Chau Chemometrics From Basics to Wavelet Transform

PCs is time-consuming. In order to speed up the calculation of PCA, a fast approximate PCA has been proposed using WT or WPT as a tool for data compression. Generally, as can the procedures in the combined methods for classification and pattern recognition, the main procedures of the fast approximate PCA can be outlined as two steps: to compress the data set and then perform the PCA. As an example, one of the algorithms for WPT compression and PCA of the set of spectral signals can be summarized as follows:

where m denotes the number of objects, n denotes the number of variables, xij denotes an element the dataset Xm×n , and x¯j is the mean of the j th column calculated as

1 m

x¯j = xij (5.84) m i =1

2.Decompose the variance spectrum into the WPT coefficients represented by the WPT tree.

3.Search the WPT tree for the best basis.

4.Compress the coefficients according to a selected criterion.

5.Decompose all the spectra into WPT coefficients and compress them in the same way as in steps 2 and 4.

6.Perform PCA on the compressed coefficients.

Using WT or WPT as a denoising tool for CFA is another type of combination of WT and FA. An example for resolution of multicomponent chromatograms by window factor analysis (WFA) with WT preprocessing can be found in the Journal of Chemometrics [12:85--93 (1998)]. In this paper, resolution and quantitative determination of a multicomponent chromatogram with strong noise was investigated. It was proved that both the resolved chromatographic profiles and the quantitative results by WFA can be greatly improved for the noisy chromatographic data matrix by WT preprocessing. Figure 5.63a,b shows the resolved chromatograms both without and with the WT preprocessing, respectively. The dotted (elliptical) lines in the figure are the experimental chromatograms

Figure 5.65. An experimental chromatogram (a) and the reconstructed results from the WNN parameters (b).

samples based on analytical data, quantitative determination from overlapping UV--vis spectra, and compression of the IR spectrum, which can be found in the literature.

Figure 5.65 shows an example using model (b) (of Fig 5.64) for compression of a chromatogram. Curve (a) in Figure 5.65 shows a chromatogram composed of 800 sampling points, and curve (b) was obtained by reconstruction of 12 wavelet bases optimized by WNN. Thus, we can represent the chromatogram only by 36 parameters. The compression ratio is more than 20 : 1. In this example, there is a high level of noise in the original experimental chromatogram. However, in the reconstructed chromatogram, the noise is filtered out. This can be explained by the fact that a limited number of wavelet bases is used in the reconstruction, and it represents only the primary information of the chromatogram. Much more wavelet basis will be needed to represent the high-frequency noise. Therefore, if we control the parameters of the WNN properly, the method can be used for compression and denoising simultaneously.

5.6.AN OVERVIEW OF THE APPLICATIONS IN CHEMISTRY

As mentioned at the beginning of this chapter, more than 370 papers have been published on WT. Applications of WT in chemistry have been extensively studied. According to the statistics of the published papers on this topic by Dr. Alexander Kai-man Leung (please refer to his homepage

at http://fg702-6.abct.polyu.edu.hk/wavelet.html ), WT has been applied in the following areas:

• Analytical chemistry

Capillary electrophoresis (CE)

Chemiluminescence/fluorescence spectroscopy

Flow injection analysis (FIA)

High-performance liquid chromatography (HPLC)

Gas chromatography (GC)

Graphite furnace atomic absorption spectrometry (GFAA) Inductively coupled plasma atomic emission spectrometry (ICP-

AES)

Infrared spectrometry (IR)

Mass spectrometry (MS)

Nuclear magnetic resonance (NMR) spectrometry

Potentiometric titration

Photoacoustic spectroscopy (PAS)

Raman spectroscopy

Ultraviolet--visible spectrometry (UV--vis)

Voltammetry

X-ray diffraction/spectroscopy

•Chemical engineering

•Chemical physics

•Quantum chemistry

•Miscellaneous (denoising, compression, comparison studies and review papers, etc.)

In the following sections, the published papers are briefly reviewed to give the readers an overview of the applications of the WT techniques in different fields.

5.6.1. Flow Injection Analysis

As early as in 1992, Bos and Hoogendam [13] proposed using WT to minimize the effect of noise and baseline drift in flow injection analysis. When a FIA system is operated near the detection limits, it is difficult to locate

peaks and find the right baseline correction method, because weak signals are embedded in the stochastic noise. In this work, the FIA signal was transformed into a two-dimensional time--frequency form so as to obtain the denoised peak intensity optimally. The maximum peak position can be obtained by searching the wavelet coefficients of the FIA signal obtained for a sample with a relatively high concentration. By using the peak intensity at the maximum peak position, quantitative determination can be further performed. Both the simulated peaks of Gaussian and exponentially modified Gaussian (EMG) and the experimental signals of Imidazole samples and 18-crown-6 samples were studied. Results show that, for a white noise and a favorable peak shape, a signal-to-noise ratio (SNR) of 2 can be tolerated at the 5% error level. This means that a significant reduction in the detection limit can be obtained in comparison with the conventional signal processing methods.

