# Fall Recognition Based on Time-Level Decision Fusion Classification
## 3 Fall Detection System
### Time-Level Decision Fusion
#### 3.4.4 Probabilistic Fusion Method  
A probabilistic approach is employed to aggregate the results from different time-point classifiers. In the fusion stage, the \(N\) results are fused using belief. As a new measure, we define belief as the posterior probability of \(\omega^{out}=i\) (\(i=1,2,\ldots,C\)) conditioned on all time-level classifiers and combine the classifiers by using belief [39]. The posterior probability \(bel_{n}\big{(}\omega^{out}=i\big{)}\) at time levels from 1 to \(n\) is defined as follows:  
\[bel_{n}\big{(}\omega^{out}=i\big{)}=p\big{(}\omega^{out}=i\big{|}I_{1:n}\big{)} \tag{10}\]  
In this case, \(p\big{(}\omega^{out}=i\big{)}\) can be obtained from the confidence score of the \(i\)th class of a classifier, and \(I_{k}\) represents the event where the classifier produces an output using the skeleton features in the \(k\)th frame. Motivated by [39; 40], we calculate the current posterior distribution from the previous time step's posterior distribution as shown in the following equation:  
\[bel_{n}\big{(}\omega^{out}=i\big{)}=\frac{p\Big{(}I_{n}|\omega^{out}=i,I_{1:n- 1}\Big{)}p\big{(}\omega^{out}=i\big{|}I_{1:n-1}\big{)}}{p\big{(}I_{n}|I_{1:n-1 }\big{)}} \tag{11}\]  
where \(p\Big{(}I_{n}|\omega^{out}=i,I_{1:n-1}\Big{)}\) is the probability of the \(n\)th classifier when previous \(n-1\) classifiers are given. Because each time-level classifier makes inferences separately and independently from the other classifiers, we obtain  
\[bel_{n}\big{(}\omega^{out}=i\big{)}=\frac{p\big{(}I_{n}|\omega^{out}=i\big{)} p\big{(}\omega^{out}=i\big{|}I_{1:n-1}\big{)}}{p\big{(}I_{n}|I_{1:n-1}\big{)}} \tag{12}\]  
Applying [PERSON] rule to \(p\big{(}I_{n}|\omega^{out}=i\big{)}\) in (12), the following equation can be derived:  
\[p\big{(}I_{n}\big{|}\omega^{out}=i\big{)}=\frac{p\big{(}\omega^{out}=i\big{|} I_{n}\big{)}p(I_{n})}{p(\omega^{out}=i)} \tag{13}\]  
Therefore, Equation (12) becomes the following equation:  
\[bel_{n}\big{(}\omega^{out}=i\big{)}=\frac{p\big{(}\omega^{out}=i\big{|}I_{n} \big{)}p(I_{n})}{p(\omega^{out}=i)}\frac{p\big{(}\omega^{out}=i\big{|}I_{n-1} \big{)}}{p(I_{n}|I_{n-1})} \tag{14}\]  
where \(p(I_{n})\) is the probability of the \(n\)th classifier, and \(p\big{(}\omega^{out}=i\big{)}\) is the probability that, when an unknown action is given, the class of the action is \(i\). Analogously, we use the following equation for the opposite event:  
\[1-bel_{n}\big{(}\omega^{out}=i\big{)}=\frac{p\big{(}\omega^{out}\
eq i\big{|}I _{n}\big{)}p(I_{n})}{p(\omega^{out}\
eq i)}\frac{p\big{(}\omega^{out}\
eq i \big{|}I_{n-1}\big{)}}{p(I_{n}|I_{n-1})} \tag{15}\]By dividing Equation (14) by Equation (15), Equation (16) is obtained:  
\[\frac{\mathit{bel}_{n}\big{(}\omega^{out}=i\big{)}}{1-\mathit{bel}_{n}(\omega^{ out}=i)}=\frac{p\big{(}\omega^{out}=i\big{|}I_{n}\big{)}}{p(\omega^{out}\
eq i \big{|}I_{n})}\frac{p\big{(}\omega^{out}=i\big{|}I_{n-1}\big{)}}{p(\omega^{out} \
eq i\big{|}I_{n-1})}\frac{p\big{(}\omega^{out}\
eq i\big{)}}{p(\omega^{out}= i)} \tag{16}\]  
We can express Equation (16) as follows:  
\[\frac{\mathit{bel}_{n}\big{(}\omega^{out}=i\big{)}}{1-\mathit{bel}_{n}(\omega ^{out}=i)}=\frac{p\big{(}\omega^{out}=i\big{|}I_{n}\big{)}}{1-p(\omega^{out}=i \big{|}I_{n})}\frac{p\big{(}\omega^{out}=i\big{|}I_{n-1}\big{)}}{1-\mathit{p}( \omega^{out}=i\big{|}I_{n-1})}\frac{1-p\big{(}\omega^{out}=i\big{)}}{p(\omega^ {out}=i)} \tag{17}\]  
\(p\big{(}\omega^{out}=i\big{|}I_{1:n-1}\big{)}\) in Equation (17) can be expressed as \(\mathit{bel}_{n-1}\big{(}\omega^{out}=i\big{)}\) according to the definition of belief.