A hydrological time series dependence model optimization method
Patent Information
- Application Number
- CN202611134626.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-29
- Publication Date
- 2026-10-27
AI Technical Summary
[0006]针对于上述现有技术的不足，本发明的目的在于提供一种水文时间序列相依模型优选方法，以解决现有技术中因缺乏长、短相依模型的统一评价基准，且仅以相关系数最大化为优选导向极易导致模型过度拟合，难以合理平衡相依性拟合相关度的统计不确定性与模型参数复杂度，从而导致水文时间序列相依性类型界定偏差、模型优选失准以及长期预测泛化能力与可靠性降低的问题
[0054]本发明方法不仅能够客观界定水文时间序列的长、短相依性类型，而且能够利用统一的信息准则在六个候选相依模型中自动优选出最适配的相依模型；针对自回归、滑动平均及自回归滑动平均三种短相依模型，本发明对传统公式中存在的偏差和错误进行了针对性勘正，并给出了推导更为严谨、形式更为简洁的解析闭式解，这一改进从根本上保障了短相依性成分在基准计算层面的准确性，消除了由于公式计算偏误引入的累积系统误差；针对分数差分的自回归、分数差分的滑动平均及分数差分的自回归滑动平均三种长相依模型，本发明彻底摒弃了“先对长相依序列进行分数阶差分、再套用短相依模型解析公式”的间接近似做法，直接从长相依随机过程的核心波谱特征出发，推导并建立了精准的频域积分解析表达式；保障了长相依性成分表征的严谨性与完整性，避免了分数差分截断导致的统计信息流失。通过引入基于相关系数的信息准则值，有效解决了传统优选方法易因单纯追求高相关系数而导致模型结构冗余和过度拟合（过拟合）的难题，保证了所选相依模型在拟合精度与模型简约性之间达到合理平衡，从而进一步提升了中长期水文模拟和预测的泛化能力。
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of hydrological and climatic science and time series analysis technology, specifically relating to a method for optimizing a hydrological time series dependency model. Background Technology
[0002] Accurate interpretation of the characteristic information contained in the time series of hydrological variables is the cornerstone of scientifically understanding hydrophysical laws, conducting efficient water resource management, and preventing and controlling floods and droughts. Dependency, as one of the core characteristics of hydrological time series, reflects the inherent correlation between hydrological variables at different times (such as the clustering tendency of drought or flood events). Its accurate identification and modeling are crucial for hydrological forecasting, stochastic simulation, and risk assessment. The dependency of hydrological series can generally be divided into two categories: short-term dependency (short-memory) and long-term dependency (long-memory). In stochastic hydrology, autoregressive (AR) models, moving average (MA) models, and autoregressive moving average (ARMA) models are classic tools for characterizing short-term dependency; while fractional-difference autoregressive (FAR) models, fractional-difference moving average (FMA) models, and fractional-difference autoregressive moving average (FARMA) models are important means of describing long-term dependency. However, in practical applications, there are still many technical challenges in how to objectively and reasonably define the type of dependency of hydrological time series and select the most suitable dependency model for simulation analysis.
[0003] When comparing different dependency models, using only the highest correlation coefficient between the fitted dependency component and the original sequence as the selection criterion can guarantee high fitting accuracy. However, since higher-order or complex models usually have more parameters to be estimated, this "accuracy-first" selection mechanism is prone to choosing overfitting dependency models. This not only increases the burden of computation and parameter estimation but also severely weakens the model's generalization ability in actual predictions. Short dependency models and long dependency models have fundamental differences in mathematical mechanisms (long dependency models usually introduce fractional difference operators, which change the statistical properties of time series). In the existing analytical framework, it is difficult to conduct a systematic horizontal comparison of these six different dependency models under the same index, thus making it impossible to find the optimal balance between ensuring correlation (goodness of fit) and controlling model complexity (number of parameters).
