Out-of-Domain-Augmented Point Processes

• The paper introduces a novel framework that integrates out-of-domain interventions into multivariate temporal point processes using a causal definition of average treatment effect.
• It models event sequences by embedding intervention information into a Transformer-based neural point process, enabling explicit treatment assignments that modify causal relations.
• Empirical results demonstrate significantly improved prediction accuracy and unbiased ATE estimation across synthetic simulations and real-world applications such as predictive maintenance and diabetes settings.

Searching arXiv for the target paper and closely related works mentioned in the provided data. arXiv search query: (Zinat et al., 14 Jul 2025) Out-of-domain-augmented point processes are multivariate temporal point processes whose dynamics are modified by exogenous, out-of-domain interventions that occur on the same timeline as the in-domain events of primary interest. In the formulation introduced in "Uncovering Causal Relation Shifts in Event Sequences under Out-of-Domain Interventions" (Zinat et al., 14 Jul 2025), these interventions are neither absorbed into a stationary generative mechanism nor treated merely as ordinary covariates. Instead, they are modeled as explicit treatment assignments that can alter the causal relations among event types, including whether a putative cause is or is not a direct cause of an outcome under different intervention states. The resulting framework combines a causal definition of average treatment effect (ATE) for temporal point processes, an unbiased inverse-probability-weighted estimator, and a Transformer-based neural point process that encodes intervention information directly into the intensity model.

1. Conceptual definition and problem setting

The basic data object is a collection of multivariate event sequences,

$\{\mathbf{s}_1,\ldots,\mathbf{s}_n\},$

where each sequence $\mathbf{s}_k$ contains timestamped typed events

$(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$

with $t_{k,i}$ the event time, $e_{k,i}\in\mathbb{E}$ the event type, $L_k$ the sequence length, and $T$ the observation horizon (Zinat et al., 14 Jul 2025). The event types are partitioned into cause events $c$, outcome events $o$, out-of-domain interventions $v$, and other measured events $\mathbf{s}_k$0.

A temporal point process is characterized by a conditional intensity function (CIF) $\mathbf{s}_k$1, where $\mathbf{s}_k$2 denotes the history before time $\mathbf{s}_k$3. In the Hawkes-process form used for exposition, the outcome intensity is

$\mathbf{s}_k$4

where $\mathbf{s}_k$5 is the baseline intensity and $\mathbf{s}_k$6 is the excitation kernel (Zinat et al., 14 Jul 2025). The paper’s central claim is that, in real-world environments, the functional dependence embodied by $\mathbf{s}_k$7 and $\mathbf{s}_k$8 may change when exogenous interventions occur.

The defining feature of an out-of-domain-augmented point process is therefore that the intensity and causal structure depend jointly on in-domain event history and on the configuration of out-of-domain interventions active in a proximal time window. The source description gives two canonical examples. In a diabetes setting, insulin injections are treated as out-of-domain interventions when the goal is to analyze the causal effect of meal events on blood glucose outcomes. In predictive maintenance, proactive maintenance actions are out-of-domain interventions relative to the failure process (Zinat et al., 14 Jul 2025).

This setup is motivated by causal relation shift: the causal effect of $\mathbf{s}_k$9 on $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$0 changes with the intervention state $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$1. A representative example is meal intake and blood glucose. Under $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$2, a meal can increase glucose; under $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$3, insulin can reduce, negate, or reverse that effect. This suggests that a single stationary point-process mechanism is inadequate when exogenous interventions alter the operative causal regime.

2. Causal formalization beyond i.i.d. settings

The framework extends Rubin’s potential outcomes formalism to continuous-time event sequences with two binary treatment dimensions defined over a proximal history window $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$4 (Zinat et al., 14 Jul 2025). At time $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$5,

• $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$6 indicates whether the cause event occurred at least once in $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$7,
• $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$8 indicates whether the out-of-domain intervention occurred at least once in $(e_{k,i}, t_{k,i}), \quad i=1,\dots,L_k,$9,
• $t_{k,i}$0 is a binary vector for other observed events in the same window.

The potential outcome is not a scalar response but the counterfactual outcome intensity,

$t_{k,i}$1

namely the CIF of outcome event $t_{k,i}$2 under treatment assignment $t_{k,i}$3 (Zinat et al., 14 Jul 2025). This formulation departs from the classical i.i.d. setting in three ways emphasized by the source: the outcome is latent, time-indexed, and defined under two binary treatment dimensions.

