We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

In the modern era of high and infinite dimensional data, classical statistical methodology is often rendered inefficient and ineffective when confronted with such big data problems as arise in genomics, medical imaging, speech analysis, and many other areas of research. Many problems manifest when the practitioner is required to take into account the covariance structure of the data during his or her analysis, which takes on the form of either a high dimensional low rank matrix or a finite dimensional representation of an infinite dimensional operator acting on some underlying function space. Thus, novel methodology is required to estimate, analyze, and make inferences concerning such covariances. In this manuscript, we propose using tools from the concentration of measure literature–a theory that arose in the latter half of the 20th century from connections between geometry, probability, and functional analysis–to construct rigorous descriptive and inferential statistical methodology for covariance matrices and operators. A variety of concentration inequalities are considered, which allow for the construction of nonasymptotic dimension-free confidence sets for the unknown matrices and operators. Given such confidence sets a wide range of estimation and inferential procedures can be and are subsequently developed. For high dimensional data, we propose a method to search a concentration in- equality based confidence set using a binary search algorithm for the estimation of large sparse covariance matrices. Both sub-Gaussian and sub-exponential concentration inequalities are considered and applied to both simulated data and to a set of gene expression data from a study of small round blue-cell tumours. For infinite dimensional data, which is also referred to as functional data, we use a celebrated result, Talagrand’s concentration inequality, in the Banach space setting to construct confidence sets for covariance operators. From these confidence sets, three different inferential techniques emerge: the first is a k-sample test for equality of covariance operator; the second is a functional data classifier, which makes its decisions based on the covariance structure of the data; the third is a functional data clustering algorithm, which incorporates the concentration inequality based confidence sets into the framework of an expectation-maximization algorithm. These techniques are applied to simulated data and to speech samples from a set of spoken phoneme data. Lastly, we take a closer look at a key tool used in the construction of concentration based confidence sets: Rademacher symmetrization. The symmetrization inequality, which arises in the probability in Banach spaces literature, is shown to be connected with optimal transport theory and specifically the Wasserstein distance. This insight is used to improve the symmetrization inequality resulting in tighter concentration bounds to be used in the construction of nonasymptotic confidence sets. A variety of other applications are considered including tests for data symmetry and tightening inequalities in Banach spaces. An R package for inference on covariance operators is briefly discussed in an appendix chapter.

With both a heterogeneous population and a large number of unidentified persons in South Africa, an accurate method to estimate ancestry is needed. The purpose of this study was to evaluate variation in nasal aperture shape in black, white and coloured South Africans, using linear measures and geometric morphometrics (GM), the latter which includes both procrustes analysis (GPA) and elliptical fourier analysis (EFA). To test statistical significance among groups, discriminant function analysis (DFA) and principal component analysis (PCA) was used. A total of 310 (164 male, 145 female) crania of black, white and coloured South Africans were used. Thirteen standard landmarks, namely, glabella, nasion, nasale superior, dacryon, nasale inferius, alare, most inferior nasal border and subspinale, were digitised with a MicroScribe G2™ (Immersion: San Jose, CA). Five linear measures, nasion-dacryon angle (NDA), nasal breadth (NLB), nasal height (NLH), inter-orbital breadth (DKB) and nasion-dacryon subtense (NDS), were calculated. For EFA, photographs were taken in a frontal plane of skulls that had been positioned in the Frankfort horizontal plane on a craniophore. All classification accuracies for all groups were better than chance. Using linear measures and GPA, black South Africans classified 55-71% correctly, coloured classified 53-61% correctly and whites classified 85-95% correctly. Black and coloured South Africans demonstrated bell-shaped nasal apertures with nasal spines superior to the inferior nasal border. White South Africans had pear-shaped nasal apertures with a nasal spine inferior of the inferior nasal border. Using EFA black South Africans classified 62% correctly. While coloured South Africans only classified 39% correctly, which demonstrates high within group variability. Due to their unique historical development, large variation (heterogeneity) within the coloured group was expected. White South Africans had the highest correct classification accuracy of 85%. For all methods, misclassification rarely occurred between white and non-white (black and coloured) groups and most difficulties arose in distinguishing non-white groups from each other. High rates of misclassification was also noted between sex designations within a group, which suggests less or an absence of sexual dimorphism for these variables The distinct separation of white South Africans may reflect the mid-to late 20th century political and social separation of white and non-white groups in South Africa. Nasal aperture shape, alone, is less useful for separating groups such that all groups have relatively intermediate nasal aperture shapes; however the pinched nasal bone structure of white South Africans clearly separates them from the other groups. When using nasal bone and aperture landmarks, linear measures are as accurate as the modern geometric techniques in distinguishing groups. All methods are feasible to use in the estimation of ancestry on modern South Africans, with craniometry a sensible solution as the data can be rapidly collected, accurately analysed and compared to current reference samples. Copyright / Dissertation (MSc)--University of Pretoria, 2012. / Anatomy / unrestricted

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.

We consider a family of generalized Procrustes problems. In this class of problems, one aims at aligning a set of vectors to a given second set of vectors. The distance between both sets is measured in the q norm, and for the alignment, isometries with respect to the p norm are allowed. In contrast to the classical Procrustes problem with p = q = 2, we allow p and q to differ. We will see that it makes a difference whether the problem is real or cast over the complex field. Therefore, we discuss the solutions for p = 2 separately for these cases. We show that all the real cases can be solved efficiently. Most of the complex cases can up to now only be solved approximately in polynomial time, but we show the existence of polynomial time algorithms for q ∈ {2, 4, ∞}. Computational experiments illustrate the suggested algorithms.:1. Introduction 2. The (lp, lq)-Procrustes problem 3. Optimization methods for the remaining cases with p not equal to 2 4. The one-dimensional complex optimization problems with p, q unequal to 2 5. Conclusions

No description available.