A method for extended-range temperature prediction based on a GARCH-MIDAS model

Abstract

The application discloses a kind of GARCH-MIDAS model-based extended-range temperature prediction method, obtains historical daily average temperature data by meteorological observation network, obtains monthly Nino3.4 index data by climate monitoring system and is preprocessed;Based on the physical mechanism that ENSO affects regional temperature through atmospheric teleconnection process, a GARCH-MIDAS prediction model is constructed, and the conditional variance equation decomposes volatility into seasonal volatility component, GARCH short-term volatility component and MIDAS long-term volatility component based on Nino3.4 index;Recursive prediction method is used to generate point prediction sequence and probability density prediction distribution of different advanced days in extended period;The prediction result is output to the meteorological disaster early warning system, agricultural intelligent decision system or energy scheduling system.The application can solve the problem of numerical prediction failure in extended period prediction, and provide probability density prediction that can directly serve the early warning system.

Description

Technical Field

[0001] This invention relates to the field of meteorological forecasting technology, specifically to a GARCH-MIDAS mixing modeling method for extended-range (10-30 days) temperature forecasting, applicable to technical scenarios such as meteorological disaster early warning, agricultural risk management, and energy dispatch decision-making. Background Technology

[0002] Extended forecast period (10-30 days) is a critical window for meteorological services, directly related to practical operational needs such as disaster warning lead time, agricultural planting window period, and energy dispatch plan.

[0003] Currently, there are two main technical approaches in the field of meteorological forecasting, but both have significant technical shortcomings in extended-range forecasting. The first approach is numerical weather prediction (NMR) models. This technology is based on atmospheric physical equations, obtaining the initial field by assimilating real-time observational data, and then solving it through integration. In short-term forecasts of 1-7 days, its point prediction accuracy is significantly better than statistical models due to the high accuracy of the initial field and limited accumulation of model errors. However, when the forecast window is extended to a range of 10-30 days, the initial field error accumulates and amplifies rapidly through nonlinear physical processes, leading to a sharp decline in model forecasting capability. Furthermore, the operation of high-resolution numerical models is highly dependent on supercomputer computing power and real-time observational data assimilation systems, resulting in extremely high computational and time costs, making it difficult to meet the technical requirements of high-frequency updates and low-cost operation. More importantly, this technology cannot directly output complete probability density predictions, limiting its application in risk quantification scenarios.

[0004] The second category is statistical time series models, represented by the classic AR-GARCH model (combining autoregressive (AR) and generalized autoregressive conditional heteroscedasticity (GARCH)). These models can capture random fluctuations in temperature based on historical data and output probability density predictions. However, existing statistical models have two fundamental technical shortcomings: First, they rely on a single information source, failing to incorporate physical information from large-scale climate oscillations such as El Niño-Southern Oscillation (ENSO). Studies show that ENSO has a significant low-frequency modulation effect on regional temperature fluctuations through atmospheric teleconnection processes; ignoring this physical mechanism leads to a systematic bias in the model's characterization of the long-term components of temperature fluctuations. Second, they lack the ability to decompose volatility across multiple scales, failing to separate short-term disturbances (daily random shocks), seasonal fluctuations (intra-annual cyclical variations), and low-frequency climate drivers (interannual scale modulation) within the same framework, resulting in an overall bias in the estimation of volatility risk.

[0005] Therefore, existing technologies suffer from the following technical shortcomings: 1) Numerical forecasts fail during the extended period due to the accumulation of initial field errors, and it is difficult to output probability density predictions; 2) Statistical models do not integrate low-frequency climate information based on physical mechanisms, making it impossible to achieve multi-scale decomposition of volatility; 3) There is a lack of technical solutions that can simultaneously meet the requirements of extended-period prediction accuracy, probability output capability, and low computational cost. These problems severely restrict the practical value of temperature forecasting in scenarios such as meteorological disaster early warning and agricultural risk management. Summary of the Invention

[0006] The purpose of this invention is to provide an extended-range temperature prediction method based on the GARCH-MIDAS model. This method addresses the technical shortcomings of numerical weather prediction models, which suffer from prediction failures due to the accumulation of initial field errors during the extended range. It also addresses the limitations of existing statistical models, which fail to integrate low-frequency climate information such as ENSO based on atmospheric teleconnection physical mechanisms and cannot achieve multi-scale decomposition of volatility. By constructing a unified modeling framework that integrates short-term volatility clustering, seasonal volatility, and long-term volatility driven by climate indices, this invention achieves high-precision point prediction and well-calibrated probability density prediction of extended-range temperature, providing reliable technical support for meteorological disaster early warning, agricultural risk management, and energy dispatch decisions.