[0004] The Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) is commonly used for determining the order of a single model, but it is usually constructed based on residual variance or likelihood functions. When comparing multiple models with different structures, such as those with long and short dependencies, the applicability of traditional information criteria is limited due to differences in data transformation (such as differencing operations) and likelihood function definitions. Although standard goodness-of-fit indices such as the Nash efficiency coefficient can be considered, they lack parameter complexity penalties, cannot identify the optimal model, and hinder the distinction between short and long dependency types. Furthermore, traditional evaluation methods rarely integrate the statistical uncertainty of the correlation coefficient itself with the model parameter penalty to construct a unified evaluation index, thus making it difficult to effectively support the optimization of dependency models.
[0005] In summary, there is an urgent need for a framework that can integrate short-dependent and long-dependent models into a unified analysis and optimization framework. This framework should be able to effectively measure the correlation between dependent components and the original sequence and their statistical uncertainty, while also reasonably penalizing the complexity of model parameters to avoid overfitting. Summary of the Invention
[0006] To address the shortcomings of the existing technologies, the present invention aims to provide a method for optimizing hydrological time series dependency models. This method solves the problems in the existing technologies, such as the lack of a unified evaluation benchmark for long-term and short-term dependency models, the tendency to overfit models by focusing solely on maximizing correlation coefficients, the difficulty in reasonably balancing the statistical uncertainty of dependency fitting correlation with the complexity of model parameters, and the resulting deviations in defining hydrological time series dependency types, inaccurate model optimization, and reduced long-term forecast generalization ability and reliability.
[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0008] The present invention provides a method for optimizing a hydrological time series dependency model, comprising the following steps:
[0009] 1) Six candidate dependency models were used to fit the hydrological time series respectively. Dependent components in;
[0010] 2) Calculate the dependent component sequence and hydrological time series obtained from fitting each candidate dependent model. The correlation coefficient between them;
[0011] 3) Using the correlation coefficients obtained in step 2), calculate the information criterion value based on the correlation coefficients for each candidate dependency model;
[0012] 4) Compare the information criterion values based on the correlation coefficient obtained in step 3), and determine the candidate dependency model with the smallest information criterion value based on the correlation coefficient as the model describing the hydrological time series. The optimal dependency model for the dependent components in the model.
[0013] Furthermore, the six candidate dependency models in step 1) include: autoregressive model, moving average model, autoregressive moving average model, fractional difference autoregressive model, fractional difference moving average model and fractional difference autoregressive moving average model.
[0014] Further, step 2) specifically includes:
[0015] 21) Using an autoregressive model to fit hydrological time series The short-dependency components in the data were analyzed, and the short-dependency component sequences fitted by the autoregressive model were compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0016] ;
[0017] in, For the autoregressive model Autoregressive coefficients of order 1 Hydrological time series The Autocorrelation coefficient; The order of the autoregressive model;
[0018] 22) Fitting hydrological time series using a moving average model The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0019] ;
[0020] in, For the moving average model The moving average coefficient; The order of the moving average model;
[0021] 23) The hydrological time series were fitted using an autoregressive moving average model. The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the autoregressive moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0022] ;
[0023] in, For the first autoregressive moving average model Autoregressive coefficients; For the first autoregressive moving average model The moving average coefficient; is the autoregressive order of the autoregressive moving average model; Let be the order of the moving average in the autoregressive moving average model. For the first Each impulse response coefficient is obtained recursively from the parameters, as follows:
[0024] , ;
[0025] In the formula, The impulse response coefficient is... Let be the ordinal number of the impulse response coefficient, and ;
[0026] 24) Estimating hydrological time series using the recalibrated range R / S analysis method Given the Hearst coefficients H, the fractional difference order d is calculated using the following expression:
[0027] d = H - 0.5;
[0028] 25) A fractional difference autoregressive model was used to fit the hydrological time series. The length-dependent components in the data are fitted by an autoregressive model with fractional differences, and the length-dependent component sequences are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0029] ;
[0030] in, Let be the frequency response function of the autoregressive operator. Let be the order of the fractional differencing autoregressive model, e be the base of the natural logarithm, and i be the imaginary unit, satisfying the definition. , The independent variable is the continuous angular frequency, and the integration interval is... It represents a complete fundamental period of a discrete-time series in the frequency domain. This represents the frequency shift operator in the discrete-time Fourier transform. Indicates taking the modulus of a complex number;
[0031] 26) A fractional difference moving average model was used to fit the hydrological time series. The length-dependent components in the data are fitted by a fractional difference moving average model to the length-dependent component sequence and the hydrological time series. Correlation coefficient between The expression is as follows:
[0032] ;
[0033] in, Let be the frequency response function of the moving average operator. The order of the fractional difference moving average model;
[0034] 27) A fractional difference autoregressive moving average model was used to fit the hydrological time series. The length-dependent components in the data were compared with the hydrological time series using an autoregressive moving average model fitted by fractional differences. Correlation coefficient between The expression is as follows:
[0035] ;
[0036] in, Let be the autoregressive order of the fractional differencing autoregressive moving average model. is the moving average order of the fractional difference autoregressive moving average model.
