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

9510348拉维尚克摘要 调查研究贝叶斯建模和推理的时间序列与无限方差稳定的创新，解决方法问题，以及应用到真实的世界数据。研究无穷方差稳定过程的理由来自于它在天文学、经济学、工程学、金融学和物理学等多个应用领域的有用性的经验证据。稳定过程建模领域有几个开放的和具有挑战性的问题，解决。现有的经典估计和推断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。研究人员开发了一种新的方法，用于对自回归分数积分移动平均过程生成的数据进行建模，该过程具有稳定的创新，这些创新表征了“无限方差”时间序列中的长记忆和短记忆行为。使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生成样本来实现推理和预测。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，表达式的完整的条件分布的参数的稳定过程的特征函数与吉布斯抽样算法或其变种相结合，以产生样本所需的后验通过近似它由一个合适的建议密度。此外，算法是有用的，直接从稳定的过程中产生的样本的作用进行了研究，将其纳入贝叶斯框架。各种边际和联合后验分布以及这些分布的摘要功能进行了分析，以及允许模型选择和预测的预测分布的表征。 研究人员研究时间序列数据的建模和预测，这些数据可能比通常情况下具有更多的极端值。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，物理，经济和金融和建模数量，如恒星的引力场，核反应堆的温度分布，应力晶格，年降雨量，股票价格等目前可用的建模和预测方法很少，效率低下。研究人员开发了新的创新方法建模的假设下，数据是由一个流行的和有用的时间序列过程，并将先验信息纳入建模以及。

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.