Behavioral Profiling Ensemble (BPE)

• BPE is a framework that uses engineered time aggregation features to capture evolving entity behavior while strictly preventing information leakage.
• It employs various window schemes—trailing, event-count, gap, and bucketized—to compute statistics like log-impression counts and smoothed event rates for robust predictive modeling.
• The ensemble approach is applied in CTR prediction, customer churn forecasting, and real-time anomaly detection, emphasizing both high statistical efficiency and online-safe feature construction.

Time aggregation features for XGBoost are engineered temporal statistics that summarize historical activity up to a reference timestamp for use as predictive covariates within tree-based models. The principal motivation is to capture evolving entity behavior (such as user, device, or session state) while strictly respecting causality—ensuring that no information from the prediction timepoint or future is included in the feature computation. This methodology is central to production systems in domains including click-through rate (CTR) prediction, customer churn forecasting, event-detection in streams, and device monitoring, enabling XGBoost to exploit structured entity histories with high statistical efficiency while obeying strict out-of-time and no-leakage protocols (Pinchuk, 15 Jan 2026, Gregory, 2018).

1. Formal Definitions and Typology of Time Aggregation Features

Let $h\in\mathbb{N}$ denote the integer-valued time index (typically hours or days), and $e$ an entity key (such as device, user, or site) with value $v$. For an arbitrary event type, define the event count over the interval $[a,b)$ as

$$I_{v,[a,b)} = \sum_{t=a}^{b-1} \mathbf{1}\left[\text{event at time } t \text{ for entity } e=v\right].$$

Analogously, for positive-labeled events (e.g., clicks), define $C_{v,[a,b)}$.

The primary classes of time aggregation feature are:

• Trailing (moving) windows: Use the interval $[h-w,h)$, summarizing the $w$ most recent time units prior to $h$.
• Event-count windows: Use the $N$ most recent events for the entity, regardless of time elapsed.
• Gap windows: Introduce an exclusion gap (e.g., $[h-(w+1),h-1)$) to separate label and historical intervals.
• Bucketized windows: Partition the history into disjoint, variable-size intervals (e.g., logarithmically increasing).
• Calendar-aligned windows: Align aggregation boundaries to calendar units (e.g., full previous days or weeks).

All approaches strictly enforce a no-lookahead constraint: at time $h$, only events for $t < h$ are available for aggregation (Pinchuk, 15 Jan 2026).

2. Mathematical Specification and Feature Construction

For each window type and entity, several canonical statistics are computed and encoded:

Feature Name	Formula	Notes
Log-impression count	$F^{imp}_{v,w}(h) = \log(1 + I_{v,[h-w,h)})$	Stabilizes wide variation in counts
Smoothed event rate	$$F^{rate}_{v,w}(h) = \dfrac{C_{v,[h-w,h)} + \alpha}{I_{v,[h-w,h)} + \alpha + \beta}$$	Smoothing (e.g., $\alpha=1$, $\beta=10$)
Event50 window	Use most recent $N=50$ events, compute as above on these	Adapts for low-activity entities

For event-count and bucketized schemes, the same logic applies over variable-length or non-consecutive time intervals. Gap windows exclude the immediate past hour to decouple the target from history. Calendar-aligned features use $[h-24, h)$ or similar for day-scale, but boundaries are strictly prior to $h$ (Pinchuk, 15 Jan 2026).

In customer churn settings, analogous aggregation includes sums, means, min/max, and trends of customer activity—over sliding windows, calendar windows, or with exponential decay—for each user at reference date $t_0$ (Gregory, 2018).

3. Methodological Protocol and Feature Set Assembly

Time aggregation feature construction is shaped by several rigorous methodological constraints:

• No leakage: All aggregation windows are anchored before the prediction time to avoid target contamination.
• Window tuples: Multiple lookback lengths are used (e.g., (1, 6, 24, 48, 168) hours) to capture both short-range recency and long-term patterns.
• Entity coverage: Features are typically computed for a fixed set of high-cardinality categorical keys (e.g., device_id, app_id, site_id).
• Feature matrix: For each instance (e.g., impression), the final feature vector concatenates $$(\text{keys})\times(\text{stats})\times(\text{windows})$$ features, with tree-based models such as XGBoost handling the resulting sparse, wide representations efficiently (Pinchuk, 15 Jan 2026).

