Chau Chemometrics From Basics to Wavelet Transform

Figure 5.55. Detail coefficients of the simulated noisy NMR spectrum obtained by WT decomposition.

in order to avoid the effect of noise. Figure 5.54, curves (c) and (d) are obtained by multiplying d2 and d3, respectively, with k2 = 60 and k3 = 10. It is clear that the SNR of the results is improved.

Computational Details of Example 5.13

1.Generate the signal with 700 data points [Figure 5.52 (a)] using Lorentzian equations.

2.Make a wavelet filter---Symmlet4.

3.Set resolution level J = 4.

4.Perform WT to obtain the c and d components with the improved algorithm.

5.Perform reconstruction with multiplying the d1 and d2 with a factor 55.0.

6.Display Figures 5.52 and 5.53.

A more detailed discussion of Example 5.13 can be found in a paper published in Applied Spectroscopy [54:731--738 (2000)]. In this paper, an experimental NMR spectrum was also processed by the method. Figure 5.56 shows two enlarged parts of the experimental and the reconstructed spectra. It is clear that the resolution of the spectra is greatly improved by this method.

Figure 5.56. An experimental NMR spectrum (a) and the reconstructed spectrum by multiplying d1 and d2 with k1 = k2 = 10 (b).

5.4.5.Resolution Enhancement by Using Wavelet Packet Transform

WPT differs from WT with respect to the decomposition tree. In WT, only the approximation coefficients on each scale are used for further decomposition, but in WPT, the further decomposition is applied to both the approximation and detail coefficients. Therefore, for resolution enhancement, the resolving ability of WPT should be stronger than that of WT, because there will be more decomposed components representing the information with different frequencies. Consequently, it is easy for us to select a component that represents the desired high-resolution information.

The procedures for resolution enhancement of analytical signals are almost the same as in methods A and B proposed above. The only difference is to use the Equations (5.42)--(5.44) instead of Equations (5.32)-- (5.34), for decomposition and reconstruction computation.

Figure 5.57 shows the experimental chromatograms of six samples containing six rare-earth ions. Concentrations of the samples are listed in Table 5.5. In order to extract the chromatographic information of each component from the overlapping chromatograms in Figure 5.57, we can subject them to WPT decomposition and obtain all wjp first. Then we can select a coefficient component to represent the desired high-resolution information. Figure 5.58 shows the selected w63 coefficients of the six chromatograms. Finally, we can estimate a baseline by linking the minimum point at both sides of every peak. After subtracting the baseline, we can obtain the results shown in Figure 5.59. It can be seen that all six peaks in the six chromatograms are well resolved except for the second peak in

Figure 5.57. Experimental chromatograms of mixed rare-earth solution samples.

chromatogram (d), which is slightly distorted. Using the resolved peaks, we can investigate the linearity of the extracted signals by examining the relationship between the peak area and concentration. Figure 5.60 shows the calibration curves of the six components. It can be seen that all six curves are satisfactory.

5.4.6. Comparison between Wavelet Transform and Fast Fourier Transform for Resolution Enhancement

As we have seen, deconvolution by Fourier transform can also be used for resolution enhancement of analytical signals. The procedures can be

Table 5.5. Compositions and Concentrations (ppm) of Mixed

Rare-Earth Solution Samples

Figure 5.58. WPT coefficients w63 of the chromatograms in Figure 5.57 obtained by WPT decomposition.

3.Perform inverse transform to obtain the deconvoluted signal: gˆ (ω) g(t ).

Figure 5.59. Baseline-corrected WPT coefficients after subtracting the estimated baseline by linking the minimum point at both sides of the peak in Figure 5.58.

Figure 5.60. Calibration curves obtained from the peak area in Figure 5.59 and concentrations of each component in the samples.

Generally, for discrete signals, FFT is used for calculation, and the window functions include the box window, the triangle window, the Hanning window, and the Hamming window. In practice, the width and position of the window must be determined by trial and error, which is always time-consuming.

Figure 5.61 compares the results obtained by two FFT deconvolution methods and the WT method of the simulated and experimental NMR spectra in curves (a) of Figures 5.52 and 5.56, respectively. Curves (a1) and (b1) in Figure 5.61 are obtained by the FFT method on the basis of the abovementioned procedures, where the Hanning window is used, those coefficients out of the window are cut off. Curves (a2) and (b2) are obtained by an improved FFT method based on the abovementioned procedures, where all the coefficients remain but those coefficients in the Hanning window are amplified by a factor k . It should be noted that many trials must be done in order to obtain satisfactory results by FFT methods, because we must select a suitable window width and position. For the second method, we must determine a suitable value for the parameter k . The (a3) and (b3) curves are obtained by the WT method in exactly the same way as curves

(b) in Figures 5.52 and 5.56. In this method, only the value of the parameter k should be optimized by trial and error.

From the results of the simulated spectrum, it can be seen that we cannot obtain a satisfactory baseline by the first FFT method, because the low-frequency part of the signal is cut off. For the second FFT method and the WT method, the baseline will not change because all the information remain, and the resolution enhancement is caused by the increase of the

Figure 5.61. Comparison of the results from the simulated (a) and experimental (b) NMR spectra by FFT deconvolution (a1, b1, a2 and b2) and WT method (a3 and b3).

high-frequency part of the signal. We can easily control the resolution of the results by changing the value of the parameter k . Comparing curves (a2) and (a3) in Figure 5.61a, the improved FFT method is even superior to the WT method because there are positive sidelobes in (a3). From the results of the experimental spectrum, it can be seen that there is no significant difference among the results of the three methods, and the resolution of (b3) is slightly better than that of curves (b1) and (b2) in Figure 5.61. Furthermore, it should be noted that we cannot expect to further enhance the resolution by using a higher value of k in the second FFT method because the negative sidelobes in (b2) will increase with the value of the parameter k .

