Measured Data¶

Delay-coordinate vectors and ordinal patterns are routinely built from measured time series. This page states which theorems proved here concern those constructions, and what they do not establish.

All results below are exact statements about mathematical models. No theorem here verifies that a particular sensor, data set, neural network or residual stream satisfies their hypotheses, and none addresses noise, finite precision or statistical estimation.

Delay vectors¶

• Exact reconstruction on a model. A delay map is injective iff the observation separates orbits at that window (delayEmbedding_injective_iff_separatesOrbits). An injective delay map identifies the state space with its delay image and transports the dynamics there by conjugacy (reconstructedDynamics_eq_conj). For compact state spaces with continuous dynamics and observation, that identification is a homeomorphism onto the image (delayHomeomorph). The reconstructed dynamics is invertible only under the corresponding invertibility condition on the original dynamics.
• Window length on a finite model. When a finite separating window exists, the least one is the separating horizon and uses at most \(N - 1\) observations on \(N \ge 1\) states (separatingHorizon_le_card_sub_one); some pairs of states may never be distinguished, and then no window separates. The horizon can be computed exactly from exact finite data (horizonSearch_eq_separating_iff). A finite sample of a continuous system is not a finite state space.
• Smooth models. For a compact manifold of dimension \(d\) and an injective \(C^2\) map with injective differentials satisfying the periodic-point conditions (countably many points of period at most \(4d\), observability at points of period at most \(2d\)), the observations giving a \(C^2\) embedding with \(2d + 1\) delays form an open dense set in the \(C^2\) topology (isOpen_and_dense_setOf_isContMDiffEmbedding_delayEmbedding). This does not say that a given observation works, nor that a given map satisfies the conditions. Takens' theorem for generic pairs of a \(C^2\) diffeomorphism and a \(C^2\) observation (isOpen_and_dense_setOf_isContMDiffEmbedding_delayEmbedding_pair) says that good pairs are open and dense; it does not certify a particular pair either.

Ordinal patterns and permutation entropy¶

• Invariance. Ordinal codes and pattern entropy are unchanged by strictly increasing transformations of the observation, and relabeled (entropy unchanged on tie-free segments) by strictly decreasing ones (ordinalDelayMap_comp_strictMono, patternEntropy_comp_strictAnti). Monotone but not strictly monotone transformations can create ties: tied windows leave the tie-free ordinalDelayMap, while the empirical statistics (observedPatterns, patternEntropy) still assign a stable-sort code, with no general invariance guarantee for such a transformation.
• Bounds. For \(N > 0\), the empirical pattern entropy of \(N\) windows of length \(d\) is at most \(\log \min(d!, N)\), and at most the log of the minimal period on a periodic orbit (patternEntropy_le_log_min, patternEntropy_le_log_min_period).
• Compression, not reconstruction. An ordinal code takes at most \(d!\) values, so it cannot distinguish more than \(d!\) states (not_injective_ordinalDelayMap_of_factorial_lt). A quantity can be computed from the code iff it is constant on the code's fibers (exists_factor_iff); whether a quantity of interest has that property is a separate, target-specific question.

The relation between empirical pattern entropy and topological or metric entropy of the underlying dynamics is not formalized here.