[0037] Furthermore, the calculation formula for the information criterion value based on the correlation coefficient in step 3) is as follows:
[0038] ;
[0039] in, The correlation coefficient is the information criterion value used to evaluate and screen the optimal dependency model as a dimensionless objective decision indicator, where N is the hydrological time series. Length, Dependency component sequences fitted to each candidate dependency model and hydrological time series The square of the correlation coefficient between them For each candidate model, the order is [order]. For autoregressive models, [order is missing]. For the moving average model, For the autoregressive moving average model, For fractional differencing autoregressive models, For the fractional difference moving average model, For the fractional differencing autoregressive moving average model, ; For summing loop variables, Used to identify the first in the model The dimensional hierarchy of each estimated parameter in the parameter space. Weights are scaled to account for information capacity.
[0040] Furthermore, step 3) specifically includes:
[0041] 31) Calculate the information criterion value based on the correlation coefficient for the autoregressive model. The expression is as follows:
[0042] ;
[0043] 32) Calculate the information criterion value based on the correlation coefficient for the moving average model. The expression is as follows:
[0044] ;
[0045] 33) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model. The expression is as follows:
[0046] ;
[0047] 34) Calculate the information criterion value based on the correlation coefficient for the autoregressive model of fractional differencing. The expression is as follows:
[0048] ;
[0049] 35) Calculate the information criterion value based on the correlation coefficient for the moving average model of fractional differences. The expression is as follows:
[0050] ;
[0051] 36) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model of fractional differencing. The expression is as follows:
[0052] .
[0053] The beneficial effects of this invention are:
[0054] This invention not only objectively defines the long and short dependency types of hydrological time series, but also automatically selects the most suitable dependency model from six candidate models using a unified information criterion. For the three short dependency models—autoregressive, moving average, and autoregressive moving average—this invention specifically corrects the biases and errors in traditional formulas and provides a more rigorous and concise analytical closed-form solution. This improvement fundamentally ensures the accuracy of short dependency components at the baseline calculation level and eliminates the cumulative systematic error introduced by formula calculation bias. For the three long dependency models—fractionally differencing autoregressive, fractionally differencing moving average, and fractionally differencing autoregressive moving average—this invention completely abandons the indirect approximation approach of "first performing fractional differencing on the long dependency series and then applying the analytical formula of the short dependency model," directly deriving and establishing a precise frequency domain integral analytical expression from the core spectral characteristics of the long dependency stochastic process. This ensures the rigor and completeness of the long dependency component representation and avoids the loss of statistical information caused by fractional differencing truncation. By introducing information criteria based on correlation coefficients, the problem of model structure redundancy and overfitting caused by simply pursuing high correlation coefficients in traditional optimization methods is effectively solved. This ensures that the selected dependent model achieves a reasonable balance between fitting accuracy and model simplicity, thereby further improving the generalization ability of medium- and long-term hydrological simulation and prediction.
