Correlation integral¶

The probability, estimated from state pairs, that two independently sampled points on a trajectory or invariant measure lie within a specified distance.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3923
Origin domain: nonlinear time series and dynamical systems
Subdomain: nonlinear time series and dynamical systems

Core Idea¶

Its small-radius scaling estimates correlation dimension, but temporal dependence, finite samples, noise, embedding choices and edge effects require exclusion windows and a defensible scaling region. State vectors are reconstructed or observed, pairwise distances are compared with radius epsilon, qualifying pairs are counted and normalized, and the log slope of the resulting cumulative fraction is examined across scales. 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¶

Correlation integral belongs to nonlinear time series and dynamical systems and is useful where the analyst can specify the typed nonlinear time series and dynamical systems carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the dynamical system or time series, state vectors and embedding, norm, radius, pair exclusions and Theiler window, finite-sample normalization, invariant-measure assumption, scaling interval, correlation dimension estimate, noise floor, saturation and uncertainty are explicit. The scope is broad within that domain but bounded by the need for the dynamical system or time series, state vectors and embedding, norm, radius, pair exclusions and Theiler window, finite-sample normalization, invariant-measure assumption, scaling interval, correlation dimension estimate, noise floor, saturation and uncertainty are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the dynamical system or time series, state vectors and embedding, norm, radius, pair exclusions and Theiler window, finite-sample normalization, invariant-measure assumption, scaling interval, correlation dimension estimate, noise floor, saturation and uncertainty are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

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 Correlation integral. Correlation integral 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 and dynamical systems carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of nonlinear time series and dynamical systems because they reuse the typed nonlinear time series and dynamical systems carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, State vectors are reconstructed or observed, pairwise distances are compared with radius epsilon, qualifying pairs are counted and normalized, and the log slope of the resulting cumulative fraction is examined across scales., and type the carrier, state every parameter and convention in the definition, test that the dynamical system or time series, state vectors and embedding, norm, radius, pair exclusions and Theiler window, finite-sample normalization, invariant-measure assumption, scaling interval, correlation dimension estimate, noise floor, saturation and uncertainty are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Correlation integral Domain-specific

Parents (1) — more general patterns this builds on

Correlation integral is a kind of Measurement Prime

The proposed strict upward parent is prime:measurement.

Hierarchy path (1) — routes to 1 parentless root

Correlation integral → Measurement

Neighborhood in Abstraction Space¶

Correlation integral sits in a crowded region of the domain-specific corpus (26^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Feedback Control & Dynamical Systems (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08