用于支持物理群体和元胞自动机轨迹中涌现识别的奇异拐点识别
Singular knee identification to support emergence recognition in physical swarm and cellular automata trajectories
AI总结：
针对涌现现象缺乏统一数学定义与严格检测方法的问题，本研究基于奇异值曲线分析开发启发式涌现检验法，确定检测标准与上下界，经多类轨迹验证，奇异值拐点夹角在跨形态涌现检测中潜力最优。
摘要：
历经数十年的研究关注，涌现现象仍缺乏一个统一的数学定义，因而无法形成可应用于物理鸟群和群体系统的严格涌现检验方法，尤其是针对同时包含确定性要素（如相互作用）和测量噪声等随机扰动的系统。本研究基于包含确定性信号与高斯噪声信号的数据矩阵的奇异值曲线分析，开发了一种启发式检验方法。研究确定了最低检测标准，并开展了统计分析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，使用了多智能体、元胞自动机和生物视频中混合确定性与随机轨迹的记录数据。案例包括Cucker-Smale（库克-斯梅尔）和Vicsek（维克塞克）集群模型、高斯噪声及其积分、鸟群的观测记录以及一维元胞自动机。研究开展了包含测量噪声的集合模拟以计算统计变异，并结合随机矩阵理论的噪声界进行了讨论。结果表明，对记录轨迹的奇异拐点分析可以检测结构与噪声连续体上的分级水平。在研究考量的8种奇异值衰减指标中，奇异值拐点处的夹角在支持跨形态涌现检测方面最具潜力；噪声界的大小可作为所需样本量的指标；而噪声界内存在大部分奇异值则可作为噪声的指标。
英文摘要：
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.
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