Runoff space dependency modeling and stochastic simulation method based on R-vine copula

Abstract

The invention discloses a runoff spatial dependency modeling and stochastic simulation method based on R-vine copula, and relates to the technical field of hydrological sequence analysis. According to the method, a spatial relation constraint matrix is adopted to represent all possible vine structures, the marginal distribution linetype of observation data of a hydrometric station is estimated, goodness of fit test is carried out, an R-vine copula model is constructed, an optimal model is determined according to BIC indexes, standard dependency statistics and information gain are calculated, and a mean value fitting sequence and a random simulation sequence are generated. According to the method, the vine structure of the R-vine copula model is specified according to the intersection sequence of the main and branch flows and the spatial position relationship of the hydrometric station, a random model capable of accurately simulating a historical runoff sequence and estimating the future hydrological situation is constructed, the spatial dependency between the upstream and downstream runoff sequences and the main and branch runoff sequences is accurately described, the association state and strength of the main and branch runoff process are determined, and the real-time performance of the main and branch runoff process is improved. And the development trend and the evolution rule of the hydrological situation in the future in the changing environment can be further explored.

Description

Technical Field

[0001] The present invention belongs to the technical field of hydrological series analysis, and particularly relates to a runoff spatial dependence modeling and stochastic simulation method based on R-vine copula. Background Art

[0002] Traditional hydrological multivariate modeling and time series simulation methods assume that all variables follow the same marginal distribution, which cannot accurately describe the dependence structure between different variables and is difficult to consider both the correlation and uncertainty between variables simultaneously. The copula function can connect variables with different marginal distributions, but it lacks accuracy in constructing high-dimensional dependence structures. A highly flexible statistical tool in graph theory, Regular Vine (R-vine), is introduced into the construction of the joint distribution function of multivariate variables. The high-dimensional function based on R-vine can be decomposed into a series of Pair-copula modules, and each Pair-copula module can select an appropriate copula function according to the specific dependence structure between variables. The R-vine copula model contains a series of nodes, edges, and tree structures, where the edges of each tree represent binary copula or conditional binary copula functions. The R-vine copula function inherits the advantages of the copula function and overcomes the deficiency that the copula function cannot be applied to high dimensions. It can accurately simulate the dependence structure of any dimension, has the characteristics of flexible structure, simple parameter solution, and powerful simulation function, and is very suitable for characterizing the complex dependence structure between different variables by establishing the joint distribution function of multivariate random variables. However, the existing R-vine copula modeling method cannot specify the vine structure according to the specific dependence relationship (such as spatial relationship) between different variables, and lacks flexibility in modeling and simulating the multivariate dependence structure. Summary of the Invention

[0003] Object of the Invention: In order to more accurately and flexibly describe the spatial dependence structure between the observed data sequences of different hydrological stations and achieve accurate simulation of the observed data sequences, the present invention provides a runoff spatial dependence modeling and stochastic simulation method based on R-vine copula.

[0004] Technical Solution: The runoff spatial dependence modeling and stochastic simulation method based on R-vine copula adopted by the present invention to solve its technical problems includes the following steps:

[0005] (1) According to the confluence order of the main stream and tributaries and the spatial position relationship of hydrological stations, determine the connection and pairing relationship between the observed data of each hydrological station in the first tree of the vine structure, and use a spatial relationship constraint matrix to represent all possible vine structures.

[0006] (2) Determine the optimal marginal distribution line types of the observation data at each hydrological station, and use the probability integral transformation to convert the observation data at each hydrological station into standardized data that follows a uniform distribution on [0, 1].

[0007] (3) According to the different vine structures corresponding to the spatial relationship constraint matrix, construct all R-vine copula models that connect the standardized data at each hydrological station, and select the optimal R-vine copula model from all the R-vine copula models.

[0008] (4) Calculate the standard dependence statistics and information gain according to the determined optimal R-vine copula model to realize the evaluation of the correlation state and intensity among the observation data at each hydrological station.

