## 5 Signal Processing and Feature Computation

Table 3: List of operators for extracting of features used in sensor signal processing (e.g., gas detection). in each sweep the positions of the current set of thresholds are optimized based on the assessment function (e.g., the quality overlap measure). In the end of the optimization process, the optimum solutions are selected by the aggregating function of the assessment function and the number of thresholds used. The _Gaussian window function_ (GWF) method computes features from sensor signals based on the product of two vectors, _i.e._, a sensor signal and Gaussian function. The GWF method consists of many kernels, which are Gaussian exponential functions with different means \(\mu_{i}\), standard deviations \(\sigma_{i}\), and magnitudes \(M_{i}\), where \(i=1\), \(2\),..., \(k\). Those three parameters represent the position, width, and the height of kernels (see Figure 19). The extracted features of GWF and Gaussian kernel function [26, 37] are defined as follows:  
\[f_{i}=\sum_{s=1}^{N}x_{s}\ G(s,M_{i},\mu_{i},\sigma_{i}) \tag{27}\]  
\[G(s,M_{i},\mu_{i},\sigma_{i})=M_{i}\ e^{-\frac{1}{2}\left(\frac{s-\mu_{i}}{ \sigma_{i}}\right)^{2}} \tag{28}\]  
where \(x_{s}\) is a measurement value of sensor signal at sampled time index \(s=1\), \(2\),..., \(N\). The magnitudes of kernels is in the range from '0' to '1'. The optimization strategy of GWF is different from MLT optimization, where the number of kernels evolves according to values of \(M_{i}\). If the values of \(M_{i}\) are zero, then those kernels can be discarded. The maximum number of kernels is defined by the designer. Figure 18: Multi-level thresholding used for extracting features of slope curves of gas sensors. MLT is modified from histogram and amplitude distribution computation by non-equal range of bins. Three methods of MLT are differential, cumulative (up) and cumulative (down) modes. Table 4 gives the details of the parameter settings of PSO and GA related to the methods of feature computation in the experiments of the benchmark gas dataset [15]. Those parameter settings of PSO and GA are defined by the designer manually (as lucky parameters). However, in the automated system design, the parameters of optimisation algorithms can be defined by the nested optimization approach. Both PSO and GA employed for finding the proper configuration of extracting features using the MLT methods have proved to overcome the suboptimum solution given by expert designer as shown in Table 5. \begin{table}
\begin{tabular}{l l l} \hline \hline
**MLT-DM-GA / MLT-CM-GA** & **MLT-DM-PSO / MLT-CM-PSO** & **GWF-GA** \\ \hline Population = _20_ & Population = _20_ & Population = _20_ \\ Selection = _Roulette Wheel_ & \(\mathbf{w}_{start}\) = 0.9; \(\mathbf{w}_{end}\) = 0.4 & Selection = _Tournament_ \\ Recombination = _discrete_; P\({}_{\mathrm{c}}\) = _0.8_ & \(\mathbf{c_{1}}\) = _2; \(\mathbf{c_{2}}\) = 2_ & Recombination = _discrete_; P\({}_{\mathrm{c}}\) = _0.85_ \\ Mutation = _uniform_; P\({}_{\mathrm{m}}\) = _0.01_ & Update fcn = _floating-point_ & Mutation = _uniform_; P\({}_{\mathrm{m}}\) = _0.1_ \\ Elitism = _10\%_ & Maximum generation = _100_ & Elitism = _10\%_ \\ Maximum generation = _100_ & Assessment fcn = _NPOM_ (_k = 5_) & Maximum generation = _100_ \\ Assessment fcn = _NPOM_ (_k = 5_) & Assessment fcn = _k-NN_ (_k = 5_) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings of GA and PSO. Figure 19: Evolving Gaussian kernels used for extracting features of gas sensor responses. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Method** & \(\mathbf{q_{\mathrm{o}}}\) & _k-NN (\%) with _k = 5_ & **Thresholds or Kernels** \\ \hline MLT-DM & 0.982 & 99.17 & 13 \\ MLT-DM – GA & 0.995 & 99.67 & 9 \\ MLT-DM – PSO & 1.00 & 100 & 9 \\ MLT-CM & 0.956 & 97.17 & 5 \\ MLT-CM – GA & 0.988 & 99.50 & 5 \\ MLT-CM – PSO & 0.995 & 99.92 & 5 \\ GWF – GA & 0.991 & 98.46 & 3 \\ \hline \hline \end{tabular}
\end{table}