Runoff stochastic simulation method and system based on adaptive bandwidth kernel density estimation and Copula function

Abstract

The invention belongs to the field of random hydrology, and particularly discloses a runoff stochastic simulation method and system based on adaptive bandwidth kernel density estimation and a Copula function. The method comprises the following steps: firstly, fitting historical runoff sequence data by adopting adaptive bandwidth kernel density estimation to obtain a probability density function of flow in any time period; obtaining an edge distribution function of the traffic in any time period according to the cumulative probability density of the traffic in any time period; then constructing a conditional distribution function of the flow in the next time period under the flow condition of the current time period by using a Copula function; and finally, generating runoff flow simulation values by sequential simulation in a period-by-period manner by utilizing a conditional dependency relationship of the flows in adjacent periods in the conditional distribution function. Compared with an existing runoff stochastic simulation method based on a Copula function, the runoff stochastic simulation method is high in runoff simulation precision and stable.

Description

Technical Field

[0001] This application belongs to the field of stochastic hydrology, and more specifically, relates to a method and system for stochastic runoff simulation based on adaptive bandwidth kernel density estimation and Copula function. Background Technology

[0002] Stochastic runoff simulation is a method for generating simulated runoff sequences based on the statistical characteristics and random variation patterns of measured runoff sequences. It is of great significance in fields such as water resources planning and operation management, and climate change assessment. The key to stochastic runoff simulation is to construct a stochastic runoff simulation model that can accurately describe the statistical characteristics of runoff sequences.

[0003] The Copula function can separate the study of marginal distributions and the correlation structure between variables. It imposes no restrictions on the type of marginal distribution when performing runoff simulations, effectively capturing and simulating complex correlations between variables. Therefore, the Copula function has become a widely used stochastic runoff simulation technique in recent years. However, the Copula function requires pre-assuming a specific form of marginal distribution based on the measured runoff sequence, and the selection of different types of marginal distribution curves has a significant impact on the accuracy of the runoff simulation results. Therefore, currently, using the Copula function for stochastic runoff simulations faces the problem of difficulty in guaranteeing the stability of simulation accuracy. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the purpose of this application is to provide a method and system for stochastic runoff simulation based on adaptive bandwidth kernel density estimation and Copula functions, aiming to solve the technical problem that the accuracy and stability of existing stochastic runoff simulations based on Copula functions are difficult to guarantee.

[0005] The first aspect of this application relates to a method for stochastic runoff simulation based on adaptive bandwidth kernel density estimation and Copula functions, including: Adaptive bandwidth kernel density estimation is used to fit historical runoff sequence data to obtain the probability density function of flow for any given time period; The marginal distribution function of the flow rate in any given time period can be obtained from the cumulative probability density of the flow rate in any given time period. Based on the Copula function, the conditional distribution function of the traffic flow in the next time period is derived from the marginal distribution function under the current traffic flow conditions. By utilizing the conditional dependency relationship of flow rates in adjacent time periods in the conditional distribution function, simulated runoff flow rates are generated time-by-time through sequential simulation.

[0006] Preferably, the adaptive bandwidth in the adaptive bandwidth kernel density estimation is obtained by the following method: An adaptive bandwidth optimization model is constructed with the goal of minimizing the distance between the historical runoff distribution function and the probability density function. Initialize the particle swarm, with each particle set as a bandwidth vector composed of a set of bandwidth candidate solutions; use an improved particle swarm optimization algorithm to solve the adaptive bandwidth optimization model.

[0007] Preferably, the adaptive bandwidth optimization model specifically minimizes the weighted sum of the Euclidean geometric distance and the maximum distance between the historical runoff distribution function and the probability density function.

[0008] Preferably, in the improved particle swarm optimization algorithm, if the diversity of any dimension of the variable among all particles is lower than a threshold, the Cauchy mutation method is introduced to perturb the variable in that dimension.

[0009] Preferably, if the diversity of any dimension of the variables among all particles is below a threshold, then the Cauchy mutation method is introduced to perturb the variables in that dimension, specifically as follows:

[0010]

[0011] in, Indicates the first Diversity of dimensional variables For particle swarm scale, Indicates location, superscript Indicates the current iteration number, superscript Mean, subscript Indicates particle number, subscript The subscript indicates the dimensional index of the particle. Indicates the maximum value, subscript This represents the minimum value. The proportionality coefficient represents the control for the degree of variation. Indicates by parameters Random numbers generated by the Cauchy distribution This is the threshold for making a decision.

