batch version of LPCA (i.e. BLPCA), as well as its performance in reducing the dimension for multivariate binary-state systems. Its application in building energy end-use profile modeling is also investigated.Comment: 6 pages, 4 figures, conference submissio

Exponential Family Matrix Completion under Structural Constraints

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.Comment: 20 pages, 9 figure

Discovering Patient Phenotypes Using Generalized Low Rank Models

The practice of medicine is predicated on discovering commonalities or distinguishing characteristics among patients to inform corresponding treatment. Given a patient grouping (hereafter referred to as a p henotype ), clinicians can implement a treatment pathway accounting for the underlying cause of disease in that phenotype. Traditionally, phenotypes have been discovered by intuition, experience in practice, and advancements in basic science, but these approaches are often heuristic, labor intensive, and can take decades to produce actionable knowledge. Although our understanding of disease has progressed substantially in the past century, there are still important domains in which our phenotypes are murky, such as in behavioral health or in hospital settings. To accelerate phenotype discovery, researchers have used machine learning to find patterns in electronic health records, but have often been thwarted by missing data, sparsity, and data heterogeneity. In this study, we use a flexible framework called Generalized Low Rank Modeling (GLRM) to overcome these barriers and discover phenotypes in two sources of patient data. First, we analyze data from the 2010 Healthcare Cost and Utilization Project National Inpatient Sample (NIS), which contains upwards of 8 million hospitalization records consisting of administrative codes and demographic information. Second, we analyze a small (N=1746), local dataset documenting the clinical progression of autism spectrum disorder patients using granular features from the electronic health record, including text from physician notes. We demonstrate that low rank modeling successfully captures known and putative phenotypes in these vastly different datasets