[0055] This invention constructs a unified information criterion (CCIC) evaluation framework based on the fitting error of correlation coefficients, systematically integrating short-dependent models (AR, MA, ARMA) and long-dependent models (FAR, FMA, FARMA) under the same quantitative evaluation benchmark. It not only quantitatively considers the statistical uncertainty of correlation coefficients under limited samples, but also solves the technical bottleneck that makes it difficult to directly compare models across different types of dependence due to differences in structure and difference transformation. It effectively avoids the subjectivity and blindness in traditional model selection, thus providing more statistically rigorous methodological support for deeply revealing the evolution mechanism of hydro-climate processes and conducting highly reliable uncertainty assessments. Attached Figure Description
[0056] Figure 1 This is a flowchart of the method of the present invention;
[0057] Figure 2a A schematic diagram of an artificially generated sequence S11 with short dependencies that follow an AR(1) process;
[0058] Figure 2b A schematic diagram of an artificially generated sequence S12 with short dependencies that follow an AR(2) process;
[0059] Figure 2c A schematic diagram of an artificially generated sequence S13 with short dependencies that conform to the AR(3) process;
[0060] Figure 3a A schematic diagram of an artificially generated sequence S21 with short dependencies that follow an MA(1) process;
[0061] Figure 3b A schematic diagram of an artificially generated sequence S22 that has short dependencies following a MA(2) process;
[0062] Figure 3c A schematic diagram of an artificially generated sequence S23 with short dependencies that follow the MA(3) process;
[0063] Figure 4a A schematic diagram of an artificially generated sequence S31 with short dependencies that follow an ARMA(1,1) process;
[0064] Figure 4b A schematic diagram of an artificially generated sequence S32 with short dependencies that follow an ARMA(1,2) process;
[0065] Figure 4c A schematic diagram of an artificially generated sequence S33 with short dependencies that follow an ARMA(2,1) process;
[0066] Figure 5a A schematic diagram of an artificially generated sequence S41 that exhibits long dependency following a FAR(1) process;
[0067] Figure 5b A schematic diagram of an artificially generated sequence S42 that exhibits long dependency following a FAR(2) process;
[0068] Figure 5c A schematic diagram of an artificially generated sequence S43 that exhibits long dependency following a FAR(3) process;
[0069] Figure 6a A schematic diagram of an artificially generated sequence S51 with long dependency following an FMA(1) process;
[0070] Figure 6b A schematic diagram of an artificially generated sequence S52 that exhibits long dependency following an FMA(2) process;
[0071] Figure 6c A schematic diagram of an artificially generated sequence S53 that exhibits long dependency following an FMA(3) process;
[0072] Figure 7a A schematic diagram of an artificially generated sequence S61 that exhibits long dependency following a FARMA(1,1) process;
[0073] Figure 7bA schematic diagram of an artificially generated sequence S62 that exhibits long dependency following a FARMA(1,2) process;
[0074] Figure 7c This is a schematic diagram of an artificially generated sequence S63 that has long dependency following the FARMA(2,1) process. Detailed Implementation
[0075] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to embodiments and accompanying drawings. The content mentioned in the embodiments is not intended to limit the present invention.
[0076] Reference Figure 1 As shown, the preferred method for a hydrological time series dependency model according to the present invention comprises the following steps:
[0077] 1) Six candidate dependency models were used to fit the hydrological time series respectively. Dependent components in;
[0078] The six candidate dependency models in step 1) include: autoregressive model, moving average model, autoregressive moving average model, fractional difference autoregressive model, fractional difference moving average model and fractional difference autoregressive moving average model.
