A method for constructing a non-stationary compound dry heat index based on a GAMLSS model and a Copula function

Abstract

The application discloses a non-stationary compound dry-hot index construction method based on a GAMLSS model and a Copula function, and comprises the following steps: obtaining long-sequence grid monthly rainfall and monthly temperature data of a research area; selecting a large-scale climate index, performing moving average processing on the climate index, and screening the climate index significantly correlated with the rainfall and temperature sequence as a candidate explanatory variable through Kendall rank correlation analysis; constructing non-stationary standardized rainfall indexes and non-stationary standardized temperature indexes based on the GAMLSS model framework and taking the preferred climate index as an explanatory variable; selecting multiple Copula functions to construct a two-dimensional joint distribution of the NSPI and the NSTI, preferably selecting the Copula function according to a goodness-of-fit criterion, calculating the joint probability of a compound dry-hot event, and obtaining a non-stationary compound dry-hot index through inverse transformation of a standard normal distribution; and verifying the applicability of the constructed NCDHI by using a typical compound dry-hot event in history, so that the accuracy and reliability of the compound dry-hot event identification are improved.

Description

Technical Field

[0001] This invention relates to the field of climate change assessment and extreme hydro-meteorological event identification technology, and in particular to a method for constructing a non-stationary composite dry-heat index based on the GAMLSS model and Copula function. Background Technology

[0002] Traditional composite heat and dryness indices are typically constructed based on the stationarity assumption of meteorological time series, which assumes that statistical parameters (such as mean and variance) of time series such as rainfall and temperature do not change over time. Based on this, researchers have developed various composite heat and dryness indices. Early studies mainly used a simple combination of the Standardized Precipitation Index (SPI) and the Standardized Temperature Index (STI), identifying composite events by setting a threshold of "positive high temperature anomaly - negative rainfall anomaly." Subsequently, Copula functions, due to their ability to flexibly characterize nonlinear and asymmetric dependencies between variables, have gradually become an important approach for constructing composite heat and dryness indices. For example, Zscheischler et al., through joint analysis of SPI and STI, were the first to systematically demonstrate that the inhibitory effect of composite heat and dryness conditions on carbon flux in terrestrial ecosystems far exceeds that of a single extreme event; Hao et al., based on the concurrent probability of monthly rainfall and temperature extreme values, proposed the Standardized Composite Heat and Dryness Index (CDHI), enabling comparability analysis of composite events in different regions. Domestic scholars have also developed a series of regionally adaptable indices, such as the Standardized Temperature-Rainfall Index (STPI), tailored to the diverse climate characteristics of China.

[0003] However, all the aforementioned traditional composite dry-heat indices rely on a crucial premise: that the time series of rainfall and temperature are stationary, meaning their probability distributions do not change over time. But this assumption has gradually become invalid under the background of global warming. Milly et al. explicitly stated in *Science* that "stationarity no longer exists," pointing out that long-term changes in climate elements such as rainfall and temperature under global warming have led to significant changes in their statistical characteristics, and drought assessments based on the assumption of stationarity have a serious hidden danger of underestimating disaster risk. Subsequently, some scholars explored the sources of non-stationarity: on the one hand, from natural variability within the climate system, such as the interdecadal fluctuations of large-scale climate modes like El Niño-Southern Oscillation (ENSO), North Atlantic Oscillation (NAO), and Atlantic Multidecadal Oscillation (AMO); on the other hand, from climate change caused by human activities, including global warming caused by greenhouse gas emissions and changes in underlying surface due to land use change. Chiang et al. further found through attribution analysis that anthropogenic climate change is the dominant factor in the increased frequency of global composite dry-heat events, contributing more than twice as much as natural variability in mid-latitude regions.

