莫斯科对应矩阵计算的恢复模型
The recovery model for the calculation of correspondence matrix for Moscow
摘要：
在本文中，我们考虑的问题，恢复对应矩阵的基础上，在莫斯科的实际对应的观察。按照传统的方法，交通网络被认为是一个有向图，其边缘对应于路段和图形顶点对应于交通参与者离开或进入的区域。城市居民的数量被认为是恒定的。恢复对应矩阵的问题是计算从$i$区域到$j$区域的所有对应。为了恢复矩阵，我们建议使用城市研究中最流行的计算对应矩阵的方法之一-熵模型。在我们的工作中，我们描述的熵模型的进化理由和过渡到解决问题的熵-线性规划（ELP）在计算对应矩阵的主要思想。为了解决ELP问题，建议将其转化为对偶问题。在本文中，我们描述了几个数值最优化方法解决这个问题：Sinkhorn方法和加速Sinkhorn方法。我们提供了以下变量的费用函数的数值实验：线性成本函数和叠加的功率和对数费用函数。在这些函数中，成本是区域之间的平均时间和距离的组合，这取决于参数。计算多组参数的对应矩阵，然后计算恢复矩阵相对于已知对应矩阵的质量。我们假设恢复的对应矩阵中的噪声是高斯的，因此，我们使用标准差作为质量度量。
英文摘要：
In this paper, we consider the problem of restoring the correspondence matrix based on the observations of real correspondences in Moscow. Following the conventional approach, the transport network is considered as a directed graph whose edges correspond to road sections and the graph vertices correspond to areas that the traffic participants leave or enter. The number of city residents is considered constant. The problem of restoring the correspondence matrix is to calculate all the correspondence from the $i$ area to the $j$ area. To restore the matrix, we propose to use one of the most popular methods of calculating the correspondence matrix in urban studies -- the entropy model. In our work, we describe the evolutionary justification of the entropy model and the main idea of the transition to solving the problem of entropy-linear programming (ELP) in calculating the correspondence matrix. To solve the ELP problem, it is proposed to pass to the dual problem. In this paper, we describe several numerical optimization methods for solving this problem: the Sinkhorn method and the Accelerated Sinkhorn method. We provide numerical experiments for the following variants of cost functions: a linear cost function and a superposition of the power and logarithmic cost functions. In these functions, the cost is a combination of average time and distance between areas, which depends on the parameters. The correspondence matrix is calculated for multiple sets of parameters and then we calculate the quality of the restored matrix relative to the known correspondence matrix. We assume that the noise in the restored correspondence matrix is Gaussian, as a result, we use the standard deviation as a quality metric.
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