This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the true change-point before it can be reliably detected. Furthermore, missing components in the observed data handicap conventional approaches. The proposed method addresses these challenges by modeling the dynamic distribution underlying the data as lying close to a time-varying low-dimensional submanifold embedded within the ambient observation space. Specifically, streaming data is used to track a submanifold approximation, measure deviations from this approximation, and calculate a series of statistics of the deviations for detecting when the underlying manifold has changed in a sharp or unexpected manner. The approach described in this paper leverages several recent results in the field of high-dimensional data analysis, including subspace tracking with missing data, multiscale analysis techniques for point clouds, online optimization, and change-point detection performance analysis. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.
In this paper we propose an efficient algorithm for reducing a large mixture of Gaussians into a smaller mixture while still preserving the component structure of the original model; this is achieved by clustering (grouping) the components. The method minimizes a new, easily computed distance measure between two Gaussian mixtures that can be motivated from a suitable stochastic model and the iterations of the algorithm use only the model parameters, avoiding the need for explicit resampling of datapoints. We demonstrate the method by performing hierarchical clustering of scenery images and handwritten digits. We analyze mixture density approximation and estimation. We form a convex set of density functions by taking the convex hull of a parametric family, e.g. mixtures of the Gaussian location family. A sequence of finite mixture densities is formulated to provide a parsimonious approximation for the target density. If the target density itself is in the convex hull, we show that the approximation error goes to zero with a rate of 1 k, where k is the number of components in the approximation. If the target density is outside of the convex hull, the approximation error is equal to the best achievable error plus a term that goes to zero with a rate of 1 k. A greedy algorithm that introduces one component at each step is shown to achieve such an error rate. Similarly, a greedy estimation algorithm is provided to find such approximation for data from an arbitrary density. This algorithm estimates one mixture component at one time. We prove that such an algorithm achieves a likelihood nearly as good as the MLE (maximum likelihood estimate) over the whole convex hull. And we identify the difference as being bounded by order O(1 k), where k is the number of components in the estimate. Risks of such estimators are shown to be bounded by a sum of approximation error and estimation error. The error terms are identified. An optimal choice of k can be derived by minimizing the risk bound. Acting as a similar role as the bandwidth in non-parainetric density estimation, k controls two error terms in opposite directions. A large k reduces approximation error and increases estimation error. A MDL (minimum description length) principle is derived to provide an estimation method for k. And the estimated k is shown to achieve the risk bound as if we know the best k in advance. A new information projection theory is derived to expand the approximating class to include its information closure. We prove the existence and uniqueness of a f* in the closure of the convex hull C (in a sense we identify), such that D ( fpf* ) = infg∈CD fpg , where Dfpg is the Kullback-Leibler divergence. And log(fk) → log(f*) in L1 (f) for any sequence fk in C with Dfpfk→ infg∈CD fpg. Other characterizing properties of f* are also given.  The first unified account of the theory, methodology, and applications of the EM algorithm and its extensionsSince its inception in 1977, the Expectation-Maximization (EM) algorithm has been the subject of intense scrutiny, dozens of applications, numerous extensions, and thousands of publications. The algorithm and its extensions are now standard tools applied to incomplete data problems in virtually every field in which statistical methods are used. Until now, however, no single source offered a complete and unified treatment of the subject.The EM Algorithm and Extensions describes the formulation of the EM algorithm, details its methodology, discusses its implementation, and illustrates applications in many statistical contexts. Employing numerous examples, Geoffrey McLachlan and Thriyambakam Krishnan examine applications both in evidently incomplete data situations-where data are missing, distributions are truncated, or observations are censored or grouped-and in a broad variety of situations in which incompleteness is neither natural nor evident. They point out the algorithm's shortcomings and explain how these are addressed in the various extensions.Areas of application discussed include: Regression Medical imaging Categorical data analysis Finite mixture analysis Factor analysis Robust statistical modeling Variance-components estimation Survival analysis Repeated-measures designs For theoreticians, practitioners, and graduate students in statistics as well as researchers in the social and physical sciences, The EM Algorithm and Extensions opens the door to the tremendous potential of this remarkably versatile statistical tool. Meta-sulfonamido-benzamide derivatives of the formula: [wherein R is a hydrogen atom or a lower alkyl, cyano, or lower alkanesulfonyl group; R1 is a lower alkyl, phenyl, amino, lower alkylamino, di(lower)alkylamino, or C4-C5 alkyleneamino group; R2 is a hydrogen or halogen atom or a lower alkyl, di(lower)alkylamino, or lower alkoxy group; R3 is a hydrogen atom or a methyl or methoxy group; R4 is a hydrogen or halogen atom; R5 is a lower alkyl, lower alkenyl, C3-C6 cycloalkyl, benzyl, or halogenobenzyl group; and n is 1 or zero] or their acid addition salts, showing pharmacological activity such as anti-emetic or psychotropic activity, are provided via several routes. The important role of finite mixture models in the statistical analysis of data is underscored by the ever-increasing rate at which articles on mixture applications appear in the statistical and ge...