# Nonlinear Random Forest Classification, a Copula-Based Approach
## 2 Preliminaries and Related Works  
The application of feature selection in machine learning and data mining techniques has been extensively considered in the literature. [PERSON] et al. (2020) ([PERSON] et al., 2020) used a neural network to carry out a feature selection, while [PERSON] et al. (2020) ([PERSON] et al., 2020) used a feature selection approach in a deep neural network. [PERSON] et al. (2017) ([PERSON] et al., 2017) reviewed the feature selection techniques in data mining; see also the book of [PERSON] and [PERSON] (2012) ([PERSON] and [PERSON], 2012). For more information, we refer to [PERSON] et al. (2009) ([PERSON] et al., 2010), [PERSON] et al. (2019) ([PERSON] et al., 2019) and [PERSON] et al. (2017) ([PERSON] et al., 2017). [PERSON] et al. (2019) ([PERSON] and [PERSON], 2011) and [PERSON] et al. (2020) ([PERSON] et al., 2020) have discussed random forest-based feature selection. It is known that the dependence structure between features plays an important role in dimension reduction. [PERSON] et al. (2009) ([PERSON] et al., 2019) carried out a dimension reduction based on extreme dependence between attributes. [PERSON] et al. (2017) ([PERSON] et al., 2017) used feature selection for outcome prediction in medical sciences. [PERSON] and [PERSON] (2010) ([PERSON] and [PERSON], 2010) investigated multi-label dimensionality reduction features by maximizing the dependence between the original feature description and the associated class labels; see also [PERSON] et al. (2018) ([PERSON] et al., 2018) ([PERSON] et al., 2018). [PERSON] and [PERSON] (2011) ([PERSON] and [PERSON], 2011) analyzed a correlation-based dimension reduction. In this work, we use dependence structures between variables to find the best feature selection and construct an agglomerative information gain of random forest. We apply our algorithm to classify influenza and COVID-19 patients. [PERSON] et al. (2020) ([PERSON] et al., 2020) carried out a COVID-19 patient health prediction using a random forest algorithm. [PERSON] et al. (2020) ([PERSON] et al., 2017) applied machine learning methods to generate a computational classification model for discriminating between COVID-19 patients and influenza patients only based on clinical variables. See also [PERSON] et al. (2020) ([PERSON] et al., 2020), [PERSON] (2020) ([PERSON] et al., 2020) and [PERSON] et al. (2020) ([PERSON] et al., 2020) and references therein for more information. [PERSON] et al. (2014) ([PERSON] et al., 2014) ([PERSON] et al., 2014) applied a random forest classifier for lymph diseases. See also [PERSON] et al. (2017) ([PERSON] and [PERSON], 2017) for chronic kidney disease diagnosis using random forest; [PERSON] et al. (2017) ([PERSON] and [PERSON], 2017) for a random forest method to detect Parkinson disease; [PERSON] et al. (2016) ([PERSON] et al., 2016) for a prediction of heart disease using random forest. Additionally, a review work of [PERSON] et al. (2019) ([PERSON] and [PERSON], 2019) may be helpful regarding this subject. [PERSON] et al. (2020) ([PERSON] et al., 2020) have implemented a mutual information-based feature selection. Assume that \(F_{X_{1},X_{2},\ldots,X_{d}}\) is the joint multivariate distribution function of the random vector \(\mathbf{X}=(X_{1},X_{2},\ldots,X_{d})\) and \(F_{X_{i}}\), \(i=1,2,\ldots,d\), are the related marginal distribution functions. A grounded \(d\)-increasing uniformly marginal function \(C:[0,1]^{d}\rightarrow[0,1]\) is called a copula of \(\mathbf{X}\) whenever it couples the multivariate distribution function \(F_{X_{1},X_{2},\ldots,X_{d}}\) to its marginals \(F_{X_{i}},i=1,2,\ldots,d\), i.e.,  
\[F_{X_{1},X_{2},\ldots,X_{d}}(x_{1},x_{2},\ldots,x_{d})=C_{X}\big{(}F_{X_{1}}( x_{1}),F_{X_{2}}(x_{2}),\ldots,F_{X_{d}}(x_{d})\big{)}. \tag{1}\]  
Note that if \(\mathbf{X}\) is a continuous random vector, then the copula \(C_{\mathbf{X}}\) is unique.