Single-target and multi-target flood interval forecasting method considering interval fitting coefficient

Abstract

The invention discloses a single-target and multi-target flood interval forecasting method considering an interval fitting coefficient, and the method comprises the following steps: 1, obtaining historical runoff and rainfall data of a target research region, and carrying out the data quality inspection; step 2, screening LUBE interval forecast factors; 3, constructing an interval forecast evaluation index system; step 4, constructing an LUBE interval forecasting model based on a three-layer dual-output ANN neural network; step 5, establishing a single-target interval forecasting model based on a gravitational search algorithm GSA; step 6, establishing a multi-target interval forecasting model based on a non-dominated sorting genetic algorithm NSGA-III; step 7, determining and selecting a Pareto solution according to the preference of a decision maker, and determining model parameters to obtain a forecast result; according to the method, an interval forecast evaluation index system is enriched, and meanwhile, the method can be expanded to the single-target and multi-target research field.

Description

Technical Field

[0001] This invention relates to the field of hydrological runoff forecasting technology, and in particular to a single-objective and multi-objective flood interval forecasting method that considers interval fitting coefficients. Background Technology

[0002] In runoff interval forecasting, rationally evaluating the quality of the forecast results is a crucial step in assessing forecast quality and uncertainty. Currently, common evaluation metrics for interval forecasts include Interval Coverage Proportion (PICP) and Interval Width Proportion (PINAW). While these metrics can effectively evaluate the coverage and width of the forecast interval, due to the unique dual-output nature of interval forecasts, they cannot practically measure the degree of fit between predicted and observed values. Summary of the Invention

[0003] The purpose of this invention is to overcome the above-mentioned shortcomings and provide a single-objective and multi-objective flood interval forecasting method that considers interval fitting coefficients, enriches the interval forecasting evaluation index system, and extends to the research fields of single-objective and multi-objective forecasting.

[0004] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients, which includes the following steps:

[0005] Step 1: Obtain historical runoff and rainfall data for the target study area and conduct data quality checks;

[0006] Step 2: Filter LUBE interval forecast factors;

[0007] Step 3: Construct an evaluation index system for interval forecasts;

[0008] Step 4: Construct a LUBE interval forecast model based on a three-layer dual-output ANN neural network;

[0009] Step 5: Establish a single-target interval prediction model based on the Gravity Search Algorithm (GSA);

[0010] Step 6: Establish a multi-objective interval prediction model based on the non-dominated sorting genetic algorithm NSGA-III;

[0011] Step 7: Select the Pareto solution based on the decision-maker's preference, determine the model parameters, and then obtain the forecast results.

[0012] Preferably, step 2 specifically includes the following process:

[0013] Historical runoff (Q) for the previous 30 periods was analyzed using the mutual information MI method. t-1 ~Q t-30 ) and precipitation data (R t-1 ~Rt-30 ) and current runoff data Q t The correlation between the two variables is analyzed, and the top 10 forecast factors with the highest mutual information are selected as the final input to the model. The mutual information between two discrete random variables X and Y is defined as follows:

[0014]

[0015] In the formula, p(x,y) is the joint density function of X and Y, and p(x) and p(x) are the marginal density functions of X and Y, respectively.

[0016] Preferably, step 3 specifically includes: constructing the interval coverage index PICP, the interval width index PINAW, the interval fitting coefficient index PIFC, and the comprehensive evaluation index CWFC.

[0017] Preferably, the constructed interval coverage index PICP reflects the probability of measured data within the forecast interval, ranging from 0% to 100%, and its calculation expression is as follows:

[0018]

[0019] In the formula, n represents the sequence length of the measured data; y i U represents the i-th measured data; i and L i Let y represent the upper and lower boundaries of the i-th forecast interval, respectively; when the measured value y i When it falls within the forecast interval, c i When the measured value y equals 1, i When it falls outside the forecast interval, c i It equals 0.

[0020] Preferably, the constructed interval width index PINAW is used to limit the width of the forecast interval. While ensuring coverage, the width of the forecast interval is reduced in order to achieve accurate and reliable forecast accuracy. Its calculation expression is as follows:

[0021]

[0022] In the formula, R represents the range of variation of the target variable.

