Embedding

Time-delay embedding is the foundation of EDM. It transforms a scalar time series into a multidimensional representation that reconstructs the dynamics of the underlying system.

Lagged Embedding

Given a time series x(t), the embedding vector at time t is:

[x(t), x(t - tau), x(t - 2*tau), ..., x(t - (E-1)*tau)]

where:

• E (embedding dimension) — the number of lagged coordinates
• tau (time delay) — the spacing between lags

from edmkit.embedding import lagged_embed

embedded = lagged_embed(x, tau=2, e=3)
# Each row is [x(t), x(t-2), x(t-4)]

Choosing E and tau

The scan function performs a grid search over candidate values of E and tau, evaluating prediction skill using cross-validation:

from edmkit.embedding import scan, select

scores = scan(
    x, None,
    E=list(range(1, 11)),
    tau=[1, 2, 3, 5],
)

best_E, best_tau, best_score = select(scores, E=list(range(1, 11)), tau=[1, 2, 3, 5])

scan returns a 3D array of shape (len(E), len(tau), K) where K is the maximum number of cross-validation folds. select picks the combination that maximizes mean - SE (standard error), which favors stable performance across folds.

Customization

scan accepts pluggable components:

• split: Cross-validation strategy (default: sliding_folds)
• predict: Prediction function (default: simplex_projection)
• metric: Evaluation metric (default: mean_rho, i.e., Pearson correlation)
• n_ahead: Prediction horizon (default: 1 step)