Dynamic Data Modeling Using Simultaneous Customer–Product Clustering Based on Markov Chains

Shaghayegh Khodkameh; Mohammad Hassan Behzadi; Mohsen Rostamy Malkhalifeh

Authors

Shaghayegh Khodkameh Department of Statistics, SR.C., Islamic Azad University, Tehran, Iran. Author https://orcid.org/0009-0002-2363-8819
Mohammad Hassan Behzadi Department of Statistics, SR.C., Islamic Azad University, Tehran, Iran. Author https://orcid.org/0009-0000-2651-0461
Mohsen Rostamy Malkhalifeh Department of mathematics and Computer Science, SR.C, Islamic Azad University, Tehran, Iran. Author https://orcid.org/0000-0001-6105-7674

Keywords:

Dynamic data modeling, customer–product co, clustering, Markov chains, customer segmentation, product clustering, transition probabilities, predictive analytics

Abstract

This study aimed to develop and evaluate a dynamic data-modeling framework based on simultaneous customer–product clustering and Markov chains for identifying latent transactional structures, estimating temporal transitions, and improving prediction of evolving customer–product interactions. This quantitative, applied, longitudinal study was conducted using electronic transaction records from selected retail and online businesses in Tehran, Iran. The final analytical dataset included 1,200 customers and 480 products observed over 12 consecutive monthly periods. Customer–product interaction matrices were constructed from standardized measures of purchase frequency and transaction value. Simultaneous co-clustering was applied to identify joint customer and product groups, and alternative clustering configurations were compared using the silhouette coefficient, within-cluster dispersion, adjusted Rand stability, and reconstruction error. First-order discrete-time Markov chains were then used to estimate monthly transition probabilities among customer and product clusters. Predictive performance was evaluated using holdout validation and compared with static independent clustering, static co-clustering, and dynamic clustering without simultaneous customer–product modeling. The four-customer-cluster and four-product-cluster solution provided the strongest overall clustering structure, with the highest silhouette coefficient (0.562) and adjusted Rand stability index (0.847). Markov analysis showed substantial state persistence, with customer self-transition probabilities ranging from 0.694 to 0.805 and product self-transition probabilities ranging from 0.681 to 0.754. The proposed dynamic co-clustering model achieved customer and product cluster prediction accuracies of 0.872 and 0.849, respectively, with adjusted Rand indices of 0.793 and 0.768. It also produced the lowest prediction errors (MAE = 0.074; RMSE = 0.118), reducing RMSE by approximately 32.2% relative to static independent clustering and 21.9% relative to static simultaneous co-clustering. Integrating simultaneous customer–product clustering with Markov transition modeling provides a robust approach for representing both relational structure and temporal evolution in transactional data and improves predictive accuracy compared with static or independently modeled alternatives.

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