[0004] For addressing nonstationarity issues, the Generalized Additive Model (GAMLSS) has become a core tool for constructing nonstationary indices due to its powerful flexibility. Proposed by Rigby and Stasinopoulos in 2005, the GAMLSS model's core advantage lies in its ability to simultaneously link location parameters (such as mean), scale parameters (such as variance), and shape parameters of a distribution with covariates through power functions, cubic spline functions, and other connection functions, thereby achieving a precise characterization of the nonstationary features of a sequence. International scholars pioneered the coupling of covariates with drought indices: Rashid and Beecham incorporated the NAO index as a covariate into the GAMLSS model to construct a nonstationary standardized precipitation index for Australia, improving drought identification accuracy by 18% compared to traditional stationary indices; Xu et al. further established a global-scale framework for identifying nonstationary compound events by incorporating time trend factors and climate indices as covariates. Significant progress has also been made in domestic research. Wang et al. constructed the Nonstationary Composite Dry Heat Index (NCDHI) based on the GAMLSS model, using climate indices and anthropogenic factors as covariates, and verified its applicability in the Chinese region. Zhang et al. used the enhanced GAMLSS framework to reveal the non-stationary evolution characteristics of complex climate extreme events under future scenarios.

[0005] Despite the rapid development of non-stationary drought or high-temperature indices driven by the aforementioned research, existing methods still have three significant shortcomings in addressing climate non-stationarity. First, the non-stationarity of marginal distributions and the time-varying nature of their joint dependence structures are not considered synergistically. Most studies only construct non-stationary marginal distributions for single factors such as rainfall or temperature, or while attempting to construct non-stationary composite indices, they neglect the time-varying nature of the joint dependence structure between drought and high temperature. Xu et al.'s research shows that non-stationarity has varying impacts on risk assessment results at different stages of modeling, with the non-stationarity of marginal distributions having the greatest impact. However, this study still assumes that the dependence structure of drought and high temperature is time-invariant, while in reality, under the background of climate change, the joint dependence of the two may also change over time, i.e., there is "non-stationarity of the dependent structure." Second, fixed Copula function types lead to insufficient adaptability. Traditional methods typically pre-determine fixed Copula function types and parameters, but the dependence characteristics of drought and heat variables differ significantly across different climate zones and time periods. For example, the joint distribution characteristics of rainfall-dominant and temperature-dominant areas are drastically different, making it difficult for a unified function to reflect spatial heterogeneity. Third, the selection of climate indices lacks a systematic regional adaptation mechanism, making it difficult to characterize the spatial heterogeneity of the driving factors of hot and dry events in different regions. For example, the high-altitude cold region in the upper reaches of the Yangtze River is mainly affected by the AO and NAO indices, while the plain region in the middle and lower reaches is dominated by the WP index. A unified strategy cannot accurately characterize the non-stationary driving mechanisms of each region. Summary of the Invention

[0006] The purpose of this invention is to overcome the above-mentioned shortcomings and provide a method for constructing a non-stationary composite dry heat index based on the GAMLSS model and Copula function. This method can simultaneously consider the non-stationarity of the marginal distribution of rainfall and temperature sequences as well as the time-varying nature of their joint dependency structure. This overcomes the defects of traditional stationary composite dry heat indices under changing environments and provides reliable technical support for the accurate identification and risk assessment of composite dry heat events.

[0007] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: A method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function includes the following specific steps: S1. Obtain long-sequence gridded monthly rainfall and temperature data for the study area, and use the ADF unit root test method to test the stationarity of the rainfall and temperature sequences grid by grid. S2. Select large-scale climate indices, perform moving average processing on the climate indices, and conduct 0-12 month lag correlation analysis between the rainfall and temperature series of each grid point and the selected climate indices. Climate indices that are significantly correlated with the rainfall and temperature series are used as candidate explanatory variables for the GAMLSS model. S3. Based on the GAMLSS model framework, using the climate indices selected in step S2 as explanatory variables, construct the non-stationary standardized precipitation index NSPI and the non-stationary standardized temperature index NSTI respectively. S4. Select multiple Copula functions to construct a two-dimensional joint distribution of NSPI and NSTI, select the best Copula function according to the goodness-of-fit criterion, calculate the joint probability of combined dry heat events, and obtain the non-stationary combined dry heat index NCDHI through standard normal inverse transformation. S5. The applicability of the NCDHI constructed in step S4 is verified by using typical historical composite dry heat events, and it is compared and analyzed with the stable composite dry heat index CDHI.

