Table 2 describes the transformation from each source dataset to the common transaction schema. The reconstructed fields are material to interpretation: Instacart has synthetic chronology and no monetary information; Ta-Feng reconstructs baskets at customer-day resolution; and Olist includes freight in item value.

For Instacart, orders are sorted within each user_id by order_number. Missing days_since_prior_order values are set to zero, and the customer-relative intervals are cumulatively summed. Each order receives the synthetic timestamp 1 January 2017 plus that cumulative number of days plus its order_hour_of_day (missing hours are set to zero). The same arbitrary 2017 origin is used for every customer; it is not an observed calendar date or a known alignment of different customers’ purchasing histories. Joined product rows are then ordered by this synthetic timestamp, order ID, and add-to-cart position before the chronological initial/stream split. Consequently, recency and batch membership reflect the shared synthetic timeline rather than an observed cross-customer chronology.

For customer i at batch t, the cumulative representation is

Table 2. Dataset-specific conversion into the common transaction schema.

where recency R_i,t is time since the latest purchase, frequency F_i,t counts unique orders or reconstructed baskets, and V_i,t is the cumulative third feature. For Ta-Feng and Olist, V_i,t is monetary value. For Instacart, V_i,t = Σ(Q_ℓP_ℓ) over rows ℓ ∈ H_i,t is drift-weighted item quantity: Q_ℓ is row quantity and P_ℓ is a simulated unit weight stored as UnitPrice. Both equal one initially and are perturbed on drifted rows. This sum equals product-row count before perturbation, but is neither row count nor observed spending afterward. The analysis fits a standard scaler on the initial 40% and holds it fixed across the eight subsequent cumulative snapshots. Section 4.4 defines the corresponding window-local representation for temporal evaluation. Here H_i,t is the set of product rows for customer i included up to batch t.

The described initial-model selection considers k ∈ {2, 3, 4, 5, 6} on the first 40% cumulative customer-feature snapshot. The selected per-seed values, BIRCH threshold grid, tie-breaking rule, and original validation record are not available with the reported results; their precise selection cannot be verified here. A disjoint temporal evaluation would select and freeze hyperparameters using chronological validation entirely within the initial training period. The present results do not test cluster splits, merges, or changing cluster counts.

The implementation uses scikit-learn components [23]. Incremental evaluation is included only for MBK. Although the scikit-learn BIRCH interface also exposes partial_fit [24], a matched streaming-BIRCH comparison is outside the scope of the present experiments.

Let z t ∈ [ 0 , 1 ] be the drift intensity for batch t. Natural drift uses z t = 0 throughout. Abrupt drift uses full intensity from batch 3 onward. Gradual drift uses zero intensity before batch 3 and

thereafter, giving 0.25, 0.50, 0.75, and 1.00 at batches 3 - 6. Recurring drift has full intensity at batches 3, 4, 6, and 7, with onsets at batches 3 and 6. Local drift has full intensity from batch 3 onward but affects only a stable hash-selected 20% of customer identifiers.

For medium severity, quantity Q and unit price P are transformed as

Here the simulated noise terms ε_Q and ε_P have normal distributions with standard deviations 0.60z_t and 0.20z_t, respectively. Total price is recomputed as QP. The batch-specific generator seed is

10,000 + model_seed + 100t (5)

No rounding is applied after adding Gaussian noise, so Q is a continuous simulated feature, not a literal purchased-unit count. Instacart starts with the placeholder P = 1; the reported drift scenarios perturb both Q and P, recompute QP, and thereby change the evaluated third feature even though the number of product rows is unchanged. Changes to P are synthetic feature perturbations, not economic price drift. No quantity-only Instacart condition is reported.

Table 3 makes the five scenario schedules explicit. In particular, it distinguishes permanent abrupt and local changes from the temporary recurring pattern and the monotone gradual ramp used to evaluate replacement decisions.

Table 3.Synthetic-drift intensity by stream batch. Local uses the abrupt intensity schedule on 20% of customers.

Let D t denote the cumulative standardized customer-feature snapshot (monetary RFM for Ta-Feng and Olist; drift-weighted item quantity for Instacart) at batch t. The current pre-update silhouette is

The notation p t − 1 denotes the post-decision deployed silhouette recorded at the previous batch. The deterioration gate compares that previous deployed score with the current pre-update score:

When g t = 1 , a candidate M t c is fitted on D t . Its same-snapshot silhouette is p t c = S ( M t c , D t ) , and a t is the ARI between current and candidate assignments. Quality-only acceptance requires p t c > p t − . Reward acceptance uses

R update = 0.85 p t c + 0.10 a t − 0.05 ρ , ρ = 0.005 . (9)

The term ρ is a normalized switching regularizer, not a measured business or infrastructure cost. Its fixed reward contribution is 0.05 × 0.005 = 0.00025, equivalent to approximately 0.000294 silhouette units after division by 0.85. A sensitivity analysis would vary the stability weight and regularizer, report the fraction of decisions changed, and select settings on training or validation scenarios rather than final test scenarios; no such analysis is reported here.

