## 3 Results and Discussions  
To design and evaluate the performance of our proposed nodule detection, we randomly divided the input materials (888 CT exams of LIDC-IDRI) into k subsets where k = 5 and performed k-fold cross-validation. Unlike other validation methods that only use one training and one testing subset, k-fold cross-validation makes our proposed nodule detection more robust. Among k subsets of samples, it uses a single subset for testing and the remaining k-1 subsets for training. In this way, cross-validation iterates the training and testing processes k times so that each of the k subsets have appeared as a test set exactly once in order to validate the performance of the proposed method. Then, the final assessment of cross-validation can be obtained by calculating the average. This idea helps to provide more accurate assessments on the predictability of the proposed nodule detection on unknown samples. Moreover, k-fold cross-validation helps our proposed nodule detection prevent overfitting. Figure 9: The architecture of the optimized random forest (RF). After preparing data for cross-validation, we try to set the hyperparameters of our proposed RF. As the performance of RF strongly depends on the hyperparameters used, selecting appropriate values of the hyperparameters is very important. Hyperparameters are the values that are predefined before running the model. Generally, these values are manually set by the developer because they cannot be estimated from the data. The common hyperparameters that are applied in RF are (i) the number of decision trees in the forest, (ii) the maximum number of features selected to develop an individual tree, (iii) the maximum number of depths of an individual tree, (iv) the minimum number of samples required to split an internal node, and (v) the minimum number of samples at the leaf node. Proper adjustment of these hyperparameter values can help the RF's prediction results be more reliable and robust. In this experiment, we define the optimal hyperparameter values using the \"trials and errors.\" We initially set a possible range of values that correspond to a specific hyperparameter and perform cross-validation. The accuracy score generated by RF, using each value of the hyperparameter, is measured, and the value that can provide the highest accuracy is picked as an optimal hyperparameter. For the first hyperparameter, in regards to the number of decision trees used in RF, we define a range from 100 to 500. Then, the accuracy scores, using each number of decision trees from the range, is calculated. Subsequently, a validation curve is created to show and compare the accuracy scores that are produced using the different number of decision trees, as illustrated in Figure 9(a). From this curve, we can see that the highest accuracy value (0.9304) is reached using (230) decision trees. Hence, we choose (230) as an optimal hyperparameter value for the number of decision trees in RF. In general, most RFs set the second hyperparameter, which is the maximum number of features, as the square root of the total number of features. However, in our research, we tested using a range from 1 to 25 (total number of extracted features) and chose 15 as a hyperparameter because it could ensure the highest accuracy (0.9232), as demonstrated in Figure 9(b). As well as, the third hyperparameter, the maximum number of depths of an individual tree is selected by finding the highest accuracy using a range from 2 to 20. As we can see in Figure 9(c), the FR reached the highest accuracy (0.9285) at the depths (18); thus, we selected that number as the third hypermeter. In the same way, the next hyperparameter, the minimum number of samples to split an internal leaf node is selected as (5) by testing a range from 2 to 20, as depicted in Figure 9(d). Finally, the last hyperparameter, the minimum number of samples at the leaf node, is chosen as (2) as it could give the highest accuracy score (0.9178), as shown in Figure 9(e).