[0012] Preferably, the improved particle swarm optimization algorithm introduces a nonlinear learning factor. and The velocity and position of the new particle:

[0013]

[0014] in, Indicates speed, Indicates location, For global optimality, Best in history, superscript Indicates the time period number, subscript Indicates particle number, subscript Indicates the dimensional index of the particle. For inertial weights, and Represents a random number between 0 and 1; and Specifically:

[0015] in, for The minimum value, for The maximum value, for The minimum value, for The maximum value, Indicates the current iteration number. This indicates the maximum number of iterations.

[0016] Preferably, the marginal distribution function of the flow rate in any given time period is obtained from the cumulative probability density of the flow rate in any given time period. Specifically, the probability density of the flow rate in any given time period is derived through the probability density function, and then the marginal distribution function of the flow rate in any given time period is obtained by accumulating the obtained probability densities. The mathematical expression is as follows:

[0017] in, express Time period traffic The probability density function, express Time period traffic The marginal distribution function.

[0018] Preferably, based on the Copula function, the conditional distribution function of the traffic flow in the next time period is derived from the marginal distribution function under the current time period traffic conditions, specifically as follows:

[0019] in, For the current time period, For the next period of time, Traffic during the current time period Under the condition, the flow rate in the next time period The conditional distribution function, For Copula functions, for Time period traffic The marginal distribution function, for Time period traffic The marginal distribution function, for Time period traffic and Time period traffic Joint distribution function:

[0020] in, It is a natural constant. The parameters of the Copula function are obtained by maximizing the function. Get parameters The estimated value:

[0021]

[0022] in, For the first One observation sample, This represents the total number of observation samples.

[0023] Preferably, by utilizing the conditional dependencies between flow rates in adjacent time periods within the conditional distribution function, simulated runoff flow rates are generated sequentially time-by-time through sequential simulation, specifically as follows: (1) Let Time period runoff Known; (2) Randomly generate uniformly distributed random numbers Based on

[0024] The derivation yields Simulated flow rate at any time , for Time period traffic conditions, Time period traffic The conditional distribution function; (3) Update Repeat step (2) until a preset number of simulated runoff flow values ​​are generated.

[0025] Secondly, this application provides a stochastic runoff simulation system based on adaptive bandwidth kernel density estimation and Copula function, the stochastic runoff simulation system comprising: The first module uses adaptive bandwidth kernel density estimation to fit historical runoff sequence data to obtain the probability density function of flow for any given time period. The second module obtains the marginal distribution function of the flow rate for any given time period from the cumulative probability density of the flow rate for any given time period; The third module, based on the Copula function, derives the conditional distribution function of the traffic flow in the next time period under the current traffic flow conditions from the marginal distribution function; The fourth module utilizes the conditional dependency relationship of flow rates in adjacent time periods in the conditional distribution function to generate simulated runoff flow rates for each time period through sequential simulation.

[0026] Overall, the technical solutions conceived in this application have the following beneficial effects compared with the prior art: (1) The method of this application uses the Copula function to construct the conditional distribution function between adjacent time periods, and pre-fits the historical runoff sequence data directly with adaptive bandwidth kernel density estimation to obtain the marginal distribution function of the time period flow. This application demonstrates through experiments that the obtained distribution function can well represent the important statistical characteristics of the actual runoff sequence, such as nonlinearity, multimodality and skewness. Based on this, the runoff stochastic simulation method of this application has higher simulation accuracy and stability than the existing methods.

[0027] (2) The method of this application uses an improved particle swarm optimization algorithm to solve the adaptive bandwidth. When updating the velocity and position of the particles, a nonlinear learning factor is introduced to enhance the global search capability of the particle swarm optimization algorithm and accelerate the convergence speed of the algorithm. At the same time, in order to improve the population diversity and avoid premature convergence of the algorithm, Cauchy mutation is used to perturb the variables of the same dimension in the particles. Thus, the improved particle swarm optimization algorithm of this application can stably find the optimal adaptive bandwidth from the global data according to the optimization objective, which enhances the versatility and reliability of the stochastic runoff simulation method of this invention. Attached Figure Description

[0028] Figure 1 This is a flowchart of the stochastic runoff simulation method based on adaptive bandwidth kernel density estimation and Copula function provided in the embodiments of this application.

