Uncertain Box–Cox Regression for Modeling the Spatial Coupling of Extreme Weather Events and Economic Impacts in the Chengdu-Chongqing Region

Abstract

In the context of ongoing climate change, extreme weather events are becoming increasingly frequent and unpredictable, posing significant challenges for traditional probability-based methods. This study presents an innovative uncertainty-based Box–Cox regression framework to assess the impacts of climate change factors, specifically temperature and precipitation, on the volatility of extreme weather events in the Chengdu-Chongqing region. To address data imprecision, we establish a new estimation theorem for the Extended Least Squares Estimator (ELSE), proving its existence, uniqueness, unbiasedness, and variance consistency under uncertainty theory. The Mann–Kendall trend test is applied to detect event frequency trends, and a coupling coordination degree model is employed to evaluate the dynamic relationship between climate resources and economic development. The results show that (1) temperature has a more significant impact on the volatility of extreme weather events than precipitation; (2) the thermal resource–economy coupling degree has remained above 0.45 since 2015, indicating a strengthening relationship but suboptimal coordination; and (3) since 2014, the water resource–economy coupling degree has consistently exceeded 0.5, reaching optimal levels and highlighting the growing importance of water resources in regional development. Based on these findings, we recommend enhancing extreme weather monitoring systems, improving infrastructure resilience, optimizing climate-related resource management, and fostering regional cooperation. This study provides a rigorous theoretical and empirical basis for integrating uncertainty modeling into climate–economy analysis. Future work should further explore alternative modeling strategies and validate conclusions using extended datasets.

1. Introduction

Climate conditions, as a fundamental component of the natural environment, provide the foundation for Earth’s life-support system and play a critical role in sustainable development. In recent years, climate change has contributed to disruptions in natural ecosystems, socioeconomic volatility, and increased health risks, making it a pressing global challenge. Understanding the complex interactions between climate and human systems is essential for addressing these challenges and promoting sustainable development. However, while extensive research has examined climate change from both global and regional perspectives, relatively few studies have explored the coupling relationships between climate variables and economic development at the regional scale. This gap is particularly relevant in climate-sensitive areas such as the Chengdu–Chongqing region, where economic and environmental dynamics are closely intertwined.

On a global scale, climate change affects multiple sectors, including agriculture, the economy, fisheries, and public health. The Intergovernmental Panel on Climate Change reported that rising global temperatures have intensified competition for water resources and reduced the productivity of staple crops, particularly in arid and semi-arid regions [1]. Additionally, Soares et al. [2] found that climate change has altered the geographic distribution of food production, exacerbating food security challenges in low-income countries. From an economic perspective, Cao et al. [3] highlighted that the continued increase in carbon emissions threatens long-term global economic stability and growth. In the fisheries sector, Mondal and Lee [4] demonstrated that ocean warming and acidification have led to shifts in fish habitats and resource distributions, potentially depleting certain fisheries by 2050. Furthermore, extreme weather events such as heatwaves have been linked to higher incidences of cardiovascular and respiratory diseases worldwide [5]. These findings illustrate the far-reaching and complex implications of climate change across multiple disciplines. Collectively, these findings reveal the widespread and interrelated consequences of climate change, highlighting the urgency of conducting regional analyses to understand its localized impacts and inform targeted adaptation strategies.

In China, climate change has similarly had significant effects on the economy, ecology, and agriculture. From an economic standpoint, Ho et al. [6] found that climate-related risks influence China’s financial markets, with corporate climate risk positively correlated with bond credit spreads. Ecologically, Yang et al. [7] noted that climate change has altered China’s water cycle, leading to reduced runoff, accelerated sediment loss, and disruptions in nitrogen and phosphorus cycles, which may threaten ecosystem stability. In agriculture, studies have documented how climate change affects cropping patterns across different regions, increasing the vulnerability of agroecosystems [8,9,10]. Collectively, these studies highlight the extensive ways in which climate conditions shape China’s natural and socioeconomic systems.

For the Chengdu-Chongqing region, the interplay between climate variability and economic development is of particular concern. The region is experiencing rapid urbanization and industrialization, which increases its vulnerability to climate change [11]. Changes in precipitation and temperature patterns are expected to have profound impacts on key sectors such as agriculture, manufacturing, and transportation, all of which are critical to the region’s economic stability [12]. Additionally, the region’s diverse topography and dense population further complicate the relationship between climate and economic growth [13]. These factors make the Chengdu-Chongqing area an important focal point for studying how regional climate dynamics interact with economic development, and they highlight the need for targeted policies to mitigate the effects of climate change on both the environment and economy.

The increasing frequency and severity of extreme weather events have drawn attention to their defining characteristics and underlying mechanisms. According to the Beijing Municipal Government, extreme weather refers to low-probability meteorological occurrences that are highly destructive, unpredictable, and location-specific. Examples include extreme rainfall, strong winds, hail, lightning, heatwaves, snowfall, cold snaps, fog, sandstorms, and tornadoes (

Table 1. Criteria for judging extreme weather.

Extreme Weather	Criteria of Judgment
Extreme rainfall	Rainstorm Orange and Above Warning Signal, which means that more than 70 mm of rain is expected in 1 h, or more than 100 mm of rain in 6 h, or more than 150 mm of rain in 24 h.
High wind	Wind orange and above warning signal, that is, the average wind is expected to reach more than 10, or gusts above 11.
Hail	Hail Red Warning Signal means that the cumulative hail time is expected to be more than 30 min, the hail diameter is more than 2 cm, and the hail thickness is more than 5 cm in the land area.
Lightning	Lightning Red Warning Signal, that is, there is expected to be strong lightning activity, accompanied by more than 10 short-time winds, or short-time heavy precipitation, or hail.
Extreme heat	High Temperature Red Alert Signal, that is, the daily maximum temperature is expected to rise to 41 °C or above, or the daily maximum temperature is above 37 °C for three consecutive days.
Extreme snowfall	Blizzard Yellow and Above Warning Signal, which is expected to snow more than 6 mm in 12 h.
Cold wave	Cold Wave Orange and Above Warning Signal, that is, the daily minimum temperature is expected to drop 12 °C or more within 24 h, and the daily minimum temperature is expected to drop to 0 °C or below, the average wind is more than 6 levels of cold air activity.
Extreme low temperature	Continuous Low Temperature Yellow Warning Signal, that is, the minimum temperature is expected to be below −12 °C for three consecutive days or more.
Heavy fog	Fog Red Warning Signal, that is, it is expected that strong fog weather may occur within 2 h, visibility is less than 50 m, or fog with visibility less than 50 m has occurred and may persist.
Sandstorm	Yellow Sandstorm Warning Signal and Above, that is, the horizontal visibility of less than 1 km sandstorm weather phenomenon is expected.
Tornado	Determined by whether the event occurred or not. A small swirl of rapidly rotating air with a central wind of more than 100 m/s and a diameter of several to several hundred meters, produced in strongly unstable weather conditions.

). Globally, such events are becoming more prevalent, as seen in record-breaking winter temperatures in Europe, “flash-freeze” cold waves and tornadoes in the United States, tropical cyclones in New Zealand, flooding in Brazil, and wildfires in Canada [14,15,16,17]. According to the World Meteorological Organization, 2023 was the warmest year on record, with the early onset of high temperatures signaling an acceleration in climate change [18]. In China, patterns such as “southern flooding and northern drought” have become more pronounced, while extreme weather events are increasingly common [19]. These developments underscore the growing impact of extreme weather on societies, economies, and ecosystems worldwide.