[0009] (5) According to the determined optimal R-vine copula model, use the Rosenblatt multivariate transformation, Monte Carlo simulation, and probability integral transformation methods to generate the mean fitting sequence and random simulation sequence of the observation data at each hydrological station.

[0010] In the present invention, the spatial relationship constraint matrix is a lower triangular matrix. The spatial relationship constraint matrix reflects the vine structure information of the R-vine copula model, and each spatial relationship constraint matrix uniquely corresponds to a vine structure.

[0011] Specifically, the vine structure of an n-dimensional R-vine copula model consists of n - 1 trees T1, T2,..., T n-1 composed. Each i-th tree contains n - i + 1 nodes, and the structure connecting adjacent nodes is called an edge. The differences in the variables or conditional variables connected by the edges in the vine structure are the basis for distinguishing different vine structures. The spatial relationship constraint matrix M that completely records the connection relationships between all the variables or conditional variables corresponding to the edges in the vine structure i,j (i, j = 1,..., n) has the following characteristics:

[0012] (ⅰ) When 1 ≤ i < j ≤ n,

[0013] (ⅱ) When i = 1,..., n - 1,

[0014] In the present invention, the maximum likelihood method is applied to estimate the parameters of the alternative marginal distribution line types, and the maximum likelihood method is also used to calculate the parameter estimation of the copula function corresponding to each edge in the vine structure.

[0015] In the present invention, based on the constructed R-vine copula model, the Monte Carlo simulation method is used to calculate the mutual information and conditional mutual information between the observed data of each hydrological station, and further calculate the standard dependence statistic and information gain. The standard dependence statistic is used to accurately measure the dependence between the observed data of different hydrological stations, including the non-linear and local correlations that cannot be reflected by the common correlation coefficients; the information gain is used to evaluate the association state between the observed data of different hydrological stations.

[0016] Specifically, the formulas for calculating the mutual information and conditional mutual information based on the constructed R-vine copula model are as follows:

[0017]

[0018]

[0019] where X, Y, and Z are the observed data sequences of any three hydrological stations, I(X,Y) is the mutual information between X and Y, I(X,Y|Z) is the conditional mutual information between X and Y under the condition that Z is known, F X (x), F Y (y), and F Z (z) are the marginal distribution functions of X, Y, and Z respectively, and c is the corresponding copula function.

[0020] Specifically, the formula for calculating the standard dependence statistic based on the mutual information and conditional mutual information is:

[0021]

[0022] where I * refers to the mutual information or conditional mutual information between X and Y.

[0023] Specifically, the formula for calculating the information gain based on the mutual information and conditional mutual information is:

[0024] I(X,Y,Z) = I(X,Z|Y) - I(X,Z) = I(Y,Z|X) - I(Y,Z)

[0025] where I(X,Z|Y) is the conditional mutual information between X and Z under the condition that Y is known, I(Y,Z|X) is the conditional mutual information between Y and Z under the condition that X is known, and I(X,Z) and I(Y,Z) refer to the mutual information between X and Z and between Y and Z respectively. An information gain greater than 0 indicates that X and Y interact, that is, more information about Z can be provided after X and Y are coupled; an information gain less than 0 indicates that X and Y are redundant, that is, there is repetition between the information provided by X and Y about Z; an information gain close to 0 indicates that the addition of X (Y) does not affect the relationship between Y (X) and Z.

[0026] In the present invention, 500 sets of mutually independent random sequences are generated as the input sequences for the random simulation of measured data. The length of each random sequence is equal to the length of the observed data sequence of each hydrological station. The input sequences are substituted into the constructed R-vine copula model, and the Rosenblatt multivariate transformation method is used to fit the standardized data of the measured runoff data. The fitting results are reordered according to the ranks of the measured runoff data, and based on the corresponding marginal distribution lines, 500 sets of fitting sequences of the measured runoff data are generated using the probability integral transformation method. The means of the 500 sets of fitting sequences at each time node are calculated, and the mean fitting sequence of the measured runoff data is summarized and formed. The graphical test method is used to test the fitting effect of the measured runoff data and the corresponding mean fitting sequence. At the same time, after substituting the input sequences into the constructed R-vine copula model, 500 sets of random simulation sequences of the measured runoff data are generated using the Monte Carlo simulation and the probability integral transformation method. The short sequence method is used to compare and analyze the differences in different statistical characteristics (mean, standard deviation, skewness coefficient) between the measured runoff data and the corresponding random simulation sequences.

