Singular knee identification to support emergence recognition in physical swarm and cellular automata trajectories

TL;DR

A heuristic test based on singular value curve analysis of data matrices containing deterministic and Gaussian noise signals which indicates singular knee analysis of recorded trajectories can detect gradated levels on a continuum of structure and noise.

Abstract

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.

Figures (11)

• Figure 1: When $n_a$ distributed agents in 3D space (position and velocity) converge to a velocity-aligned flock, the macro scale can use the relative velocity bound to describe the behavior by simpler representations (fewer coordinates) than the $6 n_a$ needed for the microscale description. For a swarm having reached velocity consensus in Euclidean space, one could specify the center of mass position (3 coordinates), converged velocity (3), and inter-agent vectors ($3 n_a$), or a total of $3n_a+6$ coordinates. As $n_a$ grows large, the compression ratio approaches 50%. By recognizing that converged inter-agent vectors are constant, updates could be to the center of mass position and velocity only, or 6 states. Macroscale compressibility then provides a stronger limit for large agent numbers.
• Figure 2: Singular values of a matrix are plotted in decreasing order to get the singular value curve. The values show a knee/elbow region. Generally after the knee region, the singular values do not decrease drastically as before. Heuristically, this point is used for 'sufficient' low-rank reconstruction of a matrix.
• Figure 3: Singular values are used to determine the major axes in a PCA. In (b) the blue axis has a higher corresponding singular value than the black axis. In (a) there are no dominant singular values. In (c) though the data is random, due to the low sample size there are apparent principal axes.
• Figure 4: Singular value curve analysis: definitions of points $P_1, P_k, P_3,$ pre- and post-knee vectors $\boldsymbol{v}_1$ and $\boldsymbol{v}_2,$ and knee angle $\theta.$
• Figure 5: Example simulation trajectories (time progression is blue to green).

Theorems & Definitions (1)

• Definition II.1: Emergent property