Given its complexity and destructive potential, extreme weather has been widely studied, with researchers focusing on its triggers, impact mechanisms, and mitigation strategies. Studies indicate that extreme weather results from both nonlinear climate system processes and anthropogenic influences. Global warming has been linked to increased frequency and intensity of heatwaves and extreme rainfall events [20]. In terms of impact mechanisms, extreme weather disrupts agricultural production, critical infrastructure, and human livelihoods. For example, Ahmed et al. [21] explored strategies to improve public transportation accessibility during extreme weather, while Masanja et al. [22] developed predictive models for heatwaves to mitigate risks for fisheries and coastal communities. Adaptive strategies range from enhancing urban infrastructure resilience to optimizing power grid recovery and strengthening early-warning systems [23,24,25]. These studies provide valuable insights into understanding and managing extreme weather events.

To better quantify extreme weather risks, researchers have employed probability-based methods such as extreme value theory, Bayesian models, and stochastic climate simulations [26,27,28]. However, conventional probabilistic approaches may not fully capture the inherent uncertainties of climate phenomena, particularly in the presence of incomplete data, significant spatiotemporal variability, or evolving climate conditions. These limitations can reduce model robustness and affect the reliability of predictions.

To address these challenges, Liu [29] introduced uncertainty theory, a mathematical framework designed to model uncertainty in complex systems where traditional probability theory may be insufficient. In climate research, uncertainty theory has been applied to model extreme rainfall events [30], optimize water resource allocation under climate change [31], and improve predictions of agricultural yields [32]. These studies suggest that uncertainty theory can complement traditional probabilistic methods by accounting for additional complexities inherent in climate variability, providing a stronger foundation for extreme weather modeling.

Despite significant research on the socioeconomic impacts of extreme weather and adaptation strategies, systematic quantitative analyses of extreme weather occurrence patterns remain limited. In particular, the coupling relationships between climate factors and regional economic systems warrant further investigation. To address this gap, this study focuses on the Chengdu–Chongqing region, a climate-sensitive area with rapid economic development and complex environmental interactions. This study seeks to address the following research questions: (1) How do temperature and precipitation influence the frequency of extreme weather events in the Chengdu-Chongqing region? (2) What is the coupling effect between climate variables and economic development? A key contribution of this study is the integration of uncertainty theory into an analytical framework for extreme weather. Using an uncertain Box–Cox regression model to quantify climate-extreme weather interactions and a coupled coordination degree model to assess economic–climate resource interactions, this research seeks to provide novel insights that may support sustainable regional development strategies.

2. Materials and Methods

2.1. Study Area

The Chengdu–Chongqing region, located in southwest China, spans 27°39′–32°19′ N latitude and 101°56′–109°14′ E longitude, covering an area of approximately 186,087 ${\mathrm{km}}^2$. It encompasses the entire administrative regions of Chengdu and Chongqing, situated within Sichuan Province [33]. Figure 1 shows a general overview of the study area. The region features diverse topographical characteristics, including plains and hills within the Sichuan Basin and karst mountains in Chongqing. Major rivers such as the Yangtze River, Min River, and Jialing River provide abundant water resources, playing a crucial role in supporting both the regional economy and agricultural production [34,35]. The Chengdu–Chongqing region experiences a subtropical humid monsoon climate, with an average annual temperature ranging from 16 °C to 18 °C and annual precipitation between 900 mm and 1200 mm. In recent years, the increasing frequency of extreme weather events has significantly affected both ecosystems and economic activities in this region [36].

As the economic center of the upper Yangtze River basin, the Chengdu–Chongqing region serves as a key driver of economic growth in western China and represents a strategic core area for national economic and security development [37]. Benefiting from a well-developed transportation network and abundant natural resources, the region’s economy is primarily driven by electronic information industries, automobile manufacturing, and agriculture [38,39]. However, in the context of climate change, the region faces increasing risks from extreme weather events, which pose challenges to economic resilience and environmental stability. By selecting the Chengdu–Chongqing region as the study area, this research aims to explore the driving mechanisms of temperature and precipitation in influencing extreme weather events and to analyze the coupling effects of climate variability on the regional economic system. The findings are expected to provide a scientific basis for promoting coordinated economic and ecological development in Southwest China.

2.2. Data

This study utilizes annual temperature and precipitation data for the Chengdu–Chongqing region from 2006 to 2021, obtained from the China Meteorological Statistical Yearbook (2007–2022) and the China Economic Database. The spatial resolution of the temperature and precipitation data is 1 km × 1 km, and any missing spatial values were supplemented using inverse distance weighted (IDW) interpolation. Regional economic data, specifically gross domestic product (GDP), were sourced from the National Bureau of Statistics and the Statistical Yearbooks of Chongqing and Chengdu (2007–2022).

Since direct regional estimates of annual precipitation, annual average temperature, and regional GDP for the Chengdu–Chongqing region are not readily available, this study employs a weighted average approach to enhance accuracy. The weighted average method is commonly used to improve the representation of climate data in heterogeneous regions, particularly when spatial and population factors influence climate outcomes, as shown in studies like Zhong et al. [40] Given the influence of urbanization and population distribution on regional climate, the weighted average method integrates urban area and population factors to derive comprehensive climate indicators for the Chengdu–Chongqing region. Additionally, the GDP data for Chongqing and Chengdu are aggregated directly to estimate the total regional economic output [41].

To ensure consistency and reliability, all variables are normalized to eliminate dimensional differences. Furthermore, inverse distance weighted (IDW) interpolation is applied to correct time-series outliers, ensuring data integrity and accuracy.

2.2.1. Precipitation, Temperature, and GDP

Existing studies indicate that geographical area and population size are significant factors influencing regional climate characteristics [42]. Larger geographical areas often correspond to more complex topography and microclimate variations, while higher population densities may intensify urban heat island effects and contribute to uneven precipitation distribution. For instance, densely populated urban areas may experience higher temperatures and altered precipitation patterns due to factors such as land use, infrastructure, and human activity [43].

Given the interaction between area, population, and climate characteristics, this study employs an area-population weighting method to determine the combined influence of precipitation and temperature on regional climate variability. The rationale behind using area and population as weighting factors is based on the understanding that these factors jointly contribute to the variability in climate conditions. Larger areas typically experience a wider range of climate conditions due to diverse geographic features such as mountains, rivers, and elevation gradients, which affect temperature and precipitation patterns. On the other hand, population size reflects the degree of urbanization, which can modify local climate through urban heat island effects, energy consumption, and other human-induced changes [44]. Therefore, by weighing precipitation and temperature data according to area and population, the study accounts for the heterogeneity of climate impacts across the region, providing a more accurate representation of regional climate dynamics.

According to the statistical yearbooks of Chongqing and Chengdu, the total area of Chongqing’s administrative division is approximately 82,403 ${\mathrm{km}}^2$, with an average population of about 30 million from 2006 to 2021. Meanwhile, the total area of Chengdu’s administrative division is 14,335 ${\mathrm{km}}^2$, with an average population of approximately 16.45 million over the same period. The weighted precipitation and average annual temperature data for the Chengdu–Chongqing region from 2006 to 2021 are presented in

Table 2. Relevant data of the Chengdu-Chongqing region from 2006 to 2021.

Time (t)	Precipitation (mm)	Average Annual Temperature (°C)	GDP (100 Million Yuan)
2006	846.41	18.17	6705.13
2007	1111.35	17.85	8246.03
2008	1099.50	17.36	10,106.44
2009	951.15	17.76	11,389.68
2010	1010.50	17.27	13,954.72
2011	996.01	17.25	17,506.49
2012	954.02	16.80	20,214.97
2013	1139.72	18.10	22,478.26
2014	1238.80	17.12	24,992.21
2015	1135.01	18.40	26,702.85
2016	1208.67	17.62	29,897.11
2017	1186.88	17.42	33,997.68
2018	1163.87	17.32	37,287.74
2019	1120.38	17.30	40,616.43
2020	1333.04	17.20	42,879.43
2021	1210.71	17.45	48,039.59

, with visual representations shown in Figure 2. These results provide insights into the regional climate trends and serve as a foundation for further examining the coupling relationship between climate variability and economic development in the region.

