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## Massive Data K-means Clustering and Bootstrapping via A-optimal Subsampling

For massive data analysis, the computational bottlenecks exist in two ways. Firstly, the data could be too large that it is not easy to store and read. Secondly, the computation time could be too long. To tackle these problems, parallel computing algorithms like Divide-and-Conquer were proposed, while one of its drawbacks is that some correlations may be lost when the data is divided into chunks. Subsampling is another way to simultaneously solve the problems of the massive data analysis while taking correlation into consideration. The uniform sampling is simple and fast, but it is inefficient, see detailed discussions in Mahoney (2011) and Peng and Tan (2018). The bootstrap approach uses uniform sampling and is computing time intensive, which will be enormously challenged when data size is massive.

*k*-means clustering is standard method in data analysis. This method does iterations to find centroids, which would encounter difficulty when data size is massive. In this thesis, we propose the approach of optimal subsampling for massive data bootstrapping and massive data*k*-means clustering. We seek the sampling distribution which minimize the trace of the variance co-variance matrix of the resulting subsampling estimators. This is referred to as A-optimal in the literature. We define the optimal sampling distribution by minimizing the sum of the component variances of the subsampling estimators. We show the subsampling*k*-means centroids consistently approximates the full data centroids, and prove the asymptotic normality using the empirical process theory. We perform extensive simulation to evaluate the numerical performance of the proposed optimal subsampling approach through the empirical MSE and the running times. We also applied the subsampling approach to real data. For massive data bootstrap, we conducted a large simulation study in the framework of the linear regression based on the A-optimal theory proposed by Peng and Tan (2018). We focus on the performance of confidence intervals computed from A-optimal subsampling, including coverage probabilities, interval lengths and running times. In both bootstrap and clustering we compared the A-optimal subsampling with uniform subsampling.