[0023] Preferably, the constructed interval fitting coefficient index PIFC is used to measure the fitting accuracy between the median of the forecast interval and the measured value, and its calculation expression is as follows:

[0024]

[0025] In the formula, The measured average value of hydrological elements is expressed as follows: The PIFC range is [0,1]. The larger the value, the better the fit between the predicted median and the measured value of the surface interval, and the better the symmetry of the upper and lower boundaries of the interval.

[0026] Preferably, the constructed comprehensive evaluation index CWFC is:

[0027]

[0028] In the formula, η1, η2, η3 and μ are all constants; where μ represents the confidence level, which is equal to (1-α); the values ​​of η1, η2 and η3 represent the penalty terms when the forecast interval width and coverage, and the fitting coefficient do not meet the standards. The values ​​range from 50 to 100. The values ​​of η1, η2 and η3 can amplify the sensitivity of the corresponding indicators.

[0029] When calibrating the model parameters, γ(PICP) is a constant of 1. When validating the model, γ(PICP) is a step function, and its expression is as follows:

[0030]

[0031] Preferably, in step 4, the mathematical expression of the model is defined as follows:

[0032]

[0033] In the formula, L i U represents the lower boundary of the output interval. i x represents the upper boundary of the output interval; k The input layer's k-th node; w ij The weights between hidden layer nodes and output layer nodes; v jk These are the weight values ​​between input layer nodes and hidden layer nodes; b wi b is the threshold between hidden layer nodes and output layer nodes; vj f1 and f2 are the thresholds between input layer nodes and hidden layer nodes; f1 and f2 are activation functions; N i and N h These represent the number of nodes in the input layer and the number of nodes in the hidden layer, respectively.

[0034] Preferably, step 5 is performed as follows:

[0035] Using a gravity search algorithm, the phenomenon of universal gravitation is simulated, where particles attract each other, causing lighter particles to move towards heavier particles, thus guiding the population to search near the larger particles. According to Newton's law of universal gravitation, the gravitational force exerted on particle i by particle j in the t-th iteration is:

[0036]

[0037] In the formula: Let G(t) be the gravitational force exerted by particle j on particle i in the d-th dimension; G(t) be the gravitational constant at the t-th iteration; M ρi (t) represents the passive gravitational mass of particle i; M aj (t) represents the active gravitational mass of particle j; R ij (t) represents the Euclidean distance between particle i and particle j; ε is a small constant; These represent the position coordinates of particle j and particle i in the d-th dimension, respectively.

[0038] To ensure that the GSA algorithm has strong global search capability in the early stages and strong local search capability in the later stages, the gravitational constant G(t) is gradually decreased with the number of iterations:

[0039]

[0040] In the formula: G0 is the initial gravitational constant; a is a constant that controls the decay rate of the gravitational constant; T is the maximum number of iterations.

[0041] Preferably, step 6 is performed as follows:

[0042] Using the interval coverage rate (PICP), interval width index (PINAW), and interval fitting coefficient (PIFC) as the objective values ​​of the multi-objective optimization algorithm NSGA-Ⅲ, the Pareto optimal solution set for the target study area is obtained through optimization.

[0043] The beneficial effects of this invention are as follows: This invention proposes for the first time the Interval Fit Coefficient (PIFC) for evaluating the deviation between the median of the interval forecast and the observed value, enriching the evaluation index system for interval forecasts. It also proposes a comprehensive evaluation index (CWFC) based on the Interval Coverage Rate (PICP), the Interval Width Index (PINAW), and the Interval Fit Coefficient (PIFC). Furthermore, this invention constructs a single-objective interval forecast model based on the Gaussian Specialized Analysis (GSA) and the comprehensive evaluation index CWFC, and finally, a multi-objective interval forecast model based on the NSGA-III algorithm. The resulting Pareto non-dominated solution set can provide decision-makers with more preference choices. This invention enriches the evaluation index system for interval forecasts and can be extended to single-objective and multi-objective research fields. Attached Figure Description

[0044] Figure 1 This is a schematic diagram illustrating the change process of the objective function for single-target interval forecasting.

[0045] Figure 2 A bar chart of forecast indicators for the first-level inspection period in Jinping;

[0046] Figure 3 A schematic diagram of the 2020 flood season interval forecast process at an 80% confidence level for a single objective.