According to the data presented in Table 2 and Figure 2, the annual precipitation in the Chengdu–Chongqing region exhibited an overall fluctuating upward trend from 2006 to 2021, reaching its peak in 2020. This trend may indicate an increase in extreme precipitation events in recent years, aligning with broader patterns observed under global climate change.

The annual average temperature followed a nonlinear pattern, initially declining before experiencing a subsequent recovery. This fluctuation may be attributed to a combination of factors, including the intensification of the urban heat island (UHI) effect and increased short-term cold air activity.

Similarly, the GDP data for the Chengdu–Chongqing region are presented in Table 2 and Figure 3, providing further insights into the relationship between climate variability and economic development.

As shown in Figure 3, the GDP of the Chengdu–Chongqing region exhibited a predominantly linear increasing trend throughout the study period.

2.2.2. Number of Extreme Weather Events

Extreme weather events have been defined using various methodologies, including the absolute threshold method, percentile method, and historical ranking method [45,46,47]. Among these, the absolute threshold method is widely utilized in extreme weather research due to its simplicity, intuitive nature, and ease of implementation. This method identifies extreme weather events by comparing temperature or precipitation values against predetermined fixed thresholds, making it a practical approach for climate studies.

Given its advantages, this study adopts the absolute threshold method to define extreme weather events in the Chengdu–Chongqing region, considering its subtropical humid monsoon climate. The following extreme weather types are selected as research objects: extreme precipitation, extreme high temperature, extreme low temperature, drought, strong wind, and hail.

(1) Definition of extreme weather events

In this study, extreme weather events are defined based on regional characteristics and relevant literature, with specific threshold values as follows:

• Extreme high temperature: daily maximum temperature $\ge 40$ °C;
• Extreme low temperature: daily minimum temperature $\le -12$ °C;
• Extreme precipitation: daily precipitation $\ge 50$ mm.

The primary data sources include the China Meteorological Yearbook (2007–2022). Due to the uncertainty in extreme weather events, actual statistics require estimation based on yearbook records and relevant assumptions.

(2) Data processing methods

• Extreme heat events

According to the China Meteorological Yearbook, the number of hot summer days with daily maximum temperatures $\ge 40$ °C reached a historical high. Additionally, 22 districts and counties recorded more than 10 days of extreme high temperatures, while 30 districts and counties experienced at least one instance of extreme heat. Based on the lowest hypothesis, this study defines the number of extreme high-temperature events as 10 occurrences for that year.

• Extreme precipitation correction

The yearbook states that 52 rainstorms were recorded in Chongqing in 2006. However, “station times” refers to the number of rainstorm events observed at multiple meteorological stations, meaning the same event could be recorded multiple times at different locations, leading to potential double-counting. To correct this issue, this study applies a modified method, defining the actual number of rainstorm events as half of the recorded station times, resulting in an estimated count of between 26 and 52 rainstorm events.

• Drought event

The yearbook description states that “the city suffered from the worst drought in 100 years”. Since droughts typically affect the entire region and are not repeated occurrences within the same year, this study defines the total number of drought events as one for 2006.

• Hail event

According to the China Meteorological Yearbook, hailstorms occurred on 11 April, 4 May, and 26–27 June. Since these events took place on different dates and were independent occurrences, this study defines the total number of hail events as four in 2006.

(3) Data calculation

By integrating the above definitions, the estimated number of extreme weather events in Chongqing in 2006 is (41, 67), including extreme high-temperature events, heavy rainfall, drought, and hailstorms. The same data processing logic is applied for other years and for Chengdu, where extreme weather event data refer to records from the Panxi region of Sichuan Province.

In summary, the dataset contains inherent uncertainties, as shown in

Table 3. The number of imprecise occurrences of extreme weather in the Chengdu-Chongqing region from 2006 to 2021.

Time (t)	Times
2006	(41, 67)
2007	(72.5, 127)
2008	(58, 90)
2009	(67, 117)
2010	(63, 79)
2011	(73, 120)
2012	(56, 74)
2013	(49, 79)
2014	(75, 151)
2015	(85, 158)
2016	(104.5, 182)
2017	(51, 76)
2018	(60.5, 101)
2019	(78, 152)
2020	(91.5, 169)
2021	(65, 84)

and Figure 4, requiring careful interpretation when analyzing extreme weather trends.

In Figure 4, the red line represents the lower bound of the frequency range, the blue line denotes the upper bound, and the green line indicates the group median. These visual representations highlight the uncertainty in the frequency of extreme weather events, reflecting potential variability in data collection and estimation methods.

The illustration above provides an overview of the dataset used in this study. The research methodology employed for further analysis will be detailed in the following sections.

2.3. Methods

2.3.1. Uncertain Box–Cox Regression Model

In uncertainty theory, let $(x_1,x_2,\dots ,x_p)$ be a vector of a set of uncertain explanatory variables, y be an uncertain response variable. It is assumed that y can be expressed as:

$$y=f(x_1,x_2,\dots ,x_p\mid \mathsf{\beta })+\epsilon$$

(1)

where f is a linear or nonlinear function, $\mathsf{\beta }$ is an unknown parameter, and $\epsilon$ is an uncertain disturbance term (uncertain variable). Given that traditional regression models typically assume normality of errors, this assumption may not always hold when dealing with imprecise data. To address this limitation, Box and Cox [48] proposed the Box–Cox transformation as a preprocessing technique to normalize data. More recently, Fang et al. [49] introduced an uncertain Box–Cox model that extends this transformation to handle uncertain variables more effectively. The model is given by:

$$H(y;\tau )=f(x_1,x_2,\dots ,x_p\mid \mathsf{\beta })+\epsilon$$

(2)

where the Box–Cox transformation $H(y;\tau )$ is defined as:

$$H(y,\tau )=\left\{\begin{array}{cc}(y^{\tau }-1)/\tau ,\hfill & \tau \ne 0\hfill \\ logy,\hfill & \tau =0\hfill \end{array}\right.$$

(3)

Here, $\tau$ represents the Box–Cox transformation parameter. Since the available observations are imprecise, we denote them as $({\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip},{\tilde{y}}_i)$, where each ${\tilde{x}}_{ij}$ is assumed to be independent across observations ($i=1,2,\dots ,n$).

The Box–Cox regression model transforms a nonlinear relationship into a linear relationship by modifying the dependent (or independent) variable. This transformation allows the model to be applied within a traditional linear regression framework, enhancing its interpretability. The selection of an optimal transformation parameter $\tau$ enables automatic adaptation to the most suitable functional form. Additionally, the transformation can help reduce or eliminate heteroscedasticity, thereby improving the validity of the model assumptions.

By integrating the Box–Cox model with uncertainty theory, the proposed approach provides a more robust framework for handling fuzzy or uncertain data distributions, which are common in empirical climate and economic studies.

Compared to alternative approaches, such as fuzzy logic and robust statistics, uncertainty theory provides distinct advantages in addressing imprecise climate data. While fuzzy logic focuses on modeling ambiguity and approximate reasoning, it can be limited when dealing with stochastic data or when precise statistical interpretation is required. Robust statistics, on the other hand, are useful for handling outliers but may not be well-suited to the uncertainty inherent in environmental data, where variability is often not purely random. In contrast, uncertainty theory provides a systematic way to quantify and model the inherent imprecision in climate data, making it more suitable for this type of analysis.

It should be noted that the uncertain Box–Cox model is particularly applicable when the dependent variable is positive. The method is also useful in scenarios where nonlinear relationships or heteroscedasticity pose challenges, allowing for improved model fit through appropriate transformations.

2.3.2. Parameter Estimation

Estimating unknown parameters $\mathsf{\beta }$ and $\tau$ in the uncertain Box–Cox model presents significant challenges, as conventional least squares methods often yield suboptimal estimation accuracy due to model misspecification [49]. Building upon existing methodologies, this study proposes an extended least squares estimation (ELSE) approach that systematically enhances parameter estimation accuracy through iterative weight optimization and bias correction mechanisms.