5.5.COMBINED TECHNIQUES

By the applications of WT discussed above, it has been shown that WT is a powerful tool for compression, denoising, and resolution of analytical signals. In the following text, we will discuss the extended applications based on the combined techniques of WT and other chemometrics methods.

5.5.1. Combined Method for Regression and Calibration

Regression and calibration are the most commonly used techniques for quantitative determination in analytical chemistry. There is no difficulty in regression and calibration calculations for traditional analytical methods because only one variable is used. In modern analytical chemistry, spectroscopic methods are increasingly employed for quantitative determination. New methods such as multiple linear regression (MLR) for modeling multivariable datasets are needed. However, the dimensionality of spectral datasets is basically limited by the number of the objects studied, whereas the number of variables can easily reach a very large number. Furthermore, the high-dimensional spectral data are closely correlated and usually noisy. Therefore, methods more suitable for modeling correlated variables are proposed, such as the principal-component regression (PCR) and partial least-squares (PLS) methods.

Generally, all the information contained in the spectra can be used for the modeling; these are called full-spectrum methods. However, in many cases, a preprocessing of the experimental spectra using WT compression can offer some advantages compared to the full-spectrum methods.

A combined procedures of WT compression and PLS is illustrated in Figure 5.62, including the following steps:

1.The measured signals, denoted by X, such as spectra, are transformed into wavelet domain represented by wavelet coefficients, W.

2.The matrix W is sorted according to their contribution to the data variance and a matrix Wsorted can be obtained. Because many wavelet coefficients in W or Wsorted are usually very small, only a limited number of columns of Wsorted are needed to represent the signal X. Therefore, the Wsorted can be divided into two submatrices, Ws and Wn , containing significant (information component) and insignificant (noisy component) coefficients, respectively. This step can be skipped in many cases because the sorting will change the relative position of the coefficients and, subsequently, cause a variation of the original information.

can be calculated. Using the maximal stability of the noisy variables as a threshold,

we can cut off those coefficients in b and the corresponding wavelet coefficients in the W or Wsorted.

4.With the submatrix Ws , build the PLS model from the training set.

5.Finally, we can use the model for prediction. It should be noted that the experimental data must be processed in the same way as that of the training set used to build the model. When you use step 3a for compression, you must compress the experimental data with the same criteria. When you use step 3b for determination of the Ws , you should keep those coefficients at the same position.

This method has been successfully used in the analysis of NIR spectra of gasoline samples. Examples can be found in the book Wavelets in Chemistry [12].

5.5.2. Combined Method for Classification and Pattern Recognition

Generally, pattern recognition refers to the ability to assign an object to one of several possible categories according to the values of some measured parameters, and the classification is one of the principal goals of pattern recognition. Many methods have been proposed for classification and pattern recognition because of their importance in chemical studies. Combined methods of WT for classification and pattern recognition include two main steps: (1) compression or feature selection is performed to the original dataset using WT as a preprocessing technique; then (2) classification or pattern recognition is performed by classifiers such as the artificial neural network (ANN), the soft independent modeling of class analogy (SIMCA), and the k th nearest neighbors (KNN), in the wavelet domain.

There have been several successful examples based on the combined method of WT and conventional classifiers for classification of analytical signals. One of them is reported by Bos and Vrielink in Chemometrics and Intelligent Laboratory Systems, [23:115--122, (1994)]. In their report, identification of monoand disubstituted benzenes utilizing WT and several classifiers from IR spectra was studied. The aim of their work is to show whether the localization property of WT in both position and scale can be used to extract this information into

a concentrated form to obtain the salient features of an IR spectrum effectively. The coefficients obtained from the WT treatment of the IR spectrum were employed as inputs for an identification process that is based on the linear or nonlinear neural network classifiers. Using the concentrated form instead of the full spectra, the time to develop the classifiers is greatly reduced. Moreover, it is expected that the quality of the classifiers will improve if they are derived from smaller datasets that contain all the relevant information. From their study, it is concluded that WT coupled with Daubechies wavelet functions is a feature extracting method that can successfully reduce IR spectral data by more than 20-fold with a significant improvement in the classification process.

Another example is reported by Collantes et al. in Analytical Chemistry, [69:1392--1397 (1997)]. They employed WPT for preprocessing of HPLC data and several classifiers as potential tools for pharmaceutical fingerprinting pattern recognition. The HPLC data for each l-tryptophan sample was preprocessed by the Haar wavelet function in the WPT treatment. Then, the coefficients thus obtained in the wavelet domain were sorted in descending order. A small portion of sorted coefficients were fed as the inputs of the ANN, KNN, and SIMCA classifiers to classify the samples according to manufacturers. With this study, they concluded that WPT preprocessing provides a fast and efficient way of encoding the chromatographic data into a highly reduced set of numerical inputs for the classification models.

For purposes classification and regression, an adaptive wavelet algorithm (AWA) using higher-multiplicity wavelets was proposed. Detailed descriptions of the method can be found in Chapter 8 and Chapter 18 of the book Wavelets in Chemistry [12]. The wavelet neural network (WNN) can also be used for classification and pattern recognition, discussed in the following section.

5.5.3. Combined Method of Wavelet Transform and

Chemical Factor Analysis

There are several ways to combine WT and chemical factor analysis (CFA) for different purposes. In Section 5.3.5, for example, WT is used for background removal in order to obtain the correct rankmap of a data matrix.

Principal-component analysis (PCA) is an important and basic method in CFA. However, because the principal components (PCs) are calculated as the eigenvectors of the variance--covariance matrix, computation of the