Cardinality-Stratified Interaction Decomposition for Interpretable Pairwise and Higher-Order Structure in Transactional Basket Data

Hidetoshi Kawase; Toshihiro Ota

cs.LGSep 28, 2026

Cardinality-Stratified Interaction Decomposition for Interpretable Pairwise and Higher-Order Structure in Transactional Basket Data

Authors: Hidetoshi Kawase, Toshihiro Ota

Organizations: CyberAgent, Inc., Shibuya, Tokyo 150–6121 Japan

Abstract

Transactional basket data can reveal associations among items, but observed co-occurrence conflates item-specific relations with basket-size structure and unmodeled higher-order dependence. We introduce Cardinality-Stratified Interaction Decomposition (CSID), an interpretable framework that decomposes log-odds contrasts stratified by the number of remaining items into item-set-specific and cardinality-common components, without fitting a global joint distribution. CSID uses an information-weighted, gauge-constrained ridge projection to estimate pair and triple components and to diagnose higher-order contributions to pairwise structure. CSID is designed primarily for interpretable decomposition of association structure rather than for full-distribution prediction. In a simulation with zero pair effects, increasingly strong small-basket cardinality potentials drive ordinary Ising couplings spuriously negative, whereas CSID pair estimates remain centered near zero. Detection power rises with the magnitude of planted triple effects, and local deprojection reduces pair-coefficient RMSE from 0.244 to 0.073. Across three grocery datasets, high-information triple components are reproducible over time. In the matched cross-period partial-transfer evaluation, transferred CSID triple components show closer agreement with later-period stratified contrasts than the nodewise-symmetrized cardinality-aware higher-order pseudolikelihood comparator, with gains in weighted Lin's concordance correlation of 0.038--0.122. These results support CSID as an exploratory and interpretable decomposition framework for pairwise and higher-order association structure in transactional data.

Figures & tables

$s$	Mean size	Ordinary Ising			CSID-2
		Mean estimate	Neg. frac.	RMSE	Mean estimate	Neg. frac.	RMSE
0.0	4.27	-0.001	0.514	0.042	0.000	0.502	0.042
0.25	3.36	-0.056	0.887	0.073	-0.000	0.492	0.047
0.5	2.72	-0.137	0.997	0.146	-0.001	0.493	0.052
0.75	2.28	-0.235	1.000	0.242	-0.002	0.496	0.062
1.0	1.97	-0.361	1.000	0.367	-0.003	0.511	0.080

Table 1: Ordinary Ising and CSID-2 estimates under a cardinality-only data-generating process. Mean estimate is the Monte Carlo mean over 20 replicates of the within-replicate average across all

\binom{10}{2}=45

fitted pair coefficients. Neg. frac. and RMSE are averaged analogously from their within-replicate pair summaries; all true pair coefficients are zero.

Figure 1: Mean pairwise estimate $\widehat{\theta}_{ij}$ (left) and RMSE to the true zero vector (right) against mean basket size. Mean basket size increases from left to right; stronger cardinality penalties therefore appear farther left, and strength levels $s$ are listed in table 1 . Lines compare ordinary pairwise Ising and CSID-2, with shaded bands showing 95% Monte Carlo intervals over 20 replicates. As baskets become smaller, ordinary Ising couplings shift downward, whereas CSID-2 remains near zero with substantially lower RMSE.

$\|\theta\|$	Power (95% MC interval)	Non-planted rate	AUPRC	AUROC	Sign recovery
0.1	0.075 [0.035, 0.115]	0.032	0.133	0.526	0.719
0.2	0.194 [0.142, 0.246]	0.034	0.259	0.662	0.850
0.3	0.338 [0.235, 0.440]	0.042	0.393	0.766	0.944
0.5	0.700 [0.589, 0.811]	0.038	0.745	0.924	0.988

Table 2: Detection of randomized sparse third-order signals. Non-planted rate is the fraction of non-planted triples with

|z_{T}|\geq 2

. Monte Carlo intervals are two-sided 95%

t

intervals over the 20 replicate-level detection rates.

Figure 2: Planted-triple detection rate (power) and non-planted exceedance rate against planted effect size $|\theta|$ . The horizontal axis shows absolute planted coefficients; the vertical axis shows the fraction of fitted triples flagged. Shaded bands are 95% Monte Carlo intervals over 20 replicates ( table 2 ). Power increases with $|\theta|$ , whereas the non-planted exceedance rate remains near 3–4%.

Estimator	Gauge-aligned RMSE	Pearson $r$
CSID-2	0.244	0.693
Reweighted CSID-2	0.073	0.985

Table 3: Recovery of pair coefficients in one controlled 60,000-basket realization with third-order saturation. RMSE is computed against the generating pair coefficients after information-weighted gauge alignment; Pearson

r

is computed across all 15 item pairs.

