AI 中文总结
GRACE提出一种通用精确变点估计框架，通过联合包络演算统一处理分段形式、过渡类型与参数选择，涵盖多种现有方法，并支持并行计算。
AI 中文摘要
大多数变点程序在估计变点位置之前，会先固定分段形式和边界关系。我们提出GRACE（通用状态感知变点估计器），用于一个ℓ₀惩罚问题，该问题联合选择变点的数量和位置、分段形式、过渡类型以及连续参数。一个过渡仅通过其继承的量依赖于先前的拟合。GRACE保留以这些量为条件的累积成本，当多个量必须一起继承时使用联合包络。这产生了一种精确的条件成本演算，用于保留、释放或固定有限多个边界参数的过渡。最优分割、FPOP和CPOP作为特例出现。CPOP启发了该构造：其水平条件成本支持连续分段线性拟合，但不能表示保留输入斜率的非连续水平位移，因为端点水平的最优性不一定意味着斜率的最优性。GRACE在剪枝任一包络之前，对每个生成的线性候选在固定水平和固定斜率下进行剖析。这两个包络支持连续斜率变化、保留斜率的水平位移、恒定状态、趋势终止和恢复，以及在一次精确优化内的完全重置。在最小二乘下，具有仿射边界关系的有限维线性基模型产生二次候选成本。我们建立了通用递归的精确性，并开发了功能性、基于惩罚和基于目标界的剪枝规则。多项式和季节性特例说明了该框架，其递归允许大量的并行计算。
英文摘要
Most changepoint procedures fix segment forms and boundary relationships before estimating changepoint locations. We introduce GRACE, the General Regime-Aware Changepoint Estimator, for an $\ell_0$-penalized problem that jointly selects the number and locations of changepoints, segment forms, transition types, and continuous parameters. A transition depends on the preceding fit only through the quantities it inherits. GRACE retains accumulated cost conditional on these quantities, using joint envelopes when several must be inherited together. This yields an exact conditional-cost calculus for transitions that preserve, release, or fix finitely many boundary parameters. Optimal partitioning, FPOP, and CPOP arise as special cases. CPOP motivates the construction: its level-conditioned cost supports continuous piecewise-linear fitting but cannot represent a discontinuous level shift preserving the incoming slope, because optimality over endpoint level need not imply optimality over slope. GRACE profiles each generated linear candidate at fixed level and at fixed slope before pruning either envelope. The two envelopes support continuous slope changes, level shifts preserving slope, constant regimes, trend termination and resumption, and complete resets within one exact optimization. Under least squares, finite-dimensional linear-basis models with affine boundary relationships yield quadratic candidate costs. We establish exactness of the general recursion and develop functional, penalty-based, and objective-bound pruning rules. Polynomial and seasonal specializations illustrate the framework, whose recursion permits substantial parallel computation.
