Phase-space reconstruction

How coordinates for the attractor are obtained — from a scalar record (delay embedding, with its two free parameters) or from the full state trajectory (MDS).

Delay embedding (Takens)

The attractor is reconstructed from the scalar record by the delay map

\[ \mathbf{Y}_i = \left( x_i,\ x_{i+\zeta},\ \dots,\ x_{i+(d-1)\zeta} \right) \in \mathbb{R}^d , \]

which is an embedding (diffeomorphic to the true attractor) for \(d > 2\,d_A\) with \(d_A\) the attractor's box-counting dimension [Takens 1981]. delay_embed(x, dim, lag) builds \(\mathbf{Y}\); the two free parameters are chosen as follows.

Two delay coordinates of chaotic Lorenz63 (\(\zeta = 8\) samples).

Optimal lag: average mutual information

The AMI between the signal and its \(\zeta\)-shifted copy,

\[ I(\zeta) = \sum_{k,l} p_{kl}(\zeta)\, \ln \frac{p_{kl}(\zeta)}{p_k\, p_l}, \]

is estimated on a histogram (\(p_k\): marginal bin probabilities, \(p_{kl}\): joint probabilities of \(x_i\) falling in bin \(k\) and \(x_{i+\zeta}\) in bin \(l\)). optimal_lag returns the first local minimum of \(I(\zeta)\) — the classic compromise between independence (large \(I\) drop) and staying on the same attractor fold [Fraser & Swinney 1986].

AMI of the Lorenz63 \(x\) record; the first local minimum sets \(\zeta = 8\).

Embedding dimension: false nearest neighbours

For each point \(\mathbf{Y}_i\) in dimension \(d\) with nearest neighbour \(\mathbf{Y}_{n(i)}\) at distance \(R_d(i)\), the neighbour is declared false if unfolding to \(d+1\) stretches it too much [Kennel et al. 1992]:

\[ \frac{\left| x_{i+d\zeta} - x_{n(i)+d\zeta} \right|}{R_d(i)} > R_\mathrm{tol} \quad \text{or} \quad \frac{R_{d+1}(i)}{\sigma_x} > A_\mathrm{tol}, \]

with \(R_\mathrm{tol} = 10\), \(A_\mathrm{tol} = 2\) and \(\sigma_x\) the signal's standard deviation. false_nearest_neighbours returns the smallest \(d\) whose false-neighbour fraction drops below 1%.

False-neighbour fraction for Lorenz63: below the 1% threshold (dashed) at \(d = 3\).

Classical multidimensional scaling

From the pairwise distance matrix \(D_{ij}\) of the (subsampled) state trajectory, the Gram matrix is recovered by double centring,

\[ B = -\tfrac{1}{2}\, H\, D^{(2)}\, H, \qquad H = I - \tfrac{1}{n}\mathbf{1}\mathbf{1}^{\!\top}, \]

where \(D^{(2)}\) holds squared distances. The embedding coordinates are \(\gamma_j = \sqrt{\mu_j}\, v_j\) from the top eigenpairs \((\mu_j, v_j)\) of \(B\) — a distance-preserving 3-D view of the attractor that requires no choice of observable.

First two MDS coordinates \(\gamma_1, \gamma_2\) of the Lorenz63 state trajectory.

References are collected on the theory overview.