Theorem 1.

For the set of imprecisely observed data $({\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip},{\tilde{y}}_i)$, $i=1,2,\dots ,n$, where ${\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip},{\tilde{y}}_i$ are independent uncertain variables with regular uncertainty distributions ${\mathsf{\Phi }}_{i1},{\mathsf{\Phi }}_{i2},\dots ,{\mathsf{\Phi }}_{ip},{\mathsf{\Psi }}_i$, $(i=1,2,\dots ,n)$, respectively, the ELSE of β and τ is obtained by solving the following minimization problem:

$$\underset{\mathsf{\beta },\tau }{min}\left\{{\left\{{\displaystyle \sum _{i=1}^n}E{[H({\tilde{y}}_i;\tau )-f({\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip}|\mathsf{\beta })]}^2\right\}}^{n/2}-exp\left\{\tau {\displaystyle \sum _{i=1}^n}E[log\left({\tilde{y}}_i\right)]\right\}\right\}$$

(4)

where

$${\gamma }_{ij}^{-1}(\alpha ,{\beta }_j)=\left\{\begin{array}{cc}{\mathsf{\Phi }}_{ij}^{-1}(1-\alpha ),\hfill & \mathit{if}{\beta }_j\ge 0\hfill \\ {\mathsf{\Phi }}_{ij}^{-1}\left(\alpha \right),\hfill & \mathit{if}{\beta }_j<0\hfill \end{array}\right.$$

(5)

for $i=1,2,\dots ,n$ and $j=1,2,\dots ,p$, here ε is the error term with zero mean and finite variance.

The detailed proof is provided in Appendix A.

Following parameter estimation, the model is evaluated using residual analysis and repeated k-fold cross-validation to assess its predictive performance.

Given a set of imprecise observations $({\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip},{\tilde{y}}_i)$, the uncertain Box–Cox regression model is expressed as:

$$H(y;{\tau }^{\ast })=f(x_1,x_2,\dots ,x_p\mid {\mathsf{\beta }}^{\ast })+\epsilon$$

(6)

where ${\mathsf{\beta }}^{\ast }$ and ${\tau }^{\ast }$ are the re-estimated parameters obtained via least squares adjustment. For each observation i $(i=1,2,\dots ,n)$, the residuals are computed as:

$${\tilde{\epsilon }}_i=H({\tilde{y}}_i;{\tau }^{\ast })-f({\tilde{x}}_{i1},{\tilde{x}}_{i2},\dots ,{\tilde{x}}_{ip}\mid {\mathsf{\beta }}^{\ast })$$

(7)

Given that $\epsilon$ is an uncertain variable, its expectation and variance can be estimated as:

$$\widehat{e}=1/n{\displaystyle \sum _{i=1}^n}E\left[{\tilde{\epsilon }}_i\right]\mathrm{and}{\widehat{\sigma }}^2=1/n{\displaystyle \sum _{i=1}^n}E\left[{({\tilde{\epsilon }}_i-\widehat{e})}^2\right]$$

(8)

where ${\tilde{\epsilon }}_i$ represents the i-th residual.

The ELSE approach is computationally intensive due to the iterative nature of the weight optimization and bias correction process. Several considerations need to be addressed to ensure efficiency and robustness in the estimation process:

(1) Convergence Issues.

As with many iterative optimization methods, ELSE is susceptible to convergence issues, particularly when the model involves large datasets or highly uncertain data. Convergence can be slow, especially when the initial parameter estimates are far from the true values. To mitigate this, careful initialization of parameters and adaptive learning rates can be employed to help the algorithm converge faster. Moreover, convergence criteria should be carefully defined to avoid excessive computation when the changes in parameter estimates become minimal.

(2) Computational Intensity.

The ELSE method requires repeated evaluations of the expected values, which involves integrating over the uncertainty distributions. For large datasets or complex models, this can be computationally expensive. Efficient numerical methods, such as Monte Carlo simulations or Gaussian quadrature, can be used to approximate the required expectations, thus reducing the computational burden. Parallel processing techniques can also be implemented to handle the high computational load by distributing the calculations across multiple processors.

(3) Numerical Stability.

The presence of uncertainty in the data can introduce numerical instability during the optimization process. Small errors in the uncertain data can propagate and lead to large variations in parameter estimates. To address this, regularization techniques may be applied to stabilize the estimation process and prevent overfitting to noisy data.

(4) Model Complexity.

The complexity of the ELSE method increases with the number of uncertain variables and the dimensionality of the parameter space. As the model becomes more complex, the optimization problem may become more difficult to solve. Dimensionality reduction techniques, such as principal component analysis (PCA), can be useful in reducing the number of variables considered during optimization, improving the computational efficiency of the method.

2.3.3. Model Evaluation and Validation

To ensure model robustness, the performance of the uncertain Box–Cox regression model is assessed using repeated k-fold cross-validation, a technique widely used in machine learning for evaluating model generalization [50]. The validation process follows these steps:

(1)

The dataset is randomly shuffled and divided into k subsets.

(2)

Each subset is used once as a test set, while the remaining $k-1$ subsets serve as the training set.

(3)

The model is trained and tested iteratively, and evaluation scores are recorded.

(4)

The final performance is summarized using key evaluation metrics.

To assess model accuracy, the following evaluation indicators are used:

(1)

Mean Squared Error (MSE)

(2)

Mean Absolute Error (MAE)

(3)

Root Mean Squared Error (RMSE)

For residual analysis, a well-fitted model should exhibit no autocorrelation or heteroscedasticity and should approximately follow a normal distribution. If these assumptions hold, the model can be considered valid and effective. Additionally, lower values of MSE, MAE, and RMSE indicate a better model fit and improved predictive performance.

2.3.4. Mann-Kendall Test

The Mann-Kendall (M-K) test is a nonparametric statistical method used to detect significant trends in time series data without requiring the data to follow a specific probability distribution [51]. This test is particularly effective for identifying monotonic trends in environmental and climatic datasets, as it is robust against non-normality and can accommodate missing values.

Assuming that a time series is represented as $X_1,X_2,\dots ,X_n$, the M-K trend test is computed as follows [51]:

$$S_k={\displaystyle \sum _{k=1}^{n-1}\sum _{j=k+1}^n}\mathrm{sgn}(x_i-x_j)$$

(9)

where the sign function is defined as:

$$\mathrm{sgn}(x_i-x_j)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{x}_i-x_j>0\hfill \\ 0,\hfill & \mathrm{if}{x}_i-x_j=0\hfill \\ -1,\hfill & \mathrm{if}{x}_i-x_j<0\hfill \end{array}\right.$$

(10)

The variance of $S_k$ is calculated as:

$$Var\left(S_k\right)=[n\times (n-1)\times (2n+5)]/18$$

(11)

The standardized test statistic Z is given by:

$$Z=\left\{\begin{array}{cc}(S_k-1)/\sqrt{Var\left(S_k\right)},\hfill & \mathrm{if}{S}_k>0\hfill \\ 0,\hfill & \mathrm{if}{S}_k=0\hfill \\ (S_k+1)/\sqrt{Var\left(S_k\right)},\hfill & \mathrm{if}{S}_k<0\hfill \end{array}\right.$$

(12)

where:

(1)

$S_k$ represents the cumulative total number of event pairs where the i-th value is greater than the j-th value in the time series X.

(2)

$\mathrm{sgn}(x_i-x_j)$ is a sign function that assigns a value based on the difference between observations.

(3)

$\mathrm{Var}\left(S_k\right)$ is the variance of the test statistic $S_k$.

(4)

Z is the standardized test statistic, which follows a standard normal distribution under the null hypothesis of no trend.

A positive value of Z ($Z>0$) suggests an increasing trend, whereas a negative value ($Z<0$) indicates a decreasing trend. The significance of the detected trend is evaluated based on confidence levels ($\alpha$):

(1)

If $\left|Z\right|\ge 1.64$, the trend is significant at the 90% confidence level.

