A rainfall probability distribution unbiased estimation method based on least square fitting

Abstract

This invention belongs to the technical field of hydrometeorological extreme value theory, and provides an unbiased estimation method for rainfall probability distribution based on least squares fitting. Existing parameter estimation methods are numerous, each with its own specific applicable scope and advantages and disadvantages. The choice of method significantly affects the accuracy of generalized extreme value distribution parameter estimation. This invention comprehensively considers the advantages of five parameter estimation methods: Bayesian, GMLE, L-moments, MLE, and MPS. Based on least squares fitting, it determines the optimal weight coefficients of a single method, establishing an unbiased and efficient ensemble parameter estimation framework. By integrating multiple parameter estimation methods, the bias and variance of individual methods can be reduced, improving the robustness and reliability of parameter estimation, and providing a reference for accurately determining the rainfall frequency curve and design rainfall.

Description

Technical Field

[0001] This invention relates to the field of hydrological and meteorological extreme value theory technology, and in particular to an unbiased estimation method for rainfall probability distribution based on least squares fitting. Background Technology

[0002] In recent years, influenced by climate change, extreme rainfall events have occurred more frequently. The significant impact of these disasters on human production and lives has drawn widespread social attention, making research on extreme rainfall crucial for flood prevention and disaster reduction. Studies on heavy rainfall risk often use frequency distribution curves to describe the probability distribution of extreme rainfall events, with shape parameters being a key factor influencing the severity of the event. Generalized extreme value distribution (GEV) curves, including Gumbel, Fréchet, and Weibull extreme value distributions, can be used to fit extreme rainfall distributions. By fitting the GEV distribution, a confidence interval for the return period at a certain confidence level can be obtained, which helps assess the risk of future extreme rainfall events. Selecting the best-fitting GEV distribution allows for a more accurate understanding of event characteristics and the implementation of appropriate risk management strategies.

[0003] Choosing an unbiased and efficient parameter estimation method is crucial for accurately assessing the risk of heavy precipitation. Unbiased estimation means that the average value of the estimated values ​​obtained using a certain parameter estimation method will equal the true parameter values. These methods can reduce the expected value of the estimation error to a certain extent, making the estimation results closer to the reality. Commonly used methods for estimating the shape parameters of the GEV distribution include the linear moments method (L-moments), maximum likelihood estimation (MLE), generalized maximum likelihood estimation (GMLE), Bayesian method, and maximum product gap method (MPS). Existing parameter estimation methods all have certain limitations. For example, the L-moments method tends to fit the tail of the distribution curve, which can easily underestimate the shape parameters; the MLE method tends to maximize the probability of observed data occurring under a given model, which may lead to overestimation; the Bayesian method requires the selection of a prior distribution, and an inappropriate prior distribution selection may lead to significant bias in the estimation results. Therefore, although various parameter estimation methods exist, each has its limitations. It is necessary to consider the characteristics of the problem and the data conditions, and to propose a robust and unbiased weighted parameter estimation method that combines the advantages and applicability of various parameter methods. The least squares method finds the best function match for the data by minimizing the sum of squared errors, determines the optimal weight coefficients of a single method and restricts the range of weight coefficients to [-1,1] for unbiased and efficient estimation, and has the advantages of unique optimal solution, convenient solution and interpretability.

[0004] Since the shape parameter of the GEV distribution lacks a true value, Monte Carlo sampling is often used to create random sample sequences to evaluate the accuracy of various parameter estimation methods and assess their effectiveness. The Monte Carlo method, employing random simulation and statistical experimentation, generates a random numerical sequence from the probability distribution of a random variable by randomly selecting numbers. This sequence serves as the input variable for specific simulations and solutions. It can handle complex parameter spaces and functional relationships, and by utilizing the properties of random sampling to cover different parameter regions, it more comprehensively considers the impact of parameter variations on the model results. Summary of the Invention

[0005] The purpose of this invention is to provide an unbiased estimation method for rainfall probability distribution based on least squares fitting. This method can comprehensively consider the advantages of multiple parameter estimation methods to perform unbiased parameter estimation. Therefore, this method can exhibit stronger adaptability and flexibility in different practical application scenarios.

