## 3 Proposed System Architecture
### Decision Making Process  
We first apply a support vector machine (\(SVM\)) in order to classify the abnormal and normal events. The main reason for selecting an \(SVM\) approach is that it can work well if the training dataset is small, or occupies a high dimensional space. For the extracted aspect ratio (\(r\)), \(l_{max}\) and \(v_{hw}\) are used as inputs to \(SVM\). Then, for the extracted point distance (\(d\)), \(l_{min}\) and \(v_{hw}\) are used as inputs to \(SVM\). Distance \(D\) extends to a point as a linear discriminating line, formulated by employing an implicit function to classify events. The formula for the observed aspect ratio (\(r\)) is  
\[D(r)=a*r(l_{\max})-b*r(v_{hw})+c,\ \ \ \left\{ \begin{array}{ll}l_{1}&\text{if }D(r)\geq 0\\ l_{2}&\text{otherwise}\end{array} \right. \tag{11}\]  
where \(D(r)\) represents the distance between a point (support vector) and a linear straight line for the feature of aspect ratio \(r\). \(l_{max}\) and \(v_{hw}\) are local maximum and half width values from the observed modified difference, respectively. \(c\) means the \(SVM\) optimization value to avoid misclassifying each training example. \(l_{1}\) and \(l_{2}\) represent \"abnormal\" and \"normal\" events, respectively. Then, the feature called point distance (\(d\)), from \(VGP\) to \(C\) is formulated as,  
\[D(d)=a*d(l_{\min})-b*d(v_{hw})+c,\ \ \ \left\{ \begin{array}{ll}l_{1}&\text{if }D(d)\geq 0\\ l_{2}&\text{otherwise}\end{array} \right. \tag{12}\]  
where \(D(d)\) represents the distance between a point (support vector) and a linear straight line for the observed point distance between virtual grounding point and centroid. \(l_{min}\) and \(v_{hw}\) are local minimum and half width values from the observed modified difference, respectively. When we observe \(v_{hw}\) for abnormal and normal events, we also notice that the period of an abnormal event is longer than that of a normal event. The reason is that a person who has fallen may take time to recover. If such a person does not get up for a long time, that would indicate a dangerous situation. An evaluation of the period of an event can be used to detect a fall. Figure 9: Analysis for possibility of abnormal period by calculating half width value. (**a**) Half width value (\(v_{hw}\)) on aspect ratio. (**b**) Half width value (\(v_{hw}\)) on point distance. To do this, we define the sets of abnormal events as \(A=\{a_{1},a_{2},\,...,\,a_{k}\}\), and of normal events as \(N=\{n_{1},\,n_{2},\,...,\,n_{k}\}\). We can then compute,  
\[\alpha_{1}=MIN\big{(}v_{Inw}\big{)}\in A \tag{13}\]  
\[\alpha_{2}=MAX\big{(}v_{Inw}\big{)}\in N \tag{14}\]  
After that, the period detection (\(PD\)) of a fall for the observed aspect ratio (\(r\)) is obtained by,  
\[PD(r)=(\alpha_{1}(r)+\alpha_{2}(r))/2,\ \ \ \left\{ \begin{array}{ll}l_{1}&\text{if }v_{ Inw}>PD(r)\\ l_{2}&\text{otherwise}\end{array} \right. \tag{15}\]  
Then, the period detection (\(PD\)) for point distance (\(d\)) is computed by,  
\[PD(d)=(\alpha_{1}(d)+\alpha_{2}(d))/2,\ \ \ \left\{ \begin{array}{ll}l_{1}&\text{if }v_{ Inw}>PD(d)\\ l_{2}&\text{otherwise}\end{array} \right. \tag{16}\]  
where \(PD(r)\) and \(PD(d)\) mean period detection for abnormal and normal events for two features (aspect ratio \(r\) and point distance \(d\), respectively). \(\alpha_{1}\) represents the minimum period value for abnormal events. \(\alpha_{2}\) represents the maximum period value for a normal event. \(l_{1}\) and \(l_{2}\) are the class labels for \"abnormal\" and \"normal\" events, respectively. In setting the decision-making rules, the \"undecided\" class is nominated in order to save the failed states. For example, when a \"normal\" case is misclassified as an \"abnormal\" case, it can be considered low risk. Therefore, the decision rules are set to include the undecided class. In doing so, the rules for abnormal and normal event classification are verified with the ground truth, which refers to the information provided by direct observation.