Unit Perception Tests and Empirical Transition Structures

Unit Perception Tests and Empirical Transition Structures

1. Unit Perception Test (UPT)

A Unit Perception Test (UPT) is a single, local test of synchrony between two timeline-referenced events.

• We have two time-indexed streams, A_t and B_t, defined over a shared timeline index t (the ``tNdx`` in the Timeline Paradigm).
• A UPT evaluates a synchrony predicate S at a specific pair of time points, for example:
  • <math>S(t) = \mathbf{1}\{A_t \in C_A \wedge B_t \in C_B\}</math>
  • where C_A and C_B are value categories and \mathbf{1\{·\}} is 1 if the condition holds and 0 otherwise.
• Each UPT is therefore a binary observation: "synchrony detected" (1) or "no synchrony detected" (0).

Because of noise and limited data, any single UPT is unreliable and should be treated as a noisy measurement, not as ground truth.

2. From UPTs to State Space

To move from isolated tests to a dynamical view, we define a state space over which the process evolves.

Examples of possible state definitions:

• Binary synchrony state:
  • <math>X_t \in \{S, \neg S\}</math>, where X_t = S if S(t)=1, else \neg S.
• Multi-valued state including lag or pattern class, for example:
  • <math>X_t \in \{\text{no synchrony}, \text{lead A}, \text{lead B}, \text{locked}\}</math>.

The Timeline Paradigm provides the indexing substrate: each state X_t is attached to a specific timeline index ``tNdx``, and can be derived from the underlying category structure (rank array, category bins, and hOccurs pointers).

3. Markovian Iteration and Empirical Transition Matrices

Once we have a sequence of states (X_t), we can treat it as a Markov process and empirically estimate how often the system moves from one state to another.

For a finite state set S = {s_1, …, s_m}:

• We observe a long sequence:
  • <math>X_1, X_2, \dots, X_T</math>
• For each ordered pair (s_i, s_j), we count how often we see a one-step transition:
  • <math>N_{ij} = \#\{t \in \{1, \dots, T-1\} : X_t = s_i,\, X_{t+1} = s_j\}</math>
• The empirical transition probability from s_i to s_j is:
  • <math>\hat{P}_{ij} = \frac{N_{ij}}{\sum_k N_{ik}}</math>

Collecting all \hat{P}_{ij} into a matrix \hat{P} gives the empirical transition matrix (also called the state transition matrix).

4. Averaging Away Randomity

Why this "averages away" noise in individual UPTs:

• Each UPT contributes a single bit of information (synchrony vs no synchrony) at one time step.
• Random false detections tend to be uncorrelated over time and across many trials.
• True dynamical relationships (for example, persistent synchrony, characteristic lags, stereotyped transitions) manifest as stable patterns in the transition counts N_{ij}.

As T grows:

• The empirical matrix \hat{P} converges (under standard assumptions) toward the underlying transition probabilities.
• The Markovian iteration (repeating UPTs across many steps) yields a perceptive envelope:
  • a stable map of which states tend to follow which, with what probabilities;
  • robust to individual noisy UPTs because it aggregates many of them.

5. Relation to the Timeline Paradigm

Mapping back to the Timeline Paradigm Matrix implementation:

• The append-only timeline index ``tNdx`` defines the order of observations.
• Category bins (the cats() ragged arrays) and the rank structure provide efficient access to:
  • which times t are in category C_A (stream A) or C_B (stream B);
  • where synchrony predicates S(t) hold.
• A UPT is an evaluation of S(t) (or S(t, t') if lags are allowed) at a specific timeline index (or pair).
• The Markov chain (X_t) is built from these UPT results, and the empirical transition matrix \hat{P} is the quantitative perceptive envelope over the dynamic under study.