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arXiv 1108.3262stat.ME
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非参数动态状态空间模型的贝叶斯推断

Bayesian Inference in Nonparametric Dynamic State-Space Models

Anurag Ghosh, Soumalya Mukhopadhyay, Sandipan Roy, Sourabh Bhattacharya

备注：
This version contains much greater clarification of the look-up table idea and a theorem regarding this is also proven and included in the supplement. Will appear in Statistical Methodology
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摘要：

我们引入状态空间模型，其中观测和演化方程的泛函是未知的，并作为随时间演化的随机函数处理。因此，我们的模型是非参数的，推广了传统的参数状态空间模型。这种随机函数方法也使我们摆脱了函数形式的限制性假设，即函数形式虽然与时间有关，但具有固定的形式。假设已知的参数函数形式的传统方法是有问题的，特别是在状态空间模型中，因为假设的验证需要观察到的时间序列和潜在状态的数据;然而，后者的数据在状态空间模型中不可用。我们指定高斯过程作为先验的随机函数，并利用“查找表的方法”的\ctn{Bhattacharya07}有效地处理模型的动态结构。我们考虑单变量和多变量的情况下，使用马尔可夫链蒙特卡罗（MCMC）的方法研究的后验分布的利益。在具有挑战性的多变量的情况下，我们证明了新开发的基于变换的MCMC（TMCMC）的\ctn{Dutta11}提供了有趣的和有效的替代方案，通常的建议分布。我们用一个具有挑战性的多变量模拟数据集来说明我们的方法，其中真实的观测和演化方程是高度非线性的，并且被视为未知的。我们取得的成果是相当令人鼓舞的。此外，使用我们的高斯过程的方法，我们分析了一个真实的数据集，这也是分析了\ctn{Shumway82}和\ctn{Carlin92}使用线性假设。我们的分析表明，接近年底的时间序列，以前的作者的线性假设打破。

英文摘要：

We introduce state-space models where the functionals of the observational and the evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional parametric state-space models. This random function approach also frees us from the restrictive assumption that the functional forms, although time-dependent, are of fixed forms. The traditional approach of assuming known, parametric functional forms is questionable, particularly in state-space models, since the validation of the assumptions require data on both the observed time series and the latent states; however, data on the latter are not available in state-space models. We specify Gaussian processes as priors of the random functions and exploit the "look-up table approach" of \ctn{Bhattacharya07} to efficiently handle the dynamic structure of the model. We consider both univariate and multivariate situations, using the Markov chain Monte Carlo (MCMC) approach for studying the posterior distributions of interest. In the case of challenging multivariate situations we demonstrate that the newly developed Transformation-based MCMC (TMCMC) of \ctn{Dutta11} provides interesting and efficient alternatives to the usual proposal distributions. We illustrate our methods with a challenging multivariate simulated data set, where the true observational and the evolutionary equations are highly non-linear, and treated as unknown. The results we obtain are quite encouraging. Moreover, using our Gaussian process approach we analysed a real data set, which has also been analysed by \ctn{Shumway82} and \ctn{Carlin92} using the linearity assumption. Our analyses show that towards the end of the time series, the linearity assumption of the previous authors breaks down.

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