Abstract. This paper studies a statistical skin-color model and its adaptation. It is revealed that (1) human skin colors cluster in a small region in a color space; (2) the variance of a skin color cluster can be reduced by intensity normalization, and (3) under a certain lighting condition, a skin-color distribution can be characterized by amultivariate normal distribution in the normalized color space. We then propose an adaptive model to characterize human skin-color distributions for tracking human faces under di erent lighting conditions. The parameters of the model are adapted based on the maximum likelihood criterion. The model has been successfully applied to a real-time face tracker and other applications. 1

This paper provides conditions and procedures for deciding if patterns of independencies found in covariance and concentration matrices can be generated by a stepwise recursive process represented by some directed acyclic graph. If such an agreement is found, we know that one or several causal processes could be responsible for the observed independencies, and our procedures could then be used to elucidate the graphical structure common to these processes, so as to evaluate their compatibility against substantive knowledge of the domain. If we find that the observed pattern of independencies does not agree with any stepwise recursive process, then there are a number of different possibilities. For instance, -- some weak dependencies could have been mistaken for independencies and led to the wrong omission of edges from the covariance or concentration graphs. -- some of the observed linear dependencies reflect accidental cancellations or hide actual nonlinear relations, or -- the process responsible for the data is non-recursive, involving aggregated variables, simultenous reciprocal interactions, or mixtures of several causal processes. In order to recognize accidental independencies it would be helpful to conduct several longitudinal studies under slightly varying conditions. In such studies the covariances for the same set of variables is estimated under different conditions and the variations in the conditions would typically affect the numerical values of the parameters. But, if the data were generated by a causal process represented by some directed acyclic graph, then the basic structural properties reflected in the missing edges of that graph should remain unchanged. Under such assumptions, the pattern of independencies that is "implied" by the dag (see Definitio...

A number of authors have considered multivariate Gaussian models for marginal independence. In this paper we develop models for binary data with the same independence structure. The models can be parameterized based on Möbius inversion and maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. The approach is illustrated on a simple example. Relations to multivariate logistic and dependence ratio models are discussed.

We introduce and study a calculus for real-valued square matrices, called partial inversion, and an associated calculus for binary square matrices. The first, applied to systems of recursive linear equations, generates new sets of parameters for different types of statistical joint response models. The corresponding generating graphs are directed and acyclic. The second calculus, applied to matrix representations of independence graphs, gives chain graphs induced by such a generating graph. Chain graphs are more complex independence graphs associated with recursive joint response models. Missing edges in independence graphs coincide with structurally zero parameters in linear systems. A wide range of consequences of an assumed independence structure can be derived by partial closure, but computationally efficient algorithms still need to be developed for applications to very large graphs.

Graphical models with bi-directed edges (↔) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation algorithms. 1

A human face provides a variety of di erent communicative functions. In this paper, we present approaches for real-time face/facial feature tracking and their applications. First, we present techniques of tracking human faces. It is revealed that human skincolor can be used as a major feature for tracking human faces. An adaptive stochastic model has been developed to characterize the skin-color distributions. Based on the maximum likelihood method, the model parameters can be adapted for di erent people and different lighting conditions. The feasibility of the model has been demonstrated by the development of a realtime face tracker. We then present a top-down approach for tracking facial features such as eyes, nostrils, and lip corners. These real-time tracking techniques have been successfully applied to many applications such as eye-gaze monitoring, head pose tracking, and lip-reading. 1.

Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, orthogonal decompositions for covariance chains are derived first in explicit form. Covariance chains are also contrasted to concentration chains, for which estimation is explicit and simple. For this purpose, maximum-likelihood equations are derived first for exponential families when some parameters satisfy zero value constraints. From these equations explicit estimates are obtained, which are asymptotically efficient, and they are applied to covariance chains. Simulation results confirm the satisfactory behaviour of the explicit covariance chain estimates also in moderate-size samples.

By invoking the extended invariance principle (EXIP), we present herein a computationally efficient method that provides asymptotic (for large samples) maximum likelihood (AML) estimation for structured covariance matrices and will be referred to as the AML algorithm. A closed-form formula for estimating Hermitian Toeplitz covariance matrices that makes AML computationally simpler than most existing Hermitian Toeplitz matrix estimation algorithms is derived. Although the AML covariance matrix estimator can be used in a variety of applications, we focus on array processing in this paper. Our simulation study shows that AML enhances the performances of angle estimation algorithms, such as MUSIC, by making them very close to the corresponding Cram er--Rao bound (CRB) for uncorrelated signals. Numerical comparisons with several structured and unstructured covariance matrix estimators are also presented.

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts.

Incomplete longitudinal data are a common problem in clinical and epidemiological studies. This work was motivated by longitudinal studies of pulmonary function in young children characterized by both missing and mistimed observations as well as time-varying covariates. The objectives of this work were to develop a model and an analysis approach which 1) accommodated both missing and mistimed data and covariates which changed over time, 2) allowed for testing of hypotheses about both the fixed and random effects, and 3) were computationally feasible and practical. A generalized Mixed Effects Model was developed which generalized some assumptions used by previous authors. In particular, the restriction that the withinsubject variance is uncorrelated and homoscedastic (0'2 I) was generalized to the form 0'2Vi where Vi is any known positive definite matrix. Maximum Likelihood Estimators were derived and the EM algorithm and the Method of Scoring were used to solve the maximum likelihood equations. Randomly generated data were used in a preliminary exploration of the