[0008] Preferably, in step S1, the null hypothesis and alternative hypothesis of the ADF unit root test are: The sequence has a unit root and is non-stationary. The sequence is stationary; at the 1% significance level, if the p-value is less than the significance level, the null hypothesis is rejected, indicating that the sequence is stationary; if the p-value is greater than or equal to the significance level, the null hypothesis cannot be rejected, indicating that the sequence is non-stationary.

[0009] Preferably, in step S2, the selected climate indices include one or more combinations of the following: Southern Oscillation Index (SOI), Pacific Decadal Oscillation (PDO) Index, Arctic Oscillation (AO) Index, North Atlantic Oscillation (NAO) Index, North Atlantic Multi-Decadal Oscillation (AMO) Index, North Pacific Oscillation (NPO) Index, Western Pacific Index (WP), Pacific-North American Teleconnection (PNA) Type, Quasi-Two-Year Oscillation (QBO), El Niño (Nino3.4) Index, and Indian Ocean Dipole Index (DMI).

[0010] Preferably, in step S2, the climate index is processed by a 12-month moving average to reduce the randomness of the climate index; the lag correlation analysis adopts the Kendall rank correlation test, with a lag period of 0-12 months, and the climate indexes that are significantly correlated with the rainfall and temperature series are selected as climate change factors.

[0011] Preferably, the GAMLSS model in step S3 is defined as follows: In the formula, ; Let n be a vector of length n. For length is The regression parameter vector, for The matrix of explanatory variables, For a known Fixed design matrix, It is A vector of normally distributed random variables of dimension 1. Let j represent the j-th random effect. Let represent the dimension of the random influence factor in the j-th random effect; the distribution function of the response variable for the rainfall series is selected as the gamma distribution, and the distribution function of the response variable for the temperature series is selected as the normal distribution.

[0012] Preferably, in step S3, the GAMLSS model fitting uses the RS algorithm for parameter estimation, the model fit goodness is evaluated using the Akaike Information Criterion and the Schwarz Bayes Criterion, and the model residuals are diagnosed for normality using a worm diagram.

[0013] Preferably, in step S4, the candidate Copula functions include one or more of Clayton Copula, Frank Copula, Gumbel Copula, Gaussian Copula, and t Copula; the goodness-of-fit criteria include one or more of Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Nash Efficiency Coefficient (NSE).

[0014] Preferably, in step S4, the parameter estimation of the Copula function adopts the maximum likelihood estimation method (MLE), and its log-likelihood function is: ; In the formula, For Copula density function, and These are the marginal probability density functions of NSPI and NSTI, respectively. This is the parameter vector of the Copula function.

[0015] Preferably, the joint probability of the combined dry heat events in step S4 is defined as: ; Where x and y are the drought and high temperature thresholds, respectively; u and v are the marginal cumulative distribution function values ​​of the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI), respectively; and C is the joint distribution function of the preferred Copula. The joint probability is transformed into a uniform distribution using the Gringorten empirical formula, and then subjected to an inverse standard normal transformation to obtain the non-stationary composite dry-heat index (NCDHI). In the formula, Let G be the standard normal distribution function, and let G represent the Gringorten empirical distribution.

[0016] Preferably, in step S5, one or more typical historical complex dry heat events within the study area are selected for verification. By comparing the differences between NCDHI and stable CDHI in terms of event identification range and intensity, the sensitivity and superiority of NCDHI are evaluated.

[0017] Beneficial effects of this invention: This invention, for the first time, incorporates the non-stationarity of marginal distributions and the time-varying nature of joint dependency structures into a unified framework for constructing a composite dry-heat index. It utilizes the GAMLSS model to achieve time-varying modeling of rainfall and temperature distribution parameters, and simultaneously optimizes Copula functions on a grid-by-grid basis to capture the spatial heterogeneity of the dependency structure of dry-heat variables, overcoming the limitation of traditional methods that rely on fixed function types. Furthermore, this invention establishes a systematic climate index zoning screening mechanism, combining Kendall rank correlation analysis and stepwise regression to optimize driving factors on a grid-by-grid basis, overcoming the shortcomings of a unified covariate strategy. Validation shows that the NCDHI, while maintaining a high degree of consistency with the CDHI spatial model, can more accurately reflect the dynamic evolution of composite dry-heat events, and its robustness and universality under different climatic backgrounds have been verified by multiple historical typical events. Attached Figure Description

