Matlab Singular Value Decomposition

Matlab is a matrix oriented language that is just about perfect for most data analysis tasks. Those readers who already know Matlab will agree with that statement. Those who have not used Matlab so far, will be amazed by the ease with which rather sophisticated programs can be developed. This strength of Matlab is a weak point in Excel. While Excel does include matrix operations, they are clumsy and probably for this reason, not well known and used. An additional shortcoming of Excel is the lack of functions for Factor Analysis or the Singular Value Decomposition. Nevertheless, Excel is very powerful and allows the analysis of fairly complex data. 

Excel does not provide functions for the factor analysis of matrices. Further, Excel does not support iterative processes. Consequently, there are no Excel examples in Chapter 5, Model-Free Analyses. There are vast numbers of free add-ins available on the internet, e.g. for the Singular Value Decomposition. Alternatively, it is possible to write Visual Basic programs for the task and link them to Excel. We strongly believe that such algorithms are much better written in Matlab and decided not to include such options in our Excel collection. 

PCA is simple in Matlab. The singular value decomposition (SVD) algorithm is employed, but this should normally give equivalent results to NIPALS except diat all the PCs are calculated at once. One difference is that die scores and loadings are bodi normalised, so that for SVD... 

To summarize, we perform a singular value decomposition of the augmented formula matrix to obtain the matrices U, W, and V. With these, we use (11.2.10) to obtain a particular basis vector N for the range. From V, we form P and then use (11.2.7) to obtain all sets of stoichiometric coefficients Vy. Then we combine N and Vy into (11.2.5) to determine all sets of mole numbers that satisfy the elemental balances. Therefore, a singular value decomposition provides the number of independent reactions 91, all sets of 91 independent stoichiometric coefficients Vy, and all possible combinations of mole numbers N that satisfy the elemental balances. A computer program for performing the decomposition is contained in the book by Press et al. [9] routines for performing the decomposition are also available in MATLAB and in Mathematica . 

Other Matrix Capabilities in MATLAB 2.2.3.1 Singular Value Decomposition... 

To solve equations (7-9), an approach with direct solution of the system (8) at each step of the iterative process by the pseudo- (or generalized) inversion method is used. It is based on singular value decomposition (SVD). It is well-known that the SSVDC procedure in the Linpack library is used to calculate SVD [12]. Paper [13] presents a standard SVD procedure in Fortran-IV used in the present paper. The current MATLAB system versions have a built-in function svd(A) implementing this decomposition for an arbitrary nxm matrix A. The calculation scheme of the SVD procedure is in decomposing the matrix (9) at each step of the iterative process into the product of three matrices. 

The states in Eqn (25.2) are now being formed as linear combinations of the -step ahead predicted outputs k= 1, 2,. ..). The literature on state space identification has shown how the states can be estimated directly from the process data by certain projections. (Verhaegen, 1994 van Overschee and de Moor, 1996 Ljung and McKelvey, 1996). The MATLAB function n4sid (Numerical Algorithms for Subspace State Space System Identification) uses subspace methods to identify state space models (Matlab 2000, van Overschee and de Moor, 1996) via singular value decomposition and estimates the state x directly from the data. 

---