Decomposition parallel factor analysis

Equation 11.17 is the fundamental expression of the PARAFAC (parallel factor analysis) model [77], which is used to describe the decomposition of trilinear data sets. For nontrilinear systems, the core C is no longer a regular cube (ncr x ncc x net), and the non-null elements are spread out in different manners, depending on each particular data set. The variables ncr, ncc, and net represent the rank in the row-wise, columnwise, and tubewise augmented data matrices, respectively. Each element in the original data set can now be obtained as shown in Equation 11.18 ... 

The parallel factor analysis (PARAFAC) model [18-20] is based on a multilinear model, and is one of several decomposition methods for a multidimensional data set. A major advantage of this model is that data can be uniquely decomposed into individual contributions. Because of this, the PARAFAC model has been widely applied to 3D and also higher dimensional data in the field of chemometrics. It is known that fluorescence data is one example that corresponds well with the PARAFAC model [21]. 

Shirakawa, H. and Miyazaki, H.S. (2004) Blind spectral decomposition of single-cell fluorescence by parallel factor analysis. Biophys.J., 86, 1739-1752. 

Wise BM, Gallagher NB, Butler SW, White DD, Barna GG, A comparison of principal component analysis, multiway principal component analysis, trihnear decomposition and parallel factor analysis for fault detection in a semiconductor etch process, Journal of Chemometrics, 1999, 13,... 

Parallel factor analysis (PARAFAC) (Harshman, 1970 Bro, 1997 Amigo et al., 2010) is a technique that is ideally suited for interpreting multivariate separations data. PARAFAC is a decomposition model for multivariate data which provides three matrices. A, B and C which contain the scores and loadings for each component. The residuals, E, and the number of factors, r, are also extracted. The PARAFAC decomposition finds the best... 

Methods for simultaneous Af-way regression can be based on the decomposition of the X array by multiway methods introduced in Section 5.2 (parallel factor analysis (PARAFAC) or Tucker models) and regressing the dependent variable on the components of those models. A drawback with this approach is that the separately estimated components are not necessarily predictive for Y. This caused the development of improved algorithms for multiway regression analysis of that kind. 

In our laboratories, a cycle time of 90 sec can be achieved with a dilution factor of 1 25 for a given sample concentration, allowing the purity and identity control of two and a half 384-well microtiter plates per day. The online dilution eliminated an external step in the workflow and reduced the risks of decomposition of samples in the solvent mixture (weakly acidic aqueous solvent) required for analysis. Mao et al.23 described an example in which parallel sample preparation reduced steps in the workflow. They described a 2-min cycle time for the analysis of nefazodone and its metabolites for pharmacokinetic studies. The cycle time included complete solid phase extraction of neat samples, chromatographic separation, and LC/MS/MS analysis. The method was fully validated and proved rugged for high-throughput analysis of more than 5000 human plasma samples. Many papers published about this topic describe different methods of sample preparation. Hyotylainen24 has written a recent review. 

We have investigated the humification of straw, as an example, under constant conditions of humidity and temperature in a climatic chamber (4,18) and separated different fractions according to a modified method of Waksman s proximate analysis. The amount of nitrogen present in these processes is the factor limiting the rate of decomposition. Therefore we added nitrogen in form of ammonium nitrate in a quantity of 1% of straw dry weight to a nutrient solution in one experiment while the nutrient solution had no nitrogen in a parallel experiment. Table I shows the calculated data. 

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