AI 中文总结
本文提出H-SSBP参数化方法解决Ornstein-Uhlenbeck过程推断的稳定性与计算效率问题，通过实验验证其大幅加速效果，并在真实数据中展现应用价值。
AI 中文摘要
具有灵活多元漂移矩阵的Ornstein-Uhlenbeck过程是捕捉耦合、非对称及阻尼振荡均值回复的强大模型。然而，基于似然的推断极具挑战性，因为漂移矩阵必须保持Hurwitz稳定性，而似然与梯度评估需要重复且代价高昂地计算和求导漂移矩阵指数及相关Lyapunov方程的解。我们提出Hurwitz平滑谱块参数化（H-SSBP），它将漂移表示为应用于独立一维及二维稳定块的基变换。简单标量约束可确保Hurwitz稳定性，同时二维块在实特征值与复特征值模式间平滑变化，避免离散模型选择。H-SSBP可表示所有在复数上可对角化的实Hurwitz矩阵，并在Hurwitz锥内具有稠密的全测度支撑。其块结构将OU转移矩阵、平稳协方差与创新协方差及其反向模式导数简化为固定大小的块或块对计算，加上基变换乘法。数值实验表明，与替代方法相比，该方法实现了显著的加速，尤其在矩阵指数伴随和Lyapunov方程核的计算上。对异步金融数据和多元系统发育性状的真实数据分析，展示了所提框架的实际应用。
英文摘要
Ornstein-Uhlenbeck processes with flexible multivariate drift matrices are powerful models for capturing coupled, asymmetric, and damped-oscillatory mean reversion. However, likelihood-based inference is challenging because the drift matrix must remain Hurwitz stable, while likelihood and gradient evaluations require repeated, costly computation and differentiation of the drift matrix exponential and of solutions to the associated Lyapunov equation. We introduce the Hurwitz smooth spectral block parametrization (H-SSBP), which represents the drift as a change of basis applied to independent one- and two-dimensional stable blocks. Simple scalar constraints enforce Hurwitz stability, while the two-dimensional blocks vary smoothly between real- and complex-eigenvalue regimes, avoiding discrete model selection. The H-SSBP represents every real Hurwitz matrix diagonalizable over the complex numbers and has dense, full-measure support within the Hurwitz cone. Its block structure reduces the OU transition matrix, stationary and innovation covariances, and their reverse-mode derivatives to constant-size block or block-pair computations plus change-of-basis multiplications. Numerical experiments demonstrate dramatic speed-ups over alternatives, particularly for matrix-exponential adjoints and Lyapunov-equation kernels. Real-data analyses of asynchronous financial data and multivariate phylogenetic traits illustrate the proposed framework in practice.