Additional statistics (e.g., first-order differences, trends, min/max, exponential decay) are commonly included when empirical cross-validation supports them. In practice, feature selection is guided by wrapper methods (iterative addition, retraining, and delta-metric tracking) and post-hoc SHAP or gain-based importance metrics.

4. Comparative Evaluation of Window Schemes

Empirical comparisons on CTR and similar datasets consistently show (Pinchuk, 15 Jan 2026):

• Trailing windows (with multi-scale lengths) yield substantial gains in ROC AUC (e.g., ≈+0.007) and PR AUC (≈+0.009) over strong target encoding baselines.
• Event-count windows (e.g., event50) provide small, consistent further uplifts (≈+0.0004 ROC AUC), especially valuable in settings with highly variable entity activity rates.
• Gap and bucketized windows statistically underperform trailing windows in out-of-time, no-lookahead evaluation protocols.
• Calendar windows offer no systematic gain, performing similarly to trailing windows when lengths are set to day-scales.

Statistical significance is verified using paired tests (DeLong’s method), confirming that the observed uplifts are robust across cross-validation folds (Pinchuk, 15 Jan 2026).

5. Generalization: Time Aggregation in Other Domains

Time aggregation concepts generalize to a broad family of time-series and event-stream learning tasks. In customer churn (Gregory, 2018), features incorporate:

• Multiple window types (sliding, calendar-based, exponential decay).
• Higher-order trends: e.g., comparing short vs. long window means to extract accelerating/decelerating usage.
• Aggregates for transactional signals (cancelations, spend), relative recency (days since last event), and cohort-normalized rates.

In signal processing domains, windowed or segment-based statistical summarization is a ubiquitous approach—applied to intrinsic time-scale decomposed signals (Sami et al., 2021) or via non-overlapping sliding windows with frequency-domain transforms (Sha et al., 2022)—though typically with explicit focus on different moments (mean, variance, skewness, kurtosis) depending on the needed signal characterization.

6. Implementation Guidance and Best Practices

Key practical recommendations, as validated in recent literature (Pinchuk, 15 Jan 2026, Gregory, 2018), include:

• Default window design: Employ trailing windows with a log-count and smoothed rate statistic for each key, using a tuple of lengths covering both short-scale and long-scale recency.
• Supplementation: Add a single event-count window (e.g., event50) for keys/entities with highly variable traffic when marginal metric increases are worthwhile.
• Strict temporal anchoring: Enforce that no event at or after the prediction time is ever included in feature computation.
• Feature selection: Use iterative wrappers and cross-validation for empirical feature selection; prune highly correlated or low-importance aggregates to reduce computational burden.
• Avoid gap and bucket schemes unless supported by domain evidence; they consistently underperform in canonical CTR and churn prediction settings under strict evaluation.

7. Impact and Current Limitations

The systematic incorporation of time aggregation features in XGBoost pipelines has produced strong, reproducible improvements in predictive accuracy for major time-series forecast and classification tasks, particularly in the presence of high-cardinality entities and nonstationary behaviors. The main limitation is the computational and implementation complexity required to maintain strict “online-safe” semantics at industrial scale, especially under high-velocity data streams and large entity vocabularies. Cross-domain generalization requires careful adjustment of aggregation window structure and statistics to align with domain signal characteristics and label frequency (Pinchuk, 15 Jan 2026, Gregory, 2018, Sha et al., 2022).

A plausible implication is that further extensibility—such as representation learning directly on entity time series or hybrid table‐neural architectures—may eventually supplant handcrafted aggregation, but current evidence strongly supports time aggregation windows as a competitive, interpretable, and computationally efficient design for XGBoost and related GBDT models (Pinchuk, 15 Jan 2026).