PAC Bayesian bounds for light cone, neighborhood, and trajectories data

光锥、邻域和轨迹数据是分析的时空数据集的重要类型，例如，在气候和地球科学、医学、经济学和智能交通中。确定考虑这种时空数据集的特征序列相关性的监督学习算法对于所提出的算法的预测能力是一个非常重要的开放问题。该项目旨在通过确定PAC贝叶斯界限，为时空数据的监督学习算法奠定数学基础。首先，假设观测数据同分布，$\theta$-lex或$\eta$-弱相关，对后者进行分析。这些概念似乎是关键的学习任务的泛化误差的范围，假设时空随机场产生的数据确定。因此，PAC贝叶斯边界的形状广泛类的平稳随机场，即混合移动平均和界限字段，检查，和发现的界限的泛化误差的紧密性进行了研究。这一分析为建立适用于光锥、邻域和轨迹数据的线性预测器的模型选择策略提供了基础。其次，考虑了受噪声污染的一般时空数据集，而不限制其生成。在这个设置中，PAC贝叶斯绑定到开发适用于非参数回归模型。这个界应该允许确定一个模型选择策略学习Gaussian processes.In目前的文献中，PAC贝叶斯界获得有界和无界损失函数，通常持有时，要求有限的指数矩，这是一个相当严格的假设，在时空框架。在这个项目中，PAC贝叶斯界确定无界和局部Lipschitz函数简单地假设有限的二阶矩。

Light cone, neighborhood, and trajectories data are important types of spatio-temporal data sets analyzed, for example, in climate and earth science, medicine, economics, and intelligent transportation. Determining supervised learning algorithms that consider the characteristic serial correlation of such spatio-temporal data sets is an open question of utmost importance for the forecasting ability of a proposed algorithm. The project aims to develop a mathematical foundation of supervised learning algorithms for spatio-temporal data via the determination of PAC Bayesian bounds. First, the latter is analyzed by assuming that the observed data are identically distributed, and $\theta$-lex or $\eta$-weakly dependent. These notions seem pivotal for determining bounds for the generalization error of a learning task, assuming that spatio-temporal random fields generate the data. Thus, the shape of PAC Bayesian bounds for broad classes of stationary random fields, namely, mixed moving average and ambit fields, is examined, and the tightness of the discovered bounds to the generalization error is investigated. This analysis should give the basis to set-up a model selection strategy for linear predictors applied to light cone, neighborhood, and trajectories data.Second, a general spatio-temporal data set contaminated by noise is considered without restricting its generation. In this set-up, the PAC Bayesian bound to be developed applies to nonparametric regression models. This bound should allow determining a model selection strategy for learning Gaussian processes.In the current literature, the PAC Bayesian bounds are obtained for bounded and unbounded loss functions and typically hold when asking for finite exponential moments, which is a quite restrictive assumption in the spatio-temporal framework. In this project, the PAC Bayesian bounds are determined for unbounded and locally Lipschitz functions simply assuming finite second moments.