Describing the probability densities of multi-object complexes by describing individual objects and their inter-object relationships leads to desirable locality without ignoring the context of an object. We describe a means of decomposing object variations into self effects and neighbor effects. We describe an approach for estimating the self and neighbor effect probability densities for each object in the complex using augmentation and prediction, supported by PGA on m-reps. We apply this method to the inter-day variation of m-reps of male pelvic organs within an individual patient.

The advancing technology for automatic segmentation of medical images should be accompanied by techniques to inform the user of the credibility of results. To the extent that this technology produces clinically acceptable segmentations for a significant fraction of cases there is a risk that the clinician will assume every result is acceptable. In the less frequent case where segmentation fails we are concerned that unless the user is alerted by the computer, she would 5A5Astill put the result to clinical use. We propose an automated method to signal suspected noncredible regions of the segmentation, triggered by outlier values of the local image match function. The user can focus her validation resources on the noncredible regions. When the local image match function is computed via a Mahalanobis dis- tance, as is the case for PCA-based matches, its value follows the chi-squared distribution. Our method signals a noncredible region wherever the probability of a chi-squared random variable being greater than the match observed is above a threshold level. ROC analysis validates our noncredibility test on m-rep segmentations of the bladder in CT images, using an image match computed by PCA on regional intensity quantile functions. We approximate ground truth as truly noncredible regions have surface distance > 5mm to a reference segmentation. We swept out ROC curves by varying the threshold level. The area under the ROC curve was 0.91. Based on this preliminary result, our method shows potential for validation in an automatic segmentation pipeline.

We study the effect of degrees of freedom on the level and power of quadratic distance based tests. The concept of an \textit{eigendepth index} is introduced and discussed in the context of selecting the optimal degrees of freedom, where optimality refers to high power. We introduce the class of diffusion kernels by the properties we seek these kernels to have and give a method for constructing them by exponentiating the rate matrix of a Markov chain. Product kernels and their spectral decomposition are discussed and shown useful for high dimensional data problems.

Bayes Factors play an increasingly important role in comparing fit in a variety of statistical models ranging from multiple regression (Carlin and Chib, 1995; Spiegelhalter and Smith, 1982) to mixture models (Richardson and Green, 1997). Full Bayesian analysis calculates the actual Bayes Factor that derives from an explicit prior distribution. However, often the exact Bayes Factor cannot be calculated, either because of computational limitations or due to unavailability of an appropriate prior. Instead we develop criteria based on approximations of the Bayes Factor, the most popular of which is the Schwarz (1978) Bayesian Information Criterion (BIC). The justification for the BIC rests either on large sample properties and the dominance of terms not involving the prior distribution for calculating the parameters, or on an implicit unit information prior distribution (Raftery, 1995). This research proposes two new approximations to the Bayes Factor that researchers can implement. The first approximation is justified by using a scaled unit information prior distribution that is more general than the unit information prior of the BIC. The other approximation utilizes more terms in the approximation than those used in the BIC. In a simulation of variable selection in regression and an empirical example, we found that the new scaled unit information prior approximation (SPBIC) performed better than the BIC, AIC, and several other competitive measures. IBIC, our second approximation performed the same as the SPBIC in the empirical example, but less well in the simulations. The SPBIC and IBIC deserve further attention as promising approximations to the Bayes Factor.

This research was initiated by the analysis of NCI60 cancer dataset . The dataset contains gene expression values (from cDNA arrays) corresponding to 3509 genes collected from 60 different patients diagnosed with 8 different cancer types (assumed unknown in the following discussion). The goal is to provide a model based approach for simultaneously clustering cancer types (columns) and the genes (rows) involved in differentiating these cancer types. We formulate a novel two-way mixture framework and adapt our distance-based model selection tool to determine the unknown number of row and column clusters. This methodology avoids two major pitfalls of using model-based clustering in high-dimensions. First, the two-way mixture has a considerably smaller parameter set, compared to the full multivariate analysis, making all parameters estimable. Second, unlike the complex distribution of likelihood-ratio-based tools under the composite null hypothesis of fixed row and column clusters, the distribution of our distance-based model selection tool is well defined, even for composite hypotheses. Finally, based on the geometry of pure Gaussian HDLSS data, we provide an effective visual diagnostic tool to uncover any remaining structure in the data. Through our analysis, we uncovered some interesting sets of gene clusters. But some of our cancer-type clusters did not match the initial cancer labels. On later verification we found that such discordance was due to the close similarity in symptoms and pathological test results of the two types of cancer in question.

Multivariate mixtures provide flexible methods for both fitting and partitioning high-dimensional data. Ray and Lindsay(2005) show that the topography of multivariate mixtures, in the sense of their key features as a density, can be analyzed rigorously in lower dimensions by use of a ridgeline manifold that contains all critical points as well as the ridges of the density. In addition, we have developed a new computing procedure similar to the EM algorithm that can quickly find the modes of a mixture density. This tool can be extended to examine the degree of separation between the modes based on the ridgeline separating them. These tools can be used in various ways. For one, we can take a conventional mixture analysis and cluster together those components whose contribution is actually unimodal. This cluster could then represent a single true component with a more complex distribution. We can also turn kernel density estimation into a hierarchical clustering tool in which the data points become identified with each other by their association with a common mode of the density estimator. Separate clusters must then correspond to gaps in the estimated density. The analysis is multi-scale, as different levels of smoothing provide different aggregations.

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