## 2 Materials and Methods
### Feature Selection and Analysis  
The process of identifying small sets of features useful for diagnostic purposes, namely the selection feature process, is a challenging task in radiomics studies. The aim is to obtain the smallest possible set of features, considered as optimal set for achieving a good predictive performance, thus leading to exclusion of non-reproducible, redundant, and irrelevant features from the dataset. In this study, we want to investigate whether the same optimal set of radiomic features is obtained from three sets of images, namely original PET, PET co-registered with T1-weighted MRI, and PET co-registered with FLAIR MRI. In that way, the robustness of radiomics features can be evaluated after co-registration. [PERSON]'s rank correlation coefficient, which belongs to the filter method group [25], is used to assess whether there is any association between two observed features and to estimate the strength of this relationship. In this way, it is possible to eliminate all features whose level of correlation is above a user-specified threshold. For two sets of variables x and y, each raw score xi and y is converted to ranks Xi and Yi. The [PERSON]'s correlation coefficient is then used on the ranked variables which can be expressed mathematically by the following formula:  
\[\rho=1-\frac{6\sum\mathrm{d}_{\mathrm{i}}^{2}}{\mathrm{n}(\mathrm{n}^{2}-1)} \tag{2}\]  
where di = Xi - Yi is the difference between ranks and n is the number of paired observations. The coefficient ranges from \(-1\) to \(+1\), where negative values indicate that y decreases with x and positive values indicate that y increases with x. In other words, a correlation of 1.0 indicates a perfect positive correlation, and a correlation of \(-1.0\) indicates a perfect negative correlation. Particularly, the strength of association between the two variables is considered very strong if the coefficient ranges from 0.8 to 1, moderate from 0.6 to 0.7, weak from 0.3 to 0.5, and very weak when less than 0.2. When the correlation coefficient is equal to 0, the two variables are independent from one another [32] (see Table 1).  
In our study, a threshold of 0.8 was selected such that a number of high correlated features will be extracted. Consequently, a list of features that are not correlated, i.e., have the correlation coefficient lower than the chosen threshold, is obtained. After applying this correlation-based method, observation of different radiomic features from three different sets of images is obtained to evaluate the robustness of radiomics features following co-registration interference.  
\begin{table}
\begin{tabular}{c c} \hline
**Correlation Coefficient** & **Degree of Association** \\ \hline \(\pm 1\) & Perfect \\ \hline \(\pm 0.9\) & Very Strong \\ \hline \(\pm 0.8\) & Very Strong \\ \hline \(\pm 0.7\) & Moderate \\ \hline \(\pm 0.6\) & Moderate \\ \hline \(\pm 0.5\) & Fair \\ \hline \(\pm 0.4\) & Fair \\ \hline \(\pm 0.3\) & Fair \\ \hline \(\pm 0.2\) & Poor \\ \hline \(\pm 0.1\) & Poor \\ \hline \(0\) & None \\ \hline \end{tabular}
\end{table}
Table 1: The Spearman’s rank correlation coefficient ranges from -1 to 1 indicating various degrees of association between radiomics features.