[0027] Preferably, the symmetric mean absolute percentage error index (sMAPE) is used to determine the marginal distribution lines of the observed data of each hydrological station, and the Cramer-von Mises test is used to perform the goodness-of-fit test on the selected optimal marginal distribution function.

[0028] Preferably, the Bayesian information criterion (BIC) is used to select the optimal R-vine copula model from all the R-vine copula models.

[0029] Beneficial effects: The runoff spatial dependence modeling and random simulation method based on R-vine copula of the present invention has the following beneficial effects:

[0030] (1) According to the confluence order of the main stream and tributaries and the spatial position relationship of hydrological stations, the present invention specifies the vine structure of the R-vine copula model, accurately depicts the spatial dependence between the upstream and downstream, and the runoff sequences of the main stream and tributaries, and clarifies the correlation state and intensity of the runoff processes of the main stream and tributaries.

[0031] (2) The present invention constructs a random model that can accurately simulate historical runoff sequences and predict future hydrological situations, which helps to further explore the future development trends and evolution laws of hydrological situations under changing environments, and provides a theoretical basis and technical support for basin water resources management, flood control, and water ecological protection. Description of the Drawings

[0032] Figure 1 is a schematic flow chart of the runoff spatial dependence modeling and random simulation method based on R-vine copula of the present invention;

[0033] Figure 2 : is a comparison chart of the fitting results of the monthly runoff measured data of the target station and the mean fitting sequence in the embodiment;

[0034] Figure 3 is a comparison diagram of the overall distribution characteristics of the measured runoff data of the target site in each month and the random simulation data in the embodiment;

[0035] Figure 4 is a mean comparison chart of the actual runoff data of the target site in each month and the random simulation data in the embodiment;

[0036] Figure 5 is a variance comparison diagram of the monthly runoff measured data and the random simulation data of the target station in the embodiment;

[0037] Figure 6 It is a comparison chart of the skewness coefficients of the measured runoff data of the target station in each month and the random simulation data in the embodiment. DETAILED DESCRIPTION

[0038] The preferred embodiments of the present invention are described in detail below in conjunction with the accompanying drawings so that the advantages and features of the present invention can be more easily understood by those skilled in the art, thereby making a clearer and more definite definition of the protection scope of the present invention.

[0039] This embodiment uses the dependency modeling of monthly runoff observation data (1906-2018) from five hydrological stations in the Colorado River Basin and the random simulation of monthly runoff at the target station as practical application cases.

[0040] A, B, C, D and E are used to refer to the five hydrological stations in the Colorado River Basin; at the same time, station B is selected as the target station for random simulation of monthly runoff series.

[0041] like Figure 1 As shown, the implementation of the present invention mainly includes the following steps:

[0042] (1) According to the confluence order of main and tributary rivers and the spatial position relationship of hydrological stations, the connection and pairing relationship between the monthly runoff observation data of stations A, B, C, D and E in the first tree of the vine structure are determined, and all possible vine structures are represented using the spatial relationship constraint matrix.

[0043] In this embodiment, the spatial relationship constraint matrix is a lower triangular matrix. The spatial relationship constraint matrix reflects the vine structure information of the R-vinecopula model. Each spatial relationship constraint matrix uniquely corresponds to a vine structure.

