# Multivariate Analysis: Theory and Geometric Interpretation

Multivariate Analysis: Theory and Geometric Interpretation David Chelidze Proper Orthogonal Decomposition (POD) We consider a scalar field , where and . POD decomposes it using orthonormal basis functions and the corresponding time coordinates , where This is equivalent to the following maximization problem: , subject to This results into the following integral eigenvalue problem , where Smooth Orthogonal Decomposition (SOD)

SOD is looking for a projective function , such that the has maximal variance subject to its minimal roughness, expressed by the following maximization problem: subject to This translates into the following generalized integral eigenvalue problem: Using the solution to the above eigenvalue problem, the scalar field can be reconstructed as: , where A set of smooth orthogonal modes modes . form a bi-orthogonal set with smooth projective Practical Calculations for POD and SOD When the field is sampled such that

solved by singular value decomposition contains the corresponding mode shapes. , where , then POD can be , where are time coordinates and The corresponding SOD problem can be solved by generalized singular value decomposition: where: are unitary matrices, columns of are smooth orthogonal modes; columns of are smooth orthogonal coordinates; columns of are smooth projective modes; are smooth orthogonal values, and

. are diagonal matrices, and Proper Orthogonal Decomposition: Geometric Interpretation Geometric Interpretation of SOD SOD identifies the subspaces where the scalar field projection is maximally smooth Smooth orthogonal modes (SOMs) are not orthogonal to each other but are linearly independent

SOMs span smooth modal subspaces Smooth projective modes (SPMs) form a biorthonormal set with SOMs and are used to obtain smooth orthogonal coordinates (SOCs) SOCs are orthogonal to each other (i.e., their covariance matrix is diagonal) SOCs are invariant under invertible linear

coordinate transform SOD Geometric Interpretation Given a cloud of trajectory points, we identify its center of mass. Then a point and its velocity can be visualized by two vectors in the figure Now we identify the first SPM by maximizing the ratio of the projections of the velocity and the position onto this mode

This is followed by the similar maximization for the second SPM and so on Smooth orthogonal modes (SOMs) form a bi-orthonormal set with SPMs and are used to obtain describe the points in the cloud

## Recently Viewed Presentations

• Shells, Shell Scripting, and the UNIX File System ... desktop manger, window manager, etc. on a UNIX system). UMBC Shells On the UMBC GL network, the default UNIX shell is tcsh - Turbo C SHell. ... Shell Scripting, and the...
• Guidelines for the Faith Formation and Development of Catholic Students. ... Schools' Pastoral Care systems explained. P2: Role of Chaplain is outlined, however it does not mention that the Chaplain is not always a member of a school's staff ......
• روش/ ساختار/مدل .... پیشنهادی. عنوان. پروژه. نام دانشجو و استاد راهنما. دانشکده مهندسی برق و کامپیوتر، دانشگاه تهران
• Today was a special day. My College Station Noon Lions Club volunteered to haul luggage for campers at the Texas Lions Camp in Kerrville. The camp is 70 years old and is funded totally by Lions clubs in Texas. Campers'...
• King Abdullah University of Science and Technology (KAUST) Bottom Pressure Variability in the Red Sea. ... In particular, the along-coast and cross-shelf SSE gradients play key roles in the along- and cross-shelf momentum equations, and the direct estimation of these...
• Internet haute vitesse ACN + Voix pour les entreprises Disponible dans les territoires de Telus et de Bell Aliant Processus de réclamation de la carte récompense Les clients qui optent pour le service DT Express ou l'adaptateur téléphonique avec un...
• McDonald's McDonald's enstand nicht in Italien, sondern wurde von den Brüdern Dick und Mac McDonald's im 1971 gegründet . Es war in Zusammenarbeit mit Walt Disney. Heute lieben die Italiener MC, weil der Service schnell und technologisch ist. Die Preise...
• Heather Jenks Associate Director, Library Services * BONUS+ BONUS+ is a resource-sharing initiative that allows post graduate students and staff at the ANU giving access to over 7 million books from participating university libraries within Australia and New Zealand.