[0079] 2) Calculate the dependent component sequence and hydrological time series obtained from fitting each candidate dependent model. The correlation coefficient between them; specifically including:
[0080] 21) Using an autoregressive model to fit hydrological time series The short-dependency components in the data were analyzed, and the short-dependency component sequences fitted by the autoregressive model were compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0081] ;
[0082] in, For the autoregressive model Autoregressive coefficients of order 1 Hydrological time series The Autocorrelation coefficient; The order of the autoregressive model;
[0083] 22) Fitting hydrological time series using a moving average model The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0084] ;
[0085] in, For the moving average model The moving average coefficient; The order of the moving average model;
[0086] 23) The hydrological time series were fitted using an autoregressive moving average model. The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the autoregressive moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0087] ;
[0088] in, For the first autoregressive moving average model Autoregressive coefficients; For the first autoregressive moving average model The moving average coefficient; is the autoregressive order of the autoregressive moving average model; Let be the order of the moving average in the autoregressive moving average model. For the first Each impulse response coefficient is obtained recursively from the parameters, as follows:
[0089] , ;
[0090] In the formula, The impulse response coefficient is... Let be the ordinal number of the impulse response coefficient, and ;
[0091] 24) Estimating hydrological time series using the recalibrated range R / S analysis method Given the Hearst coefficients H, the fractional difference order d is calculated using the following expression:
[0092] d = H - 0.5;
[0093] 25) A fractional difference autoregressive model was used to fit the hydrological time series. The length-dependent components in the data are fitted by an autoregressive model with fractional differences, and the length-dependent component sequences are compared with the hydrological time series. Correlation coefficient between The expression is as follows:
[0094] ;
[0095] in, Let be the frequency response function of the autoregressive operator. Let be the order of the fractional differencing autoregressive model, e be the base of the natural logarithm (Euler's constant), and i be the imaginary unit, satisfying the definition. , The independent variable is the continuous angular frequency, and the integration interval is... It represents a complete fundamental period of a discrete-time series in the frequency domain. This represents the frequency shift operator in the discrete-time Fourier transform. Indicates taking the modulus of a complex number;
[0096] 26) A fractional difference moving average model was used to fit the hydrological time series. The length-dependent components in the data are fitted by a fractional difference moving average model to the length-dependent component sequence and the hydrological time series. Correlation coefficient between The expression is as follows:
[0097] ;
[0098] in, Let be the frequency response function of the moving average operator. The order of the fractional difference moving average model;
[0099] 27) A fractional difference autoregressive moving average model was used to fit the hydrological time series. The length-dependent components in the data were compared with the hydrological time series using an autoregressive moving average model fitted by fractional differences. Correlation coefficient between The expression is as follows:
[0100] ;
[0101] in, Let be the autoregressive order of the fractional differencing autoregressive moving average model. is the moving average order of the fractional difference autoregressive moving average model.
[0102] 3) Using the correlation coefficients obtained in step 2), calculate the information criterion value based on the correlation coefficients for each candidate dependency model;
[0103] The formula for calculating the information criterion value based on the correlation coefficient is as follows:
[0104] ;
[0105] in, The correlation coefficient is the information criterion value used to evaluate and screen the optimal dependency model as a dimensionless objective decision indicator, where N is the hydrological time series. Length, Dependency component sequences fitted to each candidate dependency model and hydrological time series The square of the correlation coefficient between them For each candidate model, the order is [order]. For autoregressive models, [order is missing]. For the moving average model, For the autoregressive moving average model, For fractional differencing autoregressive models, For the fractional difference moving average model, For the fractional differencing autoregressive moving average model, ; For summing loop variables, Used to identify the first in the model The dimensional hierarchy of each estimated parameter in the parameter space. Weights are scaled to account for information capacity.
[0106] Specifically, the information criterion values based on correlation coefficients for the six candidate dependency models are calculated as follows:
[0107] 31) Calculate the information criterion value based on the correlation coefficient for the autoregressive model. The expression is as follows:
[0108] ;
[0109] 32) Calculate the information criterion value based on the correlation coefficient for the moving average model. The expression is as follows:
[0110] ;
[0111] 33) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model. The expression is as follows:
[0112] ;
[0113] 34) Calculate the information criterion value based on the correlation coefficient for the autoregressive model of fractional differencing. The expression is as follows:
[0114] ;
[0115] 35) Calculate the information criterion value based on the correlation coefficient for the moving average model of fractional differences. The expression is as follows:
[0116] ;
[0117] 36) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model of fractional differencing. The expression is as follows:
[0118] .