(2)

If $\left|Z\right|\ge 1.96$, the trend is significant at the 95% confidence level.

(3)

If $\left|Z\right|\ge 2.58$, the trend is significant at the 99% confidence level.

This method provides a robust statistical framework for detecting climate and environmental trends, making it widely applicable in hydrology, meteorology, and ecological studies.

2.3.5. Coupling Coordination Degree Method

The coupling coordination degree is commonly used to analyze and evaluate the coupling relationship between climate resources and the local economy [52]. Initially, the entropy method is applied to calculate the comprehensive evaluation index for both climate resources and the local economy. Suppose the climate resources or local economy involves m indicators, each covering n years of data, resulting in the matrix X:

$$X_{ij}=\left[\begin{array}{ccc}{X}_{11}& \dots & X_{1n}\\ ⋮& \ddots & ⋮\\ X_{m1}& \dots & X_{mn}\end{array}\right]$$

(13)

The entropy $e_i$, diversity coefficient $g_i$, and comprehensive weight $W_i$ of the i-th index are calculated as follows:

$$e_i=-\left(1/logn\right){\displaystyle \sum _{i=1}^n}{p}_{ij}log\left(p_{ij}\right)$$

(14)

$$g_i=1-e_i$$

(15)

$$W_i=g_i/{\displaystyle \sum _{i=1}^m}{g}_i$$

(16)

where $p_{ij}$ represents the proportion corresponding to the data of the i-th index in the j-th year, with $e_i$ taking values in the range (0, 1), and the larger $g_i$ is, the more important the index. The overall assessment index for climate resources (or local economy) is then calculated as:

$$Z_j={\displaystyle \sum _{i=1}^m}{W}_i{X}_{ij}^{\prime }$$

(17)

When $Z_j$ represents the development level of climate resources, it is denoted by $CR$; when $Z_j$ represents the development level of the local economy, it is denoted by $LE$.

The coupling coordination degree between climate resources and the local economy is calculated as:

$$C=2\sqrt{CR\times LE}/(CR+LE)$$

(18)

$$T={\delta }_{1}CR\times {\delta }_{2}LE$$

(19)

$$D=\sqrt{C\times T}$$

(20)

where C stands for the coupling degree, T is the comprehensive coordination index, ${\delta }_1$ and ${\delta }_2$ are the contribution coefficients of climate resources and the local economy, respectively. In this study, we set ${\delta }_1={\delta }_2=0.5$, and D represents the coupling coordination degree, with values in the range of $[0,1]$.

Since climate resources are closely related to environmental change and the compound benefits or development level of climate and the economy, this study refers to Liao’s [53] work to define the criteria for the coupling coordination model, as shown in

Table 4. Criteria for coupling coordination degree between climate resources and local economy.

Degree of Coupling (C)	Hierarchy of Coupling	Coupling Coordination Degree (D)	Degree of Coupling Coordination
[0, 0.3)	Low level coupling	[0, 0.1)	Extreme dissonance
		[0.1, 0.2)	Serious dissonance
		[0.2, 0.3)	Moderate disorder
[0.3, 0.5)	Antagonize	[0.3, 0.4)	Transition from low to moderate coordination
		[0.4, 0.5)	Borderline disorder
[0.5, 0.8)	Running-in	[0.5, 0.6)	Forced coordination
		[0.6, 0.7)	Primary coordination
		[0.7, 0.8)	Intermediate coordination
[0.8, 1]	High level coupling	[0.8, 0.9)	Good coordination
		[0.9, 1]	Quality coordination

.

In this context, Antagonize refers to a phase characterized by significant misalignment or conflict between resources and economic development, resulting in inefficiency and a lack of synergy. Running-in refers to the phase where resources and economic development begin to gradually adapt and coordinate, with noticeable improvement in their interaction, though the level of coordination remains low.

While the coupling coordination model provides valuable insights into the relationship between climate resources and local economic development, it does have certain limitations. One notable challenge is the use of expert scoring to generate indicators, which introduces an element of subjectivity. However, expert judgment, based on years of experience and domain knowledge, can enhance the accuracy of predictions, especially when data is limited or complex factors need to be considered.

Moreover, combining data-driven methods with expert input can mitigate some of this subjectivity. The data-driven approach offers empirical validation to expert assessments, ensuring that the model is both statistically sound and informed by expert insight. This combination improves the overall predictive power of the model.

Another limitation is that the model typically performs static analysis, treating the coupling relationship as fixed. Incorporating a dynamic approach could better reflect the changing nature of these systems over time. Additionally, while the model is effective, there is limited sensitivity analysis on weighting and parameter selection, which may impact its stability. Further exploration of these factors could strengthen the robustness of the results.

It should be noted that while the entropy method provides an effective way to calculate the weights and diversity indices, it is sensitive to the quality of input data, and slight variations in the data can influence the results. This may limit its robustness in scenarios with uncertain or noisy data. Future improvements may involve integrating complementary methods or refining the entropy-based calculations to mitigate these issues.

Despite these limitations, the coupling coordination degree model remains a valuable tool for assessing climate and economic interactions. Addressing these challenges in future iterations could improve its accuracy and adaptability for policy-making and sustainable development.

Figure 5 illustrates the basic framework of the methodology employed in this study.

3. Results

3.1. Impacts of Climate Change Factors on the Volatility of Extreme Weather Events

The fluctuations in extreme weather events are the result of multiple interacting factors. Among natural drivers, atmospheric circulation models, including phenomena like El Niño and the abnormal behavior of the Western Pacific subtropical high, significantly influence the occurrence and variation of extreme weather. Additionally, under the backdrop of global warming, climate factors such as temperature and precipitation are contributing to changes in the frequency and intensity of extreme weather events.

Human activities, especially rapid urbanization, exacerbate these impacts. Urbanization leads to an intensification of the heat island effect, altering local climates and increasing the volatility of extreme weather events. Over-exploitation of natural resources, such as forests and water, can further impact the occurrence and intensity of these events.

In this study, we focus on the fluctuating effects of climate factors, particularly precipitation and temperature, on extreme weather events. In the Chengdu-Chongqing region, factors such as heavy precipitation, intense rainfall, and high temperatures are identified as the key drivers of extreme weather variability. Therefore, we further explore the relationship between precipitation and temperature, and their correlation with the frequency of extreme weather events.

3.1.1. Research Methods and Data Sources

This study employs an uncertain Box–Cox linear regression model to analyze the relationship between precipitation, air temperature, and the frequency of extreme weather events. The model accounts for potential nonlinear relationships in the data and incorporates uncertainty in the regression analysis results. By using this model, we explore the fluctuating relationship between climate factors and extreme weather events.

The uncertain Box–Cox linear regression model is constructed to examine the relationship between precipitation and temperature with the frequency of extreme weather events. The model is formulated as follows:

$$H(\tilde{y};\tau )={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon$$

(21)

where $\tilde{y}$ is the number of extreme weather events, $x_1$ is the annual precipitation, $x_2$ is the annual average temperature, $\tau$ and ${\beta }_i(i=0,1,2)$ are unknown parameters to be estimated, $\epsilon$ is the uncertain disturbance term, and the significance level $\alpha =0.05$ is taken. This model allows for an integrated analysis of precipitation and temperature, and their fluctuating impact on extreme weather events.

3.1.2. Parameter Estimation and Model Fitting

After parameter estimation using Python 3.12.7, the results are presented in

Table 5. Parameter estimates.

Parameter Notation	Symbol of the Estimated Value	Parameter Estimates
$\tau$	$\widehat{\tau }$	0.1338
${\beta }_0$	${\widehat{\beta }}_0$	1.0038
${\beta }_1$	${\widehat{\beta }}_1$	0.0024
${\beta }_2$	${\widehat{\beta }}_2$	0.1387

.

