Statistical jump models have been recently introduced to detect persistent regimes by clustering temporal features and discouraging frequent regime changes. However, they are limited to hard clustering and thereby do not account for uncertainty in state assignments. This work presents an extension of the statistical jump model that incorporates uncertainty estimation in cluster membership. Leveraging the similarities between statistical jump models and the fuzzy c-means framework, our fuzzy jump model sequentially estimates time-varying state probabilities. Our approach offers high flexibility, as it supports both soft and hard clustering through the tuning of a fuzziness parameter, and it naturally accommodates multivariate time series data of mixed types. Through a simulation study, we evaluate the ability of the proposed model to accurately estimate the true latent-state distribution, demonstrating that it outperforms competing approaches under high cluster assignment uncertainty. We further demonstrate its utility on two empirical applications: first, by automatically identifying co-orbital regimes in the three-body problem, a novel application with important implications for understanding asteroid behavior and designing interplanetary mission trajectories; and second, on a financial dataset of five assets representing distinct market sectors (equities, bonds, foreign exchange, cryptocurrencies, and utilities), where the model accurately tracks both bull and bear market phases.

翻译：统计跳跃模型最近被引入用于通过聚类时间特征并抑制频繁的体制转换来检测持久性体制。然而，这类模型仅限于硬聚类，因而无法考虑状态分配中的不确定性。本研究提出了一种统计跳跃模型的扩展，该扩展融合了聚类隶属关系的不确定性估计。利用统计跳跃模型与模糊c均值框架之间的相似性，我们提出的模糊跳跃模型能够顺序估计时变状态概率。我们的方法具有高度灵活性：通过调整模糊度参数，既可支持软聚类也可支持硬聚类，并且能自然地适应混合类型的多元时间序列数据。通过模拟研究，我们评估了所提模型准确估计真实潜在状态分布的能力，证明其在聚类分配高度不确定的情况下优于竞争方法。我们进一步通过两个实证应用展示了其实用性：首先，在三体问题中自动识别共轨体制，这是一项具有重要应用价值的新颖尝试，对于理解小行星行为及设计星际任务轨迹具有重要意义；其次，在一个包含代表不同市场板块（股票、债券、外汇、加密货币和公用事业）的五种资产的金融数据集上，该模型能准确追踪牛市和熊市阶段。