Efficient regularised estimation methods for sparse time series models

规则化回归方法，如套索及其变种，已被广泛应用于科学的许多领域。这些方法通过利用数据中潜在的稀疏性，成功地克服了参数(P)多于观测值(N)的挑战。在时间序列的情况下，对于相对较短的时间序列，我们通常对具有大量参数的模型的拟合感兴趣。时间序列元素之间的相关性意味着，为了适应这些模型，我们需要找到通过适当的正则化来利用稀疏性的方法。这种情况的一个例子是向量自回归(VAR)过程，其中参数形成一系列矩阵。在高维设置中，参数总数可能超过可用的数据点数量。在文献中已经提出了几种方法并取得了一些理论成果，但该领域仍然非常活跃，还没有一种方法被实践者广泛应用。这个项目的目标是开发正则化回归方法，既有理论上的充分依据，又有高维的计算效率。为了实现这一目标，我们将使用最先进的稀疏优化和蒙特卡罗方法。

Regularised regression methods such as LASSO and its variants have been widely used in many areas of science. These methods successfully overcome the challenge of having more parameters (p) than observations (n) by exploiting the underlying sparsity in the data. In the case of time series, similarly we are often interested in fitting models with a large number of parameters, for relatively short time series. The dependency between the elements of the time series means that in order to fit such models, we need to find ways to exploit sparsity by appropriate regularisation. An example of such a situation is the vector autoregression (VAR) process, where the parameters form a series of matrices. In high dimensional settings, the total number of parameters can exceed the number of data points available. There have been several methods proposed and some theoretical results obtained in the literature, but the field is still very active and no method has yet been widely applied by practitioners. The goal of this project is to develop regularized regression methods that are both theoretically well founded and computationally efficient in high dimensions. In order to achieve this goal, we are going to use state-of-the-art sparse optimization and Monte Carlo methods.