[0018] Figure 1This is a flowchart of the nonstationary composite dry heat index construction method based on the GAMLSS model and Copula function of the present invention; Figure 2 This is a spatial distribution map showing the non-stationarity of the ADF unit root test of the rainfall and temperature series in the study area of ​​this invention; Figure 3 This is a spatial distribution map of the preferred climate index results of the present invention; Figure 4 The images show the normal QQ plot and worm plot of the residuals of the typical grid-point nonstationary model of this invention. Figure 5 This is a spatial pattern diagram of the optimal Copula function under the non-stationary model of the present invention; Figure 6 This is a time series comparison chart of stationary and non-stationary standardized rainfall index, standardized temperature index, and composite dry heat index at typical grid points of the present invention. Figure 7 This is a comparison of the spatial distribution of the stationary composite dry heat index (CDHI) and the non-stationary composite dry heat index (NCDHI) of this invention during a typical composite dry heat event (summer of 2006). Figure 8 This is a spatial distribution map of the monitoring results of typical complex dry heat events in the Yangtze River Basin using the Non-Stationary Complex Dry Heat Index (NCDHI) of this invention. Detailed Implementation

[0019] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0020] Example 1: As Figure 1 As shown, this invention closely integrates non-stationary time series analysis, GAMLSS statistical models, and Copula multivariate analysis theory. It is a typical interdisciplinary innovation involving the deep integration of multiple technical fields such as mathematical statistics, hydrology, and climatology, and provides a new technical solution for the accurate identification of complex dry and hot events under changing environments.

[0021] The method for constructing the nonstationary composite dry heat index based on the GAMLSS model and Copula function in this invention includes the following specific steps: S1. Obtain long-sequence gridded monthly rainfall and temperature data for the study area, and use the ADF unit root test method to test the stationarity of the rainfall and temperature sequences grid by grid. S2. Select large-scale climate indices, perform moving average processing on the climate indices, and use Kendall rank correlation analysis to screen climate indices that are significantly correlated with the precipitation and temperature series as candidate explanatory variables for the GAMLSS model; S3. Based on the GAMLSS model framework, using the climate indices selected in step S2 as explanatory variables, construct the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI). S4. Select multiple Copula functions to construct a two-dimensional joint distribution of NSPI and NSTI, select the best Copula function according to the goodness-of-fit criterion, calculate the joint probability of combined dry heat events, and obtain the non-stationary combined dry heat index (NCDHI) through standard normal inverse transformation. S5. The applicability of the NCDHI constructed in step S4 is verified by using typical historical composite dry heat events, and it is compared and analyzed with the stationary composite dry heat index (CDHI).

[0022] Preferably, in step S1, the null hypothesis and alternative hypothesis of the ADF unit root test are: (A unit root exists, and the sequence is non-stationary.) (Stationary series). At the 1% significance level, if the p-value < the significance level: reject the null hypothesis, indicating the series is stationary; if the p-value ≥ the significance level, the null hypothesis cannot be rejected, indicating the series is non-stationary. The ADF test statistic is: , Where is the estimated value of the lagged term coefficient, Its standard error. If If the value is less than the critical value, the null hypothesis is rejected and the sequence is determined to be stationary.

[0023] Preferably, in step S2, the selected climate indices include one or more combinations of the following: Southern Oscillation Index (SOI), Pacific Decadal Oscillation (PDO) Index, Arctic Oscillation (AO) Index, North Atlantic Oscillation (NAO) Index, North Atlantic Multi-Decadal Oscillation (AMO) Index, North Pacific Oscillation (NPO) Index, Western Pacific Index (WP) Index, Pacific-North American Teleconnection (PNA) Index, Quasi-Two-Year Oscillation (QBO) Index, El Niño (Nino3.4) Index, and Indian Ocean Dipole Index (DMI).

