# Attribute Relevance Score: A Novel Measure for Identifying Attribute Importance
## 3 Materials and Methods
### Maximal Information-Based Nonparametric Exploration Suite  
The Maximal Information-Based Nonparametric Exploration (MINE) suite is a collection of statistical tools designed to detect a wide range of relationships between variables in large datasets. Unlike traditional correlation measures that often focus on linear associations, MINE is tailored to capture both linear and nonlinear dependencies without making strong assumptions about the underlying data distribution. Developed by [PERSON] et al. (2011) [32], the MINE suite includes several measures, such as the maximal information coefficient (MIC), maximal asymmetry score (MAS), maximal edge value (MEV), and generalized mean information coefficient (GMIC), each designed to identify different types of associations in a data-driven manner.  
One of the primary measures in the MINE suite, the maximal information coefficient (MIC), represents a significant advancement in detecting complex associations within data. MIC is specifically designed to identify both linear and nonlinear dependencies between two variables, making it particularly valuable in scenarios where relationships are complex or nonlinear and difficult to capture with conventional methods.  
The methodology behind MIC involves optimizing the partitioning of the data space to maximize the normalized mutual information between variables (Equation (10)). It systematically explores all possible ways of dividing the data into grids and evaluates the strength of the relationship within each partition, selecting the configuration that maximizes normalized mutual information:  
\[MIC(\mathbf{x};\mathbf{y})=\max_{G\in\mathrm{Grids}(\mathbf{x};\mathbf{y})} \left\{\frac{I(\mathbf{x};\mathbf{y}\mid G)}{\log(\min\{|\mathbf{x}|,|\mathbf{ y}|\})}\right\} \tag{10}\]  
where \(I(\mathbf{x};\mathbf{y}\mid G)\) is the mutual information conditioned on a specific grid partition \(G\), and \(|\mathbf{x}|\) and \(|\mathbf{x}|\) denote the cardinalities of variables \(\mathbf{x}\) and \(\mathbf{y}\). The normalization ensures that MIC values range between 0 and 1, where 1 indicates a perfect association.  
MIC has been recognized for its ability to uncover complex patterns in data, and software tools like \"minepy\" version 1.2.6, created by [PERSON] under a GNU General Public License (GPL). Refs. [33; 34] have made it accessible for large-scale analyses. However, the exhaustive evaluation of multiple data partitions can be computationally intensive. To address this, [PERSON] et al. introduced optimizations to improve computational efficiency, especially in high-dimensional datasets.  
The MINE suite also includes the maximal asymmetry score (MAS), which extends the capability of detecting asymmetrical relationships that symmetric measures often overlook. MAS quantifies the degree of asymmetry in the relationship between two variables, offering a more nuanced perspective:  
\[MAS(\mathbf{x};\mathbf{y})=\max_{G\in\mathrm{Grids}(\mathbf{x},\mathbf{y})} \left\{\frac{|(\mathbf{x};\mathbf{y}\mid G)-I(Y;X\mid G)|}{\log(\min\{|\mathbf{ x}|,|\mathbf{y}|\})}\right\} \tag{11}\]  
where \((\mathbf{x};\mathbf{y}\mid G)\) and \(I(Y;X\mid G)\) are mutual information measures calculated in opposite directions. MAS values between 0 and 1 reflect the extent of asymmetry in the association.  
Additionally, the maximal edge value (MEV) focuses on identifying the strongest dependencies within subsets of data, which may not be uniformly strong across all points but exhibit significant relationships in specific regions:  
\[MEV(\mathbf{x};\mathbf{y})=\max_{G\in\mathrm{Grids}(\mathbf{x},\mathbf{y})} \left\{\mathrm{EdgeValue}((\mathbf{x};\mathbf{y}\mid G))\right\} \tag{12}\]  
where \(\mathrm{EdgeValue}((\mathbf{x};\mathbf{y}\mid G))\) is the mutual information at the boundaries of a grid partition \(G\), highlighting local dependencies that might be averaged out in global measures like MIC.