[0007] Technical solution: A method for extended-range temperature prediction based on the GARCH-MIDAS model, comprising the following steps:

[0008] S1. Obtain historical daily average temperature time series data of the target area through meteorological observation network, and obtain relevant low-frequency monthly climate index data through climate monitoring system, with a time span of no less than 20 years;

[0009] S2, preprocess the acquired data, and divide the preprocessed data into a training set for model training and a test set for evaluating prediction accuracy according to the proportion;

[0010] S3, based on the physical mechanism of atmospheric teleconnection, constructs the GARCH-MIDAS model;

[0011] S4. The parameters of the GARCH-MIDAS model are estimated using the quasi-maximum likelihood estimation method.

[0012] S5. For the model parameters estimated in step S4, a recursive prediction method is used to perform out-of-sample prediction on the test set to generate a daily average temperature point prediction sequence with different advance days of extension; and a probability density prediction distribution is generated through Monte Carlo simulation.

[0013] S6. Evaluate the daily average temperature point prediction sequence and probability density prediction distribution obtained in step S5, and output the prediction results to the meteorological disaster early warning system, agricultural intelligent decision-making system or energy dispatch system.

[0014] Furthermore, the conditional mean equation of the GARCH-MIDAS model includes a deterministic trend term, a seasonality term, and an autoregressive term. The conditional variance equation includes a GARCH component characterizing short-term volatility clustering, a Fourier term characterizing the seasonality of volatility, and a long-term volatility component composed of a low-frequency climate index introduced through the MIDAS method. The conditional mean equation of the GARCH-MIDAS model is constructed as follows:

[0015] ,

[0016] In the formula, t represents the date, and T t This represents the average daily temperature on day t. Used to capture long-term linear warming trends caused by global warming and regional urbanization, among which The intercept term represents the reference temperature level at the initial time t0; The slope coefficient represents the change in average temperature per unit time, expressed in °C / day. The seasonal term describes the regular fluctuations in temperature over an annual cycle; the autoregressive term... The aim is to capture the medium- to long-term memory and persistent characteristics of temperature. Represents the autoregressive coefficient of order tl, indicating the strength of the influence of the temperature on the current temperature on the past day tl. The daily average temperature observation value represents the time lag tl; L is the fixed lag order. Let be the conditional standard deviation on day t; For random disturbance terms;

[0017] Seasonal items The expression is as follows:

[0018] ,

[0019] In the formula, Let P be the day of the year, and let P represent the Fourier series.

[0020] The formula for the conditional variance equation is expressed as:

[0021] ,

[0022] In the formula, The exponential function ensures a positive seasonal constraint on volatility. Used to capture the systematic variation patterns of volatility across different seasons within a year; Measuring the impact of new information The direct impact on current fluctuations; This indicates the persistence of volatility itself; long-term volatility components The formula is expressed as:

[0023] ,

[0024] In the formula, m is a constant term, representing the baseline long-term fluctuation level of the climate index when it is in a neutral state; Represents the monthly Nino 3.4 index; The marginal effect of the Nino3.4 index on the long-term temperature fluctuation level of the target area is measured; k is the MIDAS lag order, representing from 1 to k, with a maximum of 12 months.

[0025] The Beta weighting function is expressed as follows:

[0026] ,

[0027] In the formula, , These are the shape parameters.

[0028] Furthermore, the parameters of the GARCH-MIDAS model are estimated, including: calculating the model fitting residuals, evaluating the adequacy of the model through the residual distribution shape, standard deviation and autocorrelation function, and verifying whether the model effectively captures the trend, seasonality and fluctuation clustering characteristics of the temperature series.

[0029] Furthermore, the implementation steps of the recursive prediction method are as follows:

[0030] S51 uses all historical data from the beginning to the end of year Y as the initial training set to estimate the model parameters and predict the daily average temperature in year Y+1.

[0031] S52, the temperature data of the first predicted day of year Y+1 is used to update the state variables of the model; then, the daily average temperature of the next day is predicted based on the updated state and the initially estimated parameters; in this way, each time a new daily temperature observation is obtained, it is used to update the state variables, and the temperature of the next day is recursively predicted based on the updated state; this process continues until the entire test set period is covered, thus obtaining a series of out-of-sample prediction results based on fixed parameters and real-time information state updates.

[0032] Furthermore, the probability density prediction distribution is evaluated using the probability integral transform value. The specific steps for obtaining the probability integral transform value include:

[0033] S61, from the estimated predicted distribution of the GARCH-MIDAS model, simulate a large number of daily average temperature paths for a specified future period; for each simulated path, calculate its cumulative heating degree-day (CumHDD):

[0034] ,

[0035] In the formula, T base As the reference temperature, T t This represents the average daily temperature on day t.

[0036] S62, construct its empirical distribution function based on the N CumHDD values ​​calculated from the N paths. ; S63, for each actual observed date, substitute the actual cumulative heating degree days (CumHDD) into the constructed empirical distribution function. In the middle, calculate the probability integral transform value. .

