We consider the problem of detecting multiple changes in multiple independent time series. The search for the best segmentation can be expressed as a minimization problem over a given cost function. We focus on dynamic programming algorithms that solve this problem exactly. When the number of changes is proportional to data length, an inequality-based pruning rule encoded in the PELT algorithm leads to a linear time complexity. Another type of pruning, called functional pruning, gives a close-to-linear time complexity whatever the number of changes, but only for the analysis of univariate time series. We propose a few extensions of functional pruning for multiple independent time series based on the use of simple geometric shapes (balls and hyperrectangles). We focus on the Gaussian case, but some of our rules can be easily extended to the exponential family. In a simulation study we compare the computational efficiency of different geometric-based pruning rules. We show that for small dimensions (2, 3, 4) some of them ran significantly faster than inequality-based approaches in particular when the underlying number of changes is small compared to the data length.

翻译：我们考虑在多个独立时间序列中检测多个变点的问题。最佳分割的搜索可表述为对给定代价函数的最小化问题。我们重点关注能够精确求解该问题的动态规划算法。当变点数量与数据长度成比例时，PELT算法中基于不等式的剪枝规则可实现线性时间复杂度。另一种称为函数剪枝的剪枝类型，无论变点数量多少，都能实现接近线性的时间复杂度，但仅适用于单变量时间序列分析。我们基于简单几何形状（球体和超矩形）提出了函数剪枝在多个独立时间序列中的若干扩展。我们重点关注高斯情形，但部分规则可轻松扩展至指数族分布。通过仿真研究，我们比较了不同基于几何的剪枝规则的计算效率。结果表明，在低维（2、3、4维）情况下，当变点真实数量相对于数据长度较小时，其中部分规则的运行速度显著快于基于不等式的剪枝方法。