[0024] Preferably, in step S2, the climate indices are processed using a 12-month moving average to reduce their randomness; the lag correlation analysis employs the Kendall rank correlation test, setting the lag period to 0-12 months, and selecting climate indices that are significantly correlated with rainfall and temperature series (significance level α=5%) as climate change factors. The Kendall rank correlation coefficient is defined as: , in n For sequence length, is the sign function. The selected candidate explanatory variables are further used to determine the final explanatory variables of the GAMLSS model through stepwise regression.

[0025] The GAMLSS model in step S3 is defined as follows: In the formula, ; Let n be a vector of length n. For length is The regression parameter vector, for The matrix of explanatory variables, For a known Fixed design matrix, It is A vector of normally distributed random variables of dimension 1. Let j represent the j-th random effect. Let represent the dimension of the random influence factor in the j-th random effect; the distribution function of the response variable for the rainfall series is selected as the gamma distribution, and the distribution function of the response variable for the temperature series is selected as the normal distribution. Further, in step S3, the probability density function of the gamma distribution is: Where μ is the location parameter (mean) and σ is the scale parameter. Let gamma be the gamma function. The probability density function of the normal distribution is: ,in μ For location parameters (mean). σ This is the scale parameter (standard deviation). The distribution parameter is obtained through the GAMLSS model. μ and σ All of them can be expressed as functions of explanatory variables, thereby achieving non-stationarization of parameters.

[0026] In step S3, the GAMLSS model fitting uses the Rigby-Stasinopoulos algorithm for parameter estimation, and solves for the regression parameters using iterative weighted least squares. The goodness of fit of the model is evaluated using the AIC (Akaike Information Criterion) and SBC (Schwarz Bayes Criterion). , ,in L The model likelihood function value. p The number of model parameters, n The sample size is denoted by . The smaller the AIC and SBC values, the better the model fit. The model residuals are used to diagnose normality using a worm plot. If all residual points fall within the 95% confidence interval, the model is considered to have a good fit.

[0027] In step S4, the candidate Copula functions include one or more of Clayton Copula, Frank Copula, Gumbel Copula, Gaussian Copula, and t Copula, and the distribution functions of each Copula function are as follows:

[0028] In step S4, the parameter estimation of the Copula function uses the maximum likelihood estimation (MLE) method, and its log-likelihood function is: In the formula, For Copula density function, and These are the marginal probability density functions of NSPI and NSTI, respectively. This is the parameter vector of the Copula function. The goodness-of-fit criteria include one or more of the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Nash Efficiency Coefficient (NSE), and the optimal Copula function is selected on a grid-by-grid basis.

[0029] The joint probability of the combined dry heat events in step S4 is defined as follows: ; Where x and y are the drought and high temperature thresholds, respectively; u and v are the marginal cumulative distribution function values ​​of the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI), respectively; and C is the joint distribution function of the preferred Copula. The joint probability is transformed into a uniform distribution using the Gringorten empirical formula, and then subjected to an inverse standard normal transformation to obtain the non-stationary composite dry-heat index (NCDHI). In the formula, It is the standard normal distribution function. G This represents the Gringorten empirical distribution. The formula for the Gringorten empirical distribution is: ,in i To be P Rank after ascending order n This represents the total sample size.

[0030] In step S5, one or more typical historical complex dry heat events in the study area are selected for verification. By comparing the differences between NCDHI and stable CDHI in terms of event identification range and intensity, the sensitivity and superiority of NCDHI are evaluated.

[0031] Example 2: Taking the Yangtze River Basin as an example, the non-stationary composite dry heat index construction method based on the GAMLSS model and Copula function of the present invention includes the following specific steps: S1. Obtain long-sequence gridded monthly rainfall and temperature data for the study area, and use the ADF unit root test method to test the stationarity of the rainfall and temperature sequences grid by grid. S2. Select large-scale climate indices, perform moving average processing on the climate indices, and use Kendall rank correlation analysis to screen climate indices that are significantly correlated with the precipitation and temperature series as candidate explanatory variables for the GAMLSS model; S3. Based on the GAMLSS model framework, using the climate indices selected in step S2 as explanatory variables, construct the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI). S4. Select multiple Copula functions to construct a two-dimensional joint distribution of NSPI and NSTI, select the best Copula function according to the goodness-of-fit criterion, calculate the joint probability of combined dry heat events, and obtain the non-stationary combined dry heat index (NCDHI) through standard normal inverse transformation. S5. The applicability of the NCDHI constructed in step S4 is verified by using typical historical composite dry heat events, and it is compared and analyzed with the stationary composite dry heat index (CDHI).

