Namita Lokare, Daniel Benavides, Sahil Juneja and Edgar Lobaton

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1 #analyticsx HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY ABSTRACT We propose a hierarchical activity clustering methodology, which incorporates the use of topological persistence analysis, to capture the hierarchies present in the data. The key innovations presented in this research include the hierarchical characterization of the activities over a temporal parameter as well as the characterization and parameter selection based on stability of the results using persistence analysis. RESULTS Training Hierarchical clustering is performed using the point clouds for each value of τ as the clustering parameter ϵ is increased. Regions with higher densities are clustered into a single cluster and labeling was performed by using the majority vote rule. Fig 3 shows a video of the training scheme over different values of τ. Training video Protocol DATASET Data stream Video Aggregated Persistence Diagrams Fig. 3 Regions in the diagram corresponding to the three values of ϵ that produce the most robust graphical models are shown in (d), (e) and (f). The selected models will not change if ϵ is perturbed within the specified ranges. These ranges are also highlighted as pink regions in the diagram. METHODS Assumptions: Let x(t) be the sensor observations at time t. Let γ k,τ : 0, τ R N be sensor observation trajectories over t k, t k + τ. That is, γ k,τ t x(t + t k ) We define a dissimilarity metric D(γ k1,τ, γ k2,τ) and construct point clouds as in Fig.1 Approach: We perform a filtration of the space for each by using the dissimilarity metric for a fixed τ. A multi-scale representation is obtained by considering the structure of the data over τ. Persistence homology allows us to keep a record of which clusters persist as shown in Fig.2. Fig. 1 Fig. 2 Fig. 4 Results & Analysis Classification results shown in Fig.5, show the average class accuracy. Picking a small τ (i.e., the size of the data window) and a large enough ϵ (i.e., the clustering radius) gives us a better classification accuracy. The models are stable over the bound specified in the diagrams in Fig. 4. CONCLUSION & FUTURE WORK. We show how persistence diagrams can help reduce computation time and help choose stable models for our hierarchical representations. Furthermore, we are able to characterize the stability of our results via persistence analysis. Fig. 5 Our future work will involve testing our method on other datasets and comparing it with other existing algorithm.

2 #analyticsx HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY DATASET Mocap generated skeleton Joint Position Acceleration ECG + HR

3 #analyticsx HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY HIERARCHICAL STRUCTURE OVER THE TRAINING SET 40 τ 0

4 HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY #analyticsx Constructing a point cloud from the data streams. Motion trajectories and corresponding activity labels corresponding to the x-coordinates of the left wrist and right ankle of an individual (left). Projections of the point cloud and the activity labels for different values of τ (right) A filtration of a set of points based on a specified metric (left) and its corresponding persistence diagram (right). Note that two connected components are born at ϵ (shown in blue in the persistence diagram), one of which disappears at ϵ 2. A single hole (shown in red in the persistence diagram) appears at ϵ 2 and disappears at ϵ 4

5 HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY #analyticsx Aggregated Persistence Diagrams. The points in the persistence diagrams shown in (a), (b) and (c) correspond to the connected components for all levels of τ. Regions in the diagram corresponding to the three values of ϵ that produce the most robust graphical models are shown in (d), (e) and (f). The selected models will not change if ϵ is perturbed within the specified ranges. Average performance accuracy for each class

6 HIERARCHICAL ACTIVITY CLUSTERING ANALYSIS FOR ROBUST GRAPHICAL STRUCTURE RECOVERY #analyticsx REFERENCES Lara, O.D. and Labrador, M.A., A survey on human activity recognition using wearable sensors. IEEE Communications Surveys & Tutorials, 15(3), pp Kapsouras, I. and Nikolaidis, N., Action recognition on motion capture data using a dynemes and forward differences representation. Journal of Visual Communication and Image Representation, 25(6), pp Edelsbrunner, H., Letscher, D. and Zomorodian, A., Topological persistence and simplification. Discrete and Computational Geometry, 28(4), pp Edelsbrunner, H. and Harer, J., Computational topology: an introduction. American Mathematical Soc.. Zomorodian, A. and Carlsson, G., Computing persistent homology. Discrete & Computational Geometry, 33(2), pp Ali, R., Atallah, L., Lo, B. and Yang, G.Z., 2009, June. Transitional activity recognition with manifold embedding. In 2009 Sixth International Workshop on Wearable and Implantable Body Sensor Networks (pp ). IEEE. Borg, I. and Groenen, P.J., Modern multidimensional scaling: Theory and applications. Springer Science & Business Media. Tran, T.N., Drab, K. and Daszykowski, M., Revised DBSCAN algorithm to cluster data with dense adjacent clusters. Chemometrics and Intelligent Laboratory Systems, 120, pp