[0119] 4) Compare the information criterion values based on the correlation coefficient obtained in step 3), and determine the candidate dependency model with the smallest information criterion value based on the correlation coefficient as the model describing the hydrological time series. The optimal dependency model for the dependent components in the model.
[0120] In the preferred example, to evaluate the statistical significance of model selection, the correlation coefficient between the dependent component sequence obtained by fitting the optimal dependency model and the hydrological time series is used. Compare with the correlation coefficient threshold; specifically as follows:
[0121] First, the dependency components and hydrological time series obtained from the optimal model fitting are calculated. correlation coefficient Then, a 5% significance level was selected, and the corresponding correlation coefficient threshold was calculated. ,as follows:
[0122] ;
[0123] in, This indicates that the degrees of freedom are 1 and N-2, and the significance level is... The critical value of the F-distribution at time;
[0124] Final comparison and The magnitude of the value, if If the dependency component is significant, then a fitting analysis is performed using the preferred model; if If so, it is determined to be a hydrological time series. The model contains only random components, and the preferred model has no statistical significance.
[0125] In this example, since the dependency type, theoretical parameters, and model structure of the artificially generated sequences are known, the artificially generated sequences can effectively verify the scientific validity and effectiveness of the method of this invention. However, the dependency type and actual model of measured hydrological and climate time series are often unknown, making it difficult to directly use them as a benchmark for verifying the accuracy. To systematically demonstrate the accuracy of the method of this invention in selecting the dependency model for hydrological time series, this embodiment constructs six types of artificial sequences with rigorous statistical constraints and fixed sequence lengths.
[0126] The first type of time series is an autoregressive (AR) process, denoted as S11, S12, and S13, corresponding to AR(1), AR(2), and AR(3) theoretical models under random parameters, respectively. Figure 2a , Figure 2b , Figure 2c The second type of time series is the moving average (MA) process, denoted as S21, S22, and S23, corresponding to the MA(1), MA(2), and MA(3) theoretical models under random parameters, respectively. Figure 3a , Figure 3b , Figure 3c The third type of time series is the Autoregressive Moving Average (ARMA) process, denoted as S31, S32, and S33, corresponding to the theoretical models of ARMA(1,1), ARMA(1,2), and ARMA(2,1), respectively. Figure 4a , Figure 4b , Figure 4c The fourth type of time series is a fractional difference autoregressive (FAR) process, denoted as S41, S42, and S43, which correspond to the theoretical models of FAR(1), FAR(2), and FAR(3) under random length-dependent genes, respectively. Figure 5a , Figure 5b , Figure 5c The fifth type of time series is a fractional difference moving average (FMA) process, denoted as S51, S52, and S53, which correspond to the theoretical models of FMA(1), FMA(2), and FMA(3) under random length-dependent genes, respectively. Figure 6a , Figure 6b , Figure 6c The sixth type of time series is a fractional difference autoregressive moving average (FARMA) process, denoted as S61, S62, and S63, corresponding to the theoretical models of FARMA(1,1), FARMA(1,2), and FARMA(2,1) under random length-dependent genes, respectively. Figure 7a , Figure 7b , Figure 7c The dependency model of all the time series (S11-S63) generated above was defined and optimized using the method provided by this invention. The results are shown in Table 1 (definition and optimization results of artificially generated sequences with different dependency models):
[0127] Table 1
[0128]
[0129] The dependency identification results show that, regardless of whether it is the AR, MA, and ARMA model sequences with short dependency (SRD) characteristics (S11-S33) or the FAR, FMA, and FARMA model sequences with long dependency (LRD) characteristics (S41-S63), the method of this invention can accurately determine their long or short dependency types and precisely identify their corresponding optimal dependency models. The model optimization results of all 18 test sequences are in high agreement with the theoretically preset models, and no overfitting or type misclassification occurs in any test scenario. The above results indicate that using information criteria values based on correlation coefficients for model comparison can scientifically and objectively define the dependency types of hydrological time series and automatically select the most suitable dependency model that balances fitting accuracy and model structure simplicity.