[0006] To achieve the above objectives, the present invention provides the following solution: an unbiased estimation method for rainfall probability distribution based on least squares fitting, comprising the following steps:

[0007] S1. Use multiple single-parameter estimation methods to estimate the shape parameters of the measured samples;

[0008] S2. Determine the range of shape parameters based on the union of the shape parameters estimated in S1, and generate random samples by Monte Carlo random sampling from the range of shape parameters.

[0009] S3. Calculate using a single-parameter estimation method for random samples;

[0010] S4. For the results of different single-parameter estimation methods, the least squares method is used to determine the optimal weighting coefficients of the single-parameter estimation methods, and finally, a comprehensive fitting is performed to estimate the rainfall probability distribution.

[0011] The measured samples in step S1 are based on the distribution of an extreme precipitation model, which is the GEV distribution.

[0012] Extreme precipitation values ​​are simulated using the GEV distribution; the cumulative distribution function of the time series is calculated using the following formula:

[0013]

[0014] Where μ, σ, and α represent position parameter, scaling parameter, and shape parameter, respectively;

[0015] Five single-parameter estimation methods were selected: linear moments method, maximum likelihood method, generalized maximum likelihood estimation method, Bayesian method, and maximum product interval method. These methods were used to estimate the shape parameters and obtain shape parameter estimates for different parameters.

[0016] The specific steps of generating random samples in step S2 are as follows: determine the sampling size, that is, determine the number of random sample sequences to be generated; generate a large number of random sample sequences within the selected range of GEV distribution shape parameters; and finally store the generated random sample sequences for use in different single parameter estimation methods.

[0017] Step S4 specifically includes the following steps:

[0018] Step S401: For different single parameter estimation methods, calculate the weight of the comprehensive parameter estimation method according to the least squares method;

[0019]

[0020] y j x represents the j-th sample value; ij Let represent the estimated shape parameters, where i represents the i-th parameter estimation method, j represents the j-th data point, and n represents the number of parameter estimation methods.

[0021]

[0022] S represents the sum of squared residuals; m represents the length of the observed sample.

[0023] The calculation model is as follows:

[0024]

[0025] Step S402, according to w i The rainfall probability distribution is calculated by combining the results of different single-parameter estimation methods.

[0026] The beneficial effects of this invention are as follows: The unbiased estimation method for rainfall probability distribution based on least squares fitting constructed in this invention effectively solves the problem that there is no optimal parameter estimation method and each method has its own applicable range. This method will contribute to the research on extreme rainfall and flood prevention and disaster reduction. Attached Figure Description

[0027] Figure 1 This is a flowchart of the present invention.

[0028] Figure 2 shows box plots of unbiased estimation results of the generalized extreme value distribution parameters based on five parameter estimation methods and least squares fitting for random sample data. Figure 2(a) is a box plot of the parameter estimation results with a shape parameter of 0.05; Figure 2(b) is a box plot of the parameter estimation results with a shape parameter of 0.10; Figure 2(c) is a box plot of the parameter estimation results with a shape parameter of 0.15; Figure 2(d) is a box plot of the parameter estimation results with a shape parameter of 0.20; Figure 2(e) is a box plot of the parameter estimation results with a shape parameter of 0.25; Figure 2(f) is a box plot of the parameter estimation results with a shape parameter of 0.30; Figure 2(g) is a box plot of the parameter estimation results with a shape parameter of 0.35; Figure 2(h) is a box plot of the parameter estimation results with a shape parameter of 0.40; Figure 2(i) is a box plot of the parameter estimation results with a shape parameter of 0.45; and Figure 2(j) is a box plot of the parameter estimation results with a shape parameter of 0.50. Detailed Implementation

[0029] Based on the method for estimating parameters of extreme rainstorms, this invention proposes a comprehensive method: an unbiased estimation method for the probability distribution of rainfall based on least squares fitting.

