After decades of attention, emergence continues to lack a centralized mathematical definition that leads to a rigorous emergence test applicable to physical flocks and swarms, particularly those containing both deterministic elements (eg, interactions) and stochastic perturbations like measurement noise. This study develops a heuristic test based on singular value curve analysis of data matrices containing deterministic and Gaussian noise signals. The minimum detection criteria are identified, and statistical and matrix space analysis developed to determine upper and lower bounds. This study applies the analysis to representative examples by using recorded trajectories of mixed deterministic and stochastic trajectories for multi-agent, cellular automata, and biological video. Examples include Cucker Smale and Vicsek flocking, Gaussian noise and its integration, recorded observations of bird flocking, and 1D cellular automata. Ensemble simulations including measurement noise are performed to compute statistical variation and discussed relative to random matrix theory noise bounds. The results indicate singular knee analysis of recorded trajectories can detect gradated levels on a continuum of structure and noise. Across the eight singular value decay metrics considered, the angle subtended at the singular value knee emerges with the most potential for supporting cross-embodiment emergence detection, the size of noise bounds is used as an indication of required sample size, and the presence of a large fraction of singular values inside noise bounds as an indication of noise.

翻译：经过数十年的关注，涌现现象仍缺乏一个中心化的数学定义，以推导出适用于物理集群（特别是同时包含确定性元素（如相互作用）和随机扰动（如测量噪声）的集群）的严格涌现检测方法。本研究基于对包含确定性信号与高斯噪声信号的数据矩阵进行奇异值曲线分析，开发了一种启发式检测方法。确定了最小检测标准，并发展了统计与矩阵空间分析以确定上下界。本研究将该分析应用于代表性案例，通过使用多智能体、元胞自动机及生物视频中混合确定性-随机轨迹的实测轨迹数据。案例包括Cucker Smale与Vicsek集群模型、高斯噪声及其积分、鸟类集群的观测记录，以及一维元胞自动机。通过包含测量噪声的集成仿真计算统计变异，并参照随机矩阵理论的噪声界进行讨论。结果表明，对实测轨迹的奇异值膝点分析能够检测连续统上结构与噪声的梯度水平。在考察的八种奇异值衰减度量中，奇异值膝点所张角度展现出最大潜力以支持跨具身系统的涌现检测，噪声界范围被用作所需样本量的指标，而大部分奇异值位于噪声界内则被视为噪声存在的表征。