Recurrence plot¶
A square matrix visualization marking pairs of observation times whose reconstructed system states are equal or sufficiently close under a declared metric and threshold.
Core Idea¶
A recurrence plot represents the recurrence matrix R_ij = Θ(ε−||x_i−x_j||), exposing returns, diagonal trajectories, laminar segments, transitions, and nonstationarity in state-space dynamics. State vectors are reconstructed or observed, pairwise distances are computed, and a threshold converts closeness into marks whose line structures are interpreted relative to embedding and sampling. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Recurrence plot belongs to nonlinear time series analysis and is useful where the analyst can specify the typed nonlinear time series analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate each plotted mark corresponds exactly to a state pair satisfying the declared metric, embedding, norm, and recurrence threshold. The scope is broad within that domain but bounded by the need for each plotted mark corresponds exactly to a state pair satisfying the declared metric, embedding, norm, and recurrence threshold. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making each plotted mark corresponds exactly to a state pair satisfying the declared metric, embedding, norm, and recurrence threshold the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Recurrence plot can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Recurrence plot. Recurrence plot compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed nonlinear time series analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express each plotted mark corresponds exactly to a state pair satisfying the declared metric, embedding, norm, and recurrence threshold independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of nonlinear time series analysis because they reuse the typed nonlinear time series analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, State vectors are reconstructed or observed, pairwise distances are computed, and a threshold converts closeness into marks whose line structures are interpreted relative to embedding and sampling., and type the carrier, state every parameter and convention in the definition, test that each plotted mark corresponds exactly to a state pair satisfying the declared metric, embedding, norm, and recurrence threshold, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Recurrence plot Domain-specific
Parents (1) — more general patterns this builds on
-
Recurrence plot is a kind of Recurrence Prime
The proposed strict upward parent is
prime:recurrence.
Hierarchy path (1) — routes to 1 parentless root
- Recurrence plot → Recurrence
Neighborhood in Abstraction Space¶
Recurrence plot sits in a crowded region of the domain-specific corpus (31st percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Iterative Numerical Methods & Stability (7 abstractions)
Nearest neighbors
- Correlation integral — 0.94
- Trend-stationary process — 0.91
- Seasonal subseries plot — 0.91
- Linear time-invariant system — 0.90
- Bartlett's method — 0.90
Computed from structural-signature embeddings · 2026-09-08