[0044] Specifically, the vine structure of an n-dimensional R-vine copula model consists of n - 1 trees T1, T2, ..., T n-1 which are composed of. Each i-th tree contains n - i + 1 nodes, and the structure connecting adjacent nodes is called an edge. The differences in the variables or conditional variables connected by the edges in the vine structure are the basis for distinguishing different vine structures. The spatial relationship constraint matrix M i,j (i, j = 1, ..., n) has the following properties:

[0045] (ⅰ) When 1 ≤ i < j ≤ n,

[0046] (ⅱ) When i = 1, ..., n - 1,

[0047] In this embodiment, stations A, B, and E are hydrological stations on the main stream of the Colorado River. In the first tree of all possible vine structures, stations A, B, and E need to be directly connected; stations C and D are respectively the controlled hydrological stations downstream of the Gunnison River and the Dolores River that flow into the main stream of the Colorado River successively. The confluence points of the Gunnison River and the Dolores River with the main stream of the Colorado River are both between stations B and E. Therefore, in the first tree of all possible vine structures, station C can only be directly connected to stations B and E, and station D can only be connected to station E.

[0048] In this embodiment, after determining the connection and pairing relationships among the monthly runoff observation data of stations A, B, C, D, and E in the first tree of the vine structure, list all possible connection relationships in the second, third, and fourth trees in turn, forming 5 possible vine structures, and their corresponding spatial relationship constraint matrices are shown in Table 1.

[0049] Table 1 Spatial relationship constraint matrices of 5 vine structures

[0050]

[0051] (2) Determine the optimal marginal distribution line types of the monthly runoff data of stations A, B, C, D, and E according to the symmetric mean absolute percentage error index (sMAPE), use the Cramer-von Mises test to conduct a goodness-of-fit test on the selected optimal marginal distribution function, and use the probability integral transformation to transform the monthly runoff data into standardized data that follows a uniform distribution on [0, 1].

[0052] In this embodiment, the optimal marginal distribution lines corresponding to the runoff observation data of stations A, B, C, D, and E from January to December are shown in Table 2. In the table, Norm, GEV, PIII, Gamma, Weibull, Lnorm, and Gumbel refer to the normal distribution, generalized extreme value distribution, Pearson type III, gamma distribution, Weibull distribution, lognormal distribution, and Gumbel distribution respectively. The parameter estimation of the marginal distribution line type uses the maximum likelihood method. The values in parentheses in Table 2 are the sMAPE values of the selected optimal marginal distribution line types. The P-value range of the Cramer-von Mises test is [0.0961, 0.9927], that is, all P-values are greater than the preset significance level, indicating that all the selected optimal marginal distribution line types are reasonable.

[0053] Table 2 Optimal marginal distribution lines corresponding to the runoff observation data of stations A, B, C, D, and E from January to December

[0054]

[0055] (3) According to the vine structure corresponding to the spatial relationship constraint matrix, construct an R-vine copula model that connects the standardized data of each hydrological station, and determine the optimal R-vine copula model based on the Bayesian information criterion (BIC).

[0056] In this embodiment, for the runoff observation data of stations A, B, C, D, and E from January to December, R-vine copula models are constructed according to 5 possible vine structures respectively. The Bayesian information criterion (BIC) is used as an evaluation index to select appropriate copula functions for each Pair-copula module. The parameter estimation of the copula function corresponding to each edge in the vine structure is calculated using the maximum likelihood method. The R-vine copula model with the smallest BIC value is selected as the optimal R-vine copula model. The BIC values of the optimal R-vine copula models corresponding to the runoff observation data of stations A, B, C, D, and E from January to December are shown in Table 3.

[0057] Table 3 BIC values of the optimal R-vine copula models corresponding to the runoff observation data from January to December

[0058]

[0059] (4) Calculate the standard dependence statistic and information gain according to the determined optimal R-vine copula model to realize the evaluation of the association status and intensity among the monthly runoff data of stations A, B, C, D, and E.

[0060] In this embodiment, based on the constructed R-vine copula model, the Monte Carlo simulation method is used to calculate the mutual information and conditional mutual information between the observed data of each hydrological station, and further calculate the standard dependence statistic and information gain. The standard dependence statistic is used to accurately measure the dependence between the observed data of different hydrological stations, including non-linear and local correlations that cannot be reflected by common correlation coefficients; the information gain is used to evaluate the association state between the observed data of different hydrological stations.