A Shapiro–Wilk test was conducted on the residuals, yielding a p-value of 0.3832, which is greater than 0.05, indicating that the residuals follow a normal distribution. This is further supported by the histogram of residuals and the QQ plot shown in Figure 6. The histogram reveals that the majority of residuals are concentrated around zero, with a relatively symmetrical distribution. Additionally, the points on the QQ plot align closely along the 45-degree line, further confirming that the residuals adhere to a normal distribution.

The residuals were subsequently tested for autocorrelation and heteroscedasticity. The Durbin-Watson test yielded a statistic of 2.2211, which is close to 2, indicating minimal autocorrelation between the residuals. Additionally, the residual plot (Figure 7) and residual box plot (Figure 8) show that the residuals are randomly distributed around the zero line with no discernible trend, suggesting that the residuals meet the assumption of homoscedasticity.

3.1.3. Model Diagnosis and Validation

Based on the residual analysis, we initially believe that the uncertain Box–Cox linear regression model fits the data well.

Subsequently, repeated k-fold cross-validation was performed, and the results are presented in

Table 6. Cross-validation index values.

	Value
MSE	10.1990
MAE	1.2821
RMSE	1.4075

.

The MSE (Mean Squared Error) measures the mean squared difference between predicted and actual values, with lower values indicating better prediction accuracy. As shown in Table 6, the model performs well on the test set, as indicated by the relatively low MSE. The MAE (Mean Absolute Error) measures the average absolute difference between predicted and actual values. Compared to MSE, MAE directly reflects the magnitude of errors, and its relatively small value further supports the model’s good performance.

Additionally, RMSE (Root Mean Squared Error), which is the square root of MSE, provides error values in the same units as the original data, making it easier to interpret. Despite its simplicity, RMSE retains sensitivity to large errors and, therefore, reflects the model’s effectiveness. Based on these performance metrics, the uncertain Box–Cox linear regression model demonstrates a strong ability to capture the relationship between precipitation, temperature, and the frequency of extreme weather events.

3.1.4. Result Analysis and Discussion

The results presented above indicate that the frequency of extreme weather events is positively correlated with precipitation and average temperature. Although the coefficients for both precipitation and temperature are relatively small, this does not necessarily indicate a weak relationship. In regression analysis, the magnitude of the coefficient can be influenced by the overall variability of the data, model structure, and the presence of other correlated variables. Therefore, even small coefficients can reflect meaningful relationships, particularly when multiple factors interact and data variability is significant. The positive correlation for precipitation suggests that the number of extreme precipitation events, such as rainstorms, has increased, which aligns with the observed trends. While the coefficient for temperature is larger, it still indicates a moderate effect compared to other variables, reflecting its contribution to extreme weather variability alongside precipitation. The smaller coefficient for precipitation reflects its relatively limited role in the overall composition of extreme weather events.

On the other hand, the increase in average temperature correlates with a rise in the occurrence of extreme weather, suggesting that extreme high-temperature events have become more frequent than extreme low-temperature events in the Chengdu-Chongqing region. Additionally, the increase in temperature may be accompanied by a reduction in water circulation, which can lead to a decline in extreme precipitation events. This relationship highlights a potential feedback mechanism, where extreme heat and precipitation may trigger other extreme weather events, such as droughts, strong winds, and fog.

In conclusion, both precipitation and high temperature are significant factors influencing extreme weather in this region. The observed increase in extreme weather events is largely driven by the rising frequency of extreme precipitation and extreme high-temperature events. Among these, extreme high temperatures appear to have a more substantial impact on the occurrence of extreme weather.

The predominance of extreme precipitation and extreme high temperatures in the Chengdu-Chongqing region further underscores the direct relationship between these climatic factors and extreme weather. This reinforces the relationship between precipitation, annual average temperature, and the occurrence of extreme weather events, as considered in this study. Figure 9 visually illustrates the relationship between major climate factors and extreme weather events.

Based on the data trends in the table and figures, it is evident that the average annual temperature in the Chengdu-Chongqing region has fluctuated within a certain range since 2006, remaining close to the long-term average. This fluctuation may result from the interaction between extreme high-temperature and low-temperature events. Precipitation, in contrast, shows a trend of gradual increase with fluctuations, reflecting the growing frequency of extreme precipitation or heavy rainfall events in recent years.

Although the fluctuations in the frequency of extreme weather events are increasing, the mean value remains roughly stable. The Mann-Kendall (M-K) trend test yields a statistic of Z = 0.0000, indicating that no significant trend change has occurred in the frequency of extreme weather events from 2006 to 2021. This suggests the potential presence of a complex causal relationship between temperature, precipitation, and the occurrence of extreme weather events. Despite relatively stable temperature levels during the study period and the increasing trend in precipitation, particularly in extreme precipitation events, the overall incidence of extreme weather events has remained stable in the long term. This stability may be due to limiting factors such as geographical environment and climate feedback mechanisms, which prevent large changes in the frequency of extreme events.

This highlights the inherent complexity of the climate system and the uncertainty surrounding extreme weather events.

Given the trends in precipitation and average temperature in the Chengdu-Chongqing region, coupled with the fluctuating nature of extreme weather events, it is evident that the effects of climate change on regional weather will intensify as global warming continues. Global warming is not only driving temperature anomalies but also influencing atmospheric circulation patterns. As a result, the frequency and intensity of extreme weather events are likely to increase, posing significant threats to agricultural production, ecosystems, and the livelihoods of residents.

Without timely and effective mitigation measures, extreme weather events will become more frequent and severe, leading to environmental damage and adverse effects on economic development and social stability at regional, national, and global levels. Therefore, there is an urgent need for stronger climate policies and preparedness strategies to address the impacts of global warming and to confront the climate challenges ahead.

3.1.5. Conclusions

This chapter has explored the impact of climate change factors, particularly precipitation and temperature, on the volatility of extreme weather events in the Chengdu-Chongqing region. The analysis, based on the uncertain Box–Cox regression model, has revealed that both temperature and precipitation are correlated with the occurrence of extreme weather. However, the influence of temperature on extreme weather events is more significant compared to precipitation, with extreme high-temperature events increasing in frequency, while extreme low-temperature events have decreased.

While precipitation plays a role in extreme weather, its impact is less pronounced than that of temperature. Nevertheless, the growing frequency of extreme precipitation events indicates that changes in precipitation patterns are contributing to the increasing volatility of weather in the region. The Mann-Kendall trend test did not show a clear trend in the overall frequency of extreme weather events between 2006 and 2021, suggesting that despite fluctuations in temperature and precipitation, the frequency of extreme weather events has remained relatively stable in the long run. This stability may be influenced by complex factors such as geographical environment and climate feedback mechanisms that limit dramatic changes in extreme weather occurrences.

This observation of stable extreme weather frequency amidst rising temperature and precipitation trends warrants further investigation. Possible causes include competing climate feedback mechanisms that may counteract or limit the direct effects of temperature and precipitation on extreme weather events. For example, despite rising temperatures and increased precipitation, other environmental factors (such as ocean circulation or topographical features) may help mitigate these trends. Additionally, limitations related to imprecise data, such as missing or incomplete data for certain periods, may also influence the observed stability in extreme weather frequency.

Furthermore, the cross-validation results indicate that the uncertain Box–Cox regression model is effective in predicting the frequency of extreme weather events based on climatic variables, confirming its robustness and predictive capability.

However, it is clear that the occurrence of extreme weather is a complex process influenced by both natural factors and human activities. Therefore, future research should incorporate other potential drivers of extreme weather events, such as land use changes and societal factors. Moreover, in the context of global warming, there is a pressing need for policies that address the increased frequency and intensity of extreme weather events. Efforts to mitigate the impacts of these events on socio-economic systems and the environment must be prioritized, as the effects of extreme weather continue to pose serious challenges to agriculture, ecosystems, and public health.

