Chau Chemometrics From Basics to Wavelet Transform

Figure 3.10. The resulting plot of evolving factor analysis of a three-component system. With the help of the results obtained, the points for every chemical component appearing and disappearing can be determined in this way. The only assumption of the method is that the component that first appears will first disappear in an evolving pattern.

successfully applied to detect a minor component with only 0.7% concentration ratio of the major component after correcting the heteroscedastic noise. The procedure in which this method has been utilized to carry out the factor analysis seems to be different from that of EFA, but almost the same information can be extracted. There are two additional advantages of FSMWFA over the original EFA:

1.This method can reduce the calculation time dramatically. The reason is that when the retention timepoints are large enough, which is

Figure 3.11. Illustration diagram of the fixed-size moving-window evolving factor analysis (FSMWEFA) algorithm. Instead of factor-analyzing the data matrix in a stepwise increasing way, the algorithm is conducted with a moving window. It factor-analyzes the spectra in a fixed-size window and moves the window along the chromatographic direction.

90 two-dimensional signal processing techniques in chemistry

Figure 3.12. The resulting plot of FSMWEFA of a three-component system. With the help of the results obtained, the points for every chemical component appearing and disappearing can be determined more quickly by comparison with the EFA method.

common in two-way data, the EFA will take a long time for the rank estimation, while FSMWEFA takes only a few minutes.

2.The noise level can be built up reasonably since the method is essentially a technique based on local factor analysis. If the noises are correlated slightly, the noise level will be a function of the size of the window analyzed. (see Ref. 4 for further detail).

FSMWEFA is a good tool for local factor analysis, since it collects all the information in the spectral direction. The only thing is that it depends on luck to get the right size of window for the method used to estimate the rank of local regions. Thus, it was extended to a new technique in heuristic evolving latent projections (HELP), called eigenstructure tracking analysis (ETA) in order to obtain the whole rankmap in the chromatographic direction [13]. ETA introduces an ‘‘evolving size, move window’’ by starting with a small window first and then increasing the window size in steps by one until the size exceeds the maximum number of overlapping chemical components or is sufficient in the chromatographic regions under investigation. In this way, one deduces not only the maximum resolution with respect to the selective information but also the overlapping information in the retentiontime direction. To ensure a correct rankmap, adjustment of moving-window size is sometimes crucial. In fact, with a moving window of increasing size, the sensitivity for detecting chemical signals increases, while with a moving window of decreasing size, the selectivity increases. With this in one’s mind, it will be quite helpful for correctly justifying the chemical rank and for obtaining the whole rankmap in the chromatographic direction.

3.6.2.2.Window Factor Analysis (WFA)

The window factor analysis method was developed by Malinowski. It is a self-modeling method for extracting the concentration profiles of individual

The region including (A-1) components

Figure 3.13. Illustration of the major strategy of window factor analysis.

components from evolutionary processes. The region of (A − 1) components, in which the region embracing the analyte is excluded, is used to calculate the concentration profile of the analyte. The major strategy employed in the method is illustrated in Figure 3.13.

The calculation procedure of WFA is quite simple, but the WFA principle is somewhat difficult to understand. For a two-way matrix X, it can be decomposed first by PCA in such a way as

Here A is the number of chemical components in the system. With the use of matrices T and C, Pt , and St span the same linear space such that

where βij and αij are coefficients of the corresponding linear combinations Suppose that χ is a submatrix that contains only (A − 1) components in matrix X as shown in Figure 3.13. Thus, on decomposing this submatrix, the (A−1) orthogonal principal components can be derived via the following

procedure:

A−1

χ = To Pot + E = tio poti + E i =1

92 two-dimensional signal processing techniques in chemistry

It should be mentioned that the superscript o is utilized to express these orthogonal vectors in order to distinguish tio and ti . As in Equation (3.22), we have

Yet, Equations (3.22) and (3.23) differ from each other. In Equation (3.23), the linear space includes only an (A − 1)-dimensional subspace of X; that is, it has only (A − 1) components. In fact, Equation (3.23) can be extended into an A-dimensional space, which can be accomplished by finding a new vector, say, poA, which is orthogonal with all the vectors poj ( j = 1, . . . , A−1). Then, we have

Owing to the orthogonality of potj , κij can be easily determined via

κij = pti poj

The problem here is how to obtain poA. Summing up all pti terms in Equation (3.24) gives

In fact, Equation (3.26) provides a means of computing the vector potA because all the variables on the RHS of the equation are known or obtainable via PCA, while ( κin ) on the LHS of the equation is only a normalized constant for vector potA .

