## 2 Materials and Methods
### Definition of Simplicial and Temporal Filtration  
Given the point cloud data, we then constructed a continuous shape on top of the data to highlight the underlying topology and geometry. The process to build such a shape is through mathematical filtration, which is often a simplicial complex or a nested family of simplicial complexes that reflects the innate structure of the point cloud data at different scales [8]. If we considered all the points in the point cloud data, each with a coordinate of their locations in certain embedding, then they each occupied a spherical space with the same radius \(\epsilon\) around them, which is called a nerve ball. If two nerve balls overlapped or contacted each other, then we considered an edge to be formed between them in this graph. Filtration is a process for tuning the parameter \(\epsilon\) from \(0\) to \(\infty\) and recording the families of the simplicial complexes generated through the increasingly connected (or \"complex\") graph. Usually, the challenge is to extract relevant and useful information about the shape of the data through defining such simplicial complexes from the graph (generated through the filtration process). The Rips-Vietoris complex is one of the common choices in practice for computing the topological invariants of point clouds, which are defined as follows: given the vertex set \(\mathcal{Z}\), for each pair of vertices \(a\) and \(b\), edge a-b is included in the Rips-Vietoris complex \(C(\mathcal{Z},t)\) if \(d(a,b)\leq t\), and a higher dimensional simplex is included in \(C(\mathcal{Z},t)\) if all of its edges are included. Since \(C(\mathcal{Z},t)\in C(\mathcal{Z},t^{\prime})\) whenever \(t\leq t^{\prime}\), the filtered Rips-Vietoris complex is a filtered simplicial complex as well as the maximal simplicial complex that can be built on top of its \(1-\)skeleton, and thus a clique complex or a flag complex is formed.Unlike conventional low-dimensional topological data analysis, we computed the simplices at a high dimension count (up to seven) during the entire filtration process. To record the number of cliques, we computed the filtered simplicial complexes and recorded their cumulative counts across the entire filtration process. Since the topological data analysis usually only considers the graph constructed by the spatial proximity (i.e., the distance matrix) between the data points in the low-dimensional embedding, it is not clear how to incorporate timestamp information for meaningful inference and visualization when facing the time series data streams. One approach would be to simply consider the time stamp as the metadata for post hoc labeling of the topological representations. Another alternative would be to consider time as an additional dimension in the filtration process. We present temporal filtration as the following: alongside the conventional sweeping of the parameter \(\epsilon\) from \(0\) to \(\infty\), we set another parameter \(\tau\) to indicate a hard constraint in edge forming between two points. Alternatively and intuitively, temporal filtration is equivalent to conventional filtration by using the composite norm:  
\[d((x,t_{x}),(y,t_{y}))=max(\frac{1}{\epsilon^{*}}|y-x|,\frac{1}{\tau^{*}}|t_{y }-t_{x}|) \tag{1}\]  
where \(\epsilon^{*}\) and \(\tau^{*}\) are directly related to the spatial threshold \(\epsilon\) (in the feature space) and the temporal threshold \(\tau\), respectively. As a practical note from this notation, it can be used without additional specialized software. In other words, only if the time stamp difference between the two data points is within the time delay limit \(\tau\) can two nerve balls, if spatially proximal enough (less than \(\epsilon\)), form an edge in between. On the other hand, if the time stamp difference between the two data points is larger than \(\tau\), then even if they are spatially proximal enough (less than \(\epsilon\)), they cannot form an edge.