Pincus Index a measure of stochasticity
① Embedding vectors and the similarity test
For a series X = {x1, …, xN} and embedding dimension m, the i-th embedding vector consists of m consecutive samples:
[xi, xi+1, …, xi+m−1]
The distance between two vectors is the Chebyshev distance d = maxk |xi+k − xj+k|; two vectors are “similar” when d < r.
② Approximate entropy, ApEn
ApEn(m, r, N) = −
1N − m
∑
N−mi=1
ln(
Cm+1i(r)Cmi(r)
)
Cmi(r) is the fraction of vectors in the m-dimensional embedding space lying within r of the i-th vector. Smaller ApEn ⇒ more repetitive, more predictable.
③ Removing the tolerance dependence: MaxApEn
MaxApEn = maxr
| ApEn(m+1, r) − ApEn(m, r) |
The paper scans 50 logarithmically spaced tolerances over r ∈ [10−10, 101] and uses the largest change as the entropy level of the series, avoiding the need to pick a single r.
④ Comparison with a random baseline: PI
PI =
MaxApEnoriginalMaxApEnshuffled
（shuffled: median over 100 random permutations）
PI → 1 means the original series is already as irregular as a random shuffle; smaller PI means more regularity and predictability. The paper computes it once for the time series and once for the energy series, then averages the two; the error is propagated from their shuffle standard deviations (± gives the 1σ error):
PI =
PIt + PIE2
±
√¼ (σt2 + σE2)
⑤ References
- S. M. Pincus, Proc. Natl. Acad. Sci. U.S.A. 88, 2297 (1991).
DOI ↗
- A. Delgado-Bonal, Sci. Rep. 9, 12761 (2019).
DOI ↗
Lyapunov Exponent a measure of chaos
① Definition
Two orbits initially separated by |δ0| in phase space separate approximately exponentially:
|δ(t)| ≃ |δ0| eλt
λ > 0 ⇒ chaotic (sensitive to initial conditions); λ < 0 ⇒ stable / periodic.
② Reconstructing phase space from a scalar series
Delay embedding reconstructs the orbit: xi = [xi, xi+τ, …, xi+(m−1)τ]. The paper uses m = 10 (embedding dimension), matrix dimension 4 and τ = 1.
③ The Eckmann et al. (1986) algorithm
- For each orbit point xi, find at least 8 nearest neighbours using the Chebyshev distance;
- Fit the local Jacobian Ti by least squares so that Ti(xj − xi) ≈ xj+s − xi+s;
- QR-decompose Ti Qi−1 successively and accumulate ln |Rkk|;
- Average along the orbit and divide by the step τ and the spacing s to obtain the full Lyapunov spectrum.
The largest entry of the spectrum is the maximum Lyapunov exponent (MLE). The same averaging over the time and energy series is applied:
LE =
λt + λE2
④ Constructing the series
Δti = (ti+1 − ti) × 86400 s,
ΔEi = Ei+1 − Ei
Both series are min–max normalised to [0, 1] before being fed to the algorithms above, so a linear rescaling of the time or energy unit does not affect the result.
⑤ References
- C. Schölzel, Nonlinear measures for dynamical systems, Zenodo (2019).
DOI ↗
- J. P. Eckmann, S. Oliffson Kamphorst, D. Ruelle, and S. Ciliberto, Phys. Rev. A 34, 4971 (1986).
DOI ↗