Dataset	Baskets	H1 / H2	Selected categories	$n_{111}$ min.
Complete Journey	72,936	35,087 / 37,849	20 predefined product_category values	0
Ta-Feng	90,258	44,564 / 45,694	Top 20 two-digit PRODUCT_SUBCLASS codes	10
Instacart	373,133	179,784 / 193,349	Top 20 aisles (30,000-user subsample)	30

Table 4: Datasets and temporal splits after category restriction. The column

n_{111}

min. gives the minimum total triple co-occurrence count

\sum_{r}n_{111,r}

required to retain a CSID-3 candidate in H1 ( eq. 26 ); pairs were retained without an analogous cutoff.

Dataset	Pearson (LOCO range)	Sign agreement	Negative top-20 / expectation
Complete Journey	0.572 [0.539, 0.616]	0.677	6 / 1.40
Ta-Feng	0.739 [0.549, 0.768]	0.729	11 / 1.59
Instacart	0.723 [0.664, 0.742]	0.747	9 / 1.40

Table 5: H1–H2 reproducibility of high-information CSID-3 effects. The LOCO range is the minimum and maximum Pearson correlation over the 20 omitted-category evaluations; the models are not refitted. The top information quartile is reselected after each category omission. The reported expectation is

k^{2}/N

, the expected overlap of two independently selected sets of

k=\min(20,N)

triples among the

N

evaluated triples.

Dataset	Triples	Layers	Baseline	CA-HOPL	CSID-3	Gap	LOCO range
Complete Journey	285	4,401	0.356	0.403	0.442	0.038	[0.025, 0.049]
Ta-Feng	251	2,472	0.107	0.358	0.480	0.122	[0.083, 0.141]
Instacart	285	3,812	0.390	0.549	0.597	0.049	[0.039, 0.054]

Table 6: Continuous agreement with H2 third-order contrasts for the top H1-information quartile. Baseline, CA-HOPL, and CSID-3 report information-weighted CCC on identical triples and layers; Gap is CSID-3 minus CA-HOPL. The LOCO range is the minimum and maximum gap over the 20 omitted-category evaluations; the models are not refitted.

	Weighted correlation $\rho_{\omega}$			Bias correction $C_{b,\omega}$
Dataset	Baseline	CA-HOPL	CSID-3	Baseline	CA-HOPL	CSID-3
Complete Journey	0.499	0.524	0.520	0.71	0.77	0.85
Ta-Feng	0.225	0.454	0.512	0.48	0.79	0.94
Instacart	0.495	0.619	0.637	0.79	0.89	0.94

Table 7: Decomposition of weighted CCC for the common-effect baseline, CA-HOPL, and CSID-3 in the top H1-information quartile. Following

\rho_{c,\omega}=\rho_{\omega}C_{b,\omega}

\rho_{\omega}

is the weighted Pearson correlation between H2 contrasts and each reconstruction, and

C_{b,\omega}

measures agreement in mean and scale.

Figure 3: Information-weighted Lin’s concordance correlation coefficient (CCC) between H2 stratified third-order contrasts and the common-effect baseline, CA-HOPL, and CSID-3 transfer reconstructions, plotted against the fraction of H2-evaluable triples retained by H1 information. Each panel shows one dataset; all methods use identical triples, layers, H2 common-effect reconstruction, and information weights. No H2 effect-size filter is applied. CSID-3 CCC exceeds CA-HOPL at the primary 25% fraction in every dataset.

Dataset	H1 common	Full CSID-3	Gain	Partial CSID-3
Complete Journey	0.032	0.158	0.127	0.442
Ta-Feng	-0.115	0.268	0.383	0.480
Instacart	0.127	0.418	0.291	0.597

Table 8: Information-weighted CCC for H1-only reconstruction of H2 contrasts in the top H1-information quartile. H1 common uses

\widehat{\beta}_{3,r}^{\mathrm{H1}}

alone, Full CSID-3 uses eq. 57 , Gain is their difference, and Partial CSID-3 repeats the primary reconstruction that estimates the common component from H2.

Figure 4: Conditional-odds contrasts $\Lambda_{ij\ell}=\log OR_{ij\mid x_{\ell}=1}-\log OR_{ij\mid x_{\ell}=0}$ for eight H1-ranked, H2-evaluable triples in each dataset. Each panel shows one dataset; rows list the focal pair and moderator and are ordered by the observed H2 contrast. Circles with horizontal intervals give observed H2 values with approximate 95% cell-count delta-method reference intervals; these intervals do not account for customer clustering or selection of the displayed triples. Triangles give the common-effect baseline and squares give the CSID-3 transfer. A vertical dashed line marks $\Lambda=0$ .