[0032] In step S1, monthly gridded rainfall and temperature data (spatial resolution 0.25°) for the study area from 1961 to 2021 are obtained, and the stationarity of the rainfall and temperature series is tested grid-by-grid using the ADF unit root test. The null and alternative hypotheses of the ADF unit root test are: (A unit root exists, and the sequence is non-stationary.) (Stationary series). At the 1% significance level, a p-value < 0.01 indicates a stationary series; otherwise, it is considered non-stationary. The ADF test statistic is: , Where is the estimated value of the lagged term coefficient, Its standard error. If If the value is less than the critical value, the null hypothesis is rejected, and the sequence is determined to be stationary. The test results are attached. Figure 2 As shown in the figure, 40.05% of the gridded precipitation series and 91.53% of the gridded temperature series in the entire basin exhibit non-stationary characteristics. The non-stationarity of the temperature series is more significant than that of the precipitation series, indicating that the traditional stationary assumption can no longer accurately describe the dynamic changes of climate elements, and it is necessary to construct a non-stationary index.

[0033] In step S2, eleven large-scale climate indices are selected: the Southern Oscillation Index (SOI), the Pacific Decadal Oscillation (PDO) Index, the Arctic Oscillation (AO) Index, the North Atlantic Oscillation (NAO) Index, the North Atlantic Multi-Decadal Oscillation (AMO) Index, the North Pacific Oscillation (NPO) Index, the Western Pacific Index (WP), the Pacific-North American Teleconnection Pattern (PNA), the Quasi-Two-Year Oscillation (QBO), the El Niño (Nino3.4) Index, and the Indian Ocean Dipole Index (DMI). A 12-month moving average is applied to each climate index to reduce randomness. A lag period of 0-12 months is set, and the Kendall rank correlation test (significance level α=5%) is used to screen climate indices significantly correlated with precipitation and temperature series. The Kendall rank correlation coefficient is defined as: ,in n For sequence length, Using the sign function, the selected candidate explanatory variables are further used to determine the final explanatory variables of the GAMLSS model through stepwise regression. The optimal results are attached. Figure 3 As shown in Table 1, the driving factors of the precipitation series exhibit significant spatial heterogeneity. The WP index is the precipitation driving factor with the highest proportion in the watershed (19.00%), while the AO index (13.75%) and NAO index (13.15%) have a more significant impact in the upstream high-altitude cold region. The driving factors of the temperature series, on the other hand, show a high degree of consistency, with the AMO index being the dominant factor across the entire region, accounting for as high as 56.01%.

[0034] Table 1. Statistical Table of Climate Index Optimization Results

[0035] The GAMLSS model in step S3 is defined as follows:

[0036] In the formula, ; Let n be a vector of length n. For length is The regression parameter vector, for The matrix of explanatory variables, For a known Fixed design matrix, It is A vector of normally distributed random variables of dimension 1. Let j represent the j-th random effect. Let represent the dimension of the random influence factor in the j-th random effect; the distribution function of the response variable for the rainfall series is a gamma distribution, and the distribution function of the response variable for the temperature series is a normal distribution. The probability density function of the gamma distribution is: Where μ is the location parameter (mean) and σ is the scale parameter. Let gamma be the gamma function. The probability density function of the normal distribution is: ,in μ For location parameters (mean). σ This is the scale parameter (standard deviation). The distribution parameter is obtained through the GAMLSS model. μ and σ All parameters can be expressed as functions of the explanatory variables, thus achieving parameter non-stationarization. The GAMLSS model fitting uses the Rigby-Stasinopoulos algorithm for parameter estimation, solving for the regression parameters through iterative weighted least squares. Model fit goodness of fit is evaluated using the AIC (Akaike Information Criterion) and SBC (Schwarz Bayes Criterion). , ,in L The model likelihood function value. p The number of model parameters, n The sample size is denoted by . The smaller the AIC and SBC values, the better the model fit. The model residuals are used to diagnose normality using a worm plot. If all residual points fall within the 95% confidence interval, the model is considered to have a good fit.