[0037] Compared with the prior art, the significant advantages of this invention are as follows:

[0038] 1. This invention introduces a MIDAS mixing component into the variance equation of the GARCH model, effectively integrating the monthly Nino3.4 index into the daily temperature fluctuation modeling, and for the first time achieves a quantitative characterization of the low-frequency modulation effect of ENSO on regional temperature fluctuations. Taking the 2023-2024 forecast for the Yangtze River Delta region as an example, this invention demonstrates significantly lower root mean square error (RMSE) than traditional GARCH models and professional meteorological models in extended-range forecasts exceeding 10 days in advance, solving the problems of rapid error accumulation and insufficient accuracy in numerical forecast extended-range forecasts.

[0039] 2. This invention decomposes temperature volatility into three levels: seasonal volatility, short-term GARCH volatility, and long-term MIDAS volatility, clearly separating the sources of risk at different time scales and achieving multi-scale decomposition of volatility based on physical mechanisms. Model diagnostics show that the residual squared series has no significant autocorrelation, proving that the volatility characterization is more complete and accurate, and solving the problem of inaccurate long-term risk measurement in traditional statistical models.

[0040] 3. This invention, based on the GARCH-MIDAS model and Monte Carlo simulation, generates a complete probability distribution of future temperatures, providing probability density predictions that can be directly used in early warning systems. Probability integral transformation evaluation shows that the PIT value is uniformly distributed within a unit interval and that there is no significant autocorrelation among the moments, proving that the density prediction has good statistical calibration. This prediction result can be directly input into the threshold judgment module of a meteorological disaster early warning system to achieve probability-based risk warning.

[0041] 4. The modeling framework of this invention is based on the physical mechanism of atmospheric teleconnection. Its core logic does not depend on the climate characteristics of a specific geographical region. It has low computational cost, strong regional adaptability, and good technical generalization ability. This framework can be extended to other climate regions affected by large-scale climate systems. Attached Figure Description

[0042] Figure 1 This is an overall flowchart of the method of the present invention;

[0043] Figure 2 This is a time series diagram of daily average temperature in the Yangtze River Delta region from 2020 to 2024.

[0044] Figure 3 This is a schematic diagram of the kernel density estimation of the unconditional distribution of daily average temperature in the Yangtze River Delta region from 2000 to 2024.

[0045] Figure 4 This is a schematic diagram of the kernel density estimation of the unconditional distribution of the daily average temperature residuals in the Yangtze River Delta region from 2000 to 2024.

[0046] Figure 5 This is a time series plot of the daily average temperature residuals in the Yangtze River Delta region from 2020 to 2024.

[0047] Figure 6 This is a correlation diagram of the standard residual squares of daily average temperature in the Yangtze River Delta region from 2000 to 2024.

[0048] Figure 7 This is a comparison curve of the RMSE of the GARCH-MIDAS model, the traditional GARCH model and the meteorological model in different forecast periods in the embodiment.

[0049] Figure 8 This refers to the PIT values ​​and z, z², z³, and z_t values ​​corresponding to the cumulative daily heating degree observations for each winter in the Yangtze River Delta region from 2000 to 2022. 4 The autocorrelation function graph. Detailed Implementation

[0050] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0051] like Figure 1 As shown, this invention provides an extended-range temperature prediction method based on the GARCH-MIDAS model, comprising the following steps:

[0052] Step 1, Data Acquisition: Obtain historical daily average temperature time series data for the target area through meteorological observation networks, and obtain relevant low-frequency monthly climate index data through climate monitoring systems, with a time span of no less than 20 years.

[0053] Step 2 involves preprocessing the acquired data, including removing outliers, imputing missing values, and removing data for February 29th of leap years to ensure year-to-year alignment. The preprocessed data is then divided into a training set for model training and a test set for evaluating prediction accuracy.

[0054] Step 3: Based on the physical mechanism of atmospheric teleconnection, construct the GARCH-MIDAS model. Its conditional mean equation includes a deterministic trend term, a Fourier seasonality term, and an autoregressive term. Its conditional variance equation includes a GARCH component characterizing short-term volatility clustering, a Fourier term characterizing volatility seasonality, and a long-term volatility component composed of low-frequency climate indices introduced through the MIDAS (Mixed Data Sampling) method. The conditional mean equation of the GARCH-MIDAS model is constructed as follows:

[0055] (1)

[0056] In formula (1), t represents the date, and T t This represents the average daily temperature on day t. Used to capture long-term linear warming trends caused by global warming and regional urbanization, among which The intercept term represents the reference temperature level at the initial time t0; The slope coefficient represents the change in average temperature per unit time, expressed in °C / day. The seasonal term describes the regular fluctuations in temperature over an annual cycle; the autoregressive term... The aim is to capture the medium- to long-term memory and persistence characteristics of temperature, among which Represents the autoregressive coefficient of order tl, indicating the strength of the influence of the temperature on the current temperature on the past day tl. The daily average temperature observations represent the lag period tl; based on the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), the lag order L=25; Let be the conditional standard deviation on day t, i.e., the temperature fluctuation rate; This is a random disturbance term.

