基于张量分解的热点快速检测及其在犯罪率数据中的应用
Rapid Detection of Hot-spots via Tensor Decomposition with applications to Crime Rate Data
摘要：
我们提出了一种有效的统计方法（表示为SSR张量），通过张量分解来鲁棒地快速检测时空数据集中稀疏且时间一致的热点。我们的主要思想是首先建立一个SSR模型，将张量数据分解为平滑的全局趋势均值，稀疏的局部热点和弹性。接下来，张量分解如下：基被引入来描述维内相关性，张量积被用于维间相互作用。然后，采用LASSO和融合LASSO相结合的方法对模型参数进行估计，并基于大规模凸优化思想，提出了一种高效的递推估计方法，将一般LASSO优化问题转化为正则LASSO优化问题，并采用FISTA算法进行求解，具有最快的收敛速度.最后，应用一个CANUUM过程来检测热点事件发生的时间和地点。我们在数值模拟研究和真实案例研究中比较了所提出的方法的性能，该案例研究包含一个数据集，其中包括1965-2014年期间美国大陆各州三种类型的犯罪率。在这两种情况下，所提出的SSR张量能够实现快速检测和准确定位的热点。
英文摘要：
We propose an efficient statistical method (denoted as SSR-Tensor) to robustly and quickly detect hot-spots that are sparse and temporal-consistent in a spatial-temporal dataset through the tensor decomposition. Our main idea is first to build an SSR model to decompose the tensor data into a Smooth global trend mean, Sparse local hot-spots, and Residuals. Next, tensor decomposition is utilized as follows: bases are introduced to describe within-dimension correlation, and tensor products are used for between-dimension interaction. Then, a combination of LASSO and fused LASSO is used to estimate the model parameters, where an efficient recursive estimation procedure is developed based on the large-scale convex optimization, where we first transform the general LASSO optimization into regular LASSO optimization and apply FISTA to solve it with the fastest convergence rate. Finally, a CUSUM procedure is applied to detect when and where the hot-spot event occurs. We compare the performance of the proposed method in a numerical simulation study and a real-world case study, which contains a dataset including a collection of three types of crime rates for U.S. mainland states during the year 1965-2014. In both cases, the proposed SSR-Tensor is able to achieve the fast detection and accurate localization of the hot-spots.
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