History-Conditioned Local Projector

• History-conditioned local projector is a data-driven operator that infers fine-grained transition statistics in observed Markov trajectories while accounting for hidden memory.
• The methodology empirically constructs and normalizes history-conditioned transition histograms, then uses spectral analysis of eigenvalues to quantify memory timescales.
• This approach enables the detection of deviations from the Markov property and guides state splitting to better represent hidden dynamics in complex systems.

A history-conditioned local projector is a data-driven operator on observed discrete-time Markov trajectories, which infers fine-grained transition statistics between observed (coarse-grained) states while accounting for the memory effects induced by hidden or unobservable degrees of freedom. Originally introduced in the context of Markov-state holography, this analysis allows one to empirically assess local deviations from the Markov property, detect evidence of hidden-state topology, and quantify the timescale of memory due to hidden-path decorrelations from projected high-dimensional dynamics (Zhao et al., 14 Mar 2025).

1. Definitions and Notational Framework

Let the full (microscopic) state space be $$S_\mathrm{full} = S_\mathrm{obs} \times S_\mathrm{hidden}$$, where $S_\mathrm{obs} = \{1,2,\dots,n\}$ is the space of directly observed (“lumped”) states, and $S_\mathrm{hidden}(i)$ denotes the set of microstates projected onto observed state $i$: $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$.

A length-$k$ observation history at time $t$ is $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$. For each observed transition $j \to i$ at times $t \to t+1$, and immediately preceding history $S_\mathrm{obs} = \{1,2,\dots,n\}$0, define

• $S_\mathrm{obs} = \{1,2,\dots,n\}$1: number of observed $S_\mathrm{obs} = \{1,2,\dots,n\}$2 transitions with history $S_\mathrm{obs} = \{1,2,\dots,n\}$3,
• $S_\mathrm{obs} = \{1,2,\dots,n\}$4,
• Empirical history-conditioned transition probability:

$S_\mathrm{obs} = \{1,2,\dots,n\}$5

These transition probabilities form the entries of the history-conditioned local projector.

2. Mathematical Construction

For each ordered transition $S_\mathrm{obs} = \{1,2,\dots,n\}$6 and history $S_\mathrm{obs} = \{1,2,\dots,n\}$7, the local projector-matrix entry is defined as

$S_\mathrm{obs} = \{1,2,\dots,n\}$8

Alternatively, in model-based terms (where $S_\mathrm{obs} = \{1,2,\dots,n\}$9 are microstates reached upon entering $S_\mathrm{hidden}(i)$0):

$S_\mathrm{hidden}(i)$1

Here $S_\mathrm{hidden}(i)$2 is a path-sum over hidden trajectories compatible with $S_\mathrm{hidden}(i)$3.

Given the microscopic splitting-probability matrix $S_\mathrm{hidden}(i)$4 and the sub-block $S_\mathrm{hidden}(i)$5 (within $S_\mathrm{hidden}(i)$6), the probability for the first exit from $S_\mathrm{hidden}(i)$7 is:

$S_\mathrm{hidden}(i)$8

and the history-projected transition probability becomes:

$S_\mathrm{hidden}(i)$9

Fixing $i$0, the matrix $i$1 with entries $i$2 can be analyzed spectrally. The eigenvalues satisfy $i$3, $i$4.

3. Data-Driven Estimation and Algorithm

Empirical construction proceeds by traversing a time series $i$5:

$S_\mathrm{obs} = \{1,2,\dots,n\}$07

Normalize each slice so that $i$6. The maximal $i$7 is increased until transition histograms converge, measured by the total-variation distance

$i$8

declaring convergence when $i$9 (e.g., $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$0).

4. Memory Quantification and Spectral Analysis

The subleading eigenvalue $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$1 of $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$2 determines the local memory timescale:

$$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$3

As $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$4 increases, $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$5 and $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$6, characterizing the number of steps over which hidden-path correlations decay. Large $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$7 (i.e., long $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$8) indicates slow decay and persistent hidden-path memory, signifying insufficient Markovianity at the observed level.

5. Canonical Example: Three-State System with Hidden Path

Consider observed states $$S_\mathrm{hidden}(i)=\{\alpha : \pi_\mathrm{obs}(\alpha)=i\}$$9, $k$0, $k$1, where $k$2 is actually two hidden microstates $k$3. The transition scheme is $k$4, $k$5, $k$6, $k$7, $k$8. The microscopic splitting probabilities are $k$9, $t$0, $t$1 to $t$2 each with $t$3, $t$4.

For $t$5 (no history), $t$6. For $t$7 (history of one step), there are two histories, $t$8 (from $t$9) and $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$0 (from $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$1):

$h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$2

where $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$3 are arrival fractions at $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$4 from $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$5 or $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$6. Thus, $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$7 is a diagonal matrix with $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$8, and eigenvalues $h_k = (s_{t-k+1}, \dots, s_t) \in S_\mathrm{obs}^k$9, $j \to i$0. The timescale is then $j \to i$1.

As $j \to i$2, path weights $j \to i$3 approach their stationary distribution, $j \to i$4, and $j \to i$5.

6. Testing Markov Property and Inferring Hidden Structure

The local history-conditioned projector provides a direct test for hidden memory. If, for state $j \to i$6, $j \to i$7 becomes independent of $j \to i$8 for all $j \to i$9, then $t \to t+1$0 is Markov of order $t \to t+1$1 locally; any persistent $t \to t+1$2-dependence reflects hidden-path memory or missing microstates.

If $t \to t+1$3 shows multiple bars or more than one significant eigenvalue, one proposes splitting $t \to t+1$4 into $t \to t+1$5, assigning rates so that $t \to t+1$6 match these bars. Recomputing $t \to t+1$7 with the relump validates the split if Markovianity improves.

7. Practical Considerations and Limitations

Building all histograms up to maximal history $t \to t+1$8 has computational complexity $t \to t+1$9 and storage cost $S_\mathrm{obs} = \{1,2,\dots,n\}$00 in the worst case. Reliable estimation of $S_\mathrm{obs} = \{1,2,\dots,n\}$01 generally requires $S_\mathrm{obs} = \{1,2,\dots,n\}$02 per history; since history classes grow as $S_\mathrm{obs} = \{1,2,\dots,n\}$03, $S_\mathrm{obs} = \{1,2,\dots,n\}$04 must be kept modest or histories coarsened by binning. Severe undersampling arises exponentially with $S_\mathrm{obs} = \{1,2,\dots,n\}$05. Convergence proofs require at least one observable Markov state; application to continuous observables depends on bin width $S_\mathrm{obs} = \{1,2,\dots,n\}$06 choices.

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For detailed derivations and further implementation guidance, see (Zhao et al., 14 Mar 2025).