# Feature Interaction in Terms of Prediction Performance
## 2 Proposed Interaction Measure  
The terms \"prediction performance\" and \"feature interaction\" are discussed below prior to introducing the proposed method. In this study, two types of prediction tasks, namely, regression and classification, were considered. In the regression task, the target value of a prediction is a continuous value. In the classification task, the target value of a prediction is a categorical value. The classification accuracy used to measure classification performance is defined as follows:  
\[Acc=\frac{\text{Number of correctly predicted cases}}{\text{Total number of predicted cases}}. \tag{5}\]  
The root mean square error (RMSE) used to measure regression performance is defined as follows:  
\[RMSE=\sqrt{\left(\sum\
olimits_{i=1}^{n}\left(Actual\ value_{i}-predicted\ value_{i}\right)^{2}\right)/n}. \tag{6}\]  
Thus, in the context of this study, the prediction performance refers to the classification accuracy or RMSE with respect to the type of prediction task. However, classification accuracy or RMSE is not the only performance measure. The proposed method is highly compatible with other measures such as the area under the curve (AUC) value. Figure 2: Example of a Shapley value plot. The feature value makes a significant contribution to the prediction if it has a phi >0, whereas a negative phi value reduces its contribution. Notations  
_M_: the fitted prediction model (regression or classification)  
_DS_: the given training dataset used to build a prediction model \(M\)  
\(F_{U}\): all features of a given dataset _DS_. \(F_{U}\) = {\(F_{1}\), \(F_{2}\), \(F_{3}\),..., \(F_{n}\)}  
\(PP_{M}(X)\): prediction performance of a given dataset \(X\) under the fitted model \(M\)  
\(DS\_Perm(F_{i})\): dataset consisting of a permuted feature \(F_{i}\in F_{U}\)  
Feature permutation involves a random shuffle of the feature values. Figure 3 illustrates a permuting feature denoted by \"weight\". Feature permutation neutralizes the influence of a given feature in a prediction model. **Definition 1. **_The reduced performance error \(Err\) obtained by removing feature \(F_{i}\) is defined as_  
\[Err(F_{i})=\left\{ \begin{array}{l}PP_{M}(DS)-PP_{M}(DS\_Perm(F_{i})),\text{ if }M\text{ is a classification model}\\ PP_{M}(DS\_Perm(F_{i}))-PP_{M}(DS),\text{ if }M\text{ is a regression model}\end{array} \right. \tag{7}\]  
In a classification task, \(PP_{M}\) represents the classification accuracy for the model \(M\). The use of a dataset with a permuted feature \(F_{i}\) (\(DS\_Perm(F_{i})\)) as opposed to the original dataset _DS_ may result in reduced accuracy. Thus, the reduced accuracy can be interpreted as an \"error\" caused by the permuted feature \(F_{i}\). A large value of \(Err(F_{i})\) indicates the importance of \(F_{i}\) in the given prediction task. A negative \(Err(F_{i})\) value suggests that feature \(F_{i}\) hinders/affects accurate prediction. In such cases, \(F_{i}\) can be removed from the training set _DS_ to improve the performance of the model. In a regression task, \(PP_{M}\) represents the RMSE value for the model \(M\). The dataset consisting of a permuted feature \(F_{i}\) (\(DS\_Perm(F_{i})\)) may result in a higher RMSE compared to the original dataset _DS_. This indicates that the equation describing \(Err(F_{i})\) is different for the regression and classification tasks. Previous studies have used the concept of \"permuted feature\" to measure the feature impact [6; 7; 8]. In this study, the \"permuted feature\" is used as a measure of performance reduction, implying that both feature impact and performance reduction refer to the same concept. Figure 3: An example of feature permutation. **Definition 2**. : _The reduced performance \(Err\) obtained by removing the feature set \(\{F_{i},F_{j}\}\subset F_{U}\) is defined as_  
\[Err\Big{(}\Big{\{}F_{i},F_{j}\Big{)}\Big{)}=\left\{