# Machine Learning Powered Microalgae Classification by Use of Polarized Light Scattering Data
## 4 Discussion
Moreover, the quantitative contributions of the elements will guide us to improve the detecting speed of the system by reducing the number of modulated \(S_{i}\), and find the optimal SOP to better characterize and classify these categories of microalgae, which is important for fast field probing applications. The 150\({}^{\circ}\) linear SOP and right-handed circular SOP are further discussed with the Mueller matrix theorem. For simplicity, we discuss the spherical particles. Theoretically, for spherical particles, the top right and bottom left elements of \(\mathbf{M}\) are zero [27]. When \(S_{i}\) is \(\left[I_{i},Q_{i},U_{i},V_{i}\right]^{T}\), \(S_{s}\) can be calculated by Equation (5),  
\[S_{s}=\left[ \begin{array}{cccc}M_{11}&M_{12}&0&0\\ M_{12}&M_{22}&0&0\\ 0&0&M_{33}&M_{34}\\ 0&0&-M_{34}&M_{44}\end{array} \right]\left[ \begin{array}{c}I_{i}\\ Q_{i}\\ U_{i}\\ V_{i}\end{array} \right]=\left[ \begin{array}{c}M_{11}I_{i}+M_{12}Q_{i}\\ M_{12}I_{i}+M_{22}Q_{i}\\ M_{33}U_{i}+M_{34}V_{i}\\ -M_{34}U_{i}+M_{44}V_{i}\end{array} \right]=I*\left[ \begin{array}{c}1\\ q\\ u\\ v\end{array} \right] \tag{5}\]  
where \(I\) is the scattered intensity and \(q\), \(u\), \(v\) are the normalized polarization parameters of \(S_{s}\). Usually, the standard SOP of incident light can be represented as \(\left[1,1/2,-\sqrt{3}/2,\ 0\right]^{T}\) for 150\({}^{\circ}\) and \(\left[1,\ 0,\ 0,\ 1\right]^{T}\) for R. Then, the Stokes vector of the scattered light is calculated as \(\left[M_{11}+M_{12}/2,\ M_{12}+M_{22}/2,-\sqrt{3}M_{33}/2,\ \sqrt{3}M_{34}/2\right]^{T}\) for 150\({}^{\circ}\) linear SOP and the Stokes vector of the scattered light is \(\left[M_{11},\ M_{12},\ M_{34},\ M_{44}\right]^{T}\) when the incident SOP is R.  
The classification performance of the incident SOP of 150\({}^{\circ}\) is obviously better than the performance of the incident SOP of R. Compared with these two derived Stokes vectors of scattered light, the Stokes vector of 150\({}^{\circ}\) linear SOP has the distinctive information of the elements \(M_{22}\) and \(M_{34}\), and the Stokes vector of R has the distinctive information of \(M_{44}\). Since the incident SOP of 150\({}^{\circ}\) has a better performance of the incident SOP of R, it seems that \(M_{22}\) and \(M_{34}\) may contain more important information than \(M_{44}\) for the classification task. However, for the contribution of the polarization parameters \(u\) and \(v\), the parameter \(u\) contributes less information compared with the parameter \(v\) for linear SOP, while the parameter \(v\) contributes less information compared with the parameter \(u\) for circular SOP. Thus, it seems that \(M_{34}\) is less useful compared with \(M_{33}\) or \(M_{44}\). The result in Figure 2 shows that non-linear SVM displays the best performance to classify these categories of microalgae based on Stokes vectors. However, this result does not explicitly claim the contributions of the specific polarization parameters, since it learns the polarization parameters of Stokes vectors in a hyper feature space. To verify the above analysis with Mueller matrix, we select two categories of microalgae to quantitatively evaluate the contribution of the polarization parameter \(v\) and the symmetry of the polarization states. Both the _Isochrysis_ and _Chattonella marina_ are easily distinguishable from other categories of microalgae. Their biological features are investigated in a previous book [28; 29]. The result is shown in Table 5; note that the relative difference is the difference between these two cases over the retrieved accuracy with [\(l\), \(q\), \(u\)]. Moreover, the precision information of these two categories of microalgae can be referred to in Tables 6 and 7, which shows the classification performance of the trained model on these two categories. The two circular SOPs of the incident light, R and L, have theoretical Stokes vectors of \(\left[M_{11},\ M_{12},\ M_{34},\ M_{44}\right]\) and \(\left[M_{11},\ M_{12},\ M_{34},-M_{44}\right]\).