Dataset	Pair	CSID-2	Reweighted CSID-2	$\Delta\theta^{(3)}$
Complete Journey	Milk – Cold cereal	0.711	1.167	0.456
Complete Journey	Soup – Crackers	0.577	1.008	0.431
Ta-Feng	51 – 54	1.067	2.626	1.559
Ta-Feng	73 – 47	-0.288	-1.029	-0.741
Instacart	Fresh herbs – Cereal	-0.639	-1.168	-0.529
Instacart	Packaged cheese – Lunch meat	0.464	0.973	0.510

Table 9: Illustrative pairwise projections showing large positive and negative third-order corrections. The rows were selected for interpretation rather than by a prespecified inferential rule.

Dataset	Correlation with H2	Correlation with full-period
Complete Journey	0.688	0.906
Ta-Feng	0.695	0.904
Instacart	0.814	0.945

Table 10: Cross-period reproducibility of third-order pairwise corrections. Pearson correlations are computed across all 190 focal pairs between corrections obtained by applying H1 triple coefficients to H2 pair cells and corrections based on H2 or full-period fits. All corrections use the same H1-fixed gauge convention, so the table compares relative correction patterns without changing the reference gauge.

Context aisles	Pair $\rho$	Triple $\rho$ (top 25%)	Deprojection $\rho$	$\Delta\mathrm{CCC}$
20	0.988	0.950	0.893	0.286
40	0.997	0.992	0.984	0.163
80	0.999	0.998	0.996	0.165
All (134)	1.000	1.000	1.000	0.172

Table 11: Instacart robustness to expansion of the surrounding aisle universe. Each row fixes the focal top-20 aisles and expands only the rest-cardinality context. Pair

\rho

compares all 190 focal

\widehat{\theta}_{ij}

to the all-aisle reference; triple

\rho

(top 25%) compares the highest-information quartile of focal

\widehat{\theta}_{T}

; deprojection

\rho

compares third-order pairwise corrections

\Delta\theta^{(3)}

. All correlations use the all-aisle analysis as reference.

\Delta\mathrm{CCC}

is the CSID-3 minus baseline improvement in information-weighted Lin’s CCC at

|z_{T}^{\mathrm{H1}}|\geq 2

Figure 5: Coefficient stability under Instacart item-universe expansion. Left: all 190 focal pair coefficients with 20 context aisles ( $\widehat{\theta}_{ij}^{(20)}$ ) versus the all-aisle reference ( $\widehat{\theta}_{ij}^{(\mathrm{all})}$ ). Right: the top information quartile of focal third-order coefficients ( $\widehat{\theta}_{T}^{(20)}$ versus $\widehat{\theta}_{T}^{(\mathrm{all})}$ ). Dashed lines mark equality; annotations give Pearson correlation.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Quantity	Default	$\varepsilon$ range	$\lambda$ range	$r\geq 1$
Complete Journey	Temporal $\rho$	0.572	[0.571, 0.572]	[0.571, 0.572]	0.647
	CSID–CA-HOPL CCC	0.038	[0.038, 0.039]	[0.037, 0.038]	0.049
	Correction $\rho$	0.688	[0.683, 0.699]	[0.688, 0.689]	0.777
Ta-Feng	Temporal $\rho$	0.739	[0.736, 0.745]	[0.738, 0.745]	0.779
	CSID–CA-HOPL CCC	0.122	[0.121, 0.122]	[0.114, 0.123]	0.123
	Correction $\rho$	0.695	[0.681, 0.721]	[0.687, 0.746]	0.763

Appendix

Table 12: Focused one-factor sensitivity analysis. Temporal

\rho

is the H1–H2 Pearson correlation in the top information quartile; CSID–CA-HOPL CCC is the partial-transfer gap; Correction

\rho

compares H1-based and H2-based pair-correction patterns. The

r\geq 1

analysis restores baskets containing one selected category before constructing the additional layer.

Aisles	Mode	Generated	Retained	Time (s)	Storage (MB)
20	Exhaustive	1,140	1,140	6.9	0.6
20	Sparse	1,140	1,140	6.9	0.6
40	Exhaustive	9,880	9,880	16.5	6.8
40	Sparse	9,880	9,880	16.5	6.8
80	Sparse	82,160	79,193	37.6	83.0
134	Sparse	250,000	156,735	49.9	178.2

Appendix

Table 13: Single-process candidate-scaling audit on 373,133 Instacart baskets, conducted on a laptop computer with an arm64 processor. Time is the sum of basket scanning, candidate construction, exact counting, contrast construction, and CSID-2/3 fitting; storage is the combined size of the retained stratified count and contrast arrays, not peak process memory. Sparse generation used frequent-pair triangles and bounded neighborhoods with a 250,000-candidate cap.

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