[0030] The unbiased estimation method for the probability distribution of rainfall based on least squares fitting mainly includes: calculating the range of shape parameters using five parameter estimation methods; evaluating the effectiveness of various parameter estimation methods using Monte Carlo sampling based on the shape parameter range; and finally using the least squares method to determine the optimal weighting coefficients for the shape parameters using a single parameter estimation method. The research approach is based on measured data from a specific site. Five parameter estimation methods are used to estimate the shape parameters of the GEV distribution. The maximum and minimum values ​​calculated by the five parameter estimation methods are determined as the upper and lower limits of the shape parameters of the GEV distribution. Then, Monte Carlo random sampling is performed to generate rainfall samples that conform to the specific shape parameter range. Multiple single parameter estimation methods are used to calculate the estimation results, and then the least squares method is used to determine the optimal weighting coefficients for different single parameter estimation methods. Specifically, the following steps are included:

[0031] The first step is to estimate the shape of the sample data using five single-parameter estimation methods based on the GEV distribution, and then obtain the range of shape parameters.

[0032] (1) Extreme precipitation model selected: GEV distribution

[0033] The GEV distribution was developed from extremum theory, combining the Gumbel, Fréchet, and Weibull series, and is also known as Type I, II, and III extremum distributions. The formula is as follows:

[0034]

[0035] Where x, μ, σ, and α represent extreme value variables, position parameters, scale parameters, and shape parameters, respectively.

[0036] The above three distribution types can be collectively referred to as the GEV distribution expression. GEV can comprehensively consider different types; therefore, this study chooses the GEV distribution to simulate extreme precipitation values. The cumulative distribution function of the time series is calculated using the following formula:

[0037]

[0038] Where μ, σ, and α represent the position parameter, scale parameter, and shape parameter, respectively.

[0039] (2) Estimate the sample using five parameter estimation methods.

[0040] Parameter estimation methods are a crucial aspect affecting design storms, especially the estimation of the shape parameters of the GEV distribution, which directly influences the derivation of the design storm. Five parameter estimation methods—L-moments, MLE, GMLE, Bayesian, and MPS—were used to estimate the parameters of the GEV distribution on measured samples, yielding the parameter range.

[0041] The second step involves using different parameter estimation methods to accurately estimate the location and scale parameters of the GEV distribution, but with significant differences in performance for the shape parameter, which has the greatest impact on the design storm. Therefore, based on the parameter estimation results of the five methods, Monte Carlo random sampling is performed with fixed values ​​for the location and scale parameters and a specified range for the shape parameter, generating a large number of random samples.

[0042] First, determine the sampling size, that is, determine the number of random sample sequences to be generated. Then, use the selected GEV distribution parameter range to generate a large number of random sample sequences. Finally, store the generated random sample sequences for subsequent application of different parameter estimation methods.

[0043] The third step is to estimate the parameters of the random sample using five parameter estimation methods.

[0044] The fourth step is to calculate the weights of the combined parameter estimation methods (the estimated value fits the true value) based on the least squares method for different parameter estimation methods.

[0045]

[0046] y j x represents the j-th sample value; ij Let represent the parameter estimate, where i represents the i-th parameter estimation method, j represents the j-th data point, and n represents the number of parameter estimation methods.

[0047]

[0048] S represents the sum of squared residuals; m represents the length of the observed sample.

[0049] The calculation model is as follows:

[0050]

[0051] According to w i The results of the comprehensive parameter estimation method are calculated.

[0052] Finally, the results of the comprehensive parameter estimation method and the single parameter estimation method are compared. The results of the five parameter estimation methods are compared with the results of the comprehensive parameter estimation method to evaluate the performance of the comprehensive parameter estimation method.

[0053] Thus, the results and evaluation of the unbiased estimation method for GEV distribution parameters based on least squares fitting were obtained, thereby completing the work of constructing an unbiased estimation method for rainfall probability distribution based on least squares fitting.

[0054] The present invention will be further illustrated by the following examples.