[0061] Specifically, the formulas for calculating the mutual information and conditional mutual information based on the constructed R-vine copula model are as follows:

[0062]

[0063] where X, Y, and Z are the observed data sequences of any three hydrological stations, I(X,Y) is the mutual information between X and Y, I(X,Y|Z) is the conditional mutual information between X and Y under the condition that Z is known, F X (x), F Y (y), and F Z (z) are the marginal distribution functions of X, Y, and Z respectively, and c is the corresponding copula function.

[0064] Specifically, the formula for calculating the standard dependence statistic based on the mutual information and conditional mutual information is:

[0065]

[0066] where I * refers to the mutual information or conditional mutual information between X and Y.

[0067] Specifically, the formula for calculating the information gain based on the mutual information and conditional mutual information is:

[0068] I(X,Y,Z) = I(X,Z|Y) - I(X,Z) = I(Y,Z|X) - I(Y,Z)

[0069] where I(X,Z|Y) is the conditional mutual information between X and Z under the condition that Y is known, I(Y,Z|X) is the conditional mutual information between Y and Z under the condition that X is known, and I(X,Z) and I(Y,Z) refer to the mutual information between X and Z and between Y and Z respectively. An information gain greater than 0 indicates that X and Y interact, that is, more information about Z can be provided after X and Y are coupled; an information gain less than 0 indicates that X and Y are redundant, that is, there is repetition between the information provided by X and Y about Z; an information gain close to 0 indicates that the addition of X (Y) does not affect the relationship between Y (X) and Z.

[0070] In this embodiment, first, the mutual information and conditional mutual information between the monthly runoff observation data of stations A, B, C, D, and E are calculated based on the determined optimal R-vine copula model, and the standard dependence statistic and information gain are further calculated.

[0071] The standard dependence statistics between the monthly runoff data of adjacent stations are shown in Table 4. The calculation results of the standard dependence statistics show that the dependence between the monthly runoff observation data of the main stream hydrological stations of the Colorado River (stations A, B, and E) is greater than that between the monthly runoff observation data of the two tributary hydrological stations (stations C and D), and the dependence between the monthly runoff observation data of the tributary hydrological station and the main stream hydrological station downstream of the confluence point (station E) is greater than that between the tributary hydrological station and the main stream hydrological station upstream of the confluence point (stations A and B).

[0072] Table 4 Standard dependence statistics between the monthly runoff data of adjacent stations

[0073]

[0074] The information gains between the monthly runoff data of adjacent stations are shown in Table 5. The calculation results of the information gains show that the correlation state between the monthly runoff observation data of the main stream hydrological stations of the Colorado River (stations A, B, and E) is redundant; except that the correlation state between the monthly runoff observation data of stations A and C has no effect, the correlation states between the remaining main stream hydrological stations and tributary hydrological stations and between the two tributary hydrological stations are all interactive. The correlation intensity between the monthly runoff data of adjacent stations is generally related to the distance between the corresponding drainage areas controlled by the hydrological stations. The farther the distance, the more significant the differences in precipitation, evaporation, and water intake in the drainage areas controlled by different hydrological stations, resulting in a smaller correlation intensity; at the same time, the correlation intensity between the monthly runoff observation data of the main stream hydrological stations is greater than that between the main stream hydrological stations and tributary hydrological stations and between the two tributary hydrological stations.

[0075] Table 5 Information gains between the monthly runoff data of adjacent stations

[0076]

[0077] (5) According to the determined optimal R-vine copula model, use the Rosenblatt multivariate transformation and probability integral transformation methods to generate the mean fitting sequence of the monthly runoff data of the target station (station B), and use the Monte Carlo simulation and probability integral transformation methods to generate the stochastic simulation sequence of the monthly runoff data of the target station (station B). Use the graphical test method and the short sequence method to verify the accuracy of the mean fitting sequence and the stochastic simulation sequence respectively.

