Phase reconstruction

Generate logistic samples or paste a uniformly sampled scalar series. Use decimal points, optional scientific notation, and spaces, commas or semicolons as separators. Missing/nonfinite values and literal underflow are rejected; no sample is silently removed. Delay τ is an integer number of samples. A vector contains (x[n],x[n+τ],…,x[n+(m−1)τ]); there are N−(m−1)τ vectors. The display vector needs m≥2 for a two-axis projection. FNN can still first pass its heuristic in m=1, especially for a directly observed one-dimensional map.

Both projection axes use u=(x−min x)/(max x−min x), with a common scale and equal physical units. This preserves geometry, including tiny/large amplitudes; it cannot recover resolution lost in the supplied numbers. Above m=2, choose which two coordinates to project in the plot’s menu. Identical axis selections are avoided. At most 1200 evenly distributed vectors are displayed, including the last; all valid vectors and candidate neighbours remain available to the computation.

FNN uses a common pool of N−mMax·τ vectors with an extra coordinate. Up to the chosen number of evenly spaced anchor points searches every eligible candidate in that pool. W excludes index separations≤W, independently of τ. A neighbour is false if its extra-coordinate separation divided by its previous distance exceeds R_tol, or its extended distance divided by the scalar-series standard deviation exceeds A_tol. If the previous distance is zero, an extra separation is false; identical extended vectors are not. Normalization and robust norms avoid an arbitrary distance epsilon. Ties select the earliest candidate.

The indication is the first tested m with at least 20 evaluated anchors and a rate at most the chosen cutoff, initially 10%. If no such m exists, no dimension is invented. This is a diagnostic heuristic, not a certified embedding dimension, fractal dimension or proof of determinism. Noise, sampling, quantization, temporal dependence and threshold choices affect it. Takens’ theorem has assumptions that a finite, noisy or noninvertible map example need not satisfy. Inspect rates and counts across windows and thresholds.

Inputs, pasted series, unfinished drafts, open panels and selected projection axes are saved in this browser. Reload restores completed counts and reconstructs the delay coordinates without repeating the neighbor search. Imported counts are checked against the source series and eligible-reference bounds, not independently recounted. The dashed line marks the chosen FNN threshold. PNG exports both diagrams and the projection, not the raw series.

TISEAN · False nearest neighbors

The default projection uses the first and last coordinates of the chosen vector.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.