[0029] Figure 2 This is a flowchart illustrating the particle swarm optimization algorithm for solving adaptive bandwidth, as provided in an embodiment of this application.

[0030] Figure 3 This is a flowchart of sequential simulation of runoff flow provided in the embodiments of this application.

[0031] Figure 4 This is a schematic diagram of the structure of the electronic device provided in the embodiments of this application. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0033] In this application, the terms "first" and "second," etc., are used to distinguish different objects, not to describe a specific order of objects. For example, "first particle" and "second particle," etc., are used to distinguish different particles, not to describe a specific order of particles.

[0034] In this application, the term "electrical connection" can refer to a direct circuit connection or a signal transmission via a communication protocol.

[0035] In the embodiments of this application, the terms "exemplary" or "for example" are used to indicate that something is an example, illustration, or description. Any embodiment or design that is described as "exemplary" or "for example" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design. Specifically, the use of the terms "exemplary" or "for example" is intended to present the relevant concepts in a specific manner.

[0036] In the description of the embodiments of this application, unless otherwise stated, "multiple" means two or more, for example, multiple dimensions means two or more dimensions, multiple traffic means two or more traffic, etc.

[0037] First, the technical terms involved in the embodiments of this application will be introduced.

[0038] Copula functions are a class of functions that connect a multivariate joint distribution function with its respective marginal distribution functions.

[0039] The embodiments of this application are described below with reference to the accompanying drawings. This application discloses a method for stochastic runoff simulation based on adaptive bandwidth kernel density estimation and the Copula function, such as... Figure 1 As shown, it includes the following steps: S1. Collect measured historical runoff data, perform data preprocessing on the historical runoff data, and obtain runoff sequences in standard format; In this embodiment, data preprocessing includes: filling in missing data using linear interpolation; cleaning outliers in the data using the interquartile range method in statistics; and, according to actual needs, unifying historical runoff data to a specified target time scale using the arithmetic mean method. For daily-scale runoff stochastic simulation, there should be 365 flow variables per year; for decadal-scale runoff sequence stochastic simulation, there should be 36 flow variables per year; and for monthly-scale runoff sequence stochastic simulation, there should be 12 flow variables per year. Finally, the historical runoff data were converted into a standard runoff sequence according to time sequence. Specifically:

[0040] in, For years, This represents the total number of time periods within a given target timescale within a given year.

[0041] S2. Adaptive bandwidth kernel density estimation is used to fit historical runoff sequence data to obtain the probability density function of flow for any given time period. Specifically:

[0042] in, Representing runoff sequence Traffic, For adaptive bandwidth of kernel density estimation, The sample size of the flow sequence. This is the kernel function.

[0043] The method for obtaining adaptive bandwidth includes the following sub-steps: S21. Construct an adaptive bandwidth optimization model with the goal of minimizing the distance between the historical runoff distribution function and the probability density function; the optimization model specifically minimizes the weighted sum of the Euclidean geometric distance and the maximum distance between the historical runoff distribution function and the probability density function, as follows:

[0044] in, This is the optimization model for the adaptive bandwidth. The Euclidean geometric distance between the historical runoff distribution function and the probability density function is:

[0045] The maximum distance between the historical runoff distribution function and the probability density function:

[0046] yes The weight, yes The weight, This indicates finding the minimum value. Indicates the amount of data; Represents the historical runoff distribution function With the probability density function Distance between: .

[0047] S22. Initialize the particle swarm, setting each particle as a bandwidth vector composed of a set of bandwidth candidate solutions; solve the adaptive bandwidth optimization model using an improved particle swarm optimization algorithm. Figure 2 As shown, the specific steps include the following: S221. Initialize the various parameters of the algorithm, including particle swarm size, maximum number of iterations, inertia weight, upper and lower limits of the learning factor, mutation ratio coefficient, and threshold for judging the diversity of decision variables; randomly initialize the position and velocity of each particle in the particle swarm. S222. Calculate the fitness value of each particle and update the historical best fitness value and position of the population and particles; To enhance the algorithm's global search capability and accelerate its convergence speed, a nonlinear learning factor is introduced when updating the particle's velocity and position. The improved iterative formula is as follows:

[0048]

[0049] in, Indicates speed, Indicates location, For global optimality, Best in history, superscript Indicates the time period number, subscript Indicates particle number, subscript Indicates the dimensional index of the particle. For inertial weights, and Represents a random number between 0 and 1; and The non-linear learning factor is specifically:

[0050] in, for The minimum value, for The maximum value, for The minimum value, for The maximum value, Indicates the current iteration number. This indicates the maximum number of iterations.

