# Dempster-Shafer Fusion Based on a Deep Boltzmann Machine for Blood Pressure Estimation
## 3 DBM-Based DS Fusion
### DS Fusion  
In this paper, we propose a fusion method of three DBM classifiers based on DS fusion theory. We offer a technique to handle uncertainty in the observation fusion to enhance the advantage of each classifier and compensate the weakness. The DBM classifier \(\mathbf{f}_{n}\) based on DS fusion is used to decrease the observation uncertainty through a combination of decision rules. Here, we omit the details of DS theory [14; 15; 24; 25].  
Suppose that we have an input data, \(\mathbf{x}\), and a different classier \(\mathbf{f}_{d},=1,...,D\). In addition, we suppose that each DBM classifier creates an output \(\mathbf{y}_{d}\in\mathbf{R}_{k},\mathbf{y}_{d}=\mathbf{f}(\mathbf{x})_{d}\), where \(K\) denotes the number of classes. We assume that, for each DBM classifier \(\mathbf{f}_{d}\) and each candidate class \(k\), we compute the value \(b_{k}(\mathbf{y}_{d})=b_{k}(\mathbf{f}_{d}(x))\), which is some measure of _belief_ for the proposition (\(\mathbf{y}_{d}\) is of class \(k\)). As mentioned before, we find the posterior probability of each class to calculate _belief_ given the output data. We must build a multidimensional distribution for the output vector to estimate the conditional probability distribution for all \(K\) classes. Assume that \(\mathbf{x}_{k}\) is a subset of training data equivalent to class \(k\), and \(\mathbf{\hat{y}}_{d,k}\) is the mean vector for a set \(\mathbf{f}_{d}(x_{k})\) for each \(\mathbf{f}_{d}\) with each class \(k\). Thus, \(\mathbf{\hat{y}}_{d,k}\) is a reference vector for each class. Now, we compute the \"proximity\" \(\Phi\) between \(\mathbf{\hat{y}}_{d,k}\) and \(y_{d}\) as given by:  
\[\Phi(\mathbf{\hat{y}}_{d,k},\mathbf{y}_{d})=d_{d,k}=\frac{(1+\parallel\mathbf{ \hat{y}}_{d,k}-\mathbf{y}_{d}\parallel^{2})^{-1}}{\sum_{k=1}^{K}(1+\parallel \mathbf{\hat{y}}_{d,k}-\mathbf{y}_{d})\parallel^{2})^{-1}}. \tag{16}\]  
Actually, \(\Phi(\mathbf{\hat{y}}_{d,k},\mathbf{y}_{d})\) is the maximum value that varies between 1 and 0 when the output vector matches the reference vector. Thus, \(\Phi(\mathbf{\hat{y}}_{d,k},\mathbf{y}_{d})\) must be converted _belief_\(b_{k}(\mathbf{y}_{d})\). We consider a frame of discernment \(\Omega=\{\omega_{1},...,\omega_{K}\}\), where \(\omega_{k}\) denotes the hypothesis that (\(y_{d}\) is of class \(k\)). Moreover, \(d_{d,k}\) can express _belief_\(p\)-hypothesis \(\omega_{k}\), and all \(d_{d,i}\), and \(i\
eq k\) can express _belief_\(p\)_or_\(\omega_{k}\) or _contra_\(\omega_{k}\) for any DBM classifier \(\mathbf{f}_{d}\) with each class \(k\). Here, we use \(d_{d,k}\) as a degree of support with \(\omega_{k}\). A basic probability assignment (BPA) is thus defined as  
\[\mathbf{m}_{k}(\omega_{k})=d_{d,k},\quad\mathbf{m}_{k}(\Omega_{k})=1-d_{d,k} \tag{17}\]  
where \(d_{d,i}\) is also a degree of support with \(\
eg\omega_{k}(i\
eq k)\). The fusion of these support functions with \(\
eg\omega_{k}(i\
eq k)\) denotes a separable support function with the degree of support \((1-\prod_{i\
eq k}(1-d_{d,k}))\). The corresponding BPA is given by:  
\[\mathbf{m}_{\
eg k}(\
eg\omega_{k})=(1-\prod_{i\
eq k}(1-d_{d,k})), \tag{18}\]  
\[\mathbf{m}_{\
eg k}(\Omega_{k})=1-\mathbf{m}_{\
eg k}(\
eg\omega_{k}).= \prod_{i\
eq k}(1-d_{d,k}). \tag{19}\]  
To fuse our knowledge with respect to \(\omega_{k}\), we acquire the _belief_\(\mathbf{m}_{k}\oplus\mathbf{m}_{\
eg k}\), _pro_\(\omega_{k}\), where \(\oplus\) is an orthogonal sum given by:  
\[b_{k}(\mathbf{y}_{d})=\frac{d_{d,k}(x)\prod_{i\
eq k}(1-d_{d,i})}{1-d_{d,k}[1- \prod_{i\
eq k}(1-d_{d,i}))]}. \tag{20}\]  
The _belief_ degree is acquired for each classifier. It can be combined by the DS theory of fusion and should be an orthogonal sum to obtain the final degrees of support, which are:  
\[b_{k}(\mathbf{x})=b_{k}(\mathbf{y}_{1})\oplus...\oplus b_{k}(\mathbf{y}_{D}). \tag{21}\]  
Therefore, we can assign class \(j\) to the input vector \(\mathbf{x}\) if \(b_{j}=\max_{1,...,K}b_{k}(\mathbf{x})\).