Interpolating slow-varying curves with sparse data
by Reza Hosseini
Abstract: We develop a framework to fit curves when the data are very sparse but a slow-varying pattern over time (or another variable) allows for a useful prediction. A mathematical framework is developed to show prediction can be plausible in this situation despite the sparse data structure. This is done by investigating the properties of Lipschitz functions. Deterministic bounds are found for the prediction error (of various proposed approximation methods) in terms of the Lipschitz constant and the times data are available. The approximation methods include nearest neighbour assignment, connecting data points by broken lines (chords) and fitting curves using basis functions. Moreover this work finds a non-trivial solution which is optimal and outperforms the aforementioned methods both based on theory and in simulations. The developed methods can be utilized for fitting curves to periodic data and in fact these methods can use this extra assumption to obtain better prediction error bounds. This work also finds optimal sampling times which minimize the prediction error.
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