[0037] In step S4, Clayton Copula, Frank Copula, Gumbel Copula, Gaussian Copula, and t Copula are selected as candidate functions to construct a two-dimensional joint distribution of NSPI and NSTI. The parameters of the Copula function are estimated using maximum likelihood estimation (MLE), and its log-likelihood function is: The optimal Copula function was selected on a lattice-by-lattice basis based on the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and the Nash Efficiency Coefficient (NSE). The selection results are attached. Figure 5 As shown in Table 2, under the non-stationary model, Frank Copula accounts for 47.28% of the total watershed, becoming the dominant type; Gumbel Copula accounts for 36.72%, and Clayton Copula accounts for 16.00%. Compared to the absolute dominance of Gumbel Copula (94.19%) under the stationary model, the non-stationary model reveals a more complex spatiotemporal heterogeneous dependency structure among the dry-heat variables. The joint probability of the combined dry-heat event is defined as:

[0038] In the formula, x and y are the drought and high temperature thresholds, respectively; u and v are the marginal cumulative distribution function values ​​of the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI), respectively; and C is the joint distribution function of the preferred Copula. The joint probability is transformed into a uniform distribution using the Gringorten empirical formula, and then subjected to an inverse standard normal transformation to obtain the non-stationary composite dry-heat index (NCDHI).

[0039] In the formula, It is the standard normal distribution function. G This represents the Gringorten empirical distribution.

[0040] Table 2. Statistical Table of Copula Function Optimization Results for Non-stationary Models

[0041] In step S5, multiple historical typical combined dry-heat events within the study area are selected for verification. First, the correlation between NCDHI and stationary CDHI at the global grid points is compared: as shown in the attached figure. Figure 6 As shown, the time series trends of NCDHI and CDHI for three typical grid points A (34.25°N, 92.25°E), B (29.50°N, 106.50°E), and C (30.75°N, 114.25°E) (upstream, midstream, and downstream) from 1961 to 2021 are similar. The correlation coefficient for over 80% of the grid points across the entire basin exceeds 0.6 (reaching a maximum of 0.94), and the average correlation coefficient for the entire basin is 0.71, indicating that NCDHI maintains a high degree of consistency with the stationary index in terms of spatial pattern. Secondly, the 2006 severe high-temperature drought event in the Sichuan-Chongqing region of the Yangtze River basin was selected as a typical case for verification. (See attached...) Figure 7 As shown, CDHI identified 28.0%, 58.0%, and 49.6% of the area affected by the combined dry-heat event in the Yangtze River Basin in June, July, and August, respectively; while NCDHI identified 49.1%, 64.7%, and 76.1% of the affected area, showing a month-by-month upward trend. This accurately reflects the evolution of the disaster, with drought gradually developing and high temperatures continuously overlapping. Furthermore, NCDHI identified a wider range and stronger intensity of extreme dry-heat areas. In addition, NCDHI monitoring results for several typical historical events, including 1961, 1972, 1978, 2013, and 2019, were compared month-by-month and region-by-region with actual drought records (see Tables 3 and 4). The spatial map of the identification results is shown below. Figure 8 As shown, the results indicate that the location of the dry-heat core area and the proportion of dry-heat levels identified by NCDHI are highly consistent with the actual development process of drought, verifying the robustness and universality of the method of the present invention under different climatic backgrounds.

[0042] Table 3 Statistical Table of NCDHI Composite Dry Heat Monitoring Results in the Yangtze River Basin

[0043] Table 4 Actual Records of Complex Dry Heat Occurrence in the Yangtze River Basin

