具有非均匀缺失的高维变点估计
High-dimensional changepoint estimation with heterogeneous missingness
摘要：
我们提出了一种新的方法，在部分观测，高维时间序列，经历了一个稀疏的坐标子集的平均值的同时变化的变点估计。我们的第一个方法贡献是引入一个“MissingUM”变换（流行的累积和统计的概括），它捕获了每个坐标中信号强度和缺失水平之间的相互作用。为了借力量跨越坐标，我们建议项目这些MissingUM统计沿着一个方向找到的解决方案，以惩罚优化问题的特定稀疏结构。然后，可以将变点估计为预测的单变量序列的绝对值的峰值的位置。在一个模型中，允许不同的缺失概率在不同的组件系列，我们确定的缺失和信号之间的关键相互作用是在每个坐标的信号变化的平方的加权和，与观察概率给出的权重。更具体地说，我们证明了估计和预言投影方向之间的角度，以及变点定位误差，控制与高概率的两项之和，都涉及这个加权平方和，并表示由于噪声和由于缺失分别引起的错误。下限确认我们的变点估计，我们称之为“MissInspect”，是最佳的对数因子。MissInspect方法的显著有效性进一步证明了模拟数据和覆盖新近纪的海洋学数据集。
英文摘要：
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. Our first methodological contribution is to introduce a 'MissCUSUM' transformation (a generalisation of the popular Cumulative Sum statistics), that captures the interaction between the signal strength and the level of missingness in each coordinate. In order to borrow strength across the coordinates, we propose to project these MissCUSUM statistics along a direction found as the solution to a penalised optimisation problem tailored to the specific sparsity structure. The changepoint can then be estimated as the location of the peak of the absolute value of the projected univariate series. In a model that allows different missingness probabilities in different component series, we identify that the key interaction between the missingness and the signal is a weighted sum of squares of the signal change in each coordinate, with weights given by the observation probabilities. More specifically, we prove that the angle between the estimated and oracle projection directions, as well as the changepoint location error, are 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, which we call 'MissInspect', is optimal up to a logarithmic factor. The striking effectiveness of the MissInspect methodology is further demonstrated both on simulated data, and on an oceanographic data set covering the Neogene period.
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