[0078] In this embodiment, 500 sets of independent random sequences are generated as the input sequences for the random simulation of the measured data. The length of each set of random sequences is equal to the length of the observed data sequences at each hydrological station. The input sequences are substituted into the constructed R-vine copula model, and the Rosenblatt multivariate transformation method is applied to fit the standardized data of the measured runoff data. The fitting results are reordered according to the ranks of the measured runoff data, and based on the corresponding marginal distribution curves, 500 sets of fitting sequences of the measured runoff data are generated using the probability integral transformation method. The means of the 500 sets of fitting sequences at each time node are calculated, and a mean fitting sequence of the measured runoff data is formed by summarization. The graphical test method is used to test the fitting effect of the measured runoff data and the corresponding mean fitting sequence. At the same time, after substituting the input sequences into the constructed R-vine copula model, 500 sets of random simulation sequences of the measured runoff data are generated using the Monte Carlo simulation and the probability integral transformation method. The short sequence method is used to compare and analyze the differences in different statistical characteristics (mean, standard deviation, skewness coefficient) between the measured runoff data and the corresponding random simulation sequences.

[0079] As Figure 2 shown, the fitting results of the monthly runoff measured data at the target site and the mean fitting sequence are good, indicating that the R-vine copula model constructed based on accurately depicting the spatial dependence of different runoff sequences can accurately simulate the historical runoff sequences at the target site.

[0080] As Figure 3 shown, the random simulation sequences of the monthly runoff at the target site completely cover the measured data. As Figures 4 to 6 shown, different statistical characteristics (mean, standard deviation, skewness coefficient) of the monthly runoff measured data at the target site are all located inside the box formed by the random simulation sequences, indicating that the random simulation sequences can accurately reproduce the different statistical characteristics of the measured data.

[0081] According to the above corresponding steps, the accurate modeling of the spatial dependence of the monthly runoff observed data at stations A, B, C, D, and E and the random simulation of the monthly runoff data at the target site (station B) can be realized, the correlation status and intensity of the runoff processes at different stations can be clarified, which helps to further explore the annual and interannual variation laws of the main and tributary runoff processes, summarize the future development trends and evolution laws of the hydrological regime under changing environments, and provide a theoretical basis and technical support for basin water resources management, flood prevention, and water ecological protection.

[0082] The above is only the preferred embodiment of the present invention, and it does not limit the patent scope of the present invention. Any equivalent structure or equivalent process transformation made by using the content of the specification and drawings of the present invention, or directly or indirectly applied to other related technical fields, shall be included in the patent protection scope of the present invention by the same token.

Claims

1. A runoff spatial dependence modeling and stochastic simulation method based on R-vine copula, characterized in that, It includes the following steps: (1) Determine the connection and pairing relationships among the observed data of each hydrological station in the first tree of the vine structure according to the confluence order of main and tributary rivers and the spatial position relationship of hydrological stations, and represent all possible vine structures using the spatial relationship constraint matrix; (2) Determine the optimal marginal distribution line types of the observed data of each hydrological station, and use the probability integral transformation to transform the observed data of each hydrological station into standardized data that follows a uniform distribution on [0, 1]; (3) Construct all R-vine copula models that connect the standardized data of each hydrological station according to different vine structures corresponding to the spatial relationship constraint matrix, and select the optimal R-vine copula model from all R-vine copula models; (4) Calculate the standard dependence statistic and information gain according to the determined optimal R-vine copula model to realize the evaluation of the association state and strength among the observed data of each hydrological station; (5) Generate the mean fitting sequence and random simulation sequence of the observed data of each hydrological station according to the determined optimal R-vine copula model, using the Rosenblatt multivariate transformation, Monte Carlo simulation and probability integral transformation methods.

2. A runoff spatial dependence modeling and stochastic simulation method based on the R-vine copula according to claim 1, characterized in that: The described spatial relationship constraint matrix is a lower triangular matrix. The spatial relationship constraint matrix reflects the vine structure information of the R-vine copula model, and each spatial relationship constraint matrix uniquely corresponds to a vine structure.

3. A runoff spatial dependence modeling and stochastic simulation method based on the R-vine copula according to claim 1, characterized in that: The described standard dependence statistic is used to accurately measure the dependence between the observed data of different hydrological stations, including non-linear and local correlations that cannot be reflected by common correlation coefficients, and the information gain is used to evaluate the association state between the observed data of different hydrological stations.

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