P-matrix analysis

An alternative method, often referred to as the P-matrix analysis, avoids disadvantages present with K-matrix analysis by viewing concentration as a function of spectroscopic responses. Mathematically,... 

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. 

An alternative method would be to use complete spectra and hope that discrepancies between the linear calibration model and the real world data are leveraged out by the large pool of spectral information. For example, the PLS method is said to be suitable for such a type of P-matrix analysis. However, deterioration of measured spectra is more likely to be attributed to systematic physical effects than to uncorrelated random noise. Despite excellent results obtained using full spectra PLS calibrations and predictions compared to other linear projection methods, prediction errors can stiQ be significantly higher than for... 

A commonly used measure of quality for a P-matrix analysis is the predicted residual sum of squares (PRESS) value computed by... 

Alesso, H. P., 1984, On the Relationship of Digraph Matrix Analysis to Petri Net Theory and Fault Trees, LLNL UCRL-90271 Preprint, January. 

After factoring of the P matrix, Pecora considers that the constraints are summarized by the equation CC+ = and is completely determined by . This analysis, too, is based on counting the number of complex conditions on the complex elements of C. 

The analysis of an unknown sample is carried out by multiplication of the measured spectrum y by the P-matrix... 

In the case that the original variables, the measured values y, are used for inverse calibration, there are no significant advantages of the procedure apart from the fact that no second matrix inversion has to be carried out in the analysis step see Eq. (6.87). On the contrary, it is disadvantageous that the calibration coefficients (elements of the P-matrix) do not have any physical meaning because they do not reflect the spectra of the single species. In addition, multicollinearities may appear which can make inversion of the T-matrix difficult see Eq. (6.86). 

On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called soft modeling methods are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data ... 

We now have enough information to find our Scores matrix and Loadings matrix. First of all the Loadings matrix is simply the right singular values matrix or the V matrix this matrix is referred to as the P matrix in principal components analysis terminology. The Scores matrix is calculated as... 

Fig. 6.22. Comparisons of the longitudinal splitting length, Z,p, between analysis and finite element method for graphite fiber-epoxy matrix orthotropic laminates. After Tirosh (1973).

Maizel, J. V., Lenk, R. P. (1981) Enhanced graphic matrix analysis of nucleic acid and protein sequences, Proc. Natl. Acad, Sci. USA 78 7665-7669. 

Farr-Jones S, MUjanich GP, Nadasdi L, Ramachandran J, Basus VJ. Solution structure of omega-conotoxin MVllC, a high affinity ligand of P-type calcium channels, using IH NMR spectroscopy and complete relaxation matrix analysis. J Mol Biol 1995 248(1) 106-24. 

The improvement in computer technology associated with spectroscopy has led to the expansion of quantitative infrared spectroscopy. The application of statistical methods to the analysis of experimental data is known as chemometrics [5-9]. A detailed description of this subject is beyond the scope of this present text, although several multivariate data analytical methods which are used for the analysis of FTIR spectroscopic data will be outlined here, without detailing the mathematics associated with these methods. The most conunonly used analytical methods in infrared spectroscopy are classical least-squares (CLS), inverse least-squares (ILS), partial least-squares (PLS), and principal component regression (PCR). CLS (also known as K-matrix methods) and PLS (also known as P-matrix methods) are least-squares methods involving matrix operations. These methods can be limited when very complex mixtures are investigated and factor analysis methods, such as PLS and PCR, can be more useful. The factor analysis methods use functions to model the variance in a data set. 

Ross, P.L. and Belgrader, P. (1997) Analysis of short tandem repeat polymorphisms in human DNA by matrix-assisted laser desorption/ ionization mass spectrometry. Anal. Chem., 69 (19), 3966-3972. 

The kinetics of purging volatiles in water have been studied in depth by Lin et al. [122]. Purge efficiency can be defined as the quantity of the analytes purged from a sample with a defined volume of purge gas [54]. The recovery can be calculated as the ratio of the peak area for an analyte from P T analysis to that from direct injection. The efficiency depends upon several factors such as purge volume, sample temperature, purge vessel, the matrix, and the properties of analytes. 

RIC Riccardi, C.C., Borrajo, J., Meynie, L., Fenouillot, F., and Pascault, J.-P., Thermodynamic analysis of the phase separation during the polymerization of a thermoset system into a thermoplastic matrix. I. Effect of the composition on the cloud-point curves, J. Polym. Sci. PartB Polym. Phys., 42, 1351, 2004. 

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