Conditional Independence Constraints Research Articles

Conditional independence models for seemingly unrelated regressions with incomplete data

We consider normal ≡ Gaussian seemingly unrelated regressions (SUR) with incomplete data (ID). Imposing a natural minimal set of conditional independence constraints, we find a restricted SUR/ID model whose likelihood function and parameter space factor into the product of the likelihood functions and the parameter spaces of standard complete data multivariate analysis of variance models. Hence, the restricted model has a unimodal likelihood and permits explicit likelihood inference. In the development of our methodology, we review and extend existing results for complete data SUR models and the multivariate ID problem.

Interaction structures analysed from water-quality data

Fortnightly observations of water quality parameters, discharge and water temperature along the River Elbe have been subjected to a multivariate data analysis. In a previous study [Petersen, W., Bertino, L., Callies, U., Zorita, E., 2001. Process identification by principal component analysis of river-quality data. Ecol. Model. 138, 193–213] applied principal component analysis (PCA) to show that 60% of variability in the data set can be explained through just two linear combinations of eight original variables. In the present paper more advanced multivariate methods are applied to the same data set, which are supposed to suit better interpretations in terms of the underlying system dynamics. The first method, graphical modelling, represents interaction structures in terms of a set of conditional independence constraints between pairs of variables given the values of all other variables. Assuming data from a multinormal distribution conditional independence constraints are expressed by zero partial correlations. Different graphical structures with nodes for each variable and connecting edges between them can be assessed with regard to their likelihood. The second method, canonical correlation analysis (CCA), is applied for studying the correlation structures of external forcing and water quality parameters. Results of CCA turn out to be consistent with the dominant patterns of variability obtained from PCA. The percentages of variability explained by external forcing, however, are estimated to be smaller. Fitting graphical models allows a more detailed representation of interaction structures. For instance, for given discharge and temperature correlated variations of the concentrations of oxygen and nitrate, respectively, can be modelled as being mediated by variations of pH, which is a representer for algal activity. Considerably simplified graphical models do not much affect the outcomes of both PCA and CCA, and hence it is concluded that these graphical models successfully represent the main interaction structures represented by the covariance matrix of the data. The analysed conditional independence patterns provide constraints to be satisfied by directed probabilistic networks, for instance.

Correspondence analysis and association models constrained by a conditional independence graph

The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered. Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.

Comparison of correlation coeffìcients when conditional independence constraints between variables are present

Partial correlations are the natural interaction terms to be associated with the edges of the independence graph of a multivariate normal distribution. Two edges of the graph can thus be compared by comparing the corresponding partial correlations. The comparison of dependent correlations is a well known problem in statistics but, when the variables satisfy some conditional independence relations, the maximum likelihood estimates of partial correlations are different from the sample partial correlations, so that classical results no longer bold. In this paper we analyze to what extent the classical test statistics can be applied. We show that maximum likelihood estimates of partial correlations are more efficient than sample partial correlations. Furthermore we show that the conditional independence structure of the model can be used to turn the comparison of dependent correlations into the comparison of independent parameters.

ALLOCATION OF REMOTELY SENSED DATA USING MARKOV MODELS FOR IMAGE DATA AND PIXEL LABELS

SummaryFor remotely sensed data, this paper reviews the Bayesian approach to the allocation of picture elements (pixels) to groups. Group labels are assumed a priori to be spatially correlated and, conditional on the labels, the image data are also assumed to be spatially correlated. The models considered have the property that the posterior distribution of the pixel labels given the image data inherits conditional independence constraints. Two allocation algorithms which exploit this fad are discussed. These algorithms are based on maximising the posterior distribution, and involve the use of neighbouring image and label data to update the label of any given pixel. The effect of spatial correlation in the image data on allocation performance is examined.

Recursive causal models

AbstractThe notion of a recursive causal graph is introduced, hopefully capturing the essential aspects of the path diagrams usually associated with recursive causal models. We describe the conditional independence constraints which such graphs are meant to embody and prove a theorem relating the fulfilment of these constraints by a probability distribution to a particular sort of factorisation. The relation of our results to the usual linear structural equations on the one hand, and to log-linear models, on the other, is also explained