Copula-Based Bivariate Kumaraswamy-Teissier Distributions: Modeling Temperature-Rainfall Dependence and Compound Extremes

In the high, rugged terrain of the Northwest Himalayas, the weather does not follow a simple script. Here, the air is thin, the mountains rise sharply, and the climate is a complex tapestry woven from temperature and rainfall. For decades, scientists have tried to understand how these two forces interact, but traditional methods often treated them as separate stories or forced them into rigid, predictable shapes that nature rarely obeys. In reality, a hot day might bring a sudden downpour, or a cold spell might arrive with dry winds, and these relationships change depending on the season and the specific valley. To make sense of this, researchers need tools that can bend and stretch to fit the data, rather than forcing the data to fit a box. This is where a branch of mathematics known as copulas comes in. Think of a copula as a flexible connector that allows scientists to describe how two different things move together, without having to assume they behave in the same way individually. It is a way of mapping the hidden links between variables, especially when those links become critical during extreme events like heatwaves or floods.

A team of researchers from the Indian Institute of Technology Mandi has now applied this flexible approach to the specific challenge of modeling temperature and rainfall in the Northwest Himalayas. They developed two new mathematical frameworks designed to capture the full range of behaviors seen in this region, from the monsoon-driven summers to the winter months influenced by western disturbances. The researchers combined a specific type of statistical curve, which they call the Kumaraswamy–Teissier distribution, with two different types of connectors known as the Clayton and Gumbel copulas. These connectors are special because they can describe not just how variables move together in the middle of their range, but also how they behave at the very edges—when temperatures are scorching or rainfall is torrential. To handle the fact that temperature and rainfall can sometimes move in opposite directions, the team also used rotated versions of these connectors, effectively flipping the mathematical model to capture negative relationships where one variable rises as the other falls.

The study focused on a vast area covering Jammu and Kashmir, Himachal Pradesh, and Uttarakhand, using monthly data from 1984 to 2024. Before building their models, the researchers carefully prepared the data, adjusting for the fact that some months have no rain or freezing temperatures, ensuring the numbers were ready for analysis. They then tested their new models against several existing methods to see which one could best describe the reality on the ground. The results showed that their new approach was superior. In the summer, when the monsoon dominates, the data revealed a complex picture: in the lower foothills, high temperatures tended to coincide with low rainfall, while in the higher mountains, heat and heavy rain often occurred together. In the winter, the pattern shifted, with cold temperatures frequently linked to increased rainfall across much of the region. The new models successfully captured these shifting dynamics, including the specific ways in which extreme values of temperature and rainfall tend to cluster together.

To prove their models were working, the researchers ran thousands of computer simulations, generating fake data that mimicked the real world to see if their mathematical tools could recover the original patterns. The tests confirmed that their methods were accurate and stable, even when working with limited amounts of data. When they applied the best-fitting models to the actual Himalayan data, they found that the new approach outperformed older, more rigid models. The analysis revealed that the relationship between heat and rain is not uniform; it changes from grid to grid and season to season. For instance, in the lower Himalayas during summer, the models identified a strong tendency for hot days to be dry, a pattern that older models might have missed. In contrast, other areas showed a tendency for extreme heat and extreme rain to happen at the same time.

Beyond simply describing the past, the researchers used these models to look at the future risk of extreme events. They calculated what are known as return periods, which estimate how often a specific combination of extreme heat and heavy rain might occur. They looked at scenarios where both temperature and rainfall exceed their usual limits, either individually or together. The findings suggest that in certain parts of the region, particularly in Jammu and Kashmir, the risk of these compound extremes is higher than previously thought. In the winter, for example, the models indicate that within a decade, it is likely that at least one of these variables will exceed its threshold in many areas. The study also explored conditional risks, asking questions like, "If it is raining heavily, how likely is it to be extremely hot?" The answers provided a more nuanced view of danger, showing that the presence of one extreme can either increase or decrease the likelihood of the other, depending on the location and the season.

This work offers a more realistic way to understand the hydro-climatic risks facing mountainous regions. By moving away from rigid assumptions and embracing flexible, data-driven models, the researchers have provided a clearer picture of how temperature and rainfall dance together in the Himalayas. Their approach does not just tell us that extremes happen; it explains how they are linked, offering a robust framework for assessing the risk of floods and heat stress in a changing climate. For the communities living in these high-altitude valleys, understanding these intricate connections is a vital step toward preparing for the weather of tomorrow.