空间随机游走：一种用于高阶数据的随机过程
The Spacey Random Walk: a Stochastic Process for Higher-order Data
AI总结：
本文提出空间随机游走，一种非马尔可夫过程，其稳态分布由张量特征向量给出，用于高阶数据的分析与应用。
摘要：
随机游走是应用数学中的基本模型，也是马尔可夫链的常见例子。马尔可夫链的极限稳态分布表示随机过程中每个状态被占用的时间比例。对于有限状态空间上的随机游走，计算该分布的标准方法是计算关联转移矩阵的皮尔松向量。在更高阶马尔可夫链中，存在这种皮尔松向量的代数类比，即更高阶转移概率张量的特征向量。这些向量是非负的，维度等于状态空间的维度，总和为一，并通过在更高阶马尔可夫链联合稳态分布方程中进行代数替换得出。本文提出空间随机游走，一种非马尔可夫随机过程，其稳态分布由张量特征向量给出。该过程本身是顶点增强随机游走，其离散动力学与连续动力系统相关。我们分析了这些动力学的收敛性质，并讨论了计算稳态分布的数值方法。最后，我们提供了空间随机游走模型在群体遗传学、排序和聚类数据中的多个应用，并使用该过程分析纽约出租车轨迹数据。这个例子展示了确定性的非马尔可夫结构。
英文摘要：
Random walks are a fundamental model in applied mathematics and are a common example of a Markov chain. The limiting stationary distribution of the Markov chain represents the fraction of the time spent in each state during the stochastic process. A standard way to compute this distribution for a random walk on a finite set of states is to compute the Perron vector of the associated transition matrix. There are algebraic analogues of this Perron vector in terms of transition probability tensors of higher-order Markov chains. These vectors are nonnegative, have dimension equal to the dimension of the state space, and sum to one and are derived by making an algebraic substitution in the equation for the joint-stationary distribution of a higher-order Markov chains. Here, we present the spacey random walk, a non-Markovian stochastic process whose stationary distribution is given by the tensor eigenvector. The process itself is a vertex-reinforced random walk, and its discrete dynamics are related to a continuous dynamical system. We analyze the convergence properties of these dynamics and discuss numerical methods for computing the stationary distribution. Finally, we provide several applications of the spacey random walk model in population genetics, ranking, and clustering data, and we use the process to analyze taxi trajectory data in New York. This example shows definite non-Markovian structure.
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