Autoregressive Dependency Map¶
State-history model — instantiates Lag Structure and Feedback Loop Identification
A representation of how much the current state depends on one or more earlier values of the same state variable.
An Autoregressive Dependency Map answers one narrow question: how much of today's value of a variable is explained by its own recent values? It regresses a series on lagged copies of itself — level today on level yesterday, the day before, and so on — and reports which of those own-lags carry weight and how fast that weight fades. Its defining move, the one that separates it from every cross-variable mechanism in this archetype, is that the predictors and the target are the same variable at different times: it models memory, momentum, and persistence intrinsic to the series, not the influence of some outside driver. The output is a compact statement of the series' own inertia — "the last two days matter, and their pull halves every few days."
Example¶
A water utility manages a reservoir and wants to forecast tomorrow's level so it can schedule releases. Rainfall and demand are real drivers, but before reaching for them the analyst asks how much the level simply carries itself: a full reservoir tends to stay fullish, a draining one keeps draining. She traces daily levels back three years, then regresses each day's level on the previous one, two, and three days. The one-day-back term dominates; the two-day term adds a little; the three-day term is noise. The fitted map says the level today is roughly its own recent history with a decaying tail — an AR(2) structure whose influence shrinks geometrically. That inertia alone forecasts the next few days well and, just as usefully, tells operators how long a release takes to "wash out" of the level before it stops echoing forward.
How it works¶
- Trace the series against itself. Build a table where each row is a time point and the columns are that point's own value at lag 1, 2, 3, … — a purely self-referential history trace.
- Estimate the order. Decide how many own-lags genuinely matter (the order p), typically by reading the partial autocorrelation function,[1] which isolates each lag's contribution after the shorter lags are accounted for.
- Fit and read the decay. Estimate the lag coefficients and characterize how influence fades — a stationary autoregression decays geometrically, so the map summarizes memory as a half-life rather than a fixed cutoff.
- Archive the lagged features. Persist the constructed lag columns so downstream forecasting or monitoring reuses the same self-history representation instead of rebuilding it.
Tuning parameters¶
- Order (p) — how many own-lags to include; more captures longer memory but risks fitting noise and multicollinearity among adjacent lags.
- Persistence / near-unit-root guard — how close the dominant coefficient sits to 1; a coefficient near unity means the series barely forgets, and small estimation errors then explode the implied memory.
- Differencing — whether to model levels or changes; differencing tames a trending, non-stationary series but discards level information.
- Window length — how much history to fit on; longer stabilizes estimates but assumes the memory structure hasn't shifted over that span.
- Feature-archive granularity — how many lag columns to persist for reuse; richer archives speed downstream work but bloat storage and tempt over-fitting.
When it helps, and when it misleads¶
Its strength is isolating self-persistence cleanly and cheaply: momentum, carryover, and slow decay that a snapshot ignores, expressed as a small set of own-lag weights that forecast the near future and reveal how long a shock keeps echoing in the series.
Its failure mode is that own-lag structure is easily mistaken for mechanism. A high one-day coefficient can be pure inertia, or it can be the fingerprint of an omitted outside driver whose own smoothness leaks into the series — an autoregression will happily absorb that as "memory" it does not have. Worse, a non-stationary series (a trend, a unit root) manufactures huge, spurious own-lag correlation until it is differenced.[n1] The classic misuse is reading the AR(1) coefficient as a causal loop — as if the past causes the present rather than merely predicting it. The guarding discipline is to check stationarity first, choose the order from the partial autocorrelation rather than by eye, and treat a fitted own-lag as predictive memory, never as evidence that no external driver is at work.
How it implements the components¶
state_variable_history_trace— the self-lagged table is the trace: the variable recorded against its own earlier values.dependency_order_estimation— selecting the order p is exactly the estimate of how many prior states the present depends on.memory_decay_assumption— the geometric fade of the lag coefficients encodes how fast the past is forgotten.history_feature_archive— the persisted lag columns are the reusable archive of history features.
This map does not model any second variable's influence: it does not implement cross_lag_variable_set or delay_distribution_profile — those belong to Distributed Lag Model, which spreads an external driver's influence across lags, whereas this map regresses a series only on its own history. It also does not implement feedback_loop_linkage_map — that is Causal Loop Diagram with Delay Marks.
Related¶
- Instantiates: Lag Structure and Feedback Loop Identification — it supplies the self-memory layer of the temporal-dependency map.
- Sibling mechanisms: Distributed Lag Model · Cross-Lagged Dependency Review · Impulse Response Trace · Causal Loop Diagram with Delay Marks · Recurrence Interval Histogram
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: The mechanism constructs lagged self-history, estimates the autoregressive order and coefficients, and characterizes how prior state influence decays, so its operative form is time-series modeling.
Nearest alternative: Representation, Specification & Plan — A dependency map presents the result, but fitting and inferring the lag structure are the mechanism's defining contribution.
Review outcome: Adjudicated after independent review; high confidence.
Origin Attribution¶
Primary origin: Statistics & Experimental Design
Origin pattern: Single lineage
Present-day reach: Specialized
Rationale: Time-series statistics maps a variable's dependence on its own lags through autocorrelation and partial autocorrelation.
Related originating lineages:
- Economics & Finance — Econometrics made autoregressive diagnostics routine for economic sequences.
- Mathematics — Stochastic-process theory formalizes stationarity and lag dependence.
Review resolution: Statistics and experimental design are the agreed primary lineage. Mathematical time-series theory and econometrics materially shaped lag-dependence representation, which remains a canonical specialized statistical artifact.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The partial autocorrelation function reports each lag's correlation with the present after removing the shorter lags' contribution; a partial autocorrelation that cuts off sharply after lag p is the standard signature for choosing an AR order of p. It is only meaningful once the series is made stationary. ↩
References¶
[1] Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. Time Series Analysis: Forecasting and Control (5th ed.). John Wiley & Sons (2015). Defines partial autocorrelation after shorter lags are accounted for and uses its cutoff to identify autoregressive order p. registry ↩