Mathematical Sciences: Bayesian Modeling and Inference for Time Series with Stable Innovations

摘要研究者研究了具有无限方差稳定创新的时间序列的贝叶斯建模和推理，解决了方法问题以及在现实世界数据中的应用。研究无穷方差稳定过程的理由源于其在天文学、经济学、工程学、金融学和物理学等几个应用领域的有用性的经验证据。对稳定过程建模的领域有几个开放的和具有挑战性的问题需要解决。现有的经典估计和推理方法不能同时估计定义稳定过程的参数和时间序列模型的参数。研究者开发了一种新的方法，用于建模由自回归分数积分移动平均过程生成的数据，该过程具有稳定的创新，在“无限方差”时间序列中表征长记忆和短记忆行为。推理和预测采用基于抽样的贝叶斯技术，通过马尔可夫链蒙特卡罗算法从目标后验分布中生成样本。对于稳定过程，似然的形式一般不允许有封闭的解析形式。因此，用稳定过程的特征函数表示参数的完全条件分布的表达式与Gibbs抽样算法或其变体相结合，通过适当的建议密度逼近所需的后验来生成样本。此外，研究了从稳定过程中直接生成样本的有用算法的作用，并将其纳入贝叶斯框架。分析了各种边缘和关节后验分布以及这些分布的总结特征，以及允许模型选择和预测的预测分布的特征。研究者研究时间序列数据的建模和预测，假设数据可以采取比通常情况更极端的值。有相当多的经验证据表明，这种行为在不同的应用领域，如电信、水文学、物理、经济和金融，以及模拟数量，如恒星的引力场、核反应堆的温度分布、晶体晶格的应力、年降雨量、股票价格等。目前可用的建模和预测方法很少且效率低下。研究者在假设数据是由一个流行的和有用的时间序列过程产生的，并通过将先验信息纳入建模的前提下，开发了新的创新建模方法。

9510348 Ravishanker Abstract The investigator studies Bayesian modeling and inference for time series with infinite variance stable innovations, addressing methodological problems as well as applications to real world data. The justification for the study of infinite variance stable processes stems from empirical evidence for its usefulness in several application areas such as astronomy, economics, engineering, finance and physics. The area of modeling stable processes has several open and challenging problems that are addressed. Available classical methods of estimation and inference do not simultaneously estimate the parameters defining the stable process and the parameters of the time series model. The investigator develops new methodology for modeling data generated by autoregressive fractionally integrated moving average processes with stable innovations that characterize long memory and short memory behavior in 'infinite variance' time series. Inference and prediction are implemented using sampling- based Bayesian techniques through Markov chain Monte Carlo algorithms to generate samples from the target posterior distribution. For stable processes, the form of the likelihood does not, in general, admit a closed analytical form. Hence expressions for the complete conditional distributions of the parameters in terms of the characteristic function of the stable process are combined with the Gibbs sampling algorithm or its variants to generate samples from the required posterior by approximating it by a suitable proposal density. Additionally, the role of algorithms that are useful in generating samples from stable processes directly is studied, incorporating this into the Bayesian framework. Various marginal and joint posterior distributions as well as summary features of these distributions are analyzed, as well as a characterization of predictive distributions that permit model choice and forecasting. The investigator studies modeling and forecasting for time series data assumi ng that the data can take on more extreme values than would usually be the case. There is considerable empirical evidence for this behavior in diverse areas of application such as telecommunications, hydrology, physics, economics and finance and for modeling quantities such as gravitational fields of stars, temperature distributions in nuclear reactors, stresses in crystalline lattices, annual rainfall, stock prices etc. Currently available methods for modeling and forecasting are few and inefficient . The investigator develops new innovative methodology for modeling under the assumption that the data are generated by a popular and useful time series process and by incorporating prior information into the modeling as well.