[0130] A comparative analysis of the above-mentioned hydrological time series dependency model optimization results yields the following important conclusions:
[0131] (1) Effectively solves the problem of overfitting caused by simply pursuing fitting accuracy: In traditional model comparison or optimization methods that are only guided by maximizing the correlation coefficient, high-order or mixed complex models (such as ARMA, FARMA) tend to have a higher correlation coefficient index due to their larger number of parameters and higher degrees of freedom. This can lead to the wrong selection of complex models due to overfitting when faced with real simple models (such as AR(1) or MA(1)). The method of this invention calculates the fitting error of the correlation coefficient and substitutes it into the information criterion index based on the correlation coefficient. It implements a scientific adaptive penalty on the model order (the number of parameters to be estimated), thereby accurately identifying and locking the simplest real structure model, ensuring that the model has good generalization ability and reliable extrapolation prediction.
[0132] (2) This invention breaks through the technical bottleneck of making it difficult to quantitatively compare multiple types of long and short dependency models: Long dependency models (FAR, FMA, FARMA) differ fundamentally from conventional short dependency models in terms of data dimension, scale, and likelihood function definition due to the introduction of fractional difference operators for transformation. Traditional information criteria based on residual variance or likelihood values (such as AIC, BIC) often face the dilemma of inconsistent benchmarks and ambiguous standards when comparing across dependency types. The method of this invention directly calculates the statistical uncertainty of the correlation coefficient between the fitted dependency component and the original sequence and constructs a unified parameter penalty term. For the first time, it realizes a systematic parallel horizontal comparison of six different structural models of short and long dependency categories under the same index framework, which improves the objectivity of model classification.
[0133] (3) This invention represents a methodological leap from empirical qualitative judgment to rigorous quantitative definition: Traditional methods rely heavily on researchers' subjective observation of whether the autocorrelation coefficient graph exhibits "exponential decay" or "slow decay" to qualitatively define long-term dependence. This approach is prone to serious bias when faced with finite lengths (such as shorter hydrological sequences in actual measurements) and high noise interference. The method of this invention not only achieves quantitative optimization of specific models through the minimum information criterion value, but also compares the maximum value of the optimal correlation coefficient with the significance threshold based on the F-distribution, providing a quantitative significance assessment. This effectively eliminates the systematic errors caused by subjective qualitative analysis, providing statistically rigorous methodological support for accurately interpreting the evolution mechanism of measured hydrological and climate sequences and constructing scientific stochastic simulation and medium- to long-term prediction models.
[0134] This invention has many specific applications. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.
Claims
1. A method for optimizing a hydrological time series dependency model, characterized in that, The steps are as follows: 1) Six candidate dependency models were used to fit the hydrological time series respectively. Dependent components in; 2) Calculate the dependent component sequence and hydrological time series obtained from fitting each candidate dependent model. The correlation coefficient between them; 3) Using the correlation coefficients obtained in step 2), calculate the information criterion value based on the correlation coefficients for each candidate dependency model; 4) Compare the information criterion values based on the correlation coefficient obtained in step 3), and determine the candidate dependency model with the smallest information criterion value based on the correlation coefficient as the model describing the hydrological time series. The optimal dependency model for the dependent components in the model.
2. The method for optimizing the hydrological time series dependency model according to claim 1, characterized in that, The six candidate dependency models in step 1) include: autoregressive model, moving average model, autoregressive moving average model, fractional difference autoregressive model, fractional difference moving average model, and fractional difference autoregressive moving average model.
