贲树军, 翁艺鸿. 马尔可夫过程的低秩谱估计[J]. 华南师范大学学报(自然科学版), 2022, 54(4): 101-108. doi: 10.6054/j.jscnun.2022063
引用本文: 贲树军, 翁艺鸿. 马尔可夫过程的低秩谱估计[J]. 华南师范大学学报(自然科学版), 2022, 54(4): 101-108. doi: 10.6054/j.jscnun.2022063
BI Shujun, WENG Yihong. A Low-rank Spectral Estimation of Markov Process[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(4): 101-108. doi: 10.6054/j.jscnun.2022063
Citation: BI Shujun, WENG Yihong. A Low-rank Spectral Estimation of Markov Process[J]. Journal of South China Normal University (Natural Science Edition), 2022, 54(4): 101-108. doi: 10.6054/j.jscnun.2022063

马尔可夫过程的低秩谱估计

A Low-rank Spectral Estimation of Markov Process

  • 摘要: 针对马尔可夫过程的谱估计算法利用了非负投影而导致估计矩阵不能满足低秩要求的问题,提出一个低秩谱估计算法(Low-rank Spectral Estimation Algorithm, LRSEA):首先,建立秩约束状态转移矩阵集合的局部Lipschitz型误差界,并给出满足该集合误差界不等式的近似投影矩阵; 然后,基于近似投影矩阵对现有的谱估计算法进行低秩修正,得到LRSEA算法,并为该算法建立统计误差界。通过人工合成数据实验对LRSEA算法、经验估计方法和谱估计方法进行比较,结果表明LRSEA算法的估计误差最小。最后,将LRSEA算法与k-均值聚类算法结合应用到纽约市曼哈顿岛出租车轨迹的分析问题。

     

    Abstract: As the method for spectral estimation of Markov process makes use of nonnegativity-preserving step, the spectral estimator does not necessarily satisfy low-rank condition. Motivated by this, a low-rank spectral estimation algorithm (LRSEA) is proposed. First of all, the local Lipschitzian type error bound of the rank-constrained state transition matrix set is established, and an approximate projection matrix that satisfies the error bound inequality is given. Then, using the approximate projection matrix to modify the spectral estimation method, the LRSEA is proposed, and the statistical error bound for the proposed estimation method is provided. Numerical comparisons on the synthetic data with empirical estimator and spectral estimator show that the LRSEA has the lowest estimation error. Finally, the LRSEA together with k-means algorithm is used to analyze the dataset of Manhattan taxi trips.

     

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