The character of baseline noise strongly influences several steps in the data processing procedure, including choice of the optimal filter, peak detection, peak boundary setting, precision of integration results, peak purity detection, and choice of the calibration method. Generally, noises can be classified into groups such as white noise, heteroscedastic noise, and correlated noise (1/f noise), which have different properties and subsequently have different effects on the processing of analytical signals. The second paper published in FIA is reported by Mittermayr et al. [14]. The goal of the paper is to detect correlated noise from the actual measurement in the presence of peaks without the necessity of the additional measurements of pure baselines and assumptions about the ergodicity. On the basis of the ability of WT to provide information in both time and frequency domains, the authors used the wavelet power spectral density (WPSD) to get a low-resolution equivalent to the traditional power spectral density (PSD) based on the Fourier transform (FT). Then the work demonstrated that the WT separates signal and noise in both the time and frequency domains simultaneously. An F-test is proposed to detect the presence of correlated noise, and the correlation parameter is estimated by weighted least square regression. Finally, the WSPD is applied to flow injection analysis (FIA) signals and its results are compared to the estimates obtained from data with baseline noise.

5.6.2. Chromatography and Capillary Electrophoresis

More than 30 papers have been published on adopting WT in chromatographic data processing, including smoothing, denoising [15--20], data

compression, [19,20], baseline correction [21,22], resolution of overlapping chromatograms, [23--27], and classification studies with the combined techniques [28].

Smoothing and denoising is a universal problem in signal processing. In chromatographic data processing, noise suppression is also a very common technique. It attempts to improve a chromatogram to give a higher signal-to-noise ratio (SNR). Traditionally, techniques such as the Fourier and Kalman filters and the Savitsky--Golay method are generally used for smoothing and denoising. As discussed in Section 5.1.2, WT is a very good technique for this purpose because of its ability to decompose a signal into components of different frequency. Applications similar to those described in Examples 5.4--5.7 can be found in the literature [15--17].

The work reported by Mittermayr et al. [18] used WT as a tool to improve the calibration and the detection limit of a gas chromatograph coupled with a microwave-induced plasma detector system. Symmlet wavelets and universal soft thresholding were employed for denoising. By comparing the technique with the Fourier and the Savitsky--Golay filters, they concluded that special care should be taken for the choice of the filter parameters, especially when large variations in peak widths are present. In such cases, denoising with wavelet filter was better than with the other two methods.

In References [19] and [20], two combined techniques for denoising and compression of chromatograms were proposed. The first one is the WNN introduced in Section 5.5.4. The second one is a combined technique of WT and a genetic algorithm (GA). The underlying theme of both papers was to represent an analytical signal by linear combination of wavelet functions

where N is the number of elementary functions, ψai ,bi is a set of elementary functions defined by dilation with ai and translation with bi , ci is the corresponding coefficient, and fˆ can be regarded as an approximation of the original signal f . The genetic algorithm is applied to find all the parameters ai , bi , and ci that fit the original signal best. B2-spline, B3-spline, Marr, and Morlet basis functions were studied. The B2-spline was found to be in computation and precise in representation. The number of the elementary function, N, is an important parameter in the method because it determines the compression ratio and the effect of denoising. However, the parameter depends on the complexity of the signals being analyzed and must be determined by manual trials.

References [21] and [22] and Sections 5.3.3 and 5.3.5 (in this chapter) discuss methods for removal of baseline or background from 1D and 2D chromatograms. WT-based techniques for resolution enhancement of overlapping chromatograms are discussed in Sections 5.4.4 and 5.4.5. Both WT and WPT can be used for this purpose. These techniques are described in further detail in References [23--25]. In Reference [26], the technique is applied to plant hormone analysis. Plant hormones eluted as overlapping chromatographic peaks were quantitatively determined with the help of WT resolution. Application of CWT for resolution enhancement was also reported [27]. Compared with the DWT, CWT is a more redundant transform. It tends to reinforce the traits of a signal and makes all information more visible because of its redundancy. Therefore, CWT has much more capable of extracting subtle information from seriously overlapping signals. Furthermore, in CWT, the parameters a and b vary continuously; we can choose the exact value of a to depict the component of a certain frequency band in which we are interested, whereas in DWT, we cannot adjust the analytical window precisely to meet a particular need. In this study, the magnitude of each coefficient is represented by graytone and the coefficient matrix was plotted in a graytone graph. Darker spots correspond to larger coefficients. The highest-resolution chromatogram can be obtained by visual inspection of the graytone graph.

Application of WPT combined with classifiers was also studied by Collantes et al. [28] In their work, several computer-based classifiers were evaluated as potential tools for pharmaceutical fingerprinting, which was based on analysis of HPLC trace organic impurity patterns using WPT compression.