[0051] To improve population diversity and avoid premature convergence of the algorithm, if the diversity of any dimension of the variable among all particles is below a threshold, the Cauchy mutation method is introduced to perturb the variable in that dimension. The diversity of dimensional variables is specifically as follows:

[0052] If the following conditions are met:

[0053] Then, the Cauchy mutation method is introduced to perturb the variables in the aforementioned dimension:

[0054] in, Indicates the first Diversity of dimensional variables For particle swarm scale, Indicates location, superscript Indicates the current iteration number, superscript Mean, subscript Indicates particle number, subscript The subscript indicates the dimensional index of the particle. Indicates the maximum value, subscript This represents the minimum value. The proportionality coefficient represents the control for the degree of variation. Indicates by parameters Random numbers generated by the Cauchy distribution This is the threshold for making a decision.

[0055] Calculate the fitness value of each particle. If the updated particle fitness value is lower than the fitness value of the historical best position, then update the particle's current position to the historical best position. If the updated particle fitness value is lower than the fitness value corresponding to the global best position of all particles, then update the particle's global best position.

[0056] S223, determine whether the iteration stopping condition of the particle swarm optimization algorithm is met; if it is met, stop the particle swarm iteration, the model training is completed, and the adaptive bandwidth of all data is obtained by outputting the particle at the best historical position; if it is not met, continue the particle swarm iteration.

[0057] In this embodiment, a preset number of iterations for the particle swarm is set, and iteration stops once the preset number of iterations is reached.

[0058] S3. The marginal distribution function of the flow rate in any given time period is obtained from the cumulative probability density of the flow rate. Specifically, the probability density of the flow rate in any given time period is derived through the probability density function, and then the marginal distribution function of the flow rate in any given time period is obtained by accumulating the obtained probability densities. The mathematical expression is as follows:

[0059] in, That is flow The marginal distribution function.

[0060] S4. Based on the Copula function, derive the conditional distribution function of the traffic flow in the next time period under the current traffic flow conditions from the marginal distribution function, specifically as follows:

[0061] in, For the current time period, For the next period of time, Traffic during the current time period Under the condition, the flow rate in the next time period The conditional distribution function, For Copula functions, for Time period traffic The marginal distribution function, for Time period traffic The marginal distribution function, for Time period traffic and Time period traffic Joint distribution function:

[0062] in, It is a natural constant. The parameters of the Copula function are obtained by maximizing the log-likelihood function. Get parameters The estimated value:

[0063] in, Here is the density function of Copula:

[0064] in, For the first One observation sample, This represents the total number of observation samples.

[0065] S5. Utilizing the conditional dependencies of flow rates between adjacent time periods in the conditional distribution function, simulated runoff flow rates are generated sequentially for each time period, such as... Figure 3 As shown, the specific steps include the following: S51. Starting from the time period with known flow, let... Time period runoff Known; S52. Randomly generate uniformly distributed random numbers. ,satisfy:

[0066] Based on the above derivation, we get Simulated flow rate at any time , for Time period traffic conditions, Time period traffic The conditional distribution function; S53, Update Repeat step S52 until the required number of simulated runoff flow values ​​are generated.

[0067] The feasibility and technical effects of the method in this application will be further illustrated by a specific embodiment: Using the decadal-scale historical hydrological data of the Huatan Hydrological Station in the Jinsha River Basin as the subject of this study, the Huatan Hydrological Station is located in Ningnan County, Sichuan Province. It is connected to the Wudongde Reservoir upstream and the Baihetan Reservoir downstream. It is an important hydrological observation station in the Jinsha River Basin, providing key data support for the study of hydrological characteristics and the design of water conservancy projects in the basin. It is also an important node for hydropower project management and ecological research.

[0068] The historical runoff sequence collected from the Huadan hydrological station spans from 1959 to 2010. This invention is used to perform random simulations on the sequence data, generating a simulated runoff sequence with a scale of 1000 years. The embodiment uses the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to measure the fitting performance of the marginal distribution.

[0069] Specifically, the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are as follows:

[0070]

[0071] In the formula: Represents the likelihood function; Indicates the number of parameters. Indicates the sample size.

[0072] Tables 1 and 2 show the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for fitting the marginal distributions of runoff sequence time-period variables using the log-logistic distribution (LLD), Weibull distribution (WD), gamma distribution (GD), kernel density estimation method (KDE), and the method of this application.

