发表机构
Northwestern University（西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出相空间的相空间（PSoPS）框架，结合Takens嵌入、降维与最优传输，从时间序列提取低维拓扑基，在宏观经济数据中恢复经典定律并揭示非线性结构。
AI 中文摘要
从物理学到社会科学的定律都依赖于少数可枚举的因素来解释其所描述系统中的变化。虽然经济学、物理学和化学的正统方法依赖于少数可解析解释的定律，但追求普适性的大数据模型则投入所有可用数据。两者都在极端情况下因同一缺陷而失败：基不正确——要么正交变量太少而无法描述系统变化，要么围绕本质低维但未知的动力学产生维度爆炸。我们提出并验证了一个框架，该框架直接从任意时间序列中提取可解释、数学上合理、低维且具有动力学信息的基。结合Takens延迟嵌入定理、降维和最优传输，我们构建了一个相空间的相空间（PSoPS），所有相互关联的动力学序列都存在于其中：由多个相空间形成的重心，连同各个相空间与重心之间的传输计划族，产生了一个描述每个时间序列的关系几何和经济关系拓扑的基。由281,536个序列构建的宏观经济PSoPS是一个有效维度为7.3±0.2的单一连通对象，其中经典定律（菲利普斯曲线、奥肯定律、索洛模型）嵌入为约二维的环状子吸引子。概念重心之间的耦合是有向的，货币价格和生产率作为净源，信贷和产出作为汇，仅从几何就恢复了货币传导机制。IAAFT相位随机化替代数据在三个定律上破坏了36-43%的维度，在整个语料库上破坏了38%，因此该基是不可约非线性的。最后，我们提出了对动力学共享吸引子几何的领域以及对本质低维但在原始坐标中难以导航的系统的扩展建议。
英文摘要
Laws from physics to social science rest on a small, enumerable number of factors explaining variation in the systems they describe. While the orthodoxy of economics, physics and chemistry relies on a small number of analytically interpretable laws, large data models pursuing universality throw in all available data. Both fail in their extremes through the same flaw, an incorrect basis: too few orthogonal variables to describe system variation, or an explosion of dimensionality around fundamentally low dimensional but unknown dynamics. We propose and validate a framework that extracts an interpretable, mathematically sound, low dimensional and dynamically informed basis directly from arbitrary timeseries. Combining Takens' Delay Embedding Theorem, dimensionality reduction and optimal transport, we produce a phase space of phase spaces (PSoPS) on which all interrelated dynamical series exist: barycenters formed from many phase spaces, together with the families of transport plans between individual phase spaces and the barycenter, yield a basis describing the relational geometry of each timeseries and the topology of economic relationships. The macroeconomic PSoPS built from 281,536 series is a single connected object of effective dimension 7.3 +/- 0.2, into which the canonical laws (Phillips, Okun, Solow) embed as roughly 2-dimensional looped sub-attractors. The coupling between concept barycenters is directed, with the price of money and productivity as net sources and credit and output as sinks, recovering monetary transmission from geometry alone. IAAFT phase-randomized surrogates destroy 36-43% of the dimension across the three laws and 38% on the full corpus, so the basis is irreducibly nonlinear. We close with proposed extensions to fields whose dynamics share attractor geometries and to systems that are intrinsically low dimensional yet hard to navigate in raw coordinates.
Comments29 pages, 8 figures, plus a 5 page supplement. Code: https://github.com/maxtopel/phase-space-of-phase-spaces. Data: doi:10.5281/zenodo.22602656
