William S. Rayens

Partial least squares for discrimination

Partial least squares (PLS) was not originally designed as a tool for statistical discrimination. In spite of this, applied scientists routinely use PLS for classification and there is substantial empirical evidence to suggest that it performs well in that role. The interesting question is: why can a procedure that is principally designed for overdetermined regression problems locate and emphasize group structure? Using PLS in this manner has heurestic support owing to the relationship between PLS and canonical correlation analysis (CCA) and the relationship, in turn, between CCA and linear discriminant analysis (LDA). This paper replaces the heuristics with a formal statistical explanation. As a consequence, it will become clear that PLS is to be preferred over PCA when discrimination is the goal and dimension reduction is needed. Copyright

Iron accumulation in the striatum predicts aging-related decline in motor function in rhesus monkeys

Changes in the nigrostriatal system may be involved with the motor abnormalities seen in aging. These perturbations include alterations in dopamine (DA) release, regulation and transport in the striatum and substantia nigra, striatal atrophy and elevated iron levels in the basal ganglia. However, the relative contribution of these changes to the motor deficits seen in aging is unclear. Thus, using the rhesus monkey as a model, the present study was designed to examine several of these key alterations in the basal ganglia in order to help elucidate the mechanisms contributing to age-related motor decline. First, 32 female rhesus monkeys ranging from 4 to 32 years old were evaluated for their motor capabilities using an automated hand-retrieval task. Second, non-invasive MRI methods were used to estimate brain composition and to indirectly measure relative iron content in the striatum and substantia nigra. Third, in vivo microdialysis was used to evaluate basal and stimulus-evoked levels of DA and its metabolites in the striatum and substantia nigra of the same monkeys. Our results demonstrated significant decreases in motor performance, decreases in striatal DA release, and increases in striatal iron levels in rhesus monkeys as they age from young adulthood. A comprehensive statistical analysis relating age, motor performance, DA release, and iron content indicated that the best predictor of decreases in motor ability, above and beyond levels of performance that could be explained by age alone, was iron accumulation in the striatum. This suggests that striatal iron levels may be a biomarker of motor dysfunction in aging; and as such, can be monitored non-invasively by longitudinal brain MRI scans. The results also suggest that treatments aimed at reducing accumulation of excess iron in the striatum during normal aging may have beneficial effects on age-related deterioration of motor performance.

Structure-seeking multilinear methods for the analysis of fMRI data

In comprehensive fMRI studies of brain function, the data structures often contain higher-order ways such as trial, task condition, subject, and group in addition to the intrinsic dimensions of time and space. While multivariate bilinear methods such as principal component analysis (PCA) have been used successfully for extracting information about spatial and temporal features in data from a single fMRI run, the need to unfold higher-order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data. These additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiologically meaningful. Multiway analysis of fMRI data from multiple runs of a bilateral finger-tapping paradigm was performed using the parallel factor (PARAFAC) model. A trilinear model was fitted to a data cube of dimensions voxels by time by run. Similarly, a quadrilinear model was fitted to a higher-way structure of dimensions voxels by time by trial by run. The spatial and temporal response components were extracted and validated by comparison to results from traditional SVD/PCA analyses based on scenarios of unfolding into lower-order bilinear structures.

TWO-FACTOR DEGENERACIES AND A STABILIZATION OF PARAFAC

Abstract Mitchell and Burdick (B.C. Mitchell, D.S. Burdick, An empirical comparison of resolution methods for three-way arrays, Chemom. Intell. Lab. Syst. 20 (1993) 149–161; B.C. Mitchell, D.S. Burdick, Slowly converging PARAFAC sequences: Swamps and two-factor degeneracies, J. Chemom. 8 (1994) 155–168.) uncovered an intriguing correspondence between the existence of PARAFAC swamps and the presence of two-factor degeneracies (2FDs). This observation, coupled with the recognition that post-swamp resolutions were generally better than pre-swamp resolutions allowed the user a method of detecting when a swamp had likely been encountered and, hence, when it was safe to assume that PARAFAC had converged to a suitable resolution. Still, this correspondence alone did not suggest how one might reduce the number of PARAFAC iterations required to move through a swamp. In this paper, it is noted that serious 2FDs must produce an identifiable ill-conditioning in the least squares estimation step in PARAFAC. Moreover, a serious 2FD is only one way this ill-conditioning may occur and, hence, it is more correct to say that swamps correspond to this ill-conditioning in general, rather than to the presence of 2FDs in particular. In an attempt to reduce the number of iterations that PARAFAC spends in a swamp, a particular method of stabilization is employed and results are presented which suggest that the number iterations can often be greatly reduced.

Functional MRI studies in awake rhesus monkeys: methodological and analytical strategies.

