Expectation–maximization algorithm

About: Expectation–maximization algorithm is a research topic. Over the lifetime, 11823 publications have been published within this topic receiving 528693 citations. The topic is also known as: EM algorithm & Expectation Maximization.

Likelihood based frequentist inference when data are missing at random

Abstract: One of the most often quoted results from the original work of Rubin and Little on the classification of missing value processes is the validity of likelihood based inferences under missing at random (MAR) mechanisms. Although the sense in which this result holds was precisely defined by Rubin, and explored by him in later work, it appears to be now used by some authors in a general and rather imprecise way, particularly with respect to the use of frequentist modes of inference. In this paper an exposition is given of likelihood based frequentist inference under an MAR mechanism that shows in particular which aspects of such inference cannot be separated from consideration of the missing value mechanism. The development is illustrated with three simple setups: a bivariate binary outcome, a bivariate Gaussian outcome and a two-stage sequential procedure with Gaussian outcome and with real longitudinal examples, involving both categorical and continuous outcomes. In particular, it is shown that the classical expected information matrix is biased and the use of the observed information matrix is recommended.

Robust Cluster Analysis via Mixtures of Multivariate t-Distributions

Abstract: Normal mixture models are being increasingly used as a way of clustering sets of continuous multivariate data. They provide a probabilistic (soft) clustering of the data in terms of their fitted posterior probabilities of membership of the mixture components corresponding to the clusters. An outright (hard) clustering can be subsequently obtained by assigning each observation to the component to which it has the highest fitted posterior probability of belonging. However, outliers in the data can affect the estimates of the parameters in the normal component densities, and hence the implied clustering. A more robust approach is to fit mixtures of multivariate t-distributions, which have longer tails than the normal components. The expectation-maximization (EM) algorithm can be used to fit mixtures of t-distributions by maximum likelihood. The application of this model to provide a robust approach to clustering is illustrated on a real data set. It is demonstrated how the use of t-components provides less extreme estimates of the posterior probabilities of cluster membership.

Robust mixture modeling using the skew t distribution

Abstract: A finite mixture model using the Student's t distribution has been recognized as a robust extension of normal mixtures. Recently, a mixture of skew normal distributions has been found to be effective in the treatment of heterogeneous data involving asymmetric behaviors across subclasses. In this article, we propose a robust mixture framework based on the skew t distribution to efficiently deal with heavy-tailedness, extra skewness and multimodality in a wide range of settings. Statistical mixture modeling based on normal, Student's t and skew normal distributions can be viewed as special cases of the skew t mixture model. We present analytically simple EM-type algorithms for iteratively computing maximum likelihood estimates. The proposed methodology is illustrated by analyzing a real data example.

Elements of Statistical Computing: NUMERICAL COMPUTATION

Abstract: Statistics and computing share many close relationships. Computing now permeates every aspect of statistics, from pure description to the development of statistical theory. At the same time, the computational methods used in statistical work span much of computer science. Elements of Statistical Computing covers the broad usage of computing in statistics. It provides a comprehensive account of the most important computational statistics. Included are discussions of numerical analysis, numerical integration, and smoothing. The author give special attention to floating point standards and numerical analysis; iterative methods for both linear and nonlinear equation, such as Gauss-Seidel method and successive over-relaxation; and computational methods for missing data, such as the EM algorithm. Also covered are new areas of interest, such as the Kalman filter, projection-pursuit methods, density estimation, and other computer-intensive techniques.

Tracking deformable objects with point clouds

Abstract: We introduce an algorithm for tracking deformable objects from a sequence of point clouds. The proposed tracking algorithm is based on a probabilistic generative model that incorporates observations of the point cloud and the physical properties of the tracked object and its environment. We propose a modified expectation maximization algorithm to perform maximum a posteriori estimation to update the state estimate at each time step. Our modification makes it practical to perform the inference through calls to a physics simulation engine. This is significant because (i) it allows for the use of highly optimized physics simulation engines for the core computations of our tracking algorithm, and (ii) it makes it possible to naturally, and efficiently, account for physical constraints imposed by collisions, grasping actions, and material properties in the observation updates. Even in the presence of the relatively large occlusions that occur during manipulation tasks, our algorithm is able to robustly track a variety of types of deformable objects, including ones that are one-dimensional, such as ropes; two-dimensional, such as cloth; and three-dimensional, such as sponges. Our implementation can track these objects in real time.