Dynamic relations for sparsely sampled Gaussian processes
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In longitudinal studies, it is common to observe repeated measurements data from a sample of subjects where noisy measurements are made at irregular times, with a random number of measurements per subject. Often a reasonable assumption is that the data are generated by the trajectories of a smooth underlying stochastic process. In some cases, one observes multivariate time courses generated by a multivariate stochastic process. To understand the nature of the underlying processes, it is then of interest to relate the values of a process at one time with the value it assumes at another time, and also to relate the values assumed by different components of a multivariate trajectory at the same time or at specific times selected for each trajectory. In addition, an assessment of these relationships will allow to predict future values of an individual’s trajectories.
Derivatives of the trajectories are frequently more informative than the time courses themselves, for instance, in the case of growth curves. It is then of great interest to study the estimation of derivatives from sparse data. Such estimation procedures permit the study of time-dynamic relationships between derivatives and trajectory levels within the same trajectory and between the components of multivariate trajectories. Reviewing and extending recent work, we demonstrate the estimation of corresponding empirical dynamical systems and demonstrate asymptotic consistency of predictions and dynamic transfer functions. We illustrate the resulting prediction procedures and empirical first-order differential equations with a study of the dynamics of longitudinal functional data for the relationship of blood pressure and body mass index.
Mathematics Subject Classification (2000)62G20 62H25 62M86
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