[0057] In the conditional mean equation, the seasonal term It is characterized using Fourier series:

[0058] (2)

[0059] In formula (2), It indicates the day of the year (February 29th in leap years has been excluded). The coefficients of the Pth cosine term are used to control the amplitude pattern (such as the temperature difference between winter and summer). This represents the coefficient of the P-th order sine term, controlling the phase shift (such as the difference between heating and cooling rates). A value of 3 for P indicates a third-order Fourier series. This setting ensures a smooth transition in seasonal patterns and outperforms traditional daily dummy variable methods due to its high parameter efficiency.

[0060] The formula for the conditional variance equation is expressed as:

[0061] (3)

[0062] In formula (3), the conditional variance is... The characterization contains multiple dynamic mechanisms. As a volatility seasonality factor, an exponential function is used to ensure that the volatility seasonality component is always positive; Characterized by a third-order Fourier series, it is used to capture the systematic variation of volatility in different seasons throughout the year, such as the annual periodicity of larger temperature fluctuations in winter and relatively smaller temperature fluctuations in summer. Measuring the impact of new information The direct impact on current fluctuations; while This indicates the persistence of volatility itself.

[0063] Long-term fluctuation components The formula is expressed as:

[0064] (4)

[0065] Formula (4) incorporates low-frequency macro-climate variables into the GARCH model using the MIDAS method. m is a constant term, representing the baseline long-term fluctuation level when the climate index is in a neutral state; K is the lag order, which uses the monthly Nino3.4 indices of the past K months for weighted combination to capture the low-frequency modulation effect of ENSO on the current daily temperature fluctuation rate, through the MIDAS weighting function. ENSO information with different lag months (k=1 represents the previous month, k=12 represents 12 months) is assigned different weights, thereby flexibly characterizing the time lag structure and persistence of ENSO impact. The monthly Nino3.4 index (a measure of sea surface temperature differences in the Atlantic, Pacific, and adjacent regions) is a key indicator for quantifying the intensity of the ENSO (El Niño-Southern Oscillation) phenomenon; parameters The marginal effect of the Nino3.4 index on the long-term temperature fluctuation level of the target region was measured.

[0066] The Beta weighting function has the following mathematical form:

[0067] (5)

[0068] In formula (5) , These are the shape parameters, It is generally fixed at 1. , This function, which jointly controls the decay rate of the lag weights, flexibly defines the influence pattern of the historical K-period Nino3.4 index on the current volatility with high parameter efficiency. It is particularly important to emphasize the long-term volatility component. It only applies to the conditional variance equation and does not directly affect the conditional mean.

[0069] Step 4: Based on the training set data, the parameters of the GARCH-MIDAS model are estimated using the quasi-maximum likelihood estimation method; the fitting residuals of the GARCH-MIDAS model are calculated, and the sufficiency of the model is evaluated by the residual distribution shape, standard deviation and autocorrelation function index, to verify whether it effectively captures the trend, seasonality and fluctuation clustering characteristics of the temperature series, and to ensure the rationality and robustness of the model settings.

[0070] Step 5: Based on the model parameters estimated in Step 4, a recursive prediction method is used to perform out-of-sample prediction on the test set, generating a daily average temperature point prediction sequence with different advance days of extension, and generating a probability density prediction distribution through Monte Carlo simulation.

[0071] The recursive prediction method works as follows: using all historical data from the start date to the end of year Y as the initial training set, the model parameters are estimated once, and then the daily average temperature for year Y+1 is recursively predicted based on these fixed parameters. During the recursive prediction process, each time a real temperature observation is obtained, it is used to update the model's state variables (such as the lag sequence of the autoregressive term and the GARCH conditional variance), and the model parameters are not re-estimated. Subsequently, based on the updated state and the initially estimated parameters, the daily average temperature for the next day is predicted. This process continues until the entire test set period is covered, resulting in a series of out-of-sample predictions based on fixed parameters and real-time information updates. For different prediction step sizes, the same recursive mechanism is used: after each prediction, the actual observation value for the corresponding date is used to update the state variables before proceeding to the next prediction. This process ensures that each prediction step is based on all available actual information up to that point in time, and that the prediction time point completely corresponds to the date of the weather forecast, guaranteeing the fairness of the comparison.

[0072] To systematically evaluate the point prediction performance of the prediction method of this invention, the root mean square error (RMSE) is used to evaluate the prediction accuracy: the root mean square error reflects the overall deviation between the predicted value and the actual observed value, and the smaller the root mean square error, the higher the prediction accuracy.

[0073] (6)

[0074] In formula (6), n is the total number of samples involved in the error calculation. The actual temperature value or true observed value for the i-th sample. The model predicts the temperature value for the i-th sample.

[0075] Step 6 evaluates the daily average temperature point prediction sequence and probability density prediction distribution obtained in Step 5, including calculating the root mean square error (RMSE) of the point predictions, the probability integral transform value of the probability density predictions, and the autocorrelation function (ACF), to verify the model prediction accuracy and distribution calibration effect. The prediction results are then output to meteorological disaster early warning systems, agricultural intelligent decision-making systems, or energy dispatching systems for risk warning threshold determination, irrigation window period decisions, or power load adjustments.

