# Identification of Key Performance Indicators for T20--A Novel Hybrid Analytical Approach
## 2 Materials and Methods
### Data Analysis  
The data analysis aimed to identify KPIs for developing a predictive model for determining match outcomes. It utilised non-parametric regression methods, fitting a non-linear model, with match outcome as the response variable and PIs as the features. A dynamic programming formula analysed the match performance of both teams. For clarity, the first-batting team is referred to as \"Team A\" in the \"first innings\", and the second-batting team as \"Team B\" in the \"second innings\". On the one hand, the response variable Y assumed a value of 1 if Team A won the match (i.e., Team B lost), and a value of 0 if Team A lost the match. On the other hand, the symbol X was used to denote the matrix of feature values. Each row of the data matrix [Y, X] contained all the information for one match. Furthermore, the features in the data matrix X were divided into five groups of PIs: (1) general match performance, (2) team performance, (3) batting performance, (4) bowling performance, and (5) fielding performance. The primary tool for performance analysis in the current study was the random forest algorithm. Subsequently, lasso logistic regression was applied to enhance the predictive model, enabling it to utilise fewer predictors while achieving high prediction accuracy [34]. A brief summary of the process is shown in Figure 1.  
The random forest algorithm is an ensemble method that amalgamates several individual classification trees, with each tree being trained on a distinct bootstrapping sample [35]. The individual trees generate their predictions for a new input point, and the random forest subsequently employs a majority vote on these predictions to arrive at a final prediction [35]. This technique can significantly enhance prediction accuracy compared to using a single classification tree, as the ensemble method compensates for the instability of the individual trees [34]. The random forest model captures higher-order and non-linear interactions among PIs, as well as the influence of these interactions in making predictions [36]. While some individual KPIs may not contribute to accurate predictions when considered separately, aggregating them together could considerably improve their overall prediction accuracy [37]. An important property of the random forest algorithm is that in addition to making predictions, it can also be used to evaluate the importance of the features in the model when making the final predictions [38]. There are two standard ways of measuring variable importance in the algorithm [38]. The method used in the current analysis is based on the  
Figure 1: A summary of the process of statistical data analyses. random permutation method [38]. To obtain an importance score for a feature using this method, the algorithm first calculates the out-of-bag errors for making predictions with the original dataset. It then modifies the dataset by randomly permuting the values of that feature and calculates the out-of-bag prediction errors again. The mean squared difference between the two sets of errors provides the importance score [34]. The importance score in the random forest measures the differences in prediction accuracy with and without a feature [38], providing a quantitative rating of the feature's usefulness in the model. To determine the KPIs, importance scores were calculated for each feature in the random forest model, and the top-rated features were selected. A common issue in using the random forest for feature selection is that it often picks more features than necessary, especially when several highly correlated features exist, with only one being sufficient for the prediction model. In order to reduce the potential for selecting unnecessary features, when using the random forest model, lasso logistic regression was used to fine-tune the feature selection [34].