Data-driven ntsa

Model-based ntsa asks for equations; this route only asks for a signal — a scalar record \(x(t)\) sampled at a fixed dt, no equations, no dynamodels.Model, not even a state dimension. That is enough to embed the attractor, estimate its correlation dimension, obtain a Lyapunov exponent from the data itself, and classify the regime — everything ntsa.characterize computes except the two things that genuinely need equations: the full QR Lyapunov spectrum and bifurcation sweeps.

Walkthrough

Take a single measured channel — here Lorenz63's \(x(t)\), generated once and then treated as a plain array with no memory of the equations that produced it:

from dynamodels.physical import Lorenz63
from ntsa.tools import run_long

model = Lorenz63(rho=28.)
t, y, psi = run_long(model, t_run=40.)
x = y[:, 0]   # pretend this is all you have: no equations, no psi

Wrap it in a DataSeries and run the equation-free pipeline:

from ntsa.data import DataSeries

ds = DataSeries(x, dt=model.dt, label='Lorenz63 (measured)')
res = ds.analyze()

res has the same keys as one row of ntsa.characterize, filled in from the signal alone:

Key	Comes from	Meaning
zeta, dim	optimal_lag, false_nearest_neighbours	the delay-embedding parameters
D2	correlation_dimension	Grassberger-Procaccia dimension — the data-only substitute for \(D_{KY}\)
lambda1, lambda1_std	lyapunov.rosenstein_lyapunov	leading Lyapunov exponent, estimated from the embedded trajectory itself
regime, evidence	classify_regime	fixed point / limit cycle / quasiperiodic / chaotic, from the signal's own statistics
gamma	classical_mds	a 3-D embedding of the attractor, for plotting

For this exact record, res['lambda1'] comes out at \(1.22 \pm 0.16\) against the true \(0.906\) — Rosenstein's estimate from a single 40-time-unit record, not the QR-converged value model-based ntsa gets from the equations — and res['regime'] is 'chaotic', recovered with no access to \(\sigma\), \(\rho\), \(\beta\), or the equations, only the measured \(x(t)\). res['spectrum'] stays None: a full spectrum needs equations, which this route does not have.

Draw the same 8-panel row ntsa.characterize draws for a model:

ds.characterize('figs/lorenz63_from_data.pdf')

The same Lorenz63 case as Model-based ntsa's baselines, characterized from its measurements alone.

When only some of the state is measured

Pass the full record as Y (any simultaneous multivariate measurement, not necessarily the physical state) to use it for the MDS panel instead of the scalar delay embedding — e.g. several probes, or the physical state itself if it happens to be available for that panel only:

ds = DataSeries(x, dt=model.dt, Y=psi, label='Lorenz63 (measured, full state for MDS)')

Reference

See the ntsa.data API reference for the full parameter list and every method.