[0047] Figure 4 A schematic diagram illustrating the process of changing the Pareto optimal solution for a periodic multi-objective problem;

[0048] Figure 5 A schematic diagram of the 2020 flood season interval forecast process of Jinping Level I under multiple objectives. Detailed Implementation

[0049] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0050] Example 1: A method for single-objective and multi-objective flood interval forecasting considering interval fitting coefficients, comprising the following steps:

[0051] Step 1: Obtain historical runoff and rainfall data for the target study area and conduct data quality verification.

[0052] Step 2: Screening LUBE interval forecast factors. Historical runoff (Q) over the previous 30 periods is analyzed using the mutual information (MI) method. t-1 ~Q t-30 ) and precipitation data (R t-1 ~R t-30 ) and current runoff data Q t The correlation between the variables is analyzed, and the top 10 predictors with the highest mutual information are selected as the final inputs to the model. Generally, the mutual information between two discrete random variables X and Y can be defined as:

[0053]

[0054] In the formula, p(x,y) is the joint density function of X and Y, and p(x) and p(x) are the marginal density functions of X and Y, respectively.

[0055] Step 3: Construct an evaluation index system for interval forecasts.

[0056] Interval Coverage Ratio (PICP): This reflects the probability that measured data falls within the forecast interval, ranging from 0% to 100%. Its calculation formula is as follows:

[0057]

[0058] In the formula, n represents the sequence length of the measured data; y i U represents the i-th measured data; i and L i Let y represent the upper and lower boundaries of the i-th forecast interval, respectively; when the measured value y i When it falls within the forecast interval, c i When the measured value y equals 1, i When it falls outside the forecast interval, c i=0; When making inter-regional forecasts, the PICP value should generally be greater than or equal to the given confidence level in order to ensure a high inter-regional forecast coverage.

[0059] Interval Width Index (PINAW): Used to limit the width of the forecast interval. While ensuring coverage, the width of the forecast interval is minimized as much as possible in order to achieve accurate and reliable forecast accuracy.

[0060]

[0061] In the formula, R represents the range of variation of the target variable.

[0062] Interval Fit Coefficient (PIFC): Used to measure the fitting accuracy between the median of the forecast interval and the measured value.

[0063]

[0064] In the formula, The measured average value of hydrological elements is expressed as follows: The PIFC range is [0,1]. The larger the value, the better the fit between the predicted median and the measured value of the surface interval, and the better the symmetry of the upper and lower boundaries of the interval.

[0065] Comprehensive evaluation index CWFC:

[0066]

[0067] In the formula, PICP, PINAW, and PIFC represent the coverage, width, and fitting coefficient of the forecast interval, respectively; η1, η2, η3, and μ are all constants. μ can usually be represented as the confidence level, which is equal to (1-α); the values ​​of η1, η2, and η3 represent the penalty terms when the forecast interval width, coverage, and fitting coefficient do not meet the standards. These values ​​are often relatively large, usually ranging from 50 to 100. Larger values ​​of η1, η2, and η3 can amplify the sensitivity of the corresponding indicators.

[0068] When calibrating the model parameters, γ(PICP) is a constant of 1. When validating the model, γ(PICP) is a step function, and its expression is as follows:

[0069]

[0070] Step 4: Construct a LUBE interval prediction model based on a three-layer dual-output ANN neural network. The mathematical expressions for the input and output of the three-layer ANN neural network are defined as follows:

[0071]

[0072] In the formula, L i U represents the lower boundary of the output interval.i x represents the upper boundary of the output interval; k The input layer's k-th node; w ij The weights between hidden layer nodes and output layer nodes; v jk These are the weight values ​​between input layer nodes and hidden layer nodes; b wi b is the threshold between hidden layer nodes and output layer nodes; vj f1 and f2 are the thresholds between input layer nodes and hidden layer nodes; f1 and f2 are activation functions; N i and N h These represent the number of nodes in the input layer and the number of nodes in the hidden layer, respectively.

[0073] Step 5: Establish a single-target interval prediction model based on the Gravity Search Algorithm (GSA).

