Mixture of Joint Nonhomogeneous Markov Chains to Cluster and Model Water Consumption Behavior Sequences

The emergence of smart meters has fostered the collection of massive data that support a better understanding of consumer behaviors and better management of water resources and networks. The main focus of this article is to analyze consumption behavior over time; thus, we first identify the main weekly consumption patterns. This approach allows each meter to be represented by a categorical series, where each category corresponds to a weekly consumption behavior. By considering the resulting consumption behavior sequences, we propose a new methodology based on a mixture of nonhomogeneous Markov models to cluster these categorical time series. Using this method, the meters are described by the Markovian dynamics of their cluster. The latent variable that controls cluster membership is estimated alongside the parameters of the Markov model using a novel classification expectation maximization algorithm. A specific entropy measure is formulated to evaluate the quality of the estimated partition by considering the joint Markovian dynamics. The proposed clustering model can also be used to predict future consumption behaviors within each cluster. Numerical experiments using real water consumption data provided by a water utility in France and gathered over 19 months are conducted to evaluate the performance of the proposed approach in terms of both clustering and prediction. The results demonstrate the effectiveness of the proposed method.