An Integration of Modified Uninformative Variable Elimination and Wavelet Packet Transform for Variable Selection

Apr 01, 2011
Volume 26, Issue 4

The wavelet packet transform (WPT) combined with the modified uninformative variable elimination (MUVE) method (WPT–MUVE) is proposed to select variables for multivariate calibration of spectral data. In this approach, MUVE is used to select informative variables in the wavelet packet decomposition domain. The proposed method was applied to near-infrared (NIR) reflectance spectroscopy data for analysis of protein and fat in milk powder samples, and the performance was compared to full spectrum partial least squares (PLS), conventional uninformative variable elimination (UVE), and the MUVE method. Using the proposed method, a model with fewer variables and better prediction performance was obtained.

Chemometrics plays an important role in analytical chemistry, especially spectral analysis. Many researchers have studied multivariate calibration methods to build a quantitative model in the NIR spectral analysis. It is not always possible, however, to obtain a good calibration model if the full spectral region is used for analysis, because some regions contain information that is useless or irrelevant for the model. Furthermore, the noise and background in the spectra may worsen the predictive ability of the whole model (1). Thus, a good and robust quantitative calibration model should be built by selecting diagnostic wavelengths or variables that include only sample-specific or component-specific wavelengths instead of the full spectrum. For this aim, some new algorithms have been developed, such as the genetic algorithm (GA) (2,3), simulated annealing (SA) (4,5), moving window–partial least squares (MW-PLS) (6), iterative predictor weighting–partial least squares (IPW-PLS) (7), interval PLS (iPLS) (8,9), stepwise regression analysis (SRA) (10), successive projections algorithm (SPA) (11,12), and uninformative variable elimination (UVE) (13,14).

Modified uninformative variable elimination (MUVE) as a method for variable selection was proposed by our research group (15). The method uses a simulated annealing algorithm instead of adding artificial random noise to estimate an optimal cutoff threshold and optimal latent variables for the PLS model (15). The wavelet packet transform (WPT), an extension of the wavelet transform (WT) has been found to be a very efficient tool for analyzing analytical signals. With the WPT technique, the original spectral information can be represented only by a small number of coefficients in WPT decomposition (16–21). Therefore, if the WPT technique is combined with MUVE, a less complex and efficient model can be obtained.

In our work, a combination of MUVE and WPT (MUVE–WPT) was used to select the spectral feature for multivariate calibration of spectral analysis. In MUVE–WPT, MUVE was used to select informative variables in the WPT decomposition domain. For the application of the two methods, calibration of NIR spectra and the routine ingredients (protein and fat) in milk powder samples were investigated.

Theory and Algorithm


The WPT has been found to be a very efficient tool in processing analytical signals, especially in compression of spectral data (20,21). It offers more flexibility for analytical signal representation and feature extraction. In the case of the discrete wavelet transform (DWT), the signal decomposition is unique, but with WPT, decomposition of the original signal leads to redundancy, so attention must be paid to the best-basis selection criteria. A simple algorithm to find the best decomposition tree for a given signal was proposed by Coifman and Wickerhauser (22). This algorithm can lead to the optimal tree if the cost, C, is minimized. The cost is the Shannon entropy. The entropy cost for best-basis selection criterion is straightforward for the individual signals, but not for the set of signals. Definition of the best-basis for the data set depends on selection of the relevant features. In multivariate calibration, the features of the variance spectrum are defined as shown in equation 1 (20):

where m is the number of objects, n is the number of variables, x ij is an element of the data set X(m × n), and x j is the mean of the jth column, calculated as

In this study, the best-basis provides a compact representation of spectral data in time and frequency. Therefore, in the WPT–MUVE, the coefficients in best-basis can be used to replace the original spectra for variable selection. The main steps of WPT–MUVE can be summarized as follows:

1. Calculate the variance spectrum.

2. Decompose the variance spectrum by WPT.

3. Search the WPT tree for the best-basis according to the Shannon entropy criterion.

4. Expand all spectra into the best-basis.

5. Process the coefficients in the best-basis by MUVE.

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