3. The method for optimizing the hydrological time series dependency model according to claim 1, characterized in that, Step 2) specifically includes: 21) Using an autoregressive model to fit hydrological time series The short-dependency components in the data were analyzed, and the short-dependency component sequences fitted by the autoregressive model were compared with the hydrological time series. Correlation coefficient between The expression is as follows: ； in, For the autoregressive model Autoregressive coefficients of order 1 Hydrological time series The Autocorrelation coefficient; The order of the autoregressive model; 22) Fitting hydrological time series using a moving average model The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows: ； in, For the moving average model The moving average coefficient; The order of the moving average model; 23) The hydrological time series were fitted using an autoregressive moving average model. The short-dependency components in the data are calculated, and the short-dependency component sequences fitted by the autoregressive moving average model are compared with the hydrological time series. Correlation coefficient between The expression is as follows: ； in, For the first autoregressive moving average model Autoregressive coefficient of order; For the first autoregressive moving average model The moving average coefficient; is the autoregressive order of the autoregressive moving average model; Let be the order of the moving average in the autoregressive moving average model. For the first Each impulse response coefficient is obtained recursively from the parameters, as follows: , ； In the formula, The impulse response coefficient is... Let be the ordinal number of the impulse response coefficient, and ; 24) Estimating hydrological time series using the recalibrated range R / S analysis method Given the Hearst coefficients H, the fractional difference order d is calculated using the following expression: d = H - 0.5; 25) A fractional difference autoregressive model was used to fit the hydrological time series. The length-dependent components in the data are fitted by an autoregressive model with fractional differences, and the length-dependent component sequences are compared with the hydrological time series. Correlation coefficient between The expression is as follows: ； in, Let be the frequency response function of the autoregressive operator. Let be the order of the fractional differencing autoregressive model, e be the base of the natural logarithm, and i be the imaginary unit, satisfying the definition. , The independent variable is the continuous angular frequency, and the integration interval is... It represents a complete fundamental period of a discrete-time series in the frequency domain. This represents the frequency shift operator in the discrete-time Fourier transform. Indicates taking the modulus of a complex number; 26) A fractional difference moving average model was used to fit the hydrological time series. The length-dependent components in the data are fitted by a fractional difference moving average model to the length-dependent component sequence and the hydrological time series. Correlation coefficient between The expression is as follows: ； in, Let be the frequency response function of the moving average operator. The order of the fractional difference moving average model; 27) A fractional difference autoregressive moving average model was used to fit the hydrological time series. The length-dependent components in the data were compared with the hydrological time series using an autoregressive moving average model fitted by fractional differences. Correlation coefficient between The expression is as follows: ； in, Let be the autoregressive order of the fractional differencing autoregressive moving average model. is the moving average order of the fractional difference autoregressive moving average model.
4. The method for optimizing the hydrological time series dependency model according to claim 3, characterized in that, The formula for calculating the information criterion value based on the correlation coefficient in step 3) is as follows: ； in, The correlation coefficient is the information criterion value used to evaluate and screen the optimal dependency model as a dimensionless objective decision indicator, where N is the hydrological time series. Length, Dependency component sequences fitted to each candidate dependency model and hydrological time series The square of the correlation coefficient between them For each candidate model, the order is [order]. For autoregressive models, [order is missing]. For the moving average model, For the autoregressive moving average model, For fractional differencing autoregressive models, For the fractional difference moving average model, For the fractional differencing autoregressive moving average model, ; For summing loop variables, Used to identify the first in the model The dimensional hierarchy of each estimated parameter in the parameter space. Weights are scaled to account for information capacity.
5. The method for optimizing the hydrological time series dependency model according to claim 4, characterized in that, Step 3) specifically includes: 31) Calculate the information criterion value based on the correlation coefficient for the autoregressive model. The expression is as follows: ； 32) Calculate the information criterion value based on the correlation coefficient for the moving average model. The expression is as follows: ； 33) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model. The expression is as follows: ； 34) Calculate the information criterion value based on the correlation coefficient for the autoregressive model of fractional differencing. The expression is as follows: ； 35) Calculate the information criterion value based on the correlation coefficient for the moving average model of fractional differences. The expression is as follows: ； 36) Calculate the information criterion value based on the correlation coefficient for the autoregressive moving average model of fractional differencing. The expression is as follows: 。