For a fixed intervention state $t_{k,i}$4, the ATE of the cause on the outcome intensity is defined as

$t_{k,i}$5

Using the abbreviation $t_{k,i}$6, the quantity is written as $t_{k,i}$7 (Zinat et al., 14 Jul 2025). The interpretation is a per-unit-time average treatment effect of cause occurrence on the outcome CIF, conditional on the intervention level. This is explicitly not an i.i.d. sample average; it is a process-level average over continuous time.

The paper links this ATE to process independence in multivariate point processes, following Didelez (2005) as cited in the source description. A set of events is a direct cause of event $t_{k,i}$8 if the CIF of $t_{k,i}$9 functionally depends on their history and is process independent of all other events given that set, excluding degenerate cases. Within this framework, Theorem 1 states that if $e_{k,i}\in\mathbb{E}$0 is a direct cause of $e_{k,i}\in\mathbb{E}$1 when $e_{k,i}\in\mathbb{E}$2, and not a direct cause when $e_{k,i}\in\mathbb{E}$3, then

$e_{k,i}\in\mathbb{E}$4

(Zinat et al., 14 Jul 2025). The notion of causal relation shift is therefore operationalized as variation in $e_{k,i}\in\mathbb{E}$5 across intervention states.

3. Identification assumptions and unbiased ATE estimation

To address non-random cause occurrence and confounding, the framework introduces a joint propensity score,

$e_{k,i}\in\mathbb{E}$6

which represents the probability of being treated via the cause under intervention level $e_{k,i}\in\mathbb{E}$7, conditional on proximal covariate history (Zinat et al., 14 Jul 2025).

The estimator is derived under three assumptions adapted to time windows within a point process. The first is SUTVA in the temporal setting: for each assignment pair $e_{k,i}\in\mathbb{E}$8, there is a single version of the population outcome $e_{k,i}\in\mathbb{E}$9, and assignment in one time window does not affect outcomes of other windows. The second is unconfoundedness: $L_k$0 The third is overlap: there exists $L_k$1 such that

$L_k$2

All three are standard causal assumptions, but here they are reformulated at the level of continuous-time process windows rather than independent units (Zinat et al., 14 Jul 2025).

The inverse-probability weight is

$L_k$3

and the ATE estimator is

$L_k$4

Operationally, the expectation is approximated via empirical averaging over sequences and time grids, while $L_k$5 is supplied by the fitted neural point-process model (Zinat et al., 14 Jul 2025).

Under the three assumptions, Theorem 2 states

$L_k$6

so the estimator is unbiased (Zinat et al., 14 Jul 2025). The source also gives a nonparametric estimator of the propensity score by event duration ratio, interpreted as the proportion of time during which a given covariate pattern is paired with $L_k$7. A plausible implication is that, in this framework, the quality of ATE estimation depends jointly on valid identification assumptions and on accurate recovery of the latent intensity surface.

4. Intensity modeling under intervention-augmented dynamics

The paper adopts a Hawkes-like generative viewpoint in simulation and a neural point-process parameterization in estimation. In the synthetic setup, out-of-domain interventions actively modify Hawkes parameters on the fly (Zinat et al., 14 Jul 2025). Three intervention modes are described.

Under baseline intervention, the outcome baseline intensity is changed, $L_k$8, when both $L_k$9 and $T$0. Under cause intervention, the cause-to-outcome kernel is altered, $T$1. Under covariate intervention, a covariate-to-outcome kernel such as $T$2 is modified. In this sense, the CIF

$T$3

becomes piecewise dependent on intervention state (Zinat et al., 14 Jul 2025).

The neural model does not hard-code those parametric switches. Instead, it learns $T$4 as a function of event-type embeddings, time encodings, and intervention embeddings. This is a decisive feature of out-of-domain-augmented point processes in the paper’s sense: augmentation occurs simultaneously at the level of data generation, model input representation, causal estimand definition, and empirical evaluation.

The source description emphasizes that out-of-domain interventions are not merely additive predictors. They are modeled as alternative treatment assignments that change the structural causal relation itself. This distinction matters because a model that only conditions on intervention indicators without allowing intervention-dependent changes in relational structure may fit predictive marginals while failing to capture causal relation shifts.

5. Transformer-based neural architecture

The proposed model is a hybrid Transformer-CNN point process that outputs intensities $T$5 for outcome event types (Zinat et al., 14 Jul 2025). Each event position receives three input components: an event type embedding $T$6, a relative time encoding $T$7 based on the inter-event time $T$8, and an out-of-domain intervention embedding derived from a binary vector over intervention types.

For intervention encoding, the model constructs a binary vector $T$9 whose $c$0-th entry is $c$1 if intervention $c$2 occurred at least once in $c$3, and $c$4 otherwise. This vector is projected into $c$5 (Zinat et al., 14 Jul 2025). Event and intervention embeddings are then fused through a learned weighted-combination module, after which temporal encoding is added to form the initial intervention-aware representation $c$6.

