# 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