In this way, the orthogonal space potj ( j = 1, . . . , A) obtained by extension can be expressed linearly by the original linear orthogonal space

ptj ( j = 1, . . . , A); that is, both orthogonal spaces can be related linearly with each other. Thus, the spectra of all the components can be formulated whether by using potj ( j = 1, . . . , A) or ptj ( j = 1, . . . , A) to give

As poA is orthogonal to all the vectors poj ( j = 1, . . . , A − 1), that is, potj poA = 0( j = 1, . . . , A − 1), it follows that

It should be mentioned that when the subscript i in sti is less than A, that is, when the spectrum of component i is included in the list of (A − 1) components, sti can be expressed linearly by these (A − 1) potj quantities [see Eq. (3.27)]. Hence, all the γiA (i = 1, . . . , A − 1) are equal to zeros except for γAA since only stA needs poA. In addition, stA can be normalized to yield

This equation indicates that if it is possible to calculate the product of the normalized poA and matrix X, then both cA and a normalized constant γAA can be determined. In this way, WFA can be apply to obtain the pure concentration profile of the Ath component. This procedure can be repeated for all other pure components to find their concentration profiles and to resolve the mixture system.

From the discussion above, the concrete algorithm for WFA consists of the following steps:

1.Find a region containing only (A − 1) components in matrix X. This can be achieved with the help of FWSEFA or EFA.

94two-dimensional signal processing techniques in chemistry

2.Use Equation (3.26), that is

to deduce potA ;

3. Use Equation (3.28) in the form of

XpoA = γAAcA

to obtain cA and then normalize cA afterward.

4.Repeat steps 1--3 until all ci (i = 1, 2, . . . , A) are determined. Then the spectra of the corresponding component can be found using the least-squares technique through

St = (Ct C)−1Ct X

The WFA procedure has been applied successfully to several datasets.

3.6.2.3. Heuristic Evolving Latent Projections (HELP)

HELP differs from EFA and WFA mainly by its emphasis on using selective information. The spectra of the analytes with selective information can be directly determined through decomposing the selective region. The concentration profile for the same analyte can also be determined by including both selective information and zero-concentration regions for this component in the resolution calculation, which has been termed full-rank resolution. Once the spectral and concentration profiles of the components with selective information are determined, the component stripping procedure can then be followed to continue to resolve the other components. This HELP method has been successfully utilized for solving several different real samples.

The HELP technique is based on the use of the ordered nature of ‘‘hyphenated’’ data and the selective regions appearing as straight-line segments in bivariate score and loading plots. Score and loadings plots have been used extensively in multivariate exploratory analysis for a long time, but their significance has been overlooked for rank estimation and resolution. There are at least four advantages in using the latent projection graph (LPG):

1.In the bivariate score plot, a straight-line segment pointing to the origin suggests selective information in the retention-time direction. As for the bivariate loadings plot, a straight-line segment pointing to the origin suggests selective information in the spectral direction. The

concept of ‘‘straight line’’ here is, of course, in the sense of least squares.

2.The evolving information of the appearance and disappearance of the chemical components in the retention-time direction can also be provided in LPG. If one can produce the three-dimensional LPG for the peak cluster with more than three components, the LPG can provide more depicting insight about the data structure.

3.Information enabling the detection of shifts of the chromatographic baseline and instrumental background is also provided in LPG. If there is an offset in the chromatogram, the points will not concentrate at the origin in the plot even if one includes the zero-component regions in the data.

4.LPG is also a very good diagnostic tool to identify the embedded peaks in chromatogram. This information is very important for resolving concentration profiles (see examples in the following section). The LPG works like a microscope to assist one to see the details of the data structure of two-way data.

The HELP method also emphasizes to use the local factor analysis. Using a method called eigenstructure tracking analysis (ETA), one can get the rankmap about the exact number of chemical species at every retention timepoint. The unique resolution of a two-dimensional dataset into chromatograms and spectra of the pure chemical constituents is carried out via local full-rank analysis in the HELP method. In general, the full resolution procedure for the HELP method can be described in the following four steps:

1.Confirm the background and correct a drifting baseline.

2.Determine the number of components, the selective region, and the zero-component region of each component through of the evolving latent projective graph and rankmap on the basis of the eigenstructure tracking analysis.

3.With the help of the selective information and the zero-component region, carry out a unique resolution of the two-dimensional data into pure chromatographic profiles and mass or optical spectra by means of local full-rank analysis.

4.Verify the reliability of the resolved result.

To clarify these points, an example is given here to illustrate how the HELP procedure works to resolve a two-component system.

Figure 3.14 shows a two-component chromatographic profile obtained from a GC-MS instrument. Figure 3.14a shows two selective regions

through which one can easily obtain the pure mass spectra. With the help of LPG and the local factor analysis technique, such as fixed-size moving-window factor analysis (FSMWFA), one can easily locate the selective regions in the chromatographic direction. Figure 3.14a,b shows such results. With the two pure spectra of the two components at hand, the overlapping peaks from the two components can then be resolved easily via least-squares treatment of

The resolved results are depicted in Figure 3.15.

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