3.2. The Coupling Relationship Between Climate Resources and the Economy in Chengdu-Chongqing Region

In recent years, the construction of the Chengdu-Chongqing Twin-City Economic Circle has become a focal point of attention within Chinese society. The goal is to transform the region into a national and even global economic and technological innovation center, a new hub for reform and opening-up, and a high-quality, livable city. Against the backdrop of global climate change, the impact of climate resources on economic development in the Chengdu-Chongqing region has become increasingly significant. Changes in climate resources not only directly affect agricultural production but also have profound long-term implications for the sustainable development of the regional economy.

Regarding thermal resources, the Chengdu-Chongqing region experiences a subtropical humid monsoon climate with relatively high average annual temperatures, which are conducive to agricultural growth. Warm climatic conditions favor agricultural diversification, particularly the cultivation of crops such as rice and rapeseed. This agricultural diversification not only drives the development of agriculture but also stimulates the growth of related industries, injecting new vitality into the region’s economy.

In terms of water resources, the region benefits from abundant precipitation, which ensures a reliable water supply for agricultural production and contributes to the growth of other industries. As shown in Table 2, plentiful rainfall supports agricultural irrigation and promotes the development of sectors such as energy and transportation, further driving economic growth.

To comprehensively explore the interaction between climate resources and the economy in the Chengdu-Chongqing region, this study uses GDP data as a representative indicator of economic development and analyzes the coupling relationship between thermal resources, water resources, and the economy. The coupling degree and coupling coordination degree between these resources and economic development from 2006 to 2021 are calculated, as shown in

Table 7. Coupling of thermal resources and economy in Chengdu-Chongqing region from 2006 to 2021.

Year	Coupling C Value	Hierarchy of Coupling	Coupling Coordination Degree D Value	Degree of Coupling Coordination
2006	0.1595	Low level coupling	0.3485	Transition from low to moderate coordination
2007	0.2079	Low level coupling	0.3100	Transition from low to moderate coordination
2008	0.1958	Low level coupling	0.2185	Moderate disorder
2009	0.2895	Low level coupling	0.3431	Transition from low to moderate coordination
2010	0.2423	Low level coupling	0.2453	Moderate disorder
2011	0.2823	Low level coupling	0.2823	Moderate disorder
2012	0.0186	Low level coupling	0.1222	Serious dissonance
2013	0.5701	Running-in	0.5942	Forced coordination
2014	0.3030	Antagonize	0.3207	Transition from low to moderate coordination
2015	0.7065	Running-in	0.7352	Intermediate coordination
2016	0.5424	Running-in	0.5430	Forced coordination
2017	0.5097	Running-in	0.5225	Forced coordination
2018	0.4929	Antagonize	0.5218	Forced coordination
2019	0.5080	Running-in	0.5487	Forced coordination
2020	0.4687	Antagonize	0.5325	Forced coordination
2021	0.6374	Running-in	0.6812	Primary coordination

and

Table 8. Coupling of water resources and economy in Chengdu-Chongqing region from 2006 to 2021.

Year	Coupling C Value	Hierarchy of Coupling	Coupling Coordination Degree D Value	Degree of Coupling Coordination
2006	0.0055	Low level coupling	0.0121	Extreme dissonance
2007	0.1894	Low level coupling	0.2666	Moderate disorder
2008	0.2387	Low level coupling	0.2894	Moderate disorder
2009	0.1734	Low level coupling	0.1761	Serious dissonance
2010	0.2596	Low level coupling	0.2656	Moderate disorder
2011	0.2951	Low level coupling	0.2953	Moderate disorder
2012	0.2769	Low level coupling	0.2816	Moderate disorder
2013	0.4910	Antagonize	0.4979	Borderline disorder
2014	0.6084	Running-in	0.6246	Primary coordination
2015	0.5441	Running-in	0.5454	Forced coordination
2016	0.6538	Running-in	0.6574	Primary coordination
2017	0.6849	Running-in	0.6850	Primary coordination
2018	0.6984	Running-in	0.6994	Primary coordination
2019	0.6818	Running-in	0.6902	Primary coordination
2020	0.9375	High level coupling	0.9388	Quality coordination
2021	0.8652	High level coupling	0.8713	Good coordination

, and Figure 10.

The results indicate an upward trend in the coupling degree between thermal resources, water resources, and economic development over the study period. This suggests that the relationship between the region’s economic development and climate resources has deepened. The data also indicates that changes in both thermal and water resources have influenced local economic development, with the coupling coordination degree generally showing fluctuating growth under the impact of climate change.

In the earlier years of the study period, the coupling coordination degree between thermal resources, water resources, and the economy was relatively low. This could be attributed to the region’s reliance on traditional industries at that time and a failure to fully harness the potential of thermal and water resources. The lack of attention to and inefficiency in the utilization of these resources, combined with insufficient policy guidance and planning, led to weak coordination between resources and economic development, preventing these resources from playing a more active role in driving economic growth.

However, from 2009 to 2018, the coupling coordination degree gradually increased, despite fluctuations. During this period, the implementation of national sustainable development strategies and an increased awareness of resource conservation led the Chengdu-Chongqing region to encourage enterprises to improve energy efficiency and promote primary industries related to thermal and water resources. However, the overall synergy remained limited, and industrial structural adjustment progressed slowly.

Since 2019, the coupling coordination degree of water resources and economic development has risen significantly and steadily, while the coordination degree between thermal resources and the economy has remained relatively stable. This change is largely due to the Chengdu-Chongqing Twin-City Economic Circle development strategy and the ecological civilization construction policies, which have provided strong policy support and guidance. Under this strategic framework, the region has increased investment in infrastructure and placed greater emphasis on the rational utilization and protection of thermal and water resources. This has been particularly important in the face of increasing extreme weather events such as heavy rainfall and extreme temperatures, where government departments have focused more on sustainable resource development and utilization, improving resource development and transportation conditions, and promoting the scaling-up of related industries.

In terms of thermal resources, the Chengdu-Chongqing region has focused on leveraging its advantages, developing specialized agriculture, tourism, and other industries that make optimal use of these resources. Specific measures include promoting green agriculture, ecotourism, and other low-carbon industries to enhance the region’s economic sustainability.

For water resources, the region has implemented strict water management policies, including the “three red lines” policy (control of total water use, control of water use efficiency, and restriction on water pollution in functional water zones). These measures have strengthened the regulation of water resource development and utilization, promoted the adoption of water-saving irrigation technologies, and implemented policies for the optimized allocation of water resources. Additionally, the government has encouraged the development of water-efficient industries and technological innovation to enhance the efficient use of water resources.

Overall, with strong support from both national and local governments, water resources have played a vital role in the region’s economic development. The relationship between thermal resources and the economy remains relatively balanced, though there is significant room for improvement in their coupling coordination. In conclusion, both thermal and water resources have had a significant impact on the region’s economic development. Moving forward, it is essential for the Chengdu-Chongqing region to further optimize the utilization of climate resources and implement sustainable environmental management practices to ensure long-term economic sustainability.

4. Conclusions and Discussion

In recent years, extreme weather events have become more frequent, and the situation regarding climate change remains concerning. Previous studies have utilized machine learning methods to analyze the spatio-temporal variation of extreme high temperatures in the Yangtze River Delta region [54]; others have explored the impact of extreme weather on ecosystem services within the Wuhan metropolitan area [55]. Additionally, some studies have applied CDI to analyze the lag effects and spatial heterogeneity of reservoir discharge under different dry and wet conditions in several basins in the southeastern Lin’an region of China [56]; others have also integrated ecological, economic, and social systems to assess the quality and influencing factors of the habitat of the red-crowned crane [57]. In contrast, this study presents the first attempt to systematically apply uncertainty theory in analyzing the Chengdu-Chongqing region, aiming to establish quantitative relationships between climate change drivers and the increasing frequency of extreme weather events. Moreover, this study employs the coupling coordination degree method to examine the relationship between thermal resources, water resources, and the local economy in the context of climate change, thereby contributing new insights to the field.

