2021Bachelor’s thesis
Attractor Reconstruction
Effect of Different Time Delays on the Predicted Results of Attractor Reconstruction
How much of a chaotic system can be recovered from a single observed signal? This thesis uses MATLAB experiments on the Lorenz system to examine how the spacing between observations changes reconstructed states and the forecasts built from them.
Supervised bySimon Lloyd
Selected embedding
Below the 5% FNN criterion for tested τ = 1–60 (§4.2, PDF pp. 17–18).
Neighbours per prediction
Shared by the local-average and weighted predictors (§3.2.2, PDF p. 13; §4.3.1, PDF p. 19).
Cross-variable choice
Preferred for this segment’s x/y/z comparison, not the smallest x error (§4.4, PDF pp. 22–23).
The question and reference data
How does the delay τ change prediction after reconstructing a chaotic attractor from a scalar time series? The experiment uses the Lorenz system and compares predictions against numerical reference values generated with the fourth-order Runge–Kutta method. The report describes a 10,000-value x reference; these are simulated observations, not weather measurements.
SourceCover, PDF p. 1; §1, PDF pp. 5–6; §3.3, PDF pp. 13–16; §4.1, PDF p. 16.
Choosing the embedding dimension
Delay coordinates combine observations separated by τ into a state vector. False nearest neighbours (FNN) test whether apparent neighbours separate when another coordinate is added. Using a 5% false-neighbour criterion, the thesis rejects two dimensions as unstable and selects three: the reported FNN proportions remain below the threshold across τ = 1–60. This is an empirical dimension choice, with subjective thresholds and substantial distance-computation costs.
Source§3 and §3.1, PDF pp. 8–10; §4.2, PDF pp. 16–18, Figs. 6–10.
Two local forecasting methods
The local-average predictor finds 20 neighbours of the latest reconstructed state and averages their next scalar values. The weighted variant instead normalizes Gaussian distance weights, giving closer neighbours more influence, with scale 5. Each predicted value is appended before the next prediction. The reported setup uses a 1,301-point segment after the initial transient and generates 100 recursive predictions. Delays up to 60 are compared using cumulative absolute differences from the numerical reference.
Source§3.2.1–3.2.2, PDF pp. 11–13; §4.3–4.3.2, PDF pp. 18–22; appendix, PDF pp. 27–29.
Why τ = 23 was preferred
For x, τ = 34 has lower reported cumulative error than τ = 23: 155.094 versus 165.717 for local averaging, and 158.060 versus 166.256 with weights. Weighting therefore does not automatically improve these comparisons. The preference for 23 comes from the cross-variable check: its curves also fit y and z reasonably in the selected segment, whereas 34 performs poorly on y. This supports a scoped practical choice, not a universal optimum.
Source§4.3.1, PDF p. 20, Figs. 15–18; §4.3.2, PDF pp. 21–22, Figs. 22–24; §4.4, PDF pp. 22–23, Figs. 25–28.
What the experiment cannot establish
The thesis notes limited data and computing resources, difficulty predicting peaks and valleys, and a changing number of usable reconstruction points as τ varies. Its y/z check is presented through plots, rather than a repeated statistical benchmark. The study does not establish transfer to noisy observations or other chaotic systems, and supplies no theoretical proof that 23 is optimal. Alternative predictors and more systematic parameter selection remain open directions.
Source§3.1–3.2, PDF pp. 10–13; §4.3.1, PDF p. 20; §4.4–4.5, PDF pp. 22–24.