## 4 Experiments
### Performance Evaluation  
To evaluate the performance of the proposed methods, 3-fold cross-validation was conducted in which variables for learning and testing were swapped. We here suppose that the abnormal \(A=\{a_{1}\), \(a_{2},\ldots\)\(a_{13}\}\), and the normal \(N=\{n_{1}\), \(n_{2},\ldots\)\(n_{7}\}\). We then classified abnormal into three groups: \(A_{1}=\{a_{1},\ldots\)\(a_{4}\}\), \(A_{2}=\{a_{5},\ldots\)\(a_{8}\}\), \(A_{3}=\{a_{9},\ldots\)\(a_{13}\}\), and also for normal: \(N_{1}=\{n_{1},\)\(n_{2}\}\), \(N_{2}=\{n_{3},\)\(n_{4}\}\), \(N_{3}=\{n_{5},\ldots\)\(n_{7}\}\). Then, we performed 3-trials with \(L_{1}\), \(L_{2}\), \(L_{3}\) representing the learning process, and \(T_{1}\), \(T_{2}\), \(T_{3}\) signifying the testing process for trials 1, 2 and 3, respectively. In trial 1, we assumed that \(L_{1}\) was \((A_{2}\cup A_{3})\cup(N_{2}\cup N_{3})\) for learning, and \(T_{1}\) was \(A_{1}\cup N_{1}\) for testing. In trial 2, \(L_{2}\) was \((A_{1}\cup A_{3})\cup(N_{1}\cup N_{3})\) for learning, and \(T_{2}\) was \(A_{2}\cup N_{2}\) for testing. In trial 3, \(L_{3}\) was \((A_{1}\cup A_{2})\cup(N_{1}\cup N_{2})\) for learning and \(T_{3}\) was \(A_{3}\cup N_{3}\) for testing. After computing the learning process for each trial, \(L_{1}\), \(L_{2}\) and \(L_{3}\), the detection rate was obtained by using the testing \(T_{1}\), \(T_{2}\) and \(T_{3}\), respectively. The overall accuracies of the system were finally computed for each of the features by applying \(SVM\) and \(PD\). There are four possible results in classifying abnormal and normal events, and the definitions and symbols are described as follows. * Detected Abnormal (\(P_{11}\)): A video includes an abnormal event, and is correctly classified into class \"Positive Abnormal.\"
* Undetected Abnormal (\(P_{12}\)): A video includes an abnormal event and is classified into class \"Negative Normal.\"  
Figure 16: Experimental results for trajectory of fall scenarios where blue line indicates \(MA\) of the person, red line shows the period of a fall event by \(v_{\text{line}}\), and orange point represents a falling point by \(I_{max}\) or \(I_{min}\). * Normal (\(N_{11}\)): A video does not include an abnormal event, and is correctly classified into class \"Negative Normal.\"
* Misdetected Normal (\(N_{12}\)): A video does not include an abnormal event, and is incorrectly classified into class \"Positive Abnormal.\"  
The precision, recall, and accuracy were used for evaluating performance, calculated as follows. \[\text{Precision}=\frac{P_{11}}{P_{11}+N_{12}}\star 100 \tag{17}\]  
\[\text{Recall}=\frac{P_{11}}{P_{11}+P_{12}}\star 100 \tag{18}\]  
\[\text{Accuracy}=\frac{P_{11}+N_{11}}{P_{11}+N_{11}}\star 100 \tag{19}\]  
where accuracy was considered by including the undecided area. The concepts for calculating precision, recall, and overall accuracy are illustrated in Figure 17.  
The detection precision achieved 93.33% by utilizing _SVM_ and _PD_ for each of the features: aspect ratio and point distance. The percentage of recall by applying _SVM_ and _PD_ were 100% and 93.33% for  
Figure 17: Illustration of performance evaluation from the classified abnormal and normal events. In this figure, (**a**) represents four different class labels, (**b**) represents the consideration for precision calculation (\(P_{11}\)/(\(P_{11}\)+\(N_{12}\))), (**c**) represents the recall calculation (\(P_{11}\)/(\(P_{11}\)+\(P_{12}\))), (**d**) represents the calculation of accuracy (\(P_{11}\)+\(N_{11}\)/\(P_{11}\)+\(N_{11}\)) where the undecided area is an empty set. the two features, respectively. Table 1 provides a comparison of our proposed detection method for abnormal and normal events versus existing approaches. The proposed system was implemented in MATLAB 2018b on an academic license using C++. All of the experiments were performed on a Microsoft Windows 10 Pro with an Intel (R) Core (TM) i7-4790 [EMAIL_ADDRESS] GHz and 8 GB RAM. Comparing runtime with existing systems is difficult, due to the various programming and optimization levels in use.