Chau Chemometrics From Basics to Wavelet Transform

3.2.1. Chemical Rank and Principal-Component Analysis (PCA)

During chromatographic elution, signals arise and vanish as the chemical constituents appear and disappear at the detector. The variations of the chemical constituents gives rise to corresponding changes in the rank of the two-way data matrix. Thus, rank of the submatrix within a retentiontime interval in the raw data matrix is equal to the number of chemical species eluting out simultaneously in the region. This provides a sound mathematical background for solving the problem. Without correct estimation of the number of chemical components (chemical rank) in the system, correct resolution seems to be impossible. However, with measurement errors present in data acquired and other pitfalls in real chemical measurements, correct chemical rank in the two-way data is not trivial at all. Thus, pretreatment of the two-way data and the methodologies for estimating the chemical rankmap sometimes become crucial.

If there were no measurement noises and other pitfalls in measurements, the mathematical rank (the number of independent variables and/or objects in the two-way data) and chemical rank (the number of chemical components in the unknown mixtures) should be the same. Thus, determination of the mathematical rank of a noise-free matrix is trivial. A simple way is to reduce the matrix to the row-echelon form by means of Gaussian elimination and account the number of nonzero rows. However, determination of the chemical rank of a measurement data matrix is a very difficult task because of (1) the presence of measurement noise and their nonassumed distributions, (2) heteroscedasticity of the noise, (3) background and baseline shift arising from the instruments, and (4) collinearity in the measurement data. Thus, in order to avoid these pitfalls, pretreatment of two-way data and local factor analysis becomes very important.

In general, principal-component analysis (or factor analysis) is used in chemometrics to solve the problem of estimating chemical rank in two-way data because it can be used to decompose the matrix into several independent and orthogonal principal components. The number of independent and orthogonal principal components corresponds to the number of the chemical species in the mixture. The mathematical formula of principal component analysis can be expressed as follows:

X = U Vt = UPt = A ui pti

i =1

Here U Vt represents singular value decomposition of the matrix X (see Chapter 5 for more detail), while the ui and vi (i = 1, 2, . . . , A) terms are the so-called score and loading vectors, which are orthogonal with each

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Figure 3.3. The chromatograms and corresponding spectra of the chemical components of the synthetic system shown in Figure 3.2: (a) chromatograms of the three components;

(b) spectra of the three components; (c) parameters used for synthesizing the spectra and chromatograms of the components.

peak without any shoulder. However, from the results obtained (Table 3.1), one can see that the first three eigenvalues are significantly larger than the remaining seven eigenvalues, which have relatively very small values. Thus, one can easily conclude that the chromatographic profile as depicted in Figure 3.2 comes from a peak cluster with three components. This example shows that principal-component analysis is really an efficient method

Table 3.1. Eigenvalues of the Simulated

Three-Component System

for estimating the number of chemical components in the two-way data matrix.

3.2.2. Zero-Component Regions and Estimation of

Noise Level and Background

As discussed before, the task of estimating the number of components becomes very difficult because of the presence of measurement noises and their unknown distributions, heteroscedasticity of the noise and background, and baseline shift from the instruments. To avoid these pitfalls, the determination of zero-component regions from the two-way data as acquired by the hyphenated instrument is of particular significance. The zero-component region is defined as the one where no chemical species elute in chromatographic development. Such a region has, by definition, a chemical rank of zero and is of prime importance for establishing the noise level in the data because there is no chemical species present at all within the region. First, if there is no spectral background and chromatographic shift, the signals obtained in the zero-component region are just the noises generated from the instrument within this time interval because the zerocomponent region plays the role of the analytical blank. With the help of analytical blank information from zero-component regions, an F-test can be established between the two submatrices, in which one is the region to be tested and the other is the region of zero-component. The reader can refer to the paper by Liang et al. [2] for more detail. Furthermore, the zerocomponent regions may be used to detect the presence of a systematic spectral background and/or baseline offset. This can be done by comparing the zero-component regions before elution of the first species and after the elution of the last species in a peak cluster. Figure 3.4 illustrates a chromatogram with three peak clusters and four zero-component regions.

From Figure 3.4, one can see that no chemical component is eluted in the zero-component regions. Thus, the spectra obtained from these regions should provide the blank information obtained from a spectral detector and should have quite similar spectral backgrounds. We will see later that it is possible for us to estimate the instrumental backgrounds and remove them utilizing the information obtained from the zero-component regions.

From the discussions above, we now focus on how we can correct the background shift in the two-way data. For convenience, raw two-way data in general can be divided into two parts: one originating from the chemical constituents in the analyzed mixture, and the other due to instrumental artifacts, called spectral background and chromatographic baseline shift here in order to distinguish them from the ‘‘random’’ noise. Thus, raw two-way

3.3. DOUBLE-CENTERING TECHNIQUE FOR

BACKGROUND CORRECTION

The double-centering technique for background correction is a classic method for dealing with the background of 2D data. In this section, we first examine the effect of combined row--column centering on a 2D matrix where the data are affected by the same chromatographic and spectral background as defined by Equation (3.6). As before, we ignore the random noise in the absorbance matrix. Then, the two-way data matrix can be expressed as

Double centering is defined as the operation of subtracting the row and column mean from the matrix X and adding the grand mean of the matrix:

Here Yij is the element at the i th row and the j th column in the data matrix after double centering. The grand mean of X is easily found to be

It should be noted that, in the zero-component regions, Xc is by definition a zero matrix, and according to Equations (3.9) and (3.10), double centering gives

The superscript 0 is used to indicate the zero-component region. Substitution of Equation (3.10) into Equation (3.11) gives

From Equation (3.12), one can find that the elements in the double-centered zero-component regions differ only columnwise and have the same values rowwise. Thus the pretreatment process removes the spectral background and chromatographic shift, say, sj and tI , respectively. From this perspective, the double-centering method can be used for dealing with these two

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factors in 2D data. However, one should also be aware that this has created zero-component regions of rank 1. This will lead to difficulties in assessing the level of random noise from these regions which is a key point in the multicomponent system. Furthermore, from Equation (3.12), we can also see that the nonnegativity constraint on the absorbances is violated in the double-centered data.

This procedure is now repeated for regions with chemical rank larger than zero. Starting from Equation (3.9) and double centering, we obtain

Here the superscript C denotes the non-zero-component regions. Inspection of Equation (3.14) reveals that double centering also removes spectral and chromatographic background in the regions where chemical components elute. Again, however, the nonnegativity of the intensity for 2D data is destroyed. Furthermore, except for retention-time regions where all chemical species elute simultaneously, the rank is increased by one by the double-centering procedure. It is because of the introduction of a new vector that is a linear combination of all the existing chemical species coeluted. The rank after double centering remains unchanged since the new vector introduced by the double-centering procedure depends linearly on the chemical component spectra.

From the preceding discussion, it can be seen that although double centering may be used to remove background via Equation (3.7), the procedure has some undesired effects, such as the introduction of artificial rank and negative absorbances. Therefore, another procedure is suggested here to correct the spectral background and chromatographic shift based on congruence analysis.

3.4. CONGRUENCE ANALYSIS AND LEAST-SQUARES FITTING

As mentioned before, the zero-component region using the spectral detector may provide the missing information about the spectral background, because there is no chemical component eluted in this region. Thus, one may detect the spectral and chromatographic baseline offset with the help of this information. Once the spectral and/or chromatographic baseline