## 2 Materials and Methods
### Fall Detection from Partially-Observed Depth-Map Video Sequences
Then, the class posteriori probability that a partially-observed video \(V\) belongs to the activity class \(\omega_{c}\) given the observed subsequences can be defined as follows:  
\[P\Big{(}\omega_{c}|\bigcup_{r=1}^{s}V_{r}[t_{r}(1):t_{r}(l)]\Big{)}\propto \sum_{r=1}^{s}\alpha_{r}P\Big{(}\omega_{c}|V_{r}[t_{r}(1):t_{r}(l)]\Big{)}, \tag{18}\]  
where \(P(\omega_{c}|V_{r}[t_{r}(1):t_{r}(l)])\) is the class posteriori probability that the \(r\)-th observed video subsequence \(V_{r}[t_{r}(1):t_{r}(l)]\) belongs to the class \(\omega_{c}\). \(\alpha_{r}\) represents the ratio between the length of the subsequence \(V_{r}[t_{r}(1):t_{r}(l)]\) and the sum of the lengths of the observed subsequences. The class of the partially-observed video \(V\) is the activity with the index \(c^{*}\) that maximizes the posteriori probability in Equation (18) and can be written as follows:  
\[c^{*}=\arg\max_{c}P\Big{(}\omega_{c}|\bigcup_{r=1}^{s}V_{r}[t_{r}(1):t_{r}(l) ]\Big{)}. \tag{19}\]  
Partially-observed depth-map video sequences have different lengths. At the same time, the unobserved video subsequences can occur at any time and have various durations. Hence, in order to estimate the posteriori probability described in Equation (18), we need to have a length-independent representation of the observed video subsequences. In this paper, we propose to represent each observed video subsequence using a histogram of MPGDs that consists of \(n\) bins. In particular, the proposed histogram-based representation (HBR) employs the \(k\)-means clustering algorithm to build a codebook that consists of \(n\) codewords using the MPGDs extracted from the fully-observed training video sequences. The number of codewords \(n\) is selected to match the number of sub-activities in our dataset, which is equal to eight. Then, for any new video subsequence, we construct a histogram  
Figure 4: A partially-observed depth-map video sequence composed of three observed video subsequence. of eight bins, where each bin is associated with a specific codeword. The value of each bin represents the number of MPGDs in the given video subsequence that belong to a specific codeword after performing the \(k\)-means clustering algorithm. Therefore, the HBR can represent any video subsequence in our dataset using a vector \(\mathbf{H}\in\mathbb{R}^{8}\). In order to estimate the class posteriori probability \(P\Big{(}\omega_{c}|\bigcup_{r=1}^{s}V_{r}[t_{r}(1):t_{r}(l)]\Big{)}\), we start by constructing the HBR for each observed video subsequence. Specifically, the HBR for the \(r\)-th observed video subsequence \(V_{r}[t_{r}(1):t_{r}(l)]\) is denoted as \(\mathbf{H}_{V_{r}[t_{r}(1):t_{r}(l)]}\). Then, the class posteriori probability in Equation (18) can be formulated as:  
\[P\Big{(}\omega_{c}|\mathbf{H}_{V_{r}[t_{1}(1):t_{1}(l)]},\cdots,\mathbf{H}_{V _{r}[t_{r}(1):t_{r}(l)]},\cdots,\mathbf{H}_{V_{s}[t_{s}(1):t_{s}(l)]}\Big{)} \propto\sum_{r=1}^{s}\alpha_{r}P(\omega_{c}|\mathbf{H}_{V_{r}[t_{r}(1):t_{r}( l)]}), \tag{20}\]  
where \(P(\omega_{c}|\mathbf{H}_{V_{r}[t_{r}(1):t_{r}(l)]})\) is the class posteriori probability given the HBR of the \(r\)-th observed subsequence \(V_{r}[t_{r}(1):t_{r}(l)]\). In this work, we propose to utilize a multi-class support vector machine (SVM) classifier with a Gaussian radial basis function (RBF) kernel [33; 34] to estimate the class posteriori probabilities of the observed video subsequences described in Equation (20). In order to train the multi-class SVM classifier, we utilize a one-against-one scheme using fully-observed video sequences. In particular, we divide each training sequence into \(\xi\) overlapped video segments, then using the extracted video segments, we construct a set of training pairs for each training video as follows:  
\[(V_{trn},\omega_{c})=\{(\mathbf{H}_{V_{trn}[t_{1}(1):t_{1}(l)]},\omega_{c}), \cdots,(\mathbf{H}_{V_{trn}[t_{j}(1):t_{j}(l)]},\omega_{c}),\cdots,(\mathbf{H }_{V_{trn}[t_{j}(1):t_{j}(l)]},\omega_{c})\}, \tag{21}\]  
where \(V_{trn}\) is a fully-observed training video of class \(\omega_{c}\).