  We develop a framework for analyzing multivariate time series using
topological data analysis (TDA) methods. The proposed methodology involves
converting the multivariate time series to point cloud data, calculating
Wasserstein distances between the persistence diagrams and using the
$k$-nearest neighbors algorithm ($k$-NN) for supervised machine learning. Two
methods (symmetry-breaking and anchor points) are also introduced to enable TDA
to better analyze data with heterogeneous features that are sensitive to
translation, rotation, or choice of coordinates. We apply our methods to room
occupancy detection based on 5 time-dependent variables (temperature, humidity,
light, CO2 and humidity ratio). Experimental results show that topological
methods are effective in predicting room occupancy during a time window. We
also apply our methods to an Activity Recognition dataset and obtained good
results.
