Extended principal component analysis and its applications to image analysis

The objectives of this research are to analyze and develop a modified Principal Component Analysis (PCA) and to develop a two-dimensional PCA with applications in image processing. PCA is a classical multivariate technique where its mathematical treatment is purely based on the eigensystem of positive-definite symmetric matrices. Its main function is to statistically transform a set of correlated variables to a new set of uncorrelated variables over $\IR\sp{n}$ by retaining most of the variations present in the original variables.^ The variances of the Principal Components (PCs) obtained from the modified PCA form a correlation matrix of the original variables. The decomposition of this correlation matrix into a diagonal matrix produces a set of orthonormal basis that can be used to linearly transform the given PCs. It is this linear transformation that reproduces the original variables. The two-dimensional PCA can be devised as a two successive of one-dimensional PCA. It can be shown that, for an $m\times n$ matrix, the PCs obtained from the two-dimensional PCA are the singular values of that matrix.^ In this research, several applications for image analysis based on PCA are developed, i.e., edge detection, feature extraction, and multi-resolution PCA decomposition and reconstruction. ^

Principal component analysis and assessment of language network activation patterns in pediatric epilepsy

This dissertation establishes a novel data-driven method to identify language network activation patterns in pediatric epilepsy through the use of the Principal Component Analysis (PCA) on functional magnetic resonance imaging (fMRI). A total of 122 subjects’ data sets from five different hospitals were included in the study through a web-based repository site designed here at FIU. Research was conducted to evaluate different classification and clustering techniques in identifying hidden activation patterns and their associations with meaningful clinical variables. The results were assessed through agreement analysis with the conventional methods of lateralization index (LI) and visual rating. What is unique in this approach is the new mechanism designed for projecting language network patterns in the PCA-based decisional space. Synthetic activation maps were randomly generated from real data sets to uniquely establish nonlinear decision functions (NDF) which are then used to classify any new fMRI activation map into typical or atypical. The best nonlinear classifier was obtained on a 4D space with a complexity (nonlinearity) degree of 7. Based on the significant association of language dominance and intensities with the top eigenvectors of the PCA decisional space, a new algorithm was deployed to delineate primary cluster members without intensity normalization. In this case, three distinct activations patterns (groups) were identified (averaged kappa with rating 0.65, with LI 0.76) and were characterized by the regions of: (1) the left inferior frontal Gyrus (IFG) and left superior temporal gyrus (STG), considered typical for the language task; (2) the IFG, left mesial frontal lobe, right cerebellum regions, representing a variant left dominant pattern by higher activation; and (3) the right homologues of the first pattern in Broca's and Wernicke's language areas. Interestingly, group 2 was found to reflect a different language compensation mechanism than reorganization. Its high intensity activation suggests a possible remote effect on the right hemisphere focus on traditionally left-lateralized functions. In retrospect, this data-driven method provides new insights into mechanisms for brain compensation/reorganization and neural plasticity in pediatric epilepsy.

Principal Component Analysis and Assessment of Language Network Activation Patterns in Pediatric Epilepsy

This dissertation establishes a novel data-driven method to identify language network activation patterns in pediatric epilepsy through the use of the Principal Component Analysis (PCA) on functional magnetic resonance imaging (fMRI). A total of 122 subjects’ data sets from five different hospitals were included in the study through a web-based repository site designed here at FIU. Research was conducted to evaluate different classification and clustering techniques in identifying hidden activation patterns and their associations with meaningful clinical variables. The results were assessed through agreement analysis with the conventional methods of lateralization index (LI) and visual rating. What is unique in this approach is the new mechanism designed for projecting language network patterns in the PCA-based decisional space. Synthetic activation maps were randomly generated from real data sets to uniquely establish nonlinear decision functions (NDF) which are then used to classify any new fMRI activation map into typical or atypical. The best nonlinear classifier was obtained on a 4D space with a complexity (nonlinearity) degree of 7. Based on the significant association of language dominance and intensities with the top eigenvectors of the PCA decisional space, a new algorithm was deployed to delineate primary cluster members without intensity normalization. In this case, three distinct activations patterns (groups) were identified (averaged kappa with rating 0.65, with LI 0.76) and were characterized by the regions of: 1) the left inferior frontal Gyrus (IFG) and left superior temporal gyrus (STG), considered typical for the language task; 2) the IFG, left mesial frontal lobe, right cerebellum regions, representing a variant left dominant pattern by higher activation; and 3) the right homologues of the first pattern in Broca's and Wernicke's language areas. Interestingly, group 2 was found to reflect a different language compensation mechanism than reorganization. Its high intensity activation suggests a possible remote effect on the right hemisphere focus on traditionally left-lateralized functions. In retrospect, this data-driven method provides new insights into mechanisms for brain compensation/reorganization and neural plasticity in pediatric epilepsy.

Principal component and second generation wavelet analysis of Treasury yield curve evolution

Prices of U.S. Treasury securities vary over time and across maturities. When the market in Treasurys is sufficiently complete and frictionless, these prices may be modeled by a function time and maturity. A cross-section of this function for time held fixed is called the yield curve; the aggregate of these sections is the evolution of the yield curve. This dissertation studies aspects of this evolution. ^ There are two complementary approaches to the study of yield curve evolution here. The first is principal components analysis; the second is wavelet analysis. In both approaches both the time and maturity variables are discretized. In principal components analysis the vectors of yield curve shifts are viewed as observations of a multivariate normal distribution. The resulting covariance matrix is diagonalized; the resulting eigenvalues and eigenvectors (the principal components) are used to draw inferences about the yield curve evolution. ^ In wavelet analysis, the vectors of shifts are resolved into hierarchies of localized fundamental shifts (wavelets) that leave specified global properties invariant (average change and duration change). The hierarchies relate to the degree of localization with movements restricted to a single maturity at the base and general movements at the apex. Second generation wavelet techniques allow better adaptation of the model to economic observables. Statistically, the wavelet approach is inherently nonparametric while the wavelets themselves are better adapted to describing a complete market. ^ Principal components analysis provides information on the dimension of the yield curve process. While there is no clear demarkation between operative factors and noise, the top six principal components pick up 99% of total interest rate variation 95% of the time. An economically justified basis of this process is hard to find; for example a simple linear model will not suffice for the first principal component and the shape of this component is nonstationary. ^ Wavelet analysis works more directly with yield curve observations than principal components analysis. In fact the complete process from bond data to multiresolution is presented, including the dedicated Perl programs and the details of the portfolio metrics and specially adapted wavelet construction. The result is more robust statistics which provide balance to the more fragile principal components analysis. ^