[0076] Density prediction quality assessment is performed using probability integral transformation (PIT). Taking density prediction for the cumulative heating degree days (CumHDD) during winter from 2000 to 2022 as an example, the PIT value of the actual observed values ​​in the predicted distribution is calculated, and its empirical distribution histogram and autocorrelation plots of each order of moments are plotted. Figure 7 The specific steps for generating the empirical distribution of daily cumulative heating degree (CumHDD) through Monte Carlo simulation include:

[0077] From the estimated predicted distribution of the GARCH-MIDAS model, simulate a large number (N=10,000) of daily average temperature paths for a specified future period (e.g., from November 1 to March 31 of the following year);

[0078] For each simulated path, calculate its cumulative heating degree-day (CumHDD):

[0079] (7)

[0080] In formula (7), T base Reference temperature;

[0081] Based on the N CumHDD values ​​calculated from the N paths, construct its empirical distribution function. ;

[0082] For each actual observed date, the true cumulative heating degree days (CumHDD) are substituted into the constructed empirical distribution function. In the middle, calculate the probability integral transform value. Evaluate the empirical distribution function Distribution generated from actual data Consistency; if the model is perfect, that is = For all y, then the sequence It should be a uniform distribution that is independently and identically distributed over a unit interval, i.e. The uniformity of the sequence is tested using a quartile histogram, and the independence is tested by calculating the autocorrelation function (ACF) of the sequence and its powers.

[0083] It should be noted that the GARCH-MIDAS model framework of this invention is constructed based on the physical mechanism of atmospheric teleconnection. Its core logic is to introduce the ENSO index through MIDAS technology to characterize the low-frequency modulation effect of large-scale climate oscillations on regional temperature fluctuations, without relying on the climate characteristics of a specific geographical region. In this embodiment, the Yangtze River Delta region is used as an application case to verify the effectiveness of the GARCH-MIDAS model in the subtropical monsoon climate zone. Those skilled in the art will understand that this modeling framework can be extended to other climate regions affected by large-scale climate systems (such as temperate monsoon regions, plateau climate zones, etc.), simply by adjusting the model parameters (such as autoregressive order, Fourier order, MIDAS lag order, etc.) according to the climate characteristics of the target region.

[0084] In summary, this invention constructs a GARCH-MIDAS model, integrates low-frequency climate indices into high-frequency temperature volatility modeling, employs a recursive prediction mechanism to ensure the real-time performance and reliability of the prediction process, and combines Monte Carlo simulation to provide complete density predictions. Experimental results show that the method of this invention significantly improves the accuracy of temperature point prediction over the extended period, has good density prediction calibration, high computational efficiency, and strong regional adaptability, possessing significant operational application value and promising prospects for widespread application.

[0085] It should be understood that the above-described embodiments are merely one specific implementation of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, or improvements made by those skilled in the art based on the technical solutions of the present invention should be included within the protection scope of the present invention.

[0086] Furthermore, the present invention also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the extended-range temperature prediction method described above. This storage medium includes, but is not limited to, various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory, random access memory, magnetic disks, or optical disks.

[0087] The memory may include non-permanent memory in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM, and the memory includes at least one memory chip.

[0088] A computer program product includes a computer program / instructions that, when executed by a processor, implement the steps of the extended-range temperature prediction method of the present invention.

[0089] Those skilled in the art will understand that the method described in this invention can be implemented through software, hardware, or a combination of both. The system can be developed and deployed using programming languages ​​such as Python and R, combined with statistical modeling libraries and deep learning frameworks.

[0090] In business application scenarios, the system described in this invention can adopt a microservice architecture, support modular deployment and horizontal expansion, and provide a visual interactive interface at the front end. Users can quickly obtain temperature point prediction and density prediction results for a specified area and a specified period through map selection, parameter configuration, etc., and support data export, report generation and other functions.

[0091] The method and system proposed in this invention have been piloted in various fields such as actual meteorological services, power load forecasting, and agricultural climate risk assessment, and have received positive feedback. In the future, it can be further expanded to more cross-industry scenarios such as urban planning, traffic management, and energy trading, leveraging its advantages in multi-scale, high-precision, and probabilistic temperature forecasting.

[0092] This embodiment uses extended-period temperature forecasting in the Yangtze River Delta region of China as an example to illustrate the specific implementation of this method. Figure 1 As shown, the method flow is as follows:

[0093] Daily average temperature (DAT) data for the Yangtze River Delta region from January 1, 2000 to December 31, 2024 were obtained from the China Financial Meteorological Index and Service Platform. Monthly Nino3.4 index data for the same period were obtained from the National Oceanic and Atmospheric Administration's Climate Prediction Center (NOAA).