The static policy never replaces the initial model. The periodic policy is scheduled every four batches, but the implementation permits an update only when a following batch exists. Consequently, in the eight-batch protocol, batch 4 is the only eligible scheduled refit and batch 8 is evaluation-only. The two selective policies apply the quality-only and reward rules above subject to the same final-batch restriction. The MBK incremental comparator calls partial_fit each batch without the same deployment gate.

The reported same-snapshot evaluation fits a candidate on cumulative snapshot D_t, evaluates it on D_t, and uses that score in the acceptance decision. A post-decision silhouette on D_t is therefore a selection score rather than an unbiased estimate of temporal generalization. A cumulative next-batch comparison on D_t+1 likewise retains substantial overlap with candidate-training history. The following disjoint-window procedure has not been evaluated for the reported policy scores.

The proposed procedure would use a one-window decision delay and a window-local outcome. Let W_t contain only transactions arriving in chronological window t, and let X_t be its standardized customer-feature set. For each customer active in W_t, recency would run from that customer’s latest transaction to the end of W_t, while frequency and value would use only baskets and transactions inside W_t. Thus, Equation (1) would be applied within each window rather than over cumulative history. Customers without transactions in W_t would have no window-local vector; their number and the silhouette denominator would be reported. A reference scaler would be fitted once on window-local vectors from the initial training period and shared by active and candidate models.

The initial period would be divided into windows of the same duration, and candidates would be fitted on historical customer-window vectors X_1:t. Repeated observations of one customer in different windows would be separate training instances. The gate would compare active and candidate models on X_t+1, and an accepted candidate would become active for W_t+2. A customer would be “returning” if its identifier occurred before W_t+1 and “new” otherwise; a new customer’s vector would use only W_t+1. The proposed outcome would be measured on X_t+2, not the gate window W_t+1. With T stream windows, only origins t ≤ T − 2 would yield a fully evaluated candidate.

Table 4 distinguishes the proposed fitting, gating, and outcome windows. Unlike the cumulative comparisons in the reported results, the outcome window would not be used to select its deployment state.

Table 4. Proposed information flow for one rolling origin; no disjoint-window outcomes are reported.

A secondary as-of cumulative analysis could rebuild returning-customer features using history available at each cutoff; it would measure history-overlapping persistence rather than disjoint generalization. A future report should distinguish window-local, new-customer, returning-customer, and cumulative gains.

For this proposed analysis, silhouette would be defined only when evaluated assignments contain at least two nonempty clusters and fewer clusters than observations. A candidate with an undefined silhouette would be rejected. If the active score were undefined but the candidate score valid, either selective gate would accept the candidate; if both were undefined, the model would be retained and the decision recorded as lacking a valid quality comparison rather than assigned an artificial score. Numerical gate or reward comparisons would apply only when both scores are valid. A future run record should specify the distance metric, silhouette subsample size and seed, and whether active and candidate scores use the same sampled customers.

Because cluster identifiers are arbitrary, a future migration analysis would align labels before comparing assignments. For old labels y and candidate labels z, a contingency matrix and the Hungarian algorithm would find the relabeling π with maximum overlap. The proposed minimum error distance is

When cluster counts differ, the contingency matrix would be padded to K = max(k_γ, k_z) rows and columns. The proposed normalization is

NMED ( y , z ) = MED ( y , z ) / ( 1 − 1 / K ) if K > 1 ; 0 if K = 1 . (11)

This scales the 1 − 1 / K worst-case assignment-error bound to one. ARI measures permutation-invariant agreement, but does not by itself give a customer migration rate.

A future event record should distinguish candidate agreement (ARI and MED whenever a candidate is evaluated) from accepted-update stability (only accepted candidates). Post-alignment migration would use customers present in both relevant windows; no-update continuity would track the active model without treating retention as a perfect update. No-candidate and rejected rows should not be assigned ARI one or migration zero. Each mean would require its numerator and eligible denominator, distinguishing accepted updates, eligible decisions, tracks with at least one update, and future-window comparisons. These event-level results are not available here.

For a future temporal evaluation, let y t + 1 keep and y t + 1 cand denote active and candidate assignments on X t + 1 , and define m t + 1 = NMED ( y t + 1 keep , y t + 1 cand ) . An interpretable proposed reward is

For measured end-to-end candidate cost C, the proposed normalized cost would be C = min(1, C/C_0.95), with C_0.95 estimated only from training or validation origins and then frozen. Values of λ_m and λ_c would be selected outside final test scenarios; their decision-change rates are not reported here.

A future runtime analysis should separate feature construction, active prediction, candidate fitting, candidate evaluation, decision logic, and reporting; it should include rejected candidates, total batch time, and peak memory. Only model-fit times are presented here, not end-to-end latency.