An online WT technique for denoising and resolution of overlapping chromatograms was proposed in References [29] and [30]. At first, a ‘‘parallel algorithm’’ calculating cj and dj for j = 1, . . . , J simultaneously with the progress of sampling was proposed, which can be described as follows:

/* Prepare hj and gj */

/* Preset the best resolution level J */ While (!stop-sampling)

/* Wait until elapsed time equals to the sampling time interval */

/* Sample the kth point of cj */ for ( j=1; j<=J; j++)

/* Graphical display of dJ,k or cJ,k and dj,k */ end;

k = k + 1; End of while;

where k runs from 1 to N (k = 1, . . . , N) and N is the total number of the sampling point.

Using this program, online denoising and resolution enhancement of high-performance liquid chromatography (HPLC) were studied. In these studies, the signal from detector was sampled by an A/D (analog-to-digital) convertor, and cj ,k and dj ,k can be obtained by the program simultaneously with the progress of sampling. With mixed samples using benzene, methyl benzene, and ethyl benzene, performance of the method in denoising and resolution enhancement was investigated in the References [29] and [30], respectively. Eight samples with low concentration were measured in the denoising study. The correlation coefficient above 0.99 was obtained for five standard samples, and the recoveries of three mixed samples lay between 94.0 and 105.0%. Comparison of the results with those obtained from the unprocessed chromatograms indicated that the quality of the chromatographic signals was improved by the online wavelet transform. In the study of resolution enhancement, quantitative calculation of three samples was also investigated, and recoveries between 96.3 and 104.5% were obtained.

Applications of WT in capillary electrophoresis analysis are similar with those methods used in chromatography studies, including smoothing, denoising, and resolution of the overlapping peaks [31--33]. Schirm et al. [33] developed a method to transfer the use of fingerprints from spectroscopy to electrophoresis by interpolation and wavelet filtering of the baseline signal. The resulting data were classified by several algorithms, including self-organizing maps (SOMs), artificial neural networks (ANNs), soft independent modeling of class analogy (SIMCA), and k nearest neighbors (KNNs). In order to test the performance of this combined approach in practice, it was applied to process the data from the quality assurance samples of pentosan polysulfate (PPS). A capillary electrophoresis method using indirect UV detection was employed. All algorithms were capable of classifying the examined PPS test batches. Even minor variations in the PPS composition not perceptible by visual inspection could be automatically detected. The entire method has been validated by classifying various unknown PPS quality assurance samples, which have been correctly identified without exception.

238application of wavelet transform in chemistry

5.6.3.Spectroscopy

The spectroscopic techniques include atomic absorption spectroscopy (AAS), atomic emission spectroscopy (AES), fluorescence spectroscopy, ultraviolet--visible (UV--vis) spectroscopy, infrared (IR) spectroscopy, nearinfrared (NIR) spectroscopy, Raman spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, X-ray spectroscopy, and photoacoustic spectroscopy (PAS). These techniques have been widely used in analytical chemistry for both qualitative and quantitative analysis. As in other analytical techniques, the raw spectral data of spectroscopic measurements are a combination of useful signals from the analyzing sample and noise from various forms of interference. Signal processing methods are commonly used to extract the useful signal from the raw experimental data.

Applications of WT-related methods have been studied extensively in spectroscopic signal processing. About 80 papers have been published on these topics. Only some of the papers are reviewed here.

Infrared spectroscopy plays an important role in the identification and characterization of chemicals. Applications of WT in IR spectroscopy can be categorized mainly into denoising [34], compression [35--37], pattern recognition [38], signal extraction [39], and variable selection for PLS regression [40].

Alsberg et al. [34] reported a comparative study in applying WT to denoise IR spectra. Six different methods, including SURE, VISU, HYBRID, MINMAX, MAD, and WPT, were applied to pure IR spectra with the addition of different levels of homoand heteroscedastic noise. Results show that at higher SNR, the wavelet denoising methods were better, especially, the HYBRID and VISU methods. However, at very low SNR, there was no significant difference between the performance of the wavelet methods and the traditional methods, such as Fourier and moving mean filtering. This study concluded that the visual quality of WT denoised infrared spectra is superior.

Several papers were published on the compression of IR spectrum utilizing WT. Reference [35] proposed a successful method for the compression of experimental infrared spectra. This method was based on FWT coupled with multiresolution signal decomposition (MRSD) as well as the optimal bit allocation quantization and Huffman coding techniques. Comparison of the results of IR spectra using several chemicals from the method proposed in Reference [35] with results from FFT and wavelet-based threshoding methods revealed that the proposed method outperforms the other two.

Leung et al. [36] used WT to reduce IR spectral data for library searching. In this work, FWT and WPT were applied to compress infrared IR spectrum for storage and spectral searching. The coefficient position-retaining (CPR)