## 2 Data Acquisition, Preparation and Feature Selection
### Data preparation and feature selection
This discrepancy can be measured with the following function,  
\[\mathcal{D}(j,k,\cdot) = \sum_{m}\sum_{p=1}^{C-1}\sum_{q=p+1}^{C}\mathcal{D}(\Gamma_{p}(j,k,m),\Gamma_{q}(j,k,m))\]  
Here the difference between every pair of classes is measured through an additive discriminant function \(\mathcal{D}\), for which we used the squared \(l^{2}\)-norm. A high value for \(\mathcal{D}\) means that that specific bin may be able to separate at least 2 classes that lie far apart. The local basis can now be selected using the following rule, if the measure on a bin \(\mathcal{D}(j,k,\cdot)\), is higher than the sum of the measures over the two bins it splits into, \(\mathcal{D}(j+1,2k,\cdot)+\mathcal{D}(j+1,2k+1,\cdot)\), then it is selected, otherwise it is split. After the local discriminant basis was created, we selected the 15 strongest coefficients, according to Fisher'sdiscriminant given by :  
\[FD=\frac{\sum_{c}\left(\overline{s}_{i}^{c}-\mbox{mean}_{c}(\overline{s}_{i}^{c}) \right)^{2}}{\sum_{c}\mbox{var}_{i}(s_{i}^{c})} \tag{2}\]  
where \(s\) are coefficients taken from a specific entry in the discriminating basis, and both the bar and var\({}_{i}\) take the mean and variance over all samples \(s_{i}\) in class \(c\) and mean\({}_{c}\) takes the mean over all classes. Essentially, this expression measures the distance between the class means and their common centre with respect to their widths, leading to high values when samples in a class lie tightly around their class centre. The 15 strongest features found using the selection procedure described above are summarised in Table 1. It is apparent that the best features were found in the lowest frequencies, where a dominant frequency can be found. Another reason why this may happen is that higher frequencies seem to be more variable in time, as was shown in Figure 1. A shift of a high frequency component will also shift its corresponding energy in the wavelet filter bank. The discrete wavelet transform with a symmlet wavelet as was used for this report has a very slow roll-off. After the down sampling step, aliasing will appear in and propagate through the filter bank. Normally, the aliasing (and phase distortion) is repaired in the synthesis step. This behaviour is not a problem for classification as long as it is consistent. However, phase delays in higher frequencies for consecutive clicks that cause subsequent energy shifts in the wavelet coefficients will act as a source of noise. Therefore, after it was confirmed that there were no high frequency features of interest, the algorithm was focussed on the lowest frequencies using a fifth order Butterworth low-pass filter at 2,000 Hz. After filtering the clicks were down sampled to remove redundant information. When this classification approach is followed on other data sets, the discriminant features should probably always first be selected from the full bandwidth to ensure optimal classification. All classifiers were trained using the characteristics of the first 50 clicks of the seven sets, reflecting the situation where the start of a recording is manually separated by an expert (or automatically when \begin{table}
\begin{tabular}{|c|c|c|c||c|c|c|c|} \hline index & split & frq band (Hz) & power & index & split & frq band (Hz) & power \\ \hline (1,2) & 5 & 1 - 1,500 & 3.5 & (1,1) & 5 & 1 - 1,500 & 3.4 \\ (1,3) & 5 & 1 - 1,500 & 3.0 & (1,4) & 5 & 1 - 1,500 & 2.3 \\ (2,11) & 5 & 1,500 - 3,000 & 1.3 & (1,31) & 5 & 1 - 1,500 & 1.2 \\ (1,9) & 5 & 1 - 1,500 & 1.0 & (1,21) & 5 & 1 - 1,500 & 0.95 \\ (1,10) & 5 & 1 - 1,500 & 0.80 & (2,8) & 5 & 1,500 - 3,000 & 0.80 \\ (1,13) & 5 & 1 - 1,500 & 0.79 & (1,18) & 5 & 1 - 1,500 & 0.79 \\ (1,14) & 5 & 1 - 1,500 & 0.76 & (1,22) & 5 & 1 - 1,500 & 0.73 \\ (1,16) & 5 & 1 - 1,500 & 0.65 & & & - & \\ \hline \end{tabular}
\end{table} 
Table 1: Selected wavelet packet coefficients for discrimination. The given index \((k,m)\) is the position \(m\) of the coefficient in frequency band \(k\) at the given split level. The splitting level starts at 1, which indicates the original signal, i.e.