## 3 Feature extraction and dimension reduction
### Traditional multivariate analysis methods  
Multivariate statistical analysis is a very efficient strategy for feature extraction. Principal component analysis (PCA) is the most widely used method for this purpose. PCA is a linear transformation that transforms original data to a new coordinate system where the greatest variance of the data lies on the first coordinate (called the first principal component), the second greatest varianceon the second coordinate, and so forth. In other words, PCA is an eigenvector-based multivariate analysis method, providing a map from an original data space to a new space with much lower dimensions [21]. With the abilities to remove co-linearity in variables, condense information of original data, and enhance signal quality, PCA has been extensively applied for sensor signal analysis such as fungal growth detection in bakery products [22], olive oil discrimination [23], and environmental quality monitoring [24]. As extension of PCA, independent component analysis (ICA) is another effective feature extraction method, extracting statistically independent components from original data space [25; 26]. Different from PCA that solely de-correlates the data, ICA has obtained great applications to sensor design [27; 28]. Linear discrimination analysis (LDA) and partial least square regression (PLSR) are also widely used as statistical feature extraction methods for sensor development [29; 30]. Different from PCA, the extractions based on LDA and PLSR are embedded in a classification or regression process that requires both independent data X and dependent data Y. The principles of these two methods will be discussed in the following \"System modeling\" section. Positive matrix factorization is another important feature extraction method, especially in the cases where the generated \"loading\" and \"score\" matrixes should only have positive values to ensure their physical meanings. A positive matrix factorization (PMF) method based on least square regression has been extensively applied to air quality analysis and control [31-33], with potential applications to a variety of environmental sensor design. _3.2 Digital signal processing based methods_  
In many cases, sensor output signal is a time series, such as a chromatogram, or an image, such as mass spectrometry - mass spectrometry (MS/MS) data, so digital signal processing and time series analysis have become powerful feature extraction methods for sensor signals. _3.2.1 Wavelet analysis_  
In contrast to the traditional fast Fourier transformation, wavelet analysis is able to examine the signal simultaneously in both time and frequency domains, so it is an excellent tool to analyze non-periodic, noisy, and intermittent signals and has spawned a number of wavelet-based methods for signal analysis and interrogation [7]. Basically, wavelet transformation aims to represent an arbitrary function by superposing a group of wavelets which are generated from a mother wavelet \(\Psi\) through dilations and translations. A wavelet function generated at scale \(a\) and location \(b\) can be described as,  
\[\Psi_{a,b}(x)=\frac{1}{\sqrt{a}}\,\Psi\!\left(\frac{x-b}{a}\right) \tag{1}\]  
Proposed by [PERSON] for digital signal processing, wavelet analysis was substantially developed for various application fields after [PERSON] constructed a set of wavelet orthonormal basis functions [34; 35]. In a recent study of sensor selection for machine olfaction design [36], discrete wavelet transformation not only significantly reduced the number of sensor variables but also yielded an almost 100% accuracy for the classification of two types of odor (coffee and soda). Wavelet transformation coupled with artificial neural networks was also successfully used for electronic tongue design [37] and environmental variable monitoring [38]. To ensure the feature extraction effect of wavelet analysis, a couple of things should be taken into account including wavelet type and decomposition level.