Through uncertain Box–Cox regression analysis, this study found a positive correlation between extreme weather occurrences and both precipitation and average temperature in the Chengdu-Chongqing region. Specifically, the partial regression coefficient for precipitation was about 0.0024, while that for temperature was about 0.1387. These findings indicate that both precipitation and high temperatures influence extreme weather events, with high temperatures having a more significant effect. Furthermore, the results from the Mann-Kendall (M-K) trend test indicate that from 2006 to 2021, the total frequency of extreme weather events remained largely stable. This stability may be attributed to limiting factors that prevent significant changes in extreme weather occurrences, highlighting the complexity of the climate system and the uncertainty in predicting extreme weather events.

In the analysis of the relationship between climate resources and the economy, it was found that the coupling degree between thermal resources and economic development was relatively low before 2013, with both being in a low-level coupling state. However, since 2013, the coupling degree between thermal resources and economic development has gradually increased, and the coordination degree has steadily improved, transitioning from mild imbalance to primary coordination. This improvement is largely due to the implementation of policies promoting sustainable development and energy efficiency, alongside the increasing attention to balancing economic growth with climate resource utilization. This suggests that the role of thermal resources in regional economic development is becoming more significant, though substantial room for improvement remains. In contrast, the coupling degree and coordination degree between water resources and economic development have consistently increased throughout the study period, gradually transitioning from low-level coupling to high-level coupling, reaching a state of high-quality coordination in recent years. This shift is largely attributed to the introduction of the “three red lines” water management policy in 2014, which has driven improvements in water resource management and efficiency, aligning water usage with sustainable economic development goals. The strong relationship between the efficient use of water resources and economic development reflects the region’s good progress in water resource management and utilization.

Based on these findings, this paper offers several policy recommendations to promote the coordinated development of the economy and natural resources in the Chengdu-Chongqing region while reducing the risks and losses caused by extreme weather. These recommendations aim to enhance regional resilience to climate change, support sustainable development, and foster long-term environmental, social, and economic benefits. By integrating climate adaptation measures into development planning, the region can mitigate future climate impacts and achieve a balanced, sustainable growth trajectory.

(1) Strengthen monitoring and early warning systems for extreme weather and improve emergency response mechanisms. Developing advanced meteorological networks to enhance the accuracy of extreme weather event monitoring will provide governments and businesses with accurate climate information, helping them respond swiftly and recover effectively. For example, enhancing the monitoring of extreme rainfall events in Chongqing, which has seen increasing rainfall in recent years, could help mitigate the risk of flooding. This supports SDG 11 (Sustainable Cities and Communities) by promoting disaster resilience and sustainable urban planning.

(2) Optimize water resources management and strengthen protection and utilization practices. Local governments should prioritize efficient water management, supported by policies like the “three red lines” water management policy, which has already been implemented to control water use, improve efficiency, and reduce pollution. Businesses and industries should adopt water-saving technologies, and municipal governments can support infrastructure for rainwater harvesting, particularly in agricultural and industrial zones. For example, Chengdu could invest in modern irrigation systems and promote the use of water-efficient technologies in farming, ensuring that the region’s agricultural sector remains sustainable amidst extreme weather. These actions contribute to SDG 6 (Clean Water and Sanitation) by improving water resource management and ensuring sustainable water usage in the region.

(3) Improve urban infrastructure and conduct climate risk assessments. Strengthening flood control, water supply systems, and designing buildings to withstand heatwaves and low temperatures will mitigate the socio-economic impacts of extreme weather. Governments should integrate climate risk assessments into urban planning and collaborate with businesses in sectors like construction to ensure new infrastructure is resilient to extreme weather. For instance, Chengdu could implement flood-resistant urban designs and enhance drainage systems to prepare for the increasing risk of extreme rainfall. Regular climate risk assessments in climate-sensitive sectors, such as agriculture and energy, should ensure that industries are prepared with targeted response strategies. This aligns with SDG 11 (Sustainable Cities and Communities) and SDG 13 (Climate Action), aiming for resilient infrastructure and climate-responsive urban planning.

(4) Promote the development and utilization of renewable energy to improve industrial structure and enhance energy efficiency. Increasing investments in solar, wind, and other renewable energy sources and promoting the transformation of energy-intensive industries will help mitigate the economic impacts of temperature changes. Governments should provide incentives for businesses to invest in renewable energy projects, such as solar farms in Chongqing, to enhance the region’s energy security and reduce dependence on traditional energy sources. Public–private partnerships could also be encouraged to scale up these renewable energy projects and further the region’s green economy. These efforts contribute to SDG 7 (Affordable and Clean Energy) by promoting the use of renewable energy and fostering clean energy solutions to combat climate change.

(5) Strengthen regional cooperation to promote coordinated development. Fostering cooperation is essential for tackling climate challenges regionally, but it requires more time and planning to be fully realized. Governments should establish a regional climate action plan involving local governments, businesses, and academic institutions to optimize resource distribution and reduce economic losses from extreme weather. For example, the Chengdu-Chongqing Twin-City Economic Circle could strengthen its focus on regional coordination for sustainable resource management, ensuring that the cities work together to share best practices and develop joint strategies for tackling climate risks. This collaborative approach would optimize resource allocation, mitigate the impacts of extreme weather, and support sustainable development across the region. This strengthens SDG 17 (Partnerships for the Goals), as regional cooperation is key to tackling climate change collectively and promoting sustainable development across the area.

In conclusion, actively addressing extreme weather and making rational use of climate resources are urgent needs to ensure human survival and sustainable social development in the face of climate change. Protecting the global ecological environment, reducing pollution, and strengthening ecosystem protection and restoration efforts are key to slowing the pace of global climate change and reducing the frequency and intensity of extreme weather events. At the same time, climate resources should be rationally utilized, with scientific planning and innovative technology ensuring that climate resources become a new driving force for economic development, facilitating a win-win situation for both regional economies and environmental protection.

While this study offers innovative insights, several avenues for future research remain. First, the Chengdu-Chongqing region is a complex and open system. This study has focused on a narrow regional scope, without considering differences across other cities and regions, including Chengdu and Chongqing. Future studies could expand to a broader area, accounting for regional differences in climate change and economic development to draw more generalizable conclusions. For instance, applying the framework to other monsoon-influenced regions, such as the Pearl River Delta, would broaden the relevance of the study and provide insights into the coupling of climate resources and economic development in different environmental contexts.

Second, this study only analyzes two climate factors—temperature and precipitation, without considering other climatic factors that may affect extreme weather and economic development, such as humidity, wind speed, and solar radiation. Excluding these variables may lead to biased results, as they can interact with temperature and precipitation, amplifying or mitigating their effects on extreme weather and economic growth. Future research could incorporate these additional climate factors and explore their synergistic effects, providing a more comprehensive assessment of the impact of climate change on extreme weather and economic development. Datasets such as remote sensing data or satellite-derived climatic data could be used to fill in these gaps and improve the robustness of the analysis.

Furthermore, the economic indicators used in this study are limited to GDP. Future research should include other economic indicators, such as per capita income, industrial structure, and climate disaster insurance coverage, to further explore the complex relationship between climate change and economic development. These additional indicators could offer a deeper understanding of how climate impacts influence different sectors of the economy.

Future studies could also employ more advanced analytical techniques, such as random forests, XGBoost, and other machine learning methods, to improve prediction accuracy for extreme weather events. These methods can handle complex, nonlinear relationships between variables, making them highly suitable for climate-economic studies. Additionally, spatial autoregressive models (SAR) could be used to capture spatial correlations between regions, while Monte Carlo simulations could help analyze the influence of uncertainty on the conclusions.

Lastly, due to limitations in data availability and knowledge, this study primarily relies on historical observational data. Future research could integrate a wider variety of data sources, such as remote sensing data and climate model predictions, to conduct a more comprehensive analysis of future trends. These sources could provide more precise and timely data, helping to better predict future climate conditions and their potential economic impacts.