[0094] The temperature data was checked and any missing values ​​were imputed; February 29th data for all leap years were removed to ensure each year has 365 days and that dates are aligned between years; the monthly Nino3.4 index was converted into a daily frequency series by assigning it to all dates of the corresponding month. Data from 2000 to 2022 was used as the training set, and data from 2023 to 2024 was used as the test set. Figure 2 This section presents a time series plot of daily average temperature in the Yangtze River Delta region over the past five years, along with a kernel density estimate of the unconditional distribution for the entire sample period (see...). Figure 3 ).

[0095] Based on the characteristics of daily average temperature data in the Yangtze River Delta region from 2000 to 2022, the parameters of the GARCH-MIDAS model are set with clear econometric basis and climatological meaning. The autoregressive order L=25 aims to capture the possible medium- and long-term memory characteristics of temperature in this region, ensuring that the GARCH-MIDAS model can fully extract the persistent information in the temperature series; the Fourier seasonality term order P=3 and the volatility seasonality term order Q=3 are both fitted with the complex periodic changes of temperature and its fluctuations within the year with high parameter efficiency through third-order Fourier series; the MIDAS lag order K=12 corresponds to the lag number of low-frequency climate indexes over the whole year, so as to fully extract the trans-annual impact of large-scale climate oscillations such as ENSO on regional temperature variability; the GARCH order is set to (1,1), which is the most classic and robust parameterization form for capturing the volatility clustering effect. Overall, this parameter combination ensures the flexibility of the GARCH-MIDAS model while taking into account estimation efficiency. It can simultaneously characterize the trend, seasonality, short-term fluctuation clusters, and long-term fluctuation components driven by climate indices of temperature in the Yangtze River Delta region, laying a reliable model foundation for subsequent density prediction.

[0096] A GARCH-MIDAS model was constructed based on the above parameter configuration, and quasi-maximum likelihood estimation was performed using the training set data. Subsequently, a systematic diagnosis of the GARCH-MIDAS model's fit was conducted to ensure its effectiveness and reliability.

[0097] Figure 4 The unconditional distribution of the standardized residuals after fitting the GARCH-MIDAS model is shown, exhibiting a unimodal shape that contrasts sharply with the bimodal distribution of the original temperature data. This indicates that the GARCH-MIDAS model has effectively extracted the deterministic components from the sequence. The standard deviation of the residuals is significantly lower than that of the original temperature, confirming that weather risk primarily stems from unpredictable short-term fluctuations, and that their variability is much smaller than the overall temperature variation. The overall goodness of fit R0 of the conditional mean model is [value missing]. 2 The accuracy rate reached 95.8%, which fully validated the effectiveness of the model design.

[0098] Figure 5 The estimated conditional variance series of daily average temperature from the GARCH-MIDAS model over the past five years was plotted, visually revealing the seasonal changes in volatility throughout the year, the short-term clustering effect influenced by extreme weather, and the long-term evolution trend driven by low-frequency climate indices.

[0099] Figure 6 Furthermore, a correlation plot of the standard residual squares of daily average temperature was provided. The results show that there is no significant autocorrelation in the residual squares sequence, proving that the sequence dependence in the conditional mean of the GARCH-MIDAS model has been fully extracted, and the GARCH-MIDAS model structure has successfully absorbed the fluctuation clustering effect in the original temperature sequence.

[0100] The above diagnostic results (see) Figure 4 , Figure 5 , Figure 6 Together, they verified the rationality of the GARCH-MIDAS model structure and parameter settings, laying the foundation for the reliability of subsequent out-of-sample predictions.

[0101] Model estimation and recursive prediction utilize Python's `arch` library and the `midasr` extension. Quasi-maximum likelihood estimation is used to estimate the GARCH-MIDAS model parameters in a single step. Fixed-parameter recursive out-of-sample prediction is employed: historical data from 2000 to 2022 is used as the initial training set to estimate the model parameters in a single step. Based on these fixed parameters, the daily average temperature for January 1, 2023, is predicted using state variables up to the prediction date (including autoregressive lags, GARCH conditional variance, etc.). After obtaining the actual observations for January 1, 2023, these are only used to update the state variables (updating the lag sequences of the autoregressive terms and the GARCH conditional variance), without re-estimating the model parameters. Subsequently, based on the updated state and the initially estimated parameters, the temperature for January 2, 2023, is predicted. This process is repeated, continuously updating the state variables, and predictions are made sequentially until December 31, 2024, generating a total of 730 days of out-of-sample predictions. The same recursive mechanism is used for different forecast periods: after each forecast, the actual observation value of the corresponding date is used to update the state variables before the next forecast is made. This ensures that each forecast is based on all available actual information up to that point in time, and that the forecast date corresponds exactly to the date the meteorological model releases its forecast, so as to ensure the fairness of the comparison.

[0102] In contrast, the GARCH model also uses this recursive forecasting method, while the meteorological model directly obtains forecast data on the number of days the same length is advanced from professional meteorological agencies.

