用可学习函数替代固定规则，更准确捕捉时间序列的非线性状态切换。
Learning Nonlinear Regime Transitions via Semi-Parametric State-Space Models
- 用核方法或样条函数学习状态转移概率，摆脱传统模型的固定形式限制。
- 在合成数据上显著提升非线性动态的恢复能力，在金融数据中实现更早的状态切换识别。
- 适合研究复杂动态系统、金融时序建模或需要灵活状态转换机制的场景。
我们提出一种半参数状态空间模型，用于具有潜在状态转换的时间序列数据。经典马尔可夫切换模型采用固定的参数化转移函数（如logistic或probit链接），当转换依赖于非线性和上下文相关效应时，灵活性受限。我们改用可学习函数 $f_0, f_1 \in \calH$，其中 $\calH$ 为再生核希尔伯特空间或样条逼近空间，并将转移概率定义为 $p_{jk,t} = \sigmoid(f(\bx_{t-1}))$。转移函数与观测参数通过广义期望最大化算法联合估计：E步使用标准前向-后向递推，M步转化为加权惩罚回归问题，权重来自平滑的占用度量。我们建立了可辨识性条件，并提供了估计器的一致性论证。合成数据实验表明，相比参数化基线，该模型能更好恢复非线性转换动态；对金融时间序列的实证研究显示，其状态分类更优，且能更早检测到状态转换事件。
原文摘要 · Abstract (English)
We develop a semi-parametric state-space model for time-series data with latent regime transitions. Classical Markov-switching models use fixed parametric transition functions, such as logistic or probit links, which restrict flexibility when transitions depend on nonlinear and context-dependent effects. We replace this assumption with learned functions $f_0, f_1 \in \calH$, where $\calH$ is either a reproducing kernel Hilbert space or a spline approximation space, and define transition probabilities as $p_{jk,t} = \sigmoid(f(\bx_{t-1}))$. The transition functions are estimated jointly with emission parameters using a generalized Expectation-Maximization algorithm. The E-step uses the standard forward-backward recursion, while the M-step reduces to a penalized regression problem with weights from smoothed occupation measures. We establish identifiability conditions and provide a consistency argument for the resulting estimators. Experiments on synthetic data show improved recovery of nonlinear transition dynamics compared to parametric baselines. An empirical study on financial time series demonstrates improved regime classification and earlier detection of transition events.
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