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A Bivariate Gaussian–Gamma Mixture Model for Rainfall Distribution and Extreme Event Analysis over Karnataka, India

This study proposes a bivariate Gaussian–Gamma mixture model to effectively capture the non-Gaussian characteristics, skewness, and extreme events of Southwest Monsoon rainfall in Karnataka, India, demonstrating superior performance over conventional Gaussian models for flood risk assessment and drought monitoring.

Original authors: KUMUDHA HR, KOKILA RAMESH, ANGEL RICHARD

Published 2026-09-29

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CC BY 4.0

Original authors: KUMUDHA HR, KOKILA RAMESH, ANGEL RICHARD

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the monsoon-dominated landscapes of India, the arrival of rain is not merely a weather event; it is the pulse that drives agriculture, fills reservoirs, and dictates the rhythm of daily life. For centuries, scientists have tried to predict these patterns using standard statistical tools that assume rainfall behaves like a bell curve, where most years are average and extreme events are rare outliers. However, the reality of the Indian monsoon is far more complex. The rain often arrives in sudden, violent bursts or lingers in long, dry spells, creating a distribution that is skewed and heavy-tailed, defying the neat symmetry of traditional models. When these standard tools are applied to regions like Karnataka, they often fail to capture the true nature of extreme floods or the depth of droughts, leaving planners with an incomplete picture of risk. To understand the monsoon's true character, researchers must look beyond simple averages and embrace models that can account for both the gentle, predictable drizzle and the chaotic, overwhelming deluge.

A team of researchers based in Karnataka has taken a significant step toward this more realistic understanding by developing a new statistical framework that treats rainfall as a mixture of two distinct behaviors. Instead of forcing all rain data into a single, rigid shape, their approach combines a standard model for moderate, everyday rainfall with a specialized model designed to handle the heavy, skewed tails of extreme events. By analyzing sixty-five years of monsoon data, from 1960 to 2025, across ten different districts, the team mapped how rain behaves in both the flood-prone coastal belts and the drought-stricken interior regions. They found that the coastal areas, battered by the Arabian Sea, experience persistent, heavy rainfall that tends to repeat itself from year to year, while the interior districts suffer from erratic, irregular patterns where wet and dry years follow no clear sequence. This distinction is crucial, as it reveals that a single rule cannot govern the entire state; the coast and the interior operate under different atmospheric moods.

The researchers applied their new method to ten representative districts, six of which are prone to flooding and four that frequently face drought. They began by cleaning and normalizing the raw rainfall numbers to remove the influence of local geography, allowing them to compare the underlying statistical shapes of the data. The results confirmed that the rain in these regions does not follow a simple, symmetrical pattern. In the coastal districts of Dakshina Kannada and Udupi, the data showed a strong tendency for heavy rain to cluster together, creating a "fat tail" in the distribution where extreme events are more common than standard models would predict. In contrast, the interior districts of Bagalkote, Chitradurga, Koppala, and Raichur displayed a different kind of irregularity, characterized by high variability and a lack of persistence, where a wet year is just as likely to be followed by a dry one. The new mixture model successfully captured these nuances, fitting the observed data much more closely than traditional methods, particularly in the regions where the risk of extreme weather is highest.

To test the reliability of their model, the team examined how often the rainfall crossed specific thresholds, such as levels that are ten or twenty percent above or below the long-term average. These "level crossings" act as a measure of how frequently a region shifts from normal conditions into flood or drought territory. The traditional models often missed the mark, either predicting too many extreme shifts or too few. The new mixture approach, however, aligned closely with what actually happened over the decades. In the coastal regions, it accurately predicted the frequent upward crossings that signal the onset of flood conditions, while in the interior, it correctly identified the erratic downward crossings that mark the arrival of drought. This ability to distinguish between the steady, persistent rain of the coast and the volatile, unpredictable rain of the interior provides a much sharper tool for assessing risk.

The study also looked at how rainfall in one year influences the next, a concept known as temporal persistence. By extending their model to look at pairs of consecutive years, the researchers discovered a stark divide in the state's climate memory. In the coastal districts, there is a strong link between years; if one year brings heavy rain, the next is likely to do the same, creating a persistent wet or dry spell that can last for years. This persistence is driven by the complex interaction of the Western Ghats and the monsoon winds. In the interior, this link is weak or non-existent, meaning the weather resets every year with little regard for what happened before. This finding suggests that while coastal communities can expect a certain continuity in their climate risks, those in the interior face a more chaotic environment where preparation must be reset annually.

Using these insights, the researchers generated a forecast for the 2026 monsoon season across the selected regions. The model predicts that the coastal districts, including Dakshina Kannada, Udupi, and Uttara Kannada, will likely receive higher-than-average rainfall, continuing the pattern of wet persistence seen in recent decades. Conversely, the drought-prone interior districts of Bagalkote, Koppala, and Raichur are expected to receive lower rainfall, reinforcing the need for careful water management in these areas. These predictions are not just guesses but are derived from the statistical behavior of the last sixty-five years, filtered through a model that understands the difference between a steady drizzle and a sudden deluge. By providing a clearer picture of where and when extreme rain is likely to occur, this work offers a vital resource for managing water resources, planning agriculture, and mitigating the risks of floods and droughts in a region where the monsoon is both a lifeline and a threat.