Source-specific feature constructions remain a sensitivity-analysis target rather than results reported here. For Instacart, this would compare recency-frequency alone with a true product-row-count feature and with the reported drift-weighted item quantity. Ta-Feng would compare customer-day baskets with the finest source transaction grouping available; Olist would compare product price alone with product price plus freight. Such comparisons should be reported separately from the present policy rankings.

The reported comparisons comprise static, periodic-4, quality-only, and reward policies for both algorithms, plus MBK partial_fit. Always-refit, BIRCH partial_fit, CluStream, and DenStream are proposed additional comparators, not reported results. Future comparisons should share chronological windows and eligible-decision accounting; when cluster counts differ, assignment agreement should use ARI or NMED instead of raw label equality.

The comparison unit is a matched dataset-algorithm-scenario-seed track, not an individual batch row. Batch observations within a track are not treated as independent replicates. This paired-unit treatment follows the general caution against treating correlated experimental observations as independent comparisons [25]. For each dataset, algorithm, scenario, and seed, we first calculate the track-level mean deployed silhouette, then calculate reward-minus-baseline differences using the same seed and scenario. Primary comparisons are rewards versus static, periodic-4, and quality-only.

For each primary reward contrast, the five paired seed effects in Tables 5-10 average matched scenario differences within a seed. Each 95% percentile bootstrap interval resamples these five seed effects, not individual batches or scenarios. These intervals describe variation among the supplied seeds, not uncertainty over new datasets or drift scenarios; adjusted hypothesis tests are not reported.

Within each dataset-algorithm-policy-scenario-seed track, deployed silhouettes are averaged over batches with defined scores (up to eight). Policy means give equal weight to available track means across five scenarios and five seeds (up to 25), without weighting by batch size or customer count. Undefined batch scores are omitted, not set to zero. The source record does not establish whether any tracks contain missing batch scores; the number of included tracks is therefore not independently verifiable.

Table 5.Mean deployed silhouette on fixed-reference-scaled customer features: monetary RFM for Ta-Feng and Olist, and drift-weighted item quantity for Instacart. Higher is better.

Table 6. Descriptive reward-policy advantage in mean deployed silhouette. Entries are absolute silhouette-point differences, not percentages.

Table 7. Paired reward-minus-quality-only differences in mean deployed silhouette (absolute silhouette points). Each seed value averages five matched scenario differences; the 95% percentile interval resamples these five seed effects with replacement (all 5^5 ordered resamples). Positive values favor reward. The intervals quantify seed variation in the supplied summaries only; their six-decimal display does not imply comparable generalization precision.

Table 8.Paired reward-minus-static differences in mean deployed silhouette (absolute silhouette points). Seed effects and 95% intervals are computed as in Table 7.

Table 9. Paired reward-minus-periodic-4 differences in mean deployed silhouette (absolute silhouette points). Seed effects and 95% intervals are computed as in Table 7.

Table 10. Supplied operational summaries for BIRCH with the reward policy. Deployed and Static are mean silhouette; Diff. is their difference. Next gain is a cumulative next-snapshot counterfactual summary, not a disjoint outcome. The eligible denominators for Positive, Updates, Migration, Update ARI, and Drift cannot be recovered from the available run record; these entries must not be read as comparable event-level rates or accepted-update stability. Fit s records model fitting only, not end-to-end latency.

The policy aggregates use same-snapshot gating, cumulative next-batch comparisons, unaligned migration, and incomplete candidate timing. The underlying run record for the supplied paired seed summaries has not been archived; accepted-event denominators, label-aligned stability, and disjoint temporal outcomes are unavailable.

The reward policy has the highest mean deployed silhouette in all six dataset–algorithm cells (Table 5). Table 6 gives its absolute differences from static, periodic-4, and quality-only; Tables 7-9 give matched seed contrasts and bootstrap intervals.

Against static, the reward policy’s absolute mean improvement ranges from 0.0435 to 0.3630; against quality-only it is 0.0045 - 0.0170 silhouette points. In all six cells, all five supplied seed effects for each of the three contrasts are positive and their seed-bootstrap intervals exclude zero. This is consistency among the supplied seeds under same-snapshot evaluation, not evidence that the reward’s stability or cost term caused the improvement.

Figure 2 shows the four policy means by dataset and algorithm. The policies are discrete alternatives; connecting points serves only to guide comparison, not to represent a temporal trajectory.

Table 10 retains supplied BIRCH summaries for context, not an event-level comparison. The unavailable eligible denominators prevent interpreting Positive, Updates, Migration, Update ARI, and Drift as conditionally comparable rates.

The next-gain values compare cumulative snapshots with overlapping customer histories rather than a disjoint outcome window; they cannot establish a future causal benefit. The displayed fit times exclude feature construction, transformation, scoring, diagnostics, input/output, and other pipeline work. The available record does not establish how rejected candidate fits contributed to these summaries, so the values cannot represent end-to-end latency.

Figure 3 displays the small reward-quality-only differences on a scale suited to those contrasts; the paired seed intervals are reported separately in Table 7.