時間依存型密度比推定による高次元変化検知

在本研究中，我们主要关注高维时间序列中的变化检测问题，更一般地说，在依赖数据集上，我们在第一年花了大部分时间开发了一种用于图形模型中变化检测的方法，其中我们输入来自两个不同分布的两组数据，随机变量之间具有不同的相互作用。在这种方法中，我们假设两个阶段之间的变化是微妙的，大多数相互作用保持不变。基于这一假设，我们在统计模型中假设稀疏性，并通过密度比估计方法学习两个图形模型之间的稀疏变化。今年，我们对这种方法进行了理论研究，并给出了所提出的变化检测方法优越性的统计保证。具体地说，我们给出了我们的变化检测方法工作的充分条件，在样本复杂度方面对变化的边缘和维数的增加。此外，上述方法本身是从两组数据中学习变化。然而，自然会问，是否有可能将这种强大的方法应用于图形模型结构本身的学习？我们的新想法是简单地学习联合分布和边际分布的乘积之间的差异。总之，我们不仅完成了提案中承诺的研究，而且还研究了所提出的方法和理论的新的（重要的）应用。

In this research, we have mainly focused on the problem of change detection in high-dimensional time-series, and more generally, on dependent dataset.We have spent most our time in the first year developing a methodology for change detection in Graphical Models, where we input two sets of data drawn from two different distributions with different interactions among random variables. In such methodology, we assumed that the changes between two stages are subtle and most of the interactions remain unchanged. As a consequence of this assumption, the sparsity is assumed in our statistical model and via Density Ratio Estimation method, the sparse changes between two Graphical Models are learned.In this year, we conducted a theoretical study for such methodology and give statistical guarantees of the superiority of the proposed change detection method. Specifically, we give the sufficient conditions that our change detection method works, in terms of sample complexity against the increasing number of changed edges and dimensions.Moreover, the above methodology itself is for learning changes from two sets of data. However, It is nature to ask that is it possible to apply such powerful method to the learning of the Graphical Model structure itself? Our new idea is simply learning the difference between the join distribution and the product of marginal distributions.To sum up, we have not only finished the research promised in the proposal, but also investigated a new (and important) application of the proposed method and theory.