Mapping correlations and coherence: adjacency-based approach to data visualization and regularity discovery

The paper defines gradient-based Stokes-like quantities I, Q, U and from these the correlation degree p and angle θ, and states that in the fully correlated case tan θ equals dp2/dp1 (its Eqs. (5)–(10) and surrounding text) . It also asserts a direct connection to a value-space correlation matrix M and says the correlation degree “reflects” lmax/(lmax + lmin) (without proof) . However, the paper provides no derivations for basic properties (range 0≤p≤1, invariance under rotations, eigenvalue interpretation), and contains minor inconsistencies (e.g., a caption stating θ = arctan(dU/dV), which inverts the slope relative to tan θ = dp2/dp1) . By contrast, the model solution rigorously derives p^2 = 1 − 4 det S/(tr S)^2 with S = Σ g_i g_i^T, establishes 0≤p≤1 and equality cases, proves θ’s relation to dp2/dp1 in the fully correlated case, shows invariance under orthogonal rotations, and gives the exact eigenvalue/principal-axis relations p = (λmax−λmin)/(λmax+λmin) and tan 2φ = U/Q. These fill the missing steps and correct ambiguities left by the paper.