[0074] This invention utilizes a gravity search algorithm. By simulating the phenomenon of universal gravitation, particles attract each other, causing lighter particles to move towards heavier particles, thus guiding the population to search near the larger particles. According to Newton's law of universal gravitation, the gravitational force exerted on particle i by particle j in the t-th iteration is:

[0075]

[0076] In the formula: Let G(t) be the gravitational force exerted by particle j on particle i in the d-th dimension; G(t) be the gravitational constant at the t-th iteration; M ρi (t) represents the passive gravitational mass of particle i; M aj (t) represents the active gravitational mass of particle j; R ij (t) represents the Euclidean distance between particle i and particle j; ε is a small constant.

[0077] To ensure that the GSA algorithm has strong global search capability in the early stages and strong local search capability in the later stages, the gravitational constant G(t) is gradually decreased with the number of iterations:

[0078]

[0079] In the formula: G0 is the initial gravitational constant; a is a constant that controls the decay rate of the gravitational constant; T is the maximum number of iterations.

[0080] Step 6: Establish a multi-objective interval prediction model based on the non-dominated sorting genetic algorithm (NSGA-Ⅲ).

[0081] This invention uses three indices—interval coverage rate (PICP), interval width index (PINAW), and interval fitting coefficient (PIFC)—as the objective values ​​of the multi-objective optimization algorithm NSGA-Ⅲ, and optimizes them to obtain the Pareto optimal solution set for the target study area.

[0082] Step 7: Select the Pareto solution based on the decision-maker's preference, determine the model parameters, and then obtain the forecast results.

[0083] Example 2: Taking the Jinping I Hydropower Station on the Yalong River as an example, this invention demonstrates the specific steps of a single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients:

[0084] (1) Collect daily inflow data of Jinping I Hydropower Station on Yalong River from 2015 to 2020, calculate the mutual information value of historical runoff data, average rainfall data and current runoff for the first 30 periods, and determine the final input factor as 10.

[0085] (2) Using the comprehensive evaluation index CWFC as the objective, GSA as the optimization algorithm, and a three-layer ANN neural network as the LUBE structure, a single-objective interval forecast model is constructed. Figure 1 The process of change of the objective function of the single-objective prediction model is shown.

[0086] (3) Set the algorithm parameters and termination conditions to obtain the theoretically optimal parameter combination for the single-objective interval forecast model. Use 2016–2019 as the calibration period and 2020 as the verification period. The corresponding interval forecast index values ​​under different objective functions and confidence levels are as follows: Figure 2 As shown, at the same time Figure 3 The inflow flood process of Jinping I Hydropower Station during the flood season (May to October) in 2020 is given.

[0087] (4) Using the interval coverage rate (PICP), interval width index (PINAW), and interval fitting coefficient (PIFC) as the objective values ​​of the multi-objective optimization algorithm NSGA-Ⅲ, the distribution of the 378 Pareto optimal solutions for the Jinping I-1 hydropower station can be obtained, and the pairwise relationship diagrams of the three indices are given, such as... Figure 4 As shown, PIFC, PICP, and PINAW do not compete with each other, which means that improving PICP and PINAW will not affect the accuracy of PIFC. Figure 5 The flood inflow process (May to October) of Jinping I Hydropower Station during the flood season in 2020 is presented.

[0088] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The scope of protection of the present invention should be defined as the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients, characterized in that: It comprises the following steps: Step 1, obtaining historical runoff and rainfall data of the target study area, and performing data quality inspection; Step 2, screening LUBE interval prediction factors; Step 3, constructing an interval prediction evaluation index system; Step 4, constructing a LUBE interval prediction model based on a three-layer double-output ANN neural network; Step 5, establishing a single-objective interval prediction model based on a gravitational search algorithm GSA; Step 6, establishing a multi-objective interval prediction model based on a non-dominated sorting genetic algorithm NSGA-III; Step 7, determining the selected Pareto solution according to the decision maker's preference, determining the model parameters, and obtaining the prediction results.

2. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 1, characterized in that: The step 2 specifically comprises the following process: The correlation between the first 30 time period historical runoff (Q t-1 ~Q t-30 ) and precipitation data (R t-1 ~R t-30 ) and the current runoff data Q t is analyzed by the mutual information MI method, and the first 10 forecast factors with the maximum mutual information are selected as the final input of the model; the mutual information of two discrete random variables X and Y is defined as: In the formula, p(x, y) is the joint density function of X and Y, and p(x) and p(x) are the marginal density functions of X and Y, respectively.

3. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 1, characterized in that: The step 3 specifically comprises: constructing an interval coverage rate index PICP, an interval width index PINAW, an interval fitting coefficient index PIFC, and a comprehensive evaluation index CWFC.

4. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 3, characterized in that: The constructed interval coverage rate index PICP reflects the probability of the measured data in the prediction interval, and its range is 0-100%, and its calculation expression is as follows: where n represents the sequence length of the measured data; y i represents the ith measured data; U i and L i respectively represent the upper boundary and the lower boundary of the ith prediction interval; c i equals 1 when the measured value y i falls within the prediction interval, and c i equals 0 when the measured value y i falls outside the prediction interval.

5. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 4, characterized in that: The constructed interval width index PINAW is used to limit the width of the prediction interval, and in the case of ensuring the coverage rate, the width of the prediction interval is reduced to achieve accurate and reliable prediction accuracy, and its calculation expression is as follows: In the formula, R is the range of the target variable.

6. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 5, characterized in that: The constructed interval fitting coefficient index PIFC is used to measure the fitting accuracy of the measured value in the prediction interval, and its calculation expression is as follows: In the formula, is the average of the observed hydrological elements, and its expression is PIFC ranges from [0, 1], and the greater the value, the higher the fitting degree of the surface interval forecast median and the observed value, and the better the symmetry of the upper and lower boundaries of the interval.

7. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 6, characterized in that: The constructed comprehensive evaluation index CWFC is: In the formula, η1, η2, η3 and μ are all constants; wherein μ represents the confidence level, that is, equal to (1-α); η1, η2 and η3 values represent the penalty items when the prediction interval width and coverage rate, fitting coefficient do not meet the standard, and the value range is between 50-100, and η1, η2 and η3 values can amplify the sensitivity of the corresponding index; When the model parameters are calibrated, γ(PICP) is a constant and is always 1, and when the model is tested, γ(PICP) is a step function, and its expression is as follows:

8. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 1, characterized in that: In the step 4, the mathematical expression of the model is defined as follows: where L i is the lower bound of the output interval; U i is the upper bound of the output interval; x k is the kth node of the input layer; w ij is the weight value between the hidden layer node and the output layer node; v jk is the weight value between the input layer node and the hidden layer node; b wi is the threshold value between the hidden layer node and the output layer node; b vj is the threshold value between the input layer node and the hidden layer node; f1 and f2 are activation functions; N i and N h represent the number of input layer nodes and the number of hidden layer nodes, respectively.

9. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 1, characterized in that: The specific process of the step 5 is as follows: By simulating the phenomenon of universal gravitation, particles attract each other, so that particles with light mass move towards the direction of particles with heavy mass, thereby guiding the population to search in the vicinity of the large mass particles; according to Newton's law of universal gravitation, the gravity of particle j on particle i at the tth iteration is: where: Gd(j, i) is the gravitational force of particle j on particle i in the dth dimension; G(t) is the gravitational constant for the tth iteration; M ρi M(i) (t) is the passive gravitational mass of particle i; M aj M(j) (t) is the active gravitational mass of particle j; R ij R(i, j) (t) is the Euclidean distance between particle i and particle j; ε is a small constant; Xi(j) (t) and Xi(i) (t) are the position coordinates of particle j and particle i in the dth dimension, respectively; In order to make the GSA algorithm have strong global search ability in the early stage and strong local search ability in the later stage, the gravitational constant G(t) gradually decreases with the iteration number: In the formula: G0 is the initial gravitational constant; a is a constant, which controls the decay rate of the gravitational constant; T is the maximum iteration number.

10. The single-objective and multi-objective flood interval forecasting method considering interval fitting coefficients according to claim 1, characterized in that: The specific process of the step 6 is as follows: Taking the interval coverage rate PICP, the interval width index PINAW and the interval fitting coefficient PIFC as the target values of the multi-objective optimization algorithm NSGA-III, the Pareto optimal solution set of the target study area is obtained by optimization.

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