Singular values

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

Hendler R W and Shrager R I 1994 Deconvolutions based on singular value decomposition and the pseudoinverse—a guide for beginners J. Blochem. Blophys. Methods 28 1-33... 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

It may look weird to treat the Singular Value Decomposition SVD technique as a tool for data transformation, simply because SVD is the same as PCA. However, if we recall how PCR (Principal Component Regression) works, then we are really allowed to handle SVD in the way mentioned above. Indeed, what we do with PCR is, first of all, to transform the initial data matrix X in the way described by Eqs. (10) and (11). 

Here W is diagonal matrix of singular values, is the transpose of the second re-sultant matrix, being actually the same as the loading matrix in PCA, and X is the matrix, which is applied for further modeling. 

Widely used methods of data transformation are Fast Fourier and Wavelet Transformations or Singular Value Decomposition... 

C. F. Moore, "AppHcation of Singular Value Decomposition to the Design, Analysis, and Control of Industrial Processes," Proceeding of American Control Conference, Boston, Mass., 1986, p. 643. 

The singular values of a complex n x m matrix A, denoted by cr,(A) are the nonnegative square-roots of the eigenvalues of A A ordered such that... 

The maximum singular value rr of A and the minimum singular value o of A are defined by... 

The singular value of the sensitivity funetion irfSfja )) and of the eomplementary sensitivity funetion (r(T(ja )) ean be displayed as Bode plots and play an important role in robust multivariable eontrol system design. 

The singular values of S determine the disturbanee attenuation, and thus a performanee speeifieation may be written... 

The singular value frequeney response G juj) is shown in Figure 9.32. 

The optimal value of TyjUi is aehieved when (7 = 0.13) and its singular value frequeney response is shown in Figure 9.34. 

Fig. 9.32 Plant singular value Bode magnitude plot.

Fig. 9.34 Singular value Bode magnitude plot of Tyi Ui (jo ) when 7 = 0.13.

Multivartable robust control using H infinity %Singular value loop shaping using the weighted mixed %sensitivity approach nug=200 ... 

After eleven iterations, hinfopt identifies that 7 in equation (9.176) has a best value of 0.13. The command sigma calculates the data for a singular value Bode diagram as shown in Figures 9.32, 9.34 and 9.35. Other information printed in the command window is given below... 

Xie D, Tropsha A, Schlick T. An efficient projection protocol for chemical databases singular value decomposition combined with truncated-newton minimization. / Chem Inf Comput Sci 2000 40 167-77. 

The scaled data matrix D is decomposed using singular value decomposition (see Bonvin and Rippin (1990), Hamer (1989), Golub and van Loan (1983)) into matrices with one containing stoichiometric information which can be processed into acceptable stoichiometry. The decomposition can be easily done by any available software packages (e.g. Dongarra etal. (1979), Press et ai, (1989)). Upon decomposing one obtains ... 

The last two singular values are of equal magnitude and much smaller than the others. This indicates that... 

Singular value decomposition (SVD) of a rectangular matrix X is a method which yields at the same time a diagonal matrix of singular values A and the two matrices of singular vectors U and V such that ... 

An important theorem of matrix algebra, called singular value decomposition (SVD), states that any nxp table X can be written as the matrix product of three terms U, A and V ... 

It can be proved that the decomposition is always possible and that the solution is unique (except for the algebraic signs of the columns of U and V) [3]. Singular value decomposition of a rectangular table is an extension of the classical work of Eckart and Young [4] on the decomposition of matrices. The decomposition of X into U, V and A is illustrated below using a 4x3 data table which has been adapted from V an Borm [5]. (A similar example has been used for the introduction to PC A in Chapter 17.)... 

The above relation holds within the limited precision of our calculations of the singular values, which in the present example is about four significant digits. 

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