[0103] To verify model performance, this implementation compares the performance of the GARCH-MIDAS model, a traditional GARCH model without MIDAS components, and the CFSv2 (Climate Forecast System Version 2) real-time weather forecast model developed by the National Center for Environmental Prediction (NCEP) on the test set (2023-2024). CFSv2 is a next-generation climate prediction system developed by the NCEP. Its forecast products integrate advanced dynamical models and data assimilation techniques, providing important references for short-term weather forecasts and serving as a core basis for extended-range and even seasonal climate predictions. This model is widely used in meteorological operations and research, and its forecasting skill represents the mainstream level of current numerical models at extended-range scales. It provides a strong external reference for the predictive capabilities of the GARCH-MIDAS model in this invention; this benchmark is referred to as the "weather model" below. The evaluation index is the root mean square error (RMSE), and the results are compared as follows: Figure 7 As shown in Table 1.

[0104] Based on actual observation data from January 1, 2000 to December 31, 2022 in the Yangtze River Delta region (Example 1), the point prediction accuracy of the method of this invention, the traditional GARCH model, and the numerical meteorological model at different lead times was compared across the entire time period. It should be noted that this invention focuses on extended-range (10-30 days) prediction; short-range (1-7 days) comparisons are only used to demonstrate the behavior of the GARCH-MIDAS model and are not considered as a basis for the scope of protection of this invention. The comparison results are shown in Table 1:

[0105] Table 1. Point prediction accuracy indices for meteorological models, GARCH models, and GARCH-MIDAS models.

[0106]

[0107] The results show that in the short-term (1-7 days lead time) forecast period, meteorological models outperformed meteorological models due to their abundant real-time data. However, in the extended forecast period (10-30 days), the RMSE of the GARCH-MIDAS model was consistently lower than that of the meteorological model, with RMSEs of 3.12℃ and 3.66℃ on day 10, decreasing to 2.81℃ by day 30, while the meteorological model rose to 4.41℃. This indicates that as the forecast lead time increases, the accuracy of meteorological models declines rapidly, while time series models that can robustly characterize trends, seasonality, and autoregressive dependencies are more advantageous in extended-term forecasts. This aligns with the background art's understanding that numerical forecasts are superior in the short term and ineffective in the extended period, further validating the technical contribution of this invention.

[0108] To visually demonstrate the forecast results, three RMSE forecast comparison curves for the daily average temperature throughout 2023-2024 were plotted (see...). Figure 7 As can be seen, in the short-term range of 1-7 days in advance, the meteorological model has the lowest RMSE, verifying its short-term advantage; in the extended-term range of 10-30 days in advance, the meteorological model's RMSE rises rapidly, the benchmark GARCH model fluctuates significantly, while the RMSE curve of the GARCH-MIDAS model of this invention remains stable and at the lowest level. This comparison fully demonstrates that the present invention has a stable and significant technical advantage in extended-term forecasting. At the same time, this also verifies that the selected CFSv2 meteorological model has good predictive performance in short-term forecasting, consistent with existing technical understanding.

[0109] In terms of density prediction calibration, the PIT assessment is conducted by predicting the cumulative heating degree days (November 1 to March 31 of the following year) for each winter from 2000 to 2022:

[0110] Based on historical data fitting models, 10,000 daily temperature paths from 2000 to 2022 were simulated using Monte Carlo simulation. Each simulated daily temperature path was converted into heating degree days, and its winter cumulative value was calculated, ultimately yielding 10,000 daily temperature paths for each of the 22 years. The value is used to construct its empirical distribution function. y represents winter.

[0111] Figure 8 The distribution characteristics and autocorrelation plot of PIT values ​​in the Yangtze River Delta region are presented. This distribution pattern indicates that the model of this invention exhibits relatively balanced calibration characteristics when predicting the cumulative heating degree-days in winter in the Yangtze River Delta region, without systematically underestimating CumHDD. The distribution of the PIT histogram in the four bins remains within the 95% confidence interval, with a mean of 0.55, only slightly higher than the ideal value of 0.5. Autocorrelation function (ACF) analysis shows no significant serial correlation detected at multiple lag orders. This demonstrates the model's good performance in terms of uniformity and calibration.

[0112] In summary, this invention, through the GARCH-MIDAS modeling framework, effectively integrates multi-scale climate information, significantly improving the accuracy and robustness of extended-range temperature forecasts. It has important theoretical and practical significance for serving the scientific pricing and risk management of weather derivatives.

[0113] The GARCH-MIDAS model framework of this invention has regional universality, and its core modeling logic does not depend on the climate characteristics of a specific geographical region. The embodiments use the Yangtze River Delta region as an application case to verify the model's effectiveness in the subtropical monsoon climate zone. It can be further extended to other climate regions; simply adjusting the model parameters according to the climate characteristics of the target region is sufficient, demonstrating its good generalization ability and cross-regional adaptability.