[0073] Table 1

[0074] Table 2

[0075] Among them, the smaller the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), the better the fitting effect.

[0076] As can be seen from Tables 1 and 2, the method of this application has the best overall performance in fitting the probability distribution of runoff margins and can more accurately reflect the data distribution. The model's universality and reliability are better than the comparative methods.

[0077] Table 3 presents the root mean square error (RMSE) and coefficient of determination (R²) between the statistical characteristics of the runoff sequences simulated by the two-dimensional kernel density estimation method (KDE), the runoff stochastic simulation method based on optimal bandwidth two-dimensional kernel density estimation (PSO-KDE), the runoff stochastic simulation method based on kernel density estimation (using the Silverman formula to calculate the bandwidth) and the Copula function (KDE-Copula), and the statistical characteristics of the runoff sequences simulated by the method of this application and the statistical characteristics of the measured runoff sequences. 2 ).

[0078] Table 3

[0079] Among them, the smaller the root mean square error (RMSE), the higher the coefficient of determination (R²). 2 The larger the value, the smaller the difference between the simulated statistical parameters and the actual statistical parameters, and the better the model simulation performance.

[0080] Among them, the root mean square error (RMSE) and the coefficient of determination (R²) 2 Specifically:

[0081]

[0082] In the formula: Indicates the number of time periods; Indicates the measured number of Statistical parameters for each time period; The simulation represents the first Statistical parameters for each time period; This represents the statistical mean.

[0083] As can be seen from Table 2, this application effectively ensures that the hydrological characteristic statistical parameters of the simulated runoff sequence are consistent with those of the measured runoff sequence, and the simulation accuracy of the model is better than that of the comparison method.

[0084] This application also implements a runoff stochastic simulation system based on adaptive bandwidth kernel density estimation and Copula function, the runoff stochastic simulation system comprising: The first module uses adaptive bandwidth kernel density estimation to fit historical runoff sequence data to obtain the probability density function of flow for any given time period. The second module obtains the marginal distribution function of the flow rate for any given time period from the cumulative probability density of the flow rate for any given time period; The third module, based on the Copula function, derives the conditional distribution function of the traffic flow in the next time period under the current traffic flow conditions from the marginal distribution function; The fourth module utilizes the conditional dependency relationship of flow rates in adjacent time periods in the conditional distribution function to generate simulated runoff flow rates for each time period through sequential simulation.

[0085] It should be understood that the above system is used to execute the methods in the above embodiments. The corresponding modules in the system are similar in implementation principle and technical effect to those described in the above methods. The working process of the system can be referred to the corresponding process in the above methods, and will not be repeated here.

[0086] Based on the methods in the above embodiments, this application also provides an electronic device, such as... Figure 4 As shown, the electronic device may include a processor, a communications interface, a memory, and a communication bus, wherein the processor, communications interface, and memory communicate with each other via the communication bus. The processor can invoke logical instructions stored in the memory to execute the methods described in the above embodiments.

[0087] Furthermore, the logical instructions in the aforementioned memory can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application.

[0088] Based on the methods in the above embodiments, this application provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0089] Based on the methods in the above embodiments, this application provides a computer program product that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0090] It is understood that the processor in the embodiments of this application can be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. A general-purpose processor can be a microprocessor or any conventional processor.

[0091] The method steps in this application embodiment can be implemented in hardware or by a processor executing software instructions. The software instructions can consist of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, portable hard disks, CD-ROMs, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor, enabling the processor to read information from and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can reside in an ASIC.

[0092] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted through the computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state disk (SSD)).

[0093] It is understood that the various numerical designations used in the embodiments of this application are merely for the convenience of description and are not intended to limit the scope of the embodiments of this application.

[0094] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of this application and is not intended to limit this application. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A stochastic runoff simulation method based on adaptive bandwidth kernel density estimation and Copula function, characterized in that, include: Adaptive bandwidth kernel density estimation is used to fit historical runoff sequence data to obtain the probability density function of flow for any given time period; The marginal distribution function of the flow rate in any given time period can be obtained from the cumulative probability density of the flow rate in any given time period. Based on the Copula function, the conditional distribution function of the traffic flow in the next time period is derived from the marginal distribution function under the current traffic flow conditions. By utilizing the conditional dependency relationship of flow rates in adjacent time periods in the conditional distribution function, simulated runoff flow rates are generated time-by-time through sequential simulation.