The encoder is a standard self-attentive Transformer with query, key, and value projections, multi-head attention, residual connections, layer normalization, and a position-wise feed-forward network. The reported hyperparameters are 2 encoder layers, 2 attention heads, hidden dimension $c$7, embedding dimension $c$8, feed-forward inner dimension $c$9, and dropout $o$0 (Zinat et al., 14 Jul 2025). The role of self-attention is to learn how intervention presence or absence changes long-range temporal dependencies.

After the Transformer encoder, a convolutional layer with kernel size $o$1, padding $o$2, ReLU activation, and number of filters equal to the number of basis functions captures local patterns (Zinat et al., 14 Jul 2025). The intensity is then parameterized by basis functions consisting of 1 unity basis and 7 Gaussian bases with dyadic spacing. For event type $o$3 at time $o$4,

$o$5

where $o$6 are basis weights and $o$7 are the basis functions. The model works in $o$8 space for numerical stability before exponentiation (Zinat et al., 14 Jul 2025).

Training combines point-process likelihood, next-event classification, and regularization. The negative log-likelihood includes the sum of log intensities at observed event times and the integrated intensity over the observation window; a cross-entropy term improves next-event type prediction; and an $o$9 penalty is applied to basis weights. Optimization uses Adam, a cyclic learning rate, gradient clipping, and model selection by 5-fold CV on NLL (Zinat et al., 14 Jul 2025). ATE estimation is performed post hoc from the learned intensity.

6. Empirical behavior, limitations, and broader context

The simulation study extends the Tick Hawkes simulator to dynamically inject out-of-domain interventions that modify $v$0 and $v$1 depending on cause and intervention windows (Zinat et al., 14 Jul 2025). The three synthetic scenarios are No OOD, Baseline OOD, and All Impact OOD. The reported configuration uses 1000 sequences per scenario, sequence length approximately Poisson with mean $v$2, 30 event types per sequence, and about 30 interventions per sequence in the Baseline and All Impact settings.

ATE estimation is evaluated by Bias, Variance, and MSE; process fitting is evaluated by NLL, RMSE, and MAE of predicted outcome occurrence times (Zinat et al., 14 Jul 2025). Against CAUSE, the paper reports substantially lower bias under both Baseline and All Impact interventions, slightly higher variance in some cases, and overall lower MSE. For process fitting, it reports drastically lower NLL, RMSE, and MAE in all OOD regimes, including No OOD.

Two real-world domains are used. In predictive maintenance on the Azure dataset, discretized sensor readings act as causes or covariates, reactive failures are outcomes, and proactive maintenance actions are out-of-domain interventions. The reported result is a significant reduction in RMSE and MAE, approximately $v$3–$v$4, with similar NLL relative to CAUSE (Zinat et al., 14 Jul 2025). In diabetes data, meals, insulin injections, activity levels, hypoglycemia symptoms, and glucose measurements are modeled as events; insulin or activity serves as the out-of-domain intervention depending on the causal query. The paper states that ATE-based conclusions, such as insulin offsetting meal-induced glucose rise, agree with medical literature, and that process-fitting metrics improve under OOD interventions (Zinat et al., 14 Jul 2025).

Several limitations are explicit. Both cause and intervention are simplified to binary “occurs at least once in window” indicators, which neglect dosage, frequency, and duration. Unconfoundedness may fail if interventions open backdoor paths or are themselves affected by unmeasured confounders. Scalability issues arise from combinatorial growth in intervention configurations, data sparsity for propensity estimation, linear memory growth from basis functions per event type, and the Transformer’s quadratic complexity in sequence length. The framework also assumes interventions are observed and timestamped; latent out-of-domain factors remain outside scope (Zinat et al., 14 Jul 2025).

The source situates the topic at the intersection of Hawkes processes, neural temporal point processes, Transformer Hawkes, causal inference in event sequences, counterfactual temporal point-process models, causal Transformers, and out-of-domain generalization work such as WILDS (Zinat et al., 14 Jul 2025). A common misconception is that any temporal covariate-augmented point process is already out-of-domain-augmented in this stronger sense. The formulation here is narrower: the augmentation is specifically about exogenous interventions that can change causal relations themselves. Another misconception is that intervention-aware embeddings imply zero-shot robustness to completely unseen interventions. The paper does not claim that; it presumes interventions are observed and encoded. Within those bounds, the framework presents out-of-domain-augmented point processes as families of stochastic processes indexed by intervention state, with causal relation shift quantified by intervention-conditional ATEs.