Joint return levels of maximum temperature and minimum relative humidity by combining copulas with an extreme value framework for bimodal data

Technical Summary: Joint Return Levels of Maximum Temperature and Minimum Relative Humidity in Brasília

Problem Statement
Climate change has intensified the frequency and severity of extreme weather events, particularly in the Federal District of Brazil (Brasília), where episodes of intense heat, prolonged droughts, and low relative humidity pose significant risks to ecosystems and public health. While previous studies have analyzed univariate extremes or bivariate dependencies using standard unimodal distributions, a critical gap exists in modeling the joint behavior of maximum temperature ($X$) and minimum relative humidity ($Y$) when the marginal data exhibit bimodal behavior. Conventional approaches, such as the Generalized Extreme Value (GEV) distribution, assume unimodality and fail to capture the heterogeneous, multi-regime nature of humidity extremes observed in Brasília. Furthermore, standard bivariate models with unimodal margins cannot adequately represent the observed joint density structure, leading to potential biases in risk estimation.

Methodology
The authors propose a novel bivariate framework that combines copula-based dependence modeling with Bimodal Generalized Extreme Value (BGEV) marginal distributions. The methodology proceeds through the following steps:

1. Marginal Modeling:

   • The study utilizes daily data from 2015 to 2024. Extreme values are extracted using a block maxima/minima approach with a block size ($N$) of 64 days, determined via the Ljung–Box test to ensure independence.
   • Maximum temperature ($X$) is modeled using the standard GEV distribution, as its empirical density is unimodal.
   • Minimum relative humidity ($Y$) is modeled using the BGEV distribution (proposed by Otiniano et al., 2025). The BGEV extends the GEV by introducing a shape parameter ($\delta$) that allows the distribution to be unimodal ($\delta=0$) or bimodal ($\delta>0$), accommodating the observed dual-regime behavior of humidity extremes.

2. Dependence Structure (Copula):

   • Following Sklar's theorem, the joint distribution is constructed by linking the marginal distributions ($F_X$ and $F_Y$) via a copula function $C$.
   • The authors evaluated 28 copula families, including Independence, Gaussian, Student-t, Frank, Clayton, Gumbel, Joe, BB1–BB8, and Tawn families, along with their rotated versions (90°, 180°, 270°).
   • Model selection was based on the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).

3. Return Level Calculation:

   • The study defines the extreme event of interest as the simultaneous occurrence of high temperatures and low humidity: $I = [X > x, Y < y]$.
   • Joint return levels are calculated using a conditional approach to solve for the curve $P(X > x, Y < y) = p$, where $p$ is the exceedance probability associated with a specific return period $T$.
   • A representative point on the return-level curve is selected by maximizing the cumulative probability of the temperature variable ($u = F_X(x)$) to emphasize the most extreme temperature scenarios consistent with the modeled dependence.

Key Results

• Marginal Fit: The BGEV distribution provided a superior fit for minimum relative humidity compared to the standard GEV. The GEV failed to capture the bimodal structure, effectively ignoring a secondary peak in the empirical density.
• Copula Selection: The Tawn Type I copula rotated by 270° was identified as the best-fitting model based on AIC and BIC values. This model successfully captured the asymmetry and tail dependence of the data.
• Joint Density Reproduction: The proposed GEV-BGEV model accurately reproduced the two characteristic peaks of the empirical joint density. In contrast, a model using GEV margins for both variables (GEV-GEV) failed to capture the secondary peak, resulting in a distorted representation of the joint structure.
• Return Level Discrepancies: The choice of marginal distribution significantly impacted the estimated return levels. The GEV-GEV model systematically overestimated the minimum relative humidity thresholds for a given return period compared to the GEV-BGEV model. For example, for a 365-day return period, the GEV-GEV model estimated a humidity threshold of 23.48%, whereas the GEV-BGEV model estimated 23.38%. While the percentage difference appears small, the authors note that in the lower tail of the distribution, such discrepancies represent substantial changes in the severity of extreme events.

Significance and Claims
The paper claims that the indiscriminate use of unimodal distributions (like GEV) for all margins in climate extreme analysis can introduce significant biases in the estimation of joint return levels. By integrating the BGEV distribution with copula modeling, the study demonstrates a more robust approach for capturing the complexity of real-world climate data, specifically where bimodality is present.

The primary significance of this work lies in its application to environmental risk monitoring and planning. The authors argue that accurate quantification of the joint occurrence of high temperatures and low humidity is essential for:

• Developing effective climate change adaptation strategies.
• Planning preventive measures against wildfires and heatwaves.
• Informing public health and water resource management in the Federal District.

The study concludes that flexible statistical frameworks, capable of modeling multimodal marginal behaviors, are essential for reliable inference in extreme value analysis, particularly as climate extremes become more frequent and complex. Future research directions suggested include incorporating covariates to account for meteorological and environmental factors influencing these joint behaviors.