[0044] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The scope of protection of the present invention should be limited to the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function, characterized in that: It includes the following specific steps: S1. Obtain long-sequence gridded monthly rainfall and temperature data for the study area, and use the ADF unit root test method to test the stationarity of the rainfall and temperature sequences grid by grid. S2. Select large-scale climate indices, perform moving average processing on the climate indices, and conduct 0-12 month lag correlation analysis between the rainfall and temperature series of each grid point and the selected climate indices. Climate indices that are significantly correlated with the rainfall and temperature series are used as candidate explanatory variables for the GAMLSS model. S3. Based on the GAMLSS model framework, using the climate indices selected in step S2 as explanatory variables, construct the non-stationary standardized precipitation index NSPI and the non-stationary standardized temperature index NSTI respectively. S4. Select multiple Copula functions to construct a two-dimensional joint distribution of NSPI and NSTI, select the best Copula function according to the goodness-of-fit criterion, calculate the joint probability of combined dry heat events, and obtain the non-stationary combined dry heat index NCDHI through standard normal inverse transformation. S5. The applicability of the NCDHI constructed in step S4 is verified by using typical historical composite dry heat events, and it is compared and analyzed with the stable composite dry heat index CDHI.

2. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S1, the null hypothesis and alternative hypothesis of the ADF unit root test are: The sequence has a unit root and is non-stationary. The sequence is stationary; at the 1% significance level, if the p-value is less than the significance level, the null hypothesis is rejected, indicating that the sequence is stationary; if the p-value is greater than or equal to the significance level, the null hypothesis cannot be rejected, indicating that the sequence is non-stationary.

3. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S2, the selected climate indices include one or more combinations of the following: Southern Oscillation Index (SOI), Pacific Decadal Oscillation (PDO) Index, Arctic Oscillation (AO) Index, North Atlantic Oscillation (NAO) Index, North Atlantic Multi-Decadal Oscillation (AMO) Index, North Pacific Oscillation (NPO) Index, Western Pacific Index (WP), Pacific-North American Teleconnection (PNA) Index, Quasi-Two-Year Oscillation (QBO) Index, El Niño (Nino3.4) Index, and Indian Ocean Dipole Index (DMI).

4. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S2, the climate index is processed by a 12-month moving average to reduce the randomness of the climate index; the lag correlation analysis adopts the Kendall rank correlation test, with a lag period of 0-12 months, and the climate indexes that are significantly correlated with the rainfall and temperature series are selected as climate change factors.

5. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: The GAMLSS model in step S3 is defined as follows: In the formula, ; Let n be a vector of length n. For length is The regression parameter vector, for The matrix of explanatory variables, For a known Fixed design matrix, It is A vector of normally distributed random variables of dimension 1. Let j represent the j-th random effect. Let represent the dimension of the random influence factor in the j-th random effect; the distribution function of the response variable for the rainfall series is selected as the gamma distribution, and the distribution function of the response variable for the temperature series is selected as the normal distribution.

6. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S3, the GAMLSS model fitting uses the RS algorithm for parameter estimation, the model fit goodness is evaluated using the Akaike Information Criterion and the Schwarz Bayes Criterion, and the model residuals are diagnosed as normal using a worm diagram.

7. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S4, the candidate Copula functions include one or more of Clayton Copula, Frank Copula, Gumbel Copula, Gaussian Copula, and t Copula; the goodness-of-fit criteria include one or more of Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Nash Efficiency Coefficient (NSE).

8. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S4, the parameter estimation of the Copula function adopts the maximum likelihood estimation method (MLE), and its log-likelihood function is: ； In the formula, For Copula density function, and These are the marginal probability density functions of NSPI and NSTI, respectively. This is the parameter vector of the Copula function.

9. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: The joint probability of the combined dry heat events in step S4 is defined as follows: ； Where x and y are the drought and high temperature thresholds, respectively; u and v are the marginal cumulative distribution function values ​​of the non-stationary standardized precipitation index (NSPI) and the non-stationary standardized temperature index (NSTI), respectively; and C is the joint distribution function of the preferred Copula. The joint probability is transformed into a uniform distribution using the Gringorten empirical formula, and then subjected to an inverse standard normal transformation to obtain the non-stationary composite dry-heat index (NCDHI). In the formula, Let G be the standard normal distribution function, and let G represent the Gringorten empirical distribution.

10. The method for constructing a nonstationary composite dry heat index based on the GAMLSS model and Copula function according to claim 1, characterized in that: In step S5, one or more typical historical complex dry heat events in the study area are selected for verification. By comparing the differences between NCDHI and stable CDHI in terms of event identification range and intensity, the sensitivity and superiority of NCDHI are evaluated.

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