Mixtures of stochastic differential equations with random effects: Application to data clustering

Abstract We consider N independent stochastic processes (Xi(t), t ∈ [0, Ti]), i = 1, . . . , N , defined by a stochastic differential equation with drift term depending on a random variable φi. The distribution of the random effect φi is a Gaussian mixture distribution, depending on unknown parameters which are to be estimated from the continuous observation of the processes Xi. The likelihood of the observation is explicit. When the number of components is known, we prove the consistency of the exact maximum likelihood estimators and use the EM algorithm to compute it. When the number of components is unknown, BIC (Bayesian Information Criterion) is applied to select it. To assign each individual to a class, we define a classification rule based on estimated posterior probabilities. A simulation study illustrates our estimation and classification method on various models. A real data analysis is performed on growth curves with convincing results.