Evolving Factor Analyses, EFA

An alternative and faster method estimates the pure spectra in a single step. The compound windows derived from an EPCA, are used to calculate a rotation matrix R by which the PCs are transformed into the pure spectra X = C = T R R V.  

Consequently, C = T R, where T is the score matrix of X. Focusing on a particular component, c, (tth column of the matrix C) one can write c, = T r, where r, is the /th column of R. Because compound i is not present in the shaded areas of c, (see Fig. 34.29), the values of c, in these areas are zero. This allows us to calculate the rotation vector r,. Therefore, all zero rows of c, are combined into a new column vector, c°, and the corresponding rows of T are combined into a new matrix T °. As a result one obtains  

The procedure is schematically shown in Fig. 34.29. Equation (34.10) represents a homogeneous system of equations with a trivial solution r, = 0. Because component / is absent in the concentration vector, this component does not contribute to the matrix T °. As a consequence the rank of T is one less than its number of rows. A non-trivial solution therefore can be calculated. The value of one element of r, is arbitrarily chosen and the other elements are calculated by a simple regression [17]. Because the solution depends on the initially chosen value, the size (scale) of the true factors remains undetermined. By repeating this procedure for all columns c, (t = 1 to p), one obtains all columns of R, the entire rotation matrix. 

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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. 

Some of the local-rank analysis methods, such as evolving-factor analysis (EFA) [27-29], are more process oriented and rely on the sequential evolution of the components as a function of time or any other variable in the data set, while others, such as fixed-size moving-window-evolving-factor analysis (FSMW-EFA) [30, 31], can be applied to processes and mixtures. EFA and FSMW-EFA are the two pioneering local-rank analysis methods and can still be considered the most representative and widely used. 

To solve a univariate deconvolution problem, approaches such as evolving factor analysis (EFA) (Maeder, 1987) or multivariate curve resolution (MCR) (Tauler Barcelo, 1993), among others (Vivo-Truyols et al., 2002 Sarkar et al., 1998 Kong et al. 2005) can be used. When these approaches are used with univariate data, the variables to be solved for are the number, positions, and abundances of each of the peaks that make up the signal. 

Construction of non-random initial estimates of matrix C or S. Local rank analysis methods, such as evolving factor analysis (EFA) [1,2], or methods based on the selection of pure variables, such as simple-to-use-interactive self-modelling mixture analysis (SIMPLISMA) [18,19], can be used for this purpose. 

Fixed-size moving-window-evolving factor analysis (FSMW-EFA). This technique, also called window factor analysis (WFA), is based on a window of a predefined number of rows or spectra, typically from three to five, which... 

FSMW-EFA—fixed-size moving-window-evolving factor analysis FT—Fourier transform... 

Fixed-size image window-evolving factor analysis (FSIW-EFA) is an evolution of the local rank algorithm fixed size moving window-EFA [111], particularly designed for the study of the local pixel complexity in images [112]. To do so, two main ideas are taken into account the need to divide the image into small areas to get local information and the need to preserve the 2D or 3D spatial... 

Qualitative exploratory analysis for example principal component analysis (PCA), fixed size image window-evolving factor analysis (FSIW-EFA). 

FSIW-EFA fixed size image window-evolving factor analysis—Surface screening for number of components GC-MS gas chromatography-mass spectrometiy... 

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