# Multiple Sensors Based Hand Motion Recognition Using Adaptive Directed Acyclic Graph
## 2 Hand Motion Recognition Method
### Decision DAGSVM  
The most primary problem with both one-versus-rest and one-versus-one support vector machine is unclassifiable regions. The binary-class linear classification model is defined as:  
\[Y_{ij}\left(x\right)=w_{ij}^{T}\varphi\left(x\right)+b, \tag{1}\]  
where \(w_{ij}\) is a \(y\)-dimensional vector. \(\varphi\left(x\right)\) is a mapping function that maps \(x\) into the \(y\)-dimensional feature space. \(b_{ij}\) is the bias term, and \(Y_{ij}\left(x\right)=-Y_{ji}\left(x\right)\). The unclassifiable region is shown in Figure 1 with three-classes, and its pairwise formulation is given by  
\[R_{i}=\left\{x|\ Y_{ij}\left(x\right)>0,j=1,2,...,n,j\
eq i\right\}. \tag{2}\]  
\begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline
**Various Methods** & **Training Time** & **SVs** & **Size** & **Advantages** & **Disadvantages** \\ \hline One-versus-rest & Short & \(K\) & Large & Simple Effective & \begin{tabular}{c} Misclassification \\ Rejected classification \\ \end{tabular} \\ \hline One-versus-one & Long & \(K*\left(k-1\right)/2\) & Moderate & \begin{tabular}{c} High accuracy \\ Fast classification \\ \end{tabular} & \
\begin{tabular}{c} Inseparable problem \\ Tend to overfit \\ \end{tabular} \\ \hline Direct MSVM & Long & NoSVs & Large & Natural optimization & Complex computation \\ \hline DAGSVM & Long & \(K*\left(k-1\right)/2\) & Large &
\begin{tabular}{c} Efficient to train and evaluate \\ Avoid misclassification \\ No rejected classification \\ \end{tabular} & Error accumulation \\ \hline \end{tabular}
\end{table}
Table 1: A summary of the common multiclass SVM methods.  
\(x\) will be classified into class \(i\) if \(x\) is in \(R_{i}\). Otherwise, \(x\) is classified by voting into the class with the most votes. For the input vector \(x\), sign function is used to get the appropriate symbol. For real \(x\),  
\[sign\left(x\right)=\left\{\begin{array}{ll}1,&x>0,\\ 0,&x=0,\\ -1,&x<0.\end{array}\right. \tag{3}\]  
Here, we define \(sign\left(x\right)=1\) when \(x=0\). \(Y_{i}\left(x\right)\) will be calculated according to the \(sign\left(x\right)\) as follows:  
\[Y_{i}\left(x\right)=\sum_{j=1,j\
eq i}^{n}sign\left(Y_{ij}\left(x\right) \right), \tag{4}\]  
and then \(x\) will be classified into the appropriate class via  
\[arg\underset{i=1,2,\ldots n}{max}Y_{i}\left(x\right). \tag{5}\]  
If \(x\in R_{i},k\
eq i,Y_{i}\left(x\right)=n-1\) and \(Y_{k}\left(x\right)<n-1\). In this case, \(x\) is classified into \(i\). However, if all \(Y_{i}\left(x\right)\
eq n-1\), plural _is_ may appear in Equation (5), and \(x\) is unclassifiable. \(Y_{i}\left(x\right)=0\)\(\left(i=1,2,3\right)\) in the brown area in Figure 1, so this region is unclassifiable.  
[PERSON] et al. proposed a decision DAG, which combined many two-class classifiers into a multiclass classifier [23]. It is a graph whose edges have an orientation and no cycles. For a \(k\)-class problem, DAG has \(k\) leaves labeled by the classes, where each of the \(k*\left(k-1\right)/2\) nodes is labeled with an element of a Boolean function. All nodes are arranged in a triangular shape with the single root node at the top, two nodes in the second layer and so on until the final layer of \(k\) leaves. DAG starts at the root node, and the binary function at a node is evaluated. According to the value of binary function, the node is classified into the left edge or right edge. DAG is done until the list contains only one class. The current state of the list is the total state of the system. After the implementation of DAG, the unclassifiable region is resolved as shown in Figure 2.  
Figure 1: Unclassifiable regions.