Statistical Analysis of Categorical Time Series through Sparse Markov Models

对类别值序列的分析最好是通过捕获序列的统计属性的模型来完成，同时要足够简单，以便统计分析是可行的。在许多情况下，这种数据的分析已经被具有短记忆或马尔可夫性质的序列简化，在某种意义上，条件概率仅取决于最近的过去。虽然马尔可夫模型被广泛应用，但通常低阶模型是合适的，因为估计的条件概率的数量随着用于条件化的过去观测数量的增加而以几何级数增长。另一个缺点涉及缺乏模型灵活性，因为可能的马尔可夫模型的数量有限。稀疏马尔可夫模型(SMM)有助于解决这两个问题，因此可以进行更好的模型拟合。虽然马尔可夫模型的理论结果很普遍，但关于SMM的理论结果相对较少，直到最近十年才被考虑。因此，与SMM理论和应用相关的知识在分析分类时间序列方面有着巨大的潜力。这是这个项目的根本目的。稀疏马尔可夫模型允许对也足够灵活的简约模型进行拟合，从而在比真值短的条件上下文引起的偏差和具有许多要估计的参数的方差之间获得更好的权衡，从而改进推理。本课题研究了稀疏马尔可夫模型的理论性质和应用，考虑了变长马尔可夫链在分类时间序列分析中的特殊情况。相关的目标是(I)发展SMM数据预测的理论、中心极限定理、正则化回归模型拟合以及导出方法和当前方法的渐近和有限样本性质的比较；(Ii)发展一种称为隐藏稀疏马尔可夫模型(HSMM)的新模型；(Iii)将有效计算马尔可夫序列模式统计分布的方法推广到稀疏和隐藏稀疏马尔可夫模型；(Iv)应用VLMC/SMM和HSMM改进各种分类时间序列的分析和推断。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Analysis of a sequence of categorical values is best done by a model that captures the statistical properties of the sequence, while being simple enough so that statistical analysis is feasible. In many cases, the analysis of such data has been simplified by the sequence having a short memory or Markov property in the sense that conditional probabilities depend on only the very recent past. Whereas Markov models are applied extensively, typically low-order models are fit because the number of estimated conditional probabilities grows geometrically as the number of past observations used for conditioning increases. Another drawback involves the lack of model flexibility, as the number of possible Markov models is limited. Sparse Markov models (SMM) help with these two problems, thus allowing better model fits. While theoretical results for Markov models are prevalent, those for SMMs are relatively rare, only being considered in the last decade. Thus, there is tremendous potential for the furthering of knowledge related to theory and applications of SMMs to analyze categorical time series. This is the fundamental aim of this project. Sparse Markov models allow the fitting of a parsimonious model that is also flexible enough so that a better trade-off is obtained between bias that arises from conditioning contexts that are shorter than truth and variance from having many parameters to estimate, thus improving inference. This project studies theoretical properties and applications of sparse Markov models; the special case of variable length Markov chains (VLMCs) to the analysis of categorical time series is considered. Related objectives are (i) To develop theory for prediction of data from SMMs, central limit theorems, model fitting through regularized regression, and comparisons of asymptotic and finite-sample properties of derived and current methods; (ii) To develop a new model called hidden sparse Markov models (HSMMs); (iii) To extend methods for efficient computation of distributions of pattern statistics in Markovian sequences to both sparse and hidden sparse Markov models; (iv) To apply VLMCs/SMMs and HSMMs to improve the analysis and inference of various categorical time series.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.