Orthogonal/orthogonality decomposition

Tabib, M.V. and Joshi, J.B. (2008) Analysis of dominant flow stmctures and their flow dynamics in chemical process equipment using snapshot proper orthogonal decomposition technique. Chem. Eng. Sci., 63 (14), 3695-3715. 

The power algorithm [21] is the simplest iterative method for the calculation of latent vectors and latent values from a square symmetric matrix. In contrast to NIPALS, which produces an orthogonal decomposition of a rectangular data table X, the power algorithm decomposes a square symmetric matrix of cross-products X which we denote by C. Note that Cp is called the column-variance-covariance matrix when the data in X are column-centered. 

Correlation between three or more parameters is very difficult to detect unless an eigenvalue decomposition of matrix A is performed. As already discussed in Chapter 8, matrix A is symmetric and hence an eigenvalue decomposition is also an orthogonal decomposition... 

Levy and Goldstein chose to tackle the problem in a different way. They based their reasoning, in one of their derivations, on the orthogonal decomposition of the space spanned by i/j, into S and S1, respectively the occupied and unoccupied subspaces. 

Figure 1. Orthogonal decomposition of a three-dimensional Hilbert space geometrical representation of the two orthogonal subspaces. 

The principle of FA and PCA consists in an orthogonal decomposition of the original n x m data matrix X into a product of two matrixes, F (nxk matrix of factor scores, common factors) and L (kxm matrix of factor loadings)... 

The estimation of unmeasured total flow rates is done by using the Q-R orthogonal decomposition of matrix D. The equation can be written as... 

From the orthogonal decomposition of the A2 matrix, the unmeasured process variables are divided in two subsets ... 

Figure 15 Linking models at various scales using ROMs and deriving lower scale specifications through an inverse optimization formulation. The ROM included at each scale is a reduced representation of the model at the scale below that could range from a set of parameters such as, for example, elementary rate constants to complex models derived from proper orthogonal decomposition and perhaps even to the full lower scale model. This is symbolized by coloring the ROM box with the same color as that of the box representing the adjacent lower scale model.

It is noted that during later stages of b3rpass transition caused by con-vecting vortex in the freestream, large coherent vortices are formed inside the shear layer. Such coherent structures have been characterized by Proper orthogonal Decomposition (POD) - a method that was originally developed... 

Sengupta, T.K. and De, S (2004). Proper orthogonal decomposition of direct numerical simulation data of by-pass transition. Comput. Struct. 82, 2693-2703. 

A general requirement for P-matrix analysis is n = rank(R). Unfortcmately, for most practical cases, the rank of R is greater than the number of components, i.e., rank(R) > n, and rank(R) = min(m, p). Thus, P-matrix analysis is associated with the problem of substituting R with an R that produces rank(R ) = n. This is mostly done by orthogonal decomposition methods, such as principal components analysis, partial least squares (PLS), or continuum regression [4]. Dimension requirements of involved matrices for these methods are m > n, and p > n. If the method of least squares is used, additional constraints on matrix dimensions are needed [4]. The approach of P-matrix analysis does not require quantitative concentration information of all constituents. Specifically, calibration samples with known concentrations of analytes under investigation satisfy the calibration needs. The method of PLS will be used in this chapter for P-matrix analysis. 

Prof. Roger Shaw (University of California, Davis) describes new developments in the analysis of the turbulent velocity fields in vegetative canopies, using proper orthogonal decomposition (a technique for eddy identification first introduced by J. Lumley in 1967). He argues that this technique complements the previous studies based on statistical models and large eddy simulation numerical methods. 

PCA involves the orthogonal decomposition of the set of process measurements along the directions that explain the maximum variation in the data. For a continuous process, the elements of the n x m data matrix... 

Separating measured data vectors or matrices into independent lower order approximations and residual terms is useful both in process performance evaluation, as variance contributions can be clearly separated, and in feedback process control, as the number of decision variables can be significantly reduced while the adverse effects of autocorrelation are eliminated. In the following two sections orthogonal decomposition approaches using Gram polynomials and principal components analysis (PCA) will be introduced. 

Dominant correlations of data are usually captured by a small number of initial eigenvectors. A simple orthogonal decomposition is accomplished by partitioning U = [UmU/j] and A = [Am A ] where M designates the number of initial dominant modes to be used for approximation while R stands for the remaining N — M) modes or the residual. Data matrix becomes Y = JM- M + R- R = Ym + Yr. For a successful approximation, YM captures significant variability trends and Yr simply represents residual random noise. Transformation in the form Y Ym uses M N + K) data entries and provides [1 — M N + K)/ NK)]100 % data compression. 

Proper orthogonal decomposition has been used to analyze ensembles of concentration and vorticity magnitude distributions in forced reacting rectangular jets... 

A RECURSIVE PROPER ORTHOGONAL DECOMPOSITION ALGORITHM FOR FLOW CONTROL PROBLEMS... 

Sahoo, D., S. Park, D. Wee, A. M. Annaswamy, and A. F. Ghoniem. 2002. A recursive proper orthogonal decomposition algorithm for flow control problems. Adaptive Control Laboratory, MIT. Technical Report 0208. 

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