Technical Summary: A Bivariate Gaussian–Gamma Mixture Model for Rainfall Distribution and Extreme Event Analysis over Karnataka, India

Problem Statement
Rainfall variability in monsoon-dominated regions like Karnataka, India, significantly influences agriculture, water resources, and disaster management. However, traditional statistical approaches often assume rainfall follows a Gaussian (normal) distribution. The authors argue that this assumption is insufficient because monsoon rainfall exhibits non-Gaussian characteristics, including skewness, heavy tails, and extreme events (both floods and droughts). While machine learning techniques (e.g., ANN) have been applied to rainfall prediction, they often prioritize forecasting accuracy over explicitly representing the underlying probability distribution and higher-order moments. Furthermore, existing studies in the region have largely been limited to univariate modeling, failing to explicitly capture the temporal dependence (persistence) of rainfall processes, which is particularly evident in coastal regions where wet and dry conditions tend to persist over successive periods.

Methodology
The study utilizes long-term Southwest Monsoon (SWM) rainfall data (1960–2025) from ten representative districts in Karnataka, categorized into flood-prone coastal regions (e.g., Dakshina Kannada, Udupi) and drought-prone interior regions (e.g., Bagalkote, Raichur).

Data Transformation: To address non-negativity and skewness, observed rainfall ( $R_i$ ) is normalized using the long-term mean ( $m_R$ ) and transformed via a logarithmic function: $Z_i = \log(R_i / m_R)$ . This stabilizes variance and allows for probabilistic modeling.
Gaussian–Gamma Mixture Model (GGM): A unified probabilistic framework is proposed where the probability density function is a weighted sum of a Gaussian component (representing moderate fluctuations) and a Gamma component (representing skewed, heavy-tailed extreme events). The model is defined as:
$f(z) = w f_G(z: \mu, \sigma^2) + (1-w) f_\Gamma(z: \alpha, \beta)$
where $w$ is the mixing weight.
Parameter Estimation: Model parameters ( $w, \mu, \sigma^2, \alpha, \beta$ ) are estimated using the Expectation–Maximization (EM) algorithm to maximize the log-likelihood of the observed data.
Model Validation: The GGM is evaluated against observed data using:
- Statistical moments (mean, standard deviation, skewness, kurtosis).
- Goodness-of-fit tests (Kolmogorov–Smirnov, Anderson–Darling, Chi-square).
- Level-crossing statistics to analyze the frequency of rainfall exceeding or falling below specific thresholds (e.g., ±10% and ±20% of normal).
Bivariate Extension: To analyze temporal dependence, the model is extended to a bivariate framework ( $Z_1, Z_2$ representing consecutive years). This incorporates joint probability densities for Gaussian–Gaussian, Gaussian–Gamma, Gamma–Gaussian, and Gamma–Gamma regimes to study persistence and cross-regime correlations.
Prediction: The conditional predictive density derived from the bivariate model is used to forecast SWM rainfall for the year 2026.

Key Contributions
The paper outlines three specific contributions:

Development of a Bivariate Mixture Framework: Moving beyond univariate analysis to jointly model distributional characteristics and temporal dependence in rainfall.
Improved Representation of Extremes: Demonstrating that the mixture model better captures skewness, kurtosis, and extreme rainfall behavior compared to conventional Gaussian models.
Regional Assessment of Temporal Dependence: Quantifying the persistence of rainfall patterns across different climatic zones (coastal vs. interior) within a unified statistical structure.

Results

Distributional Fit: The GGM provided a superior fit to the observed data compared to a single Gaussian distribution. Goodness-of-fit tests (KS, AD, Chi-square) yielded p-values above the 5% significance level for all regions, indicating no statistically significant difference between observed data and the fitted GGM.
Moment Reproduction: The model successfully reproduced observed statistical moments. Specifically, it captured the high skewness and kurtosis in coastal regions (e.g., Udupi, Dakshina Kannada) and the irregularity in interior drought-prone regions more accurately than a standard Gaussian model.
Level Crossing Analysis: The GGM estimates for upward and downward level crossings (indicating flood and drought conditions) were significantly closer to observed values than those from a pure Gaussian model, which tended to overestimate or underestimate crossing frequencies under extreme conditions.
Temporal Dependence: The bivariate analysis revealed strong empirical and cross-regime correlations in coastal regions (e.g., Udupi showed an empirical correlation of 0.84), indicating high persistence of wet/dry conditions. In contrast, interior regions showed weaker dependence structures. The Gamma–Gamma regime correlations were notably stronger in high-rainfall regions, suggesting extreme events exhibit higher persistence than moderate fluctuations.
2026 Predictions: The model predicted higher SWM rainfall for coastal districts (e.g., Uttara Kannada: 271.60 cm) and lower rainfall for interior districts (e.g., Koppala: 16.32 cm), consistent with historical persistence characteristics.

Significance and Claims
The authors claim that the proposed Gaussian–Gamma Mixture framework offers a robust statistical tool for analyzing rainfall variability and extremes in monsoon regions. By integrating mixture modeling, level-crossing statistics, and bivariate joint probability density, the approach effectively captures both the central tendency and the tail behavior of rainfall distributions. The study asserts that this framework is particularly useful for applications in flood risk assessment, drought monitoring, and water resource management. The paper concludes that the model's ability to represent non-Gaussian characteristics and temporal dependence makes it a valuable alternative to conventional Gaussian models and purely predictive machine learning approaches that lack explicit probabilistic representation of underlying distributions.

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