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Abstract:
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We propose a new method for changepoint estimation in partially-observed, high-dimensional time series that undergo a simultaneous change in mean in a sparse subset of coordinates. We introduce a 'MissCUSUM' transformation, that captures the interaction between the signal strength and the level of missingness in each coordinate. To borrow strength across the coordinates, we project these MissCUSUM statistics along a direction found as the solution to a penalised optimisation problem. The changepoint can then be estimated as the location of the peak of the projected series. In a model that allows different missing probabilities in different series, we identify that the key interaction between the missingness and the signal is a observation probabilities-weighted sum of squares of the signal change in each coordinate. More specifically, we prove that the changepoint estimation error is controlled with high probability by the sum of two terms, both involving this weighted sum of squares, and representing the error incurred due to noise and the error due to missingness respectively. A lower bound confirms that our changepoint estimator is optimal up to a logarithmic factor.
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