A hierarchical Bayesian framework is introduced for developing rich mixture models for real-valued time series, partly motivated by important applications in financial time series analysis. At the top level, meaningful discrete states are identified as appropriately quantised values of some of the most recent samples. These observable states are described as a discrete context-tree model. At the bottom level, a different, arbitrary model for real-valued time series -- a base model -- is associated with each state. This defines a very general framework that can be used in conjunction with any existing model class to build flexible and interpretable mixture models. We call this the Bayesian Context Trees State Space Model, or the BCT-X framework. Efficient algorithms are introduced that allow for effective, exact Bayesian inference and learning in this setting; in particular, the maximum a posteriori probability (MAP) context-tree model can be identified. These algorithms can be updated sequentially, facilitating efficient online forecasting. The utility of the general framework is illustrated in two particular instances: When autoregressive (AR) models are used as base models, resulting in a nonlinear AR mixture model, and when conditional heteroscedastic (ARCH) models are used, resulting in a mixture model that offers a powerful and systematic way of modelling the well-known volatility asymmetries in financial data. In forecasting, the BCT-X methods are found to outperform state-of-the-art techniques on simulated and real-world data, both in terms of accuracy and computational requirements. In modelling, the BCT-X structure finds natural structure present in the data. In particular, the BCT-ARCH model reveals a novel, important feature of stock market index data, in the form of an enhanced leverage effect.

翻译：针对实值时间序列的丰富混合模型构建，本文提出一种层次贝叶斯框架，其部分动机源于金融时间序列分析中的重要应用。在顶层，通过适当量化最近若干样本值，将具有意义的离散状态识别为可观测的上下文树模型。在底层，每个状态关联一个不同的实值时间序列任意模型（即基模型）。这定义了一个极为通用的框架，可与任意现有模型类结合使用，构建灵活且可解释的混合模型。我们将其称为贝叶斯上下文树状态空间模型（BCT-X框架）。本文引入高效算法，支持该设定下的精确贝叶斯推断与学习，特别是可识别最大后验概率（MAP）上下文树模型。这些算法可顺序更新，便于高效在线预测。通过两个具体实例展示该通用框架的实用性：以自回归（AR）模型为基模型时，得到非线性AR混合模型；以条件异方差（ARCH）模型为基模型时，得到能系统刻画金融数据中已知波动率不对称性的强大混合模型。在预测方面，BCT-X方法在模拟数据和真实数据上均优于现有最优技术，兼顾预测精度与计算效率。在建模方面，BCT-X结构能揭示数据中的自然结构。特别地，BCT-ARCH模型发现了股票市场指数数据的一种新型重要特征——增强杠杆效应。