constraints

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

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

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.

The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known families of Wishart distributions as special cases. The relations to graphical models and Bayesian statistics are indicated.

Abstract. We provide a remedy for two concerns that have dogged the use of principal components in regression: (i) principal components are computed from the predictors alone and do not make apparent use of the response, and (ii) principal components are not invariant or equivariant under full rank linear transformation of the predictors. The development begins with principal fitted components [Cook, R. D. (2007). Fisher lecture: Dimension reduction in regression (with discussion). Statist. Sci. 22 1–26] and uses normal models for the inverse regression of the predictors on the response to gain reductive information for the forward regression of interest. This approach includes methodology for testing hypotheses about the number of components and about conditional independencies among the predictors. Key words and phrases: Central subspace, dimension reduction, inverse regression, principal components. 1.

Abstract: Changes between different sets of parameters are often needed in multivariate statistical modeling such as transformations within linear regression or in exponential models. There may, for instance, be specific inference questions based on subject matter interpretations, alternative well-fitting constrained models, compatibility judgements of seemingly distinct constrained models, or different reference priors under alternative parameterizations. We introduce and discuss a partial mapping, called partial replication and relate it to a more complex mapping, called partial inversion. Both operations are used to decompose matrix operations, to explain recursion relations among sets of linear parameters, to change between different types of linear models, to approximate maximum-likelihood estimates in exponential family models under independence constraints, and to switch partially between sets of canonical and moment parameters in exponential family distributions or between sets of corresponding maximum-likelihood estimates. Key words and phrases: Exponential family, independence constraints, matrix

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