2. The method of stochastic simulation of runoff according to claim 1, characterized in that, The adaptive bandwidth in the adaptive bandwidth kernel density estimation is obtained through the following method: An adaptive bandwidth optimization model is constructed with the goal of minimizing the distance between the historical runoff distribution function and the probability density function. Initialize the particle swarm, with each particle set as a bandwidth vector composed of a set of bandwidth candidate solutions; use an improved particle swarm optimization algorithm to solve the adaptive bandwidth optimization model.

3. The method of stochastic simulation of runoff according to claim 2, characterized in that, The adaptive bandwidth optimization model specifically minimizes the weighted sum of the Euclidean geometric distance and the maximum distance between the historical runoff distribution function and the probability density function.

4. The method of stochastic simulation of runoff according to claim 2, characterized in that, In the improved particle swarm optimization algorithm, if the diversity of any dimension of the variable among all particles is lower than a threshold, the Cauchy mutation method is introduced to perturb the variable in that dimension.

5. The method of stochastic simulation of runoff according to claim 4, characterized in that, If the diversity of any dimension of the variables among all particles is below a threshold, then the Cauchy mutation method is introduced to perturb the variables in that dimension, specifically as follows: wherein, denotes the dimensionality of the is the swarm size, denotes the position, the superscript denotes the current iteration number, the superscript denotes the mean, the subscript denotes the particle index, the subscript denotes the dimension index of the particle, the subscript denotes the maximum, the subscript denotes the minimum, denotes the proportionality coefficient that controls the degree of variation, denotes a random number generated by the parameter Cauchy distribution, is the decision threshold.

6. The stochastic runoff simulation method according to claim 2, characterized in that, In the improved particle swarm optimization algorithm, a nonlinear learning factor is introduced and Velocity and position of new particles: in, Indicates speed, Indicates location, For global optimality, Best in history, superscript Indicates the time period number, subscript Indicates particle number, subscript Indicates the dimensional index of the particle. For inertial weights, and Represents a random number between 0 and 1; and Specifically: in, for The minimum value, for The maximum value, for The minimum value, for The maximum value, Indicates the current iteration number. This indicates the maximum number of iterations.

7. The stochastic runoff simulation method according to claim 1, characterized in that, The marginal distribution function of the flow rate for any given time period is obtained from the cumulative probability density of the flow rate. Specifically: in, express Time period traffic The probability density function, express Time period traffic The marginal distribution function.

8. The stochastic runoff simulation method according to claim 1, characterized in that, Based on the Copula function, the conditional distribution function of the traffic flow in the next time period is derived from the marginal distribution function under the current time period traffic conditions, specifically as follows: in, For the current time period, For the next period of time, Traffic during the current time period Under the condition, the flow rate in the next time period The conditional distribution function, For Copula functions, for Time period traffic The marginal distribution function, for Time period traffic The marginal distribution function, for Time period traffic and Time period traffic Joint distribution function: in, It is a natural constant. The parameters of the Copula function are obtained by maximizing the function. Get parameters The estimated value: in, For the first One observation sample, This represents the total number of observation samples.

9. The stochastic runoff simulation method according to claim 1, characterized in that, By utilizing the conditional dependencies of flow rates between adjacent time periods in the conditional distribution function, simulated runoff flow rates are generated sequentially time-by-time through sequential simulation, specifically as follows: (1) Let Time period runoff Known; (2) Randomly generate uniformly distributed random numbers Based on The derivation yields Simulated flow rate at any time , for Time period traffic conditions, Time period traffic The conditional distribution function; (3) Update Repeat step (2) until a preset number of simulated runoff flow values ​​are generated.

10. A stochastic runoff simulation system based on adaptive bandwidth kernel density estimation and Copula function, characterized in that, The stochastic runoff simulation system includes: The first module uses adaptive bandwidth kernel density estimation to fit historical runoff sequence data to obtain the probability density function of flow for any given time period. The second module obtains the marginal distribution function of the flow rate for any given time period from the cumulative probability density of the flow rate for any given time period; The third module, based on the Copula function, derives the conditional distribution function of the traffic flow in the next time period under the current traffic flow conditions from the marginal distribution function; The fourth module utilizes the conditional dependency relationship of flow rates in adjacent time periods in the conditional distribution function to generate simulated runoff flow rates for each time period through sequential simulation.

Images