Biswas et al. (2026) Comprehensive evaluation of state order variants of Markov chain for stochastic rainfall simulation across diverse climatic regimes of India

Identification

• Journal: Stochastic Environmental Research and Risk Assessment
• Year: 2026
• Date: 2026-04-01
• Authors: Papu Biswas, Ujjwal Saha
• DOI: 10.1007/s00477-026-03207-0

Research Groups

Water Resources Engineering, Department of Civil Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, India.

Short Summary

This study systematically evaluates various Markov chain state-order variants, including a novel Hybrid Three-state model, for stochastic daily rainfall simulation across 58 diverse climatic stations in India. It identifies the most suitable model for each station and proposes the Hybrid model as a robust, parsimonious option for nationwide application, while also assessing model suitability across Köppen climate classifications.

Objective

• To systematically evaluate and identify the most suitable state-order variant of the Markov chain model for stochastic daily rainfall simulation across 58 stations in diverse climatic regimes of India.
• To explore the relevance and performance of a computationally efficient Hybrid Three-state Markov chain model, which requires fewer transition probabilities than conventional multi-state models.
• To assess if model suitability can be generalized across regions based on Köppen climate classifications, providing a systematic framework for optimal model selection.

Study Configuration

• Spatial Scale: 58 stations across India.
• Temporal Scale: Daily rainfall sequences, derived from hourly data. Each station had at least 20 years of data.

Methodology and Data

• Models used:
  • Markov chain models: Two-state (dry/wet) First, Second, and Third order; Three-state (dry/moderate/heavy rainfall) First and Second order; Hybrid Three-state (dry/moderate/heavy rainfall) First order.
  • Rainfall amount generation: Gamma distribution (fitted to non-zero rainfall, and separately for moderate/heavy categories).
  • Extreme rainfall magnitudes: Generalized Extreme Value (GEV) distribution (fitted to annual maximum rainfall series).
  • Model selection criteria: Akaike’s Information Criterion (AIC), Bayesian Information Criterion (BIC), Efficient Determination Criterion (EDC).
  • Performance metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE).
• Data sources:
  • Hourly rainfall data from India Meteorological Department (IMD) for 58 stations (at least 20 years, with less than 10% missing data, reconstructed using a year-based interpolation approach).
  • Köppen climate classifications (Beck et al. 2023) for regional generalization.

Main Results

• For monthly mean rainfall, First-order Markov chain variants (Two-state, Hybrid, Three-state) performed well, with the Two-state First-order model being optimal for most stations.
• For monthly standard deviation of rainfall, the Hybrid First-order variant demonstrated superior performance for most stations, followed by the Three-state First-order and Three-state Second-order models. The Two-state First-order model performed poorly.
• For rainfall magnitudes of various return periods (2, 5, 10, 25, 50, 100 years), the Three-state Second-order Markov chain excelled for most stations, followed by the Hybrid State First-order model.
• The Two-state Third-order model satisfactorily reproduced the standard deviation of dry periods, indicating the benefit of higher-order memory for capturing persistent dry spells.
• The Hybrid state First-order model exhibited superior overall performance for most stations across India, effectively simulating intra-annual mean and standard deviation, as well as extreme rainfall magnitudes, while requiring only three transition probabilities.
• Information-theoretic criteria (AIC, BIC, EDC) often favored simpler two-state models, highlighting a trade-off between statistical parsimony and physical realism, where performance-based diagnostics showed the Hybrid First-order model offered a better balance.
• Model suitability varied across Köppen climate regimes: Hybrid First-order and Three-state variants performed well in Tropical monsoon (Am), Tropical savannah (Aw), and Temperate dry winter hot summer (Cwa) regimes, while Three-state variants were superior in Arid steppe hot (BSh) regimes.

Contributions

• Provides a comprehensive and pioneering performance analysis of diverse Markov chain state-order variants, including a novel Hybrid Three-state model, for stochastic rainfall simulation across 58 stations in India's climatically variable regimes.
• Identifies the most suitable Markov chain state-order variant for each station based on a suite of hydrologically and climatologically relevant statistical variables and error metrics.
• Introduces and evaluates a computationally efficient Hybrid Three-state Markov chain model, demonstrating its robustness and applicability as a potential unified and reliable variant for rainfall simulation across India due to its parameter parsimony and strong performance.
• Establishes a systematic framework for selecting optimal Markov model structures under varying Köppen climate contexts, offering regional generalization insights.
• Integrates information-theoretic model selection with hydrologically relevant performance diagnostics, revealing important trade-offs between statistical parsimony and physical realism in Markov-based rainfall simulation.

Funding

No funding was received to assist with the preparation of this manuscript.

Citation

@article{Biswas2026Comprehensive,
  author = {Biswas, Papu and Saha, Ujjwal},
  title = {Comprehensive evaluation of state order variants of Markov chain for stochastic rainfall simulation across diverse climatic regimes of India},
  journal = {Stochastic Environmental Research and Risk Assessment},
  year = {2026},
  doi = {10.1007/s00477-026-03207-0},
  url = {https://doi.org/10.1007/s00477-026-03207-0}
}