[0114] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for extended-range temperature prediction based on the GARCH-MIDAS model, characterized in that, Includes the following steps: S1. Obtain historical daily average temperature time series data of the target area through meteorological observation network, and obtain relevant low-frequency monthly climate index data through climate monitoring system, with a time span of no less than 20 years; S2, preprocess the acquired data, and divide the preprocessed data into a training set for model training and a test set for evaluating prediction accuracy according to the proportion; S3, based on the physical mechanism of atmospheric teleconnection, constructs the GARCH-MIDAS model; S4. The parameters of the GARCH-MIDAS model are estimated using the quasi-maximum likelihood estimation method. S5. For the model parameters estimated in step S4, a recursive prediction method is used to perform out-of-sample prediction on the test set to generate a daily average temperature point prediction sequence with different advance days of extension; and a probability density prediction distribution is generated through Monte Carlo simulation. S6. Evaluate the daily average temperature point prediction sequence and probability density prediction distribution obtained in step S5, and output the prediction results to the meteorological disaster early warning system, agricultural intelligent decision-making system or energy dispatch system.

2. The extended-range temperature prediction method based on the GARCH-MIDAS model according to claim 1, characterized in that, The conditional mean equation of the GARCH-MIDAS model includes a deterministic trend term, a seasonality term, and an autoregressive term. The conditional variance equation includes a GARCH component characterizing short-term volatility clustering, a Fourier term characterizing the seasonality of volatility, and a long-term volatility component composed of a low-frequency climate index introduced through the MIDAS method. The conditional mean equation of the GARCH-MIDAS model is constructed as follows: ， In the formula, t represents the date, and T t This represents the average daily temperature on day t. Used to capture long-term linear warming trends caused by global warming and regional urbanization, among which The intercept term represents the reference temperature level at the initial time t0; The slope coefficient represents the change in average temperature per unit time, expressed in °C / day. The seasonal term describes the regular fluctuations in temperature over an annual cycle; the autoregressive term... The aim is to capture the medium- to long-term memory and persistent characteristics of temperature. Represents the autoregressive coefficient of order tl, indicating the strength of the influence of the temperature on the current temperature on the past day tl. The daily average temperature observation value represents the time lag tl; L is the fixed lag order. Let be the conditional standard deviation on day t; For random disturbance terms; Seasonal items The expression is as follows: ， In the formula, Let P be the day of the year, and let P represent the Fourier series. The formula for the conditional variance equation is expressed as: ， In the formula, The exponential function ensures a positive seasonal constraint on volatility. Used to capture the systematic variation patterns of volatility across different seasons within a year; Measuring the impact of new information The direct impact on current fluctuations; This indicates the persistence of volatility itself; long-term volatility components The formula is expressed as: ， In the formula, m is a constant term, representing the baseline long-term fluctuation level of the climate index when it is in a neutral state; Represents the monthly Nino 3.4 index; The marginal effect of the Nino3.4 index on the long-term temperature fluctuation level of the target area is measured; k is the MIDAS lag order, representing from 1 to k, with a maximum of 12 months. The Beta weighting function is expressed as follows: ， In the formula, , These are the shape parameters.

3. The extended-range temperature prediction method based on the GARCH-MIDAS model according to claim 1, characterized in that, The parameters of the GARCH-MIDAS model are estimated, including: calculating the model fitting residuals, evaluating the adequacy of the model through the residual distribution shape, standard deviation and autocorrelation function, and verifying whether the model effectively captures the trend, seasonality and fluctuation clustering characteristics of the temperature series.

4. The extended-range temperature prediction method based on the GARCH-MIDAS model according to claim 1, characterized in that, The implementation steps of the recursive prediction method are as follows: S51 uses all historical data from the beginning to the end of year Y as the initial training set to estimate the model parameters and predict the daily average temperature in year Y+1. S52, the temperature data of the first predicted day of year Y+1 is used to update the state variables of the model; then, the daily average temperature of the next day is predicted based on the updated state and the initially estimated parameters; in this way, each time a new daily temperature observation is obtained, it is used to update the state variables, and the temperature of the next day is recursively predicted based on the updated state; this process continues until the entire test set period is covered, thus obtaining a series of out-of-sample prediction results based on fixed parameters and real-time information state updates.

5. The extended-range temperature prediction method based on the GARCH-MIDAS model according to claim 1, characterized in that, The probability density prediction distribution is evaluated using the probability integral transform value. The specific steps for obtaining the probability integral transform value include: S61, from the estimated predicted distribution of the GARCH-MIDAS model, simulate a large number of daily average temperature paths for a specified future period; for each simulated path, calculate its cumulative heating degree-day (CumHDD): ， In the formula, T base As the reference temperature, T t This represents the average daily temperature on day t. S62, construct its empirical distribution function based on the N CumHDD values ​​calculated from the N paths. ; S63, for each actual observed date, substitute the actual cumulative heating degree days (CumHDD) into the constructed empirical distribution function. In the middle, calculate the probability integral transform value. .

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the method as described in any one of claims 1 to 5.

7. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the computer program is executed, it implements the steps of the method as described in any one of claims 1 to 5.

8. A computer program product comprising a computer program / instructions, characterized in that, When the computer program / instructions are executed by the processor, they implement the steps of the method according to any one of claims 1 to 5.

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