本文为英国剑桥大学（作者：RogerFrigola-Alcalde）的博士论文，共109页。

时间序列数据的分析在社会科学、生物学、工程学或计量经济学等领域都很重要。本文提出了一系列学习时间序列贝叶斯非参数模型的算法，这些模型的目标是双重的。首先，它们的目标是做出预测，量化由于数据数量和质量的限制而产生的不确定性。第二，它们具有足够的灵活性，可以对高度复杂的数据建模，同时在数据不需要复杂模型时防止过度拟合。

首先，我们对基于高斯过程的时间序列模型进行了统一的文献综述；然后，我们将注意力集中在高斯过程的状态空间模型（GP-SSM）：离散时间非线性状态空间模型的贝叶斯非参数泛化。我们提出了一个新的GP-SSM公式，并对其性质提供了新的见解；然后，我们通过开发基于粒子马尔可夫链蒙特卡罗和变分推理的GP-SSM新学习算法来进一步挖掘这些见解。

最后，我们提出了一个过滤的非线性自回归模型，该模型具有简单、鲁棒、快速的学习算法，非常适合非专家在大数据集上的应用。它的主要优点是避免了具有昂贵计算代价（而且可能难以改变）的平滑步骤，这是学习非线性状态空间模型的关键部分。

The analysis of time series data isimportant in fields as disparate as the social sciences, biology, engineeringor econometrics. In this dissertation, we present a number of algorithmsdesigned to learn Bayesian nonparametric models of time series. The goal ofthese kinds of models is twofold. First, they aim at making predictions whichquantify the uncertainty due to limitations in the quantity and the quality ofthe data. Second, they are flexible enough to model highly complex data whilstpreventing overfitting when the data does not warrant complex models. We beginwith a unifying literature review on time series models based on Gaussianprocesses. Then, we centre our attention on the Gaussian Process State-SpaceModel (GP-SSM): a Bayesian nonparametric generalisation of discrete-timenonlinear state-space models. We present a novel formulation of the GP-SSM thatoffers new insights into its properties. We then proceed to exploit thoseinsights by developing new learning algorithms for the GP-SSM based on particleMarkov chain Monte Carloand variational inference. Finally, we present a filtered nonlinearauto-regressive model with a simple, robust and fast learning algorithm thatmakes it well suited to its application by non-experts on large datasets. Itsmain advantage is that it avoids the computationally expensive (and potentiallydifficult to tune) smoothing step that is a key part of learning nonlinearstate-space models.

1 引言

2 高斯过程的时间序列建模

3 高斯过程的状态空间模型：描述

4 高斯过程的状态空间模型：蒙特卡洛学习

5 高斯过程的状态空间模型：变分学习

6 过滤的自回归高斯过程模型

7 结论

附录A 近似贝叶斯推理

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