Singular value decomposition analysis

Secondly, although stable solutions covering the entire temporal range of interest are attainable, the spectra may not be well resolved that is, for a given dataset and noise, a limit exists on the smallest resolvable structure (or separation of structures) in the Laplace inversion spectrum [54]. Estimates can be made on this resolution parameter based on a singular-value decomposition analysis of K and the signal-to-noise ratio of the data [56], It is important to keep in mind the concept of the spectral resolution in order to interpret the LI results, such as DDIF, properly. 

Romo, T.D., Clarage, J.B., Sorensen, D.C., Phillips, G.N. Automatic identification of discrete substates in proteins-singular-value decomposition analysis of time-averaged crystallographic refinements. Proteins 1995, 22, 311-21. 

This is acceptable provided that the assumption is stated up front. The melt curves in the lower part of Fig. 17.3 were fit this way. Alternatively, the full model can be used, with additional steps taken to ensure the validity of the results. An example is shown in the upper part of Fig. 17.3. Here, the experiments were repeated multiple times (minimizing the measurement error), the data were fit simultaneously (using a global fitting algorithm), and the results were corroborated using a separate singular value decomposition analysis. 

Understanding Oxotransferase Reactivity in a Model System Using Singular Value Decomposition Analysis... 

Sucheta et al. [28] published the experimental proofs of the theoretical mechanism of cytochrome c oxidase functioning proposed originally be Kharkats and Volkov [15,17 19,24]. Using time-resolved optical absorption difference spectra and singular value decomposition analysis, Sucheta et al. [28] found the presence of peroxy and ferryl intermediates at room temperature during reduction of oxygen by cytochrome c oxidase and measured the rate constants. 

To gain some insight into the problem, singular value decomposition analysis was used. There are three manipulated variables reflux, reboiler heat input and sidestream flow rate. The steady-state gains between tray temperatures and the three inputs were obtained numerically. The resulting singular values are 119, 15.8, and 2.68. The large condition number indicates that the control of more than one temperature in the column will be... 

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. 

In the previous section we have developed principal components analysis (PCA) from the fundamental theorem of singular value decomposition (SVD). In particular we have shown by means of eq. (31.1) how an nxp rectangular data matrix X can be decomposed into an nxr orthonormal matrix of row-latent vectors U, a pxr orthonormal matrix of column-latent vectors V and an rxr diagonal matrix of latent values A. Now we focus on the geometrical interpretation of this algebraic decomposition. 

J. Mandel, Use of the singular value decomposition in regression analysis. Am. Statistician, 36 (1982) 15-24. 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

This is just the same matrix discussed with singular value decomposition. For data analysis, though, it is necessary to adjust the columns to have zero mean by subtracting from each entry in the column the average of the column entries. Once this is done, the loadings are the v, and satisfy... 

We now have the data necessary to calculate the singular value decomposition (SVD) for matrix A. The operation performed in SVD is sometimes referred to as eigenanal-ysis, principal components analysis, or factor analysis. If we perform SVD on the A matrix, the result is three matrices, termed the left singular values (LSV) matrix or the V matrix the singular values matrix (SVM) or the S matrix and the right singular values matrix (RSV) or the V matrix. 

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. 

There is still a long list of different interpretations for the expression Factor Analysis. All the meanings of the term can be explained on the basis of the Singular Value Decomposition. 

The Singular Value Decomposition, SVD, has superseded earlier algorithms that perform Factor Analysis, e.g. the NIPALS or vector iteration algorithms. SVD is one of the most stable, robust and powerful algorithms existing in the world of numerical computing. It is clearly the only algorithm that should be used for any calculation in the realm of Factor Analysis. 

The Singular Value Decomposition of a matrix Y into the product USV is full of rich and powerful information. The model-free analyses we discussed so far are based on the examination of the matrices of eigenvectors U and V. Evolving Factor Analysis, EFA, is primarily based on the analysis of the matrix S of singular values. 

The controllability analysis was conducted in two parts. The theoretical control properties of the three schemes were first predicted through the use of the singular value decomposition (SVD) technique, and then closed-loop dynamic simulations were conducted to analyze the control behavior of each system and to compare those results with the theoretical predictions provided by SVD. 

Nanosecond Absorption Spectroscopy Absorption apparatus, 226, 131 apparatus, 226, 152 detectors, 226, 126 detector systems, 226, 125 excitation source, 226, 121 global analysis, 226, 146, 155 heme proteins, 226, 142 kinetic applications, 226, 134 monochromators/spectrographs, 226, 125 multiphoton effects, 226, 141 nanosecond time-resolved recombination, 226, 141 overview, 226, 119, 147 probe source, 226, 124 quantum yields, 226, 139 rhodopsin, 226, 158 sample holders, 226, 133 singular value decomposition, 226, 146, 155 spectral dynamics, 226, 136 time delay generators, 226, 130. 

Henry, E. R., and Hofrichter, J. 1992. Singular value decomposition Application to analysis of experimental data. Meth. Enzymol. 210 129-92. 

Rajagopal, S., Schmidt, M., Anderson, S., and Moffat, K. 2004. Methodology for analysis of experimental time-resolved crystallographic data by singular value decomposition. Acta Cry St. D 60 860-71. 

Schmidt, M., Rajagopal, S., Ren, Z., and Moffat, K. 2003. Application of singular value decomposition to the analysis of time-resolved macromolecular x-ray data. Biophys. J. 84 2112-29. 

Mandel, J., Use of the Singular Value Decomposition in Regression Analysis The American Statistician 1982, 36, 15-24. 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

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