[0055] China was chosen as the study area, and the data used in this study was daily rainfall data from meteorological stations provided by the China Meteorological Data Network (http: / / data.cma.cn / ). Daily rainfall data sequences meeting the requirements of a time period greater than 30 years and a missing rate of less than 1% were selected, and finally, 712 meteorological stations were chosen for analysis. The specific steps are as follows:

[0056] The first step is to estimate the sample data based on the GEV distribution function using five parameter estimation methods (L-moments, MLE, GMLE, Bayesian, and MPS) to obtain the range of shape parameters.

[0057] In the second step, we generated 1000 random samples from the shape parameter range of the GEV distribution, with a sample size of 60, ensuring that the sample size is basically consistent with the data length of Chinese meteorological stations. Since the location and scale parameters of the GEV distribution calculated by the five estimation methods are similar, the Monte Carlo experiment fixed the location and scale parameters to the median of the estimation results of 712 Chinese meteorological stations, which are 60 and 20 respectively, to ensure that the generated random samples are representative of the real data. Finally, according to the shape parameter estimation range, the Monte Carlo experiment specified 10 shape parameters for Monte Carlo random sampling at intervals of 0.05 from 0.05 to 0.50, forming a total of 10 parameter combinations.

[0058] The third step involves estimating the GEV distribution parameters using five parameter estimation methods based on the 1000 random samples generated in the second step, and repeating this process 1000 times. The shape parameters of the generated random samples are also estimated using the five parameter estimation methods. The box plot is shown in Figure 2.

[0059] The fourth step is to determine the optimal weighting coefficients for a single method based on the least squares fitting, with the weighting coefficients limited to the range of [-1, 1]. The estimation results are shown in Figure 2.

[0060] Table 1. Weights of the five parameter estimation methods

[0061]

[0062]

[0063] The fifth step involves comparing different parameter estimation methods. The results of the five parameter estimation methods are compared with the results of the comprehensive parameter estimation method within the range of 0.05 to 0.50 to evaluate the performance of the comprehensive parameter estimation method. For the least squares method used to calculate the weights of the comprehensive method with weight coefficients ranging from [-1,1], the deviation of the comprehensive estimation method is significantly smaller than that of the other five parameter estimation methods.

[0064] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A method for unbiased estimation of rainfall probability distribution based on least squares fitting, characterized in that, Includes the following steps: S1. Use multiple single-parameter estimation methods to estimate the shape parameters of the measured samples; The measured samples in step S1 are based on the distribution of an extreme precipitation model. The extreme precipitation model is based on the GEV distribution; Extreme precipitation values ​​are simulated using the GEV distribution; the cumulative distribution function of the time series is calculated using the following formula: Among them, µ, σ and α These represent position parameters, scale parameters, and shape parameters, respectively. Five single-parameter estimation methods were selected: linear moments method, maximum likelihood method, generalized maximum likelihood estimation method, Bayesian method, and maximum product interval method. These methods were used to estimate the shape parameters and obtain shape parameter estimates for different parameter methods. S2. Determine the range of shape parameters based on the union of the shape parameters estimated in S1, and generate random samples by Monte Carlo random sampling from the range of shape parameters. S3. Calculate using a single-parameter estimation method for random samples; S4. For the results of different single-parameter estimation methods, the least squares method is used to determine the optimal weighting coefficient of the single-parameter estimation method, and finally, a comprehensive fitting is performed to estimate the rainfall probability distribution. Step S4 specifically includes the following steps: Step S401: For different single parameter estimation methods, calculate the weight of the comprehensive parameter estimation method according to the least squares method; y j Indicates the first j Each sample value; x ij Represents the estimated shape parameters, where i Indicates the first i One-parameter estimation method j Indicates the first j One data point; n This indicates the number of parameter estimation methods; This represents the sum of squared residuals from the fit; Indicates the length of the observed sample; The calculation model is as follows: Step S402, according to The rainfall probability distribution is calculated by combining the results of different single-parameter estimation methods.

2. The unbiased estimation method for rainfall probability distribution based on least squares fitting according to claim 1, characterized in that, The specific steps of generating random samples in step S2 are as follows: determine the sampling size, that is, determine the number of random sample sequences to be generated; generate a large number of random sample sequences within the selected range of GEV distribution shape parameters; and finally store the generated random sample sequences for use in different single parameter estimation methods.

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