Functional imaging of the non-human primate brain in awake animals is now feasible because of recent methodological advances. Here we detail our procedures for conducting functional MRI (fMRI) studies in rhesus monkeys. Our emphasis has been on analyzing drug-evoked responses within and across test groups, meaning that techniques have had to be developed for training and testing relatively large groups of animals. Group size is important as unbiased estimates are best derived from analyzing responses in multiple animals with replicate scans per animal due to partial volume errors in evaluating small brain regions and motion artifacts during scanning. While the procedures presented here were developed for mapping responses obtained from stimulating dopaminergic systems, much of the methodology is generally applicable for non-human primate fMRI studies and addresses specific problems encountered in imaging awake animals. These are (1) adapting animals to an MRI environment, (2) minimizing head movements, (3) reducing ambient scanning noise levels, and (4) developing multivariate methods of image data analysis suitable for eliciting the dynamic brain response while (5) detecting and deleting outlying observations due to motion artifacts. Procedures are demonstrated for first pre-processing and analyzing responses in a voxel-based approach in a single animal and then proceeding to analyze responses across animals and replicate scans for regions of interest. Collectively, the procedures described provide an approach for fMRI mapping of elicited responses using conventional 1.5T MR scanners.

Covariance pooling and stabilization for classification

Abstract It is well known that the quadratic discriminant rule (QD) is optimal for large, normal training sets in the sense of minimizing the overall misclassification rate. However, when the size of the training set is small compared to the number of variables, the performance of QD is degraded because it uses unstable sample mean vectors and covariance matrices. Friedman [8] and Greene and Rayens [9] recently proposed different methods for addressing the problem of unstable covariance matrices. This article details a critical comparison of these two methods, finding important strengths and weaknesses in both. In addition, a third discriminant rule which combines ideas from both of the aforementioned methods is developed and included in the comparisons.

Longitudinal functional alterations in asymptomatic women at risk for Alzheimer's disease.

Purpose. The authors sought to determine whether known alterations of brain function in normal individuals who are at high risk for Alzheimers disease (AD) worsen or stay the same after a significant interval of time. Methods. The authors used func tional magnetic resonance imaging (fMRI) to observe cortical activation during confrontation naming in 14 women with high AD risk and 10 with low risk, based on family history and apolipoprotein‐E4 allele status. They repeated the identical scan protocol in the same patients after 4 years. Results. fMRI activation in high‐AD‐risk participants was found to be further diverged from that of their low‐AD‐risk counterparts over this period. Conclusion. fMRI may report on the presence and pro gression of neuropathology in the ventral temporal cortex or in functionally connected regions in presymptomatic AD.

Partially pooled covariance matrix estimation in discriminant analysis

The Linear Discriminant Rule (LD) is theoretically justified for use in classification when the population within-groups covariance matrices are equal, something rarely known in practice. As an alternative, the Quadratic Discriminant Rule (QD) avoids assuming equal covariance matrices, but requires the estimation of a large number of parameters. Hence, the performance of QD may be poor if the training set sizes are small or moderate. In fact, simulation studies have shown that in the two-groups case LD often outperforms QD for small training sets even when the within -groups covariance matrices differ substantially. The present article shows this to be true when there are more than two groups, as well. Thus, it would seem reasonable and useful to develop a data-based method of classification that, in effect, represents a compromise between QD and LD. In this article we develop such a method based on an empirical Bayes formulation in which the within-groups covariance matrices are assumed to be outcomes of...

Dependence Properties of Generalized Liouville Distributions on the Simplex

Abstract Compositional data arise naturally in several branches of science, including chemistry, geology, biology, medicine, ecology, and manufacturing design. Thus the correct statistical analysis of this type of data is of fundamental importance. Prior to the pioneering and extensive work of Aitchison, the Dirichlet distribution provided the parametric model of choice when analyzing such data. But Aitchison and others have since pointed out that the Dirichlet distribution is appropriate only for modeling compositional vectors that exhibit forms of extreme independence. Aitchison developed his logistic normal classes partly in response to this shortcoming. Unfortunately, Aitchisons logistic normal classes do not contain the Dirichlet distribution as a special case. As a result, they exhibit interesting dependence structures but are unable to model extreme independence. The generalized Liouville family is studied in this article. This family, which contains the Dirichlet class, is shown to contain densit...

Partial least squares and compositional data: problems and alternatives

Abstract It is still widely unknown in chemometrics that the statistical analysis of compositional data requires fundamentally different tools than a similar analysis of unconstrained data. This article examines the problems that potentially occur when one performs a partial least squares (PLS) analysis on compositional data and suggests logcontrast partial least squares (LCPLS) as an alternative.