Vera-Vera J Fernando, Roldán-Nofuentes J Antonio
Department of Statistics and O.R., University of Granada, Faculty of Sciences, Fuentenueva s/n, 18071, Granada, Spain.
Math Biosci Eng. 2024 Feb 6;21(3):3631-3651. doi: 10.3934/mbe.2024160.
Time series clustering is a usual task in many different areas. Algorithms such as K-means and model-based clustering procedures are used relating to multivariate assumptions on the datasets, as the consideration of Euclidean distances, or a probabilistic distribution of the observed variables. However, in many cases the observed time series are of unequal length and/or there is missing data or, simply, the time periods observed for the series are not comparable between them, which does not allow the direct application of these methods. In this framework, dynamic time warping is an advisable and well-known elastic dissimilarity procedure, in particular when the analysis is accomplished in terms of the shape of the time series. In relation to a dissimilarity matrix, K-means clustering can be performed using a particular procedure based on classical multidimensional scaling in full dimension, which can result in a clustering problem in high dimensionality for large sample sizes. In this paper, we propose a procedure robust to dimensionality reduction, based on an auxiliary configuration estimated from the squared dynamic time warping dissimilarities, using an alternating least squares procedure. The performance of the model is compared to that obtained using classical multidimensional scaling, as well as to that of model-based clustering using this related auxiliary linear projection. An extensive Monte Carlo procedure is employed to analyze the performance of the proposed method in which real and simulated datasets are considered. The results obtained indicate that the proposed K-means procedure, in general, slightly improves the one based on the classical configuration, both being robust in reduced dimensionality, making it advisable for large datasets. In contrast, model-based clustering in the classical projection is greatly affected by high dimensionality, offering worse results than K-means, even in reduced dimension.
时间序列聚类是许多不同领域中的常见任务。诸如K均值和基于模型的聚类程序等算法被用于与数据集的多变量假设相关的情况，例如考虑欧几里得距离或观测变量的概率分布。然而，在许多情况下，观测到的时间序列长度不等和/或存在缺失数据，或者简单地说，为这些序列观测的时间段之间不可比，这使得这些方法无法直接应用。在此框架下，动态时间规整是一种可取且广为人知的弹性差异度量程序，特别是当根据时间序列的形状进行分析时。关于差异矩阵，可以使用基于全维经典多维缩放的特定程序来执行K均值聚类，对于大样本量，这可能会导致高维聚类问题。在本文中，我们基于从平方动态时间规整差异估计的辅助配置，使用交替最小二乘法，提出了一种对降维具有鲁棒性的程序。将该模型的性能与使用经典多维缩放获得的性能进行比较，以及与使用这种相关辅助线性投影的基于模型的聚类的性能进行比较。采用广泛的蒙特卡罗程序来分析所提出方法在考虑真实和模拟数据集时的性能。获得的结果表明，所提出的K均值程序总体上比基于经典配置的程序略有改进，两者在降维时都具有鲁棒性，这使得它适用于大型数据集。相比之下，经典投影中的基于模型的聚类受到高维的极大影响，即使在降维时也比K均值提供更差的结果。
