Heterogeneous Temporal Dependence

Summary

Heterogeneous temporal dependence asks how a numeric time series relates to a different temporal data type, especially a categorical event sequence. Examples include telemetry versus alerts, sales versus promotions, temperature versus calendar regimes, traffic versus holidays, and system metrics versus deployments.

The key requirement is to preserve the semantics of each input. Numeric observations, categorical events, exogenous variables, actions, control inputs, and interventions should not be collapsed into arbitrary numeric IDs or bins merely to apply a homogeneous correlation measure.

Core Distinctions

Question	Appropriate interpretation	Common overclaim
Are event labels statistically associated with numeric values?	Mixed-type dependence	“The event caused the value.”
Does event information at time $t$ predict values at $t+\tau$?	Lagged dependence / candidate leading signal	“The lag identifies causal direction.”
Does a calendar label reduce forecast uncertainty?	Context or exogenous-variable relevance	“The calendar variable is an action.”
Does one channel lead another?	Dynamic cross-channel dependence	“The leading channel is an intervention.”
Does an operator-selected change alter future state?	Action/control/intervention effect	Requires an action-conditioned or causal contract.

HUMI Anchor

HUMI is the direct source for mixed dependence between numeric observations and categorical event sequences. It avoids two lossy transformations:

• binning numeric values into symbols;
• assigning event IDs a false numeric geometry.

Instead, it estimates discrete—continuous mutual information while separating one-off numeric values from repeated, quantized values. Its optional event clustering merges labels that induce similar conditional numeric distributions.

flowchart TD
  O["numeric observations"] --> D["mixed-type dependence operator"]
  E["categorical events"] --> D
  D --> Q1["Which events carry target information?"]
  D --> Q2["At which lags is dependence strongest?"]
  D --> Q3["Which calendar contexts explain repetition?"]
  D -. "does not identify by itself" .-> C["causal intervention effect"]

HUMI is useful because quantization and repeated values are ordinary in operational time series, not edge cases. Its evidence is nevertheless incomplete: the continuous/discrete partition depends on empirical multiplicity, its normalization is heuristic, ground-truth experiments are narrow, and the official clustering implementation differs from the paper.

Relation To Event Context

Impact-Driven Event Embeddings and HUMI operate at different layers:

Source	Event representation	Target use	Main boundary
HUMI	categorical labels, optionally clustered by conditional numeric distribution	model-agnostic dependence and covariate ranking	no learned forecasting backbone or causal identification
Impact-Driven Event Embeddings	learned response/timing representation from recurring-event residuals	future-known covariates for zero-shot Chronos-2	synthetic benchmark and known recurring calendar

A natural combined pipeline is to use a mixed-type dependence operator for screening and diagnostics, then learn richer event representations only for event families that carry stable out-of-sample information. The selection split MUST precede downstream evaluation to avoid target leakage.

Relation To Multivariate Lead—Lag Structure

Learning from Leading Indicators estimates local numeric-channel lead—lag relationships inside multivariate forecasting. HUMI instead bridges categorical events and numeric observations. Both show that temporal dependence can be asymmetric, local, and lag-sensitive, but neither turns predictive dependence into causal intervention semantics.

The broader research target should support:

1. numeric—numeric channel dependence;
2. event—numeric dependence;
3. event—event co-occurrence and redundancy;
4. conditional dependence after calendar, topology, and shared-context adjustment;
5. action/control-input-conditioned transition effects.

Measurement Hygiene

A credible heterogeneous-dependence benchmark SHOULD report:

• timestamp alignment and tolerance rules;
• regular versus irregular sampling;
• event duration, overlap, and co-occurrence;
• numeric precision, rounding, and repeated-value rate;
• missingness and censoring;
• event cardinality and rare-event support;
• sample size and effective sample size under autocorrelation;
• independent selection, validation, and downstream evaluation splits;
• estimator bias, variance, confidence intervals, and permutation/null calibration;
• unconditional versus conditional MI;
• lag-search multiplicity correction;
• stability across windows, series, tenants, and regimes;
• whether the variable is context, an exogenous event, an action, a control input, or an intervention.

Dependence versus causality

For an event $E_t$ and later observation $S_{t+\tau }$,

$$I(E_t;S_{t+\tau })>0$$

means statistical dependence. It does not determine whether $E_t$ causes $S_{t+\tau }$, $S$ influences event recording, both share a common cause, or the association is induced by calendar structure, selection, or autocorrelation. Causal use needs an explicit graph or identification assumption, measured confounders, interventions or natural experiments, and an estimand tied to alternative actions.

Marginal versus conditional relevance

Marginal MI can rank two redundant covariates highly even when the second adds no information after the first. For context selection, a stronger target is incremental information such as

$$I(E_j;S\mid E_{-j},C),$$

where $C$ contains relevant calendar, topology, history, or other shared context. Estimating conditional MI reliably is harder, especially for sparse events and high-dimensional conditioning sets, but it is closer to the actual covariate-selection question.

Relation To Foundation TSFM Agenda

Agenda slot	Verdict	Evidence	Missing pieces
Context interface	partially closes	HUMI ranks categorical context; Impact-Driven Event Embeddings learns response-aware future-known covariates.	Needs out-of-sample conditional selection, open-world events, context corruption, and multivariate latent-state targets.
Time representation and event streams	partially closes	HUMI supports event-to-numeric lag scans without event-ID ordering; response embeddings add event-relative timing.	Needs irregular high-rate streams, durations, asynchronous alignment, online updates, and point-process baselines.
Native multivariate encoding	adjacent	Leading-indicator methods expose local numeric cross-channel dependence.	No unified scalable operator over thousands of numeric channels plus event types.
Causal structure	warning	The sources expose useful dependence and lag signals.	No identification from interventions, confounder adjustment, or valid counterfactual estimand.
Control and counterfactuals	insufficient evidence	Events and covariates are observed context.	Needs typed actions/control inputs and action-conditioned next-state evaluation.

Open Questions

• Can mixed-type dependence be estimated online under changing precision, event vocabularies, and latent regimes?
• Which conditional-MI approximation scales to thousands of numeric channels and sparse event types without turning into a learned black box?
• Can event clustering be learned on one set of series and transferred to new targets without target leakage?
• How should effective sample size be estimated when both the numeric series and event occurrences are autocorrelated?
• What null preserves temporal dependence while breaking only event—numeric association?
• Can dependence screens identify event/context channels that improve latent-state accessibility, not only forecast error?
• When does an event become an action or intervention in the data contract, and what additional evidence is required for counterfactual prediction?

Related Pages

• HUMI
• Context-Aided Forecasting
• Causal Time Series
• Time-Series Foundation Models
• Time-Series Benchmark Hygiene
• Foundation Time-Series Model Research Agenda
• Terminology