Sparse data probability distribution modeling method applied to geotechnical parameter data acquisition

Abstract

The invention relates to the technical field of geotechnical parameter analysis, and discloses a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition, which comprises the following steps of: establishing a sparse data set according to acquired sample data and sample quantity; based on the sparse data set, adopting a Gaussian function as a kernel function, and obtaining an approximate probability density function and an approximate cumulative distribution function through a kernel density estimation algorithm; obtaining an empirical cumulative distribution function of the sparse data set; obtaining a maximum absolute difference through a difference value between the approximate cumulative distribution function and the empirical cumulative distribution function; taking a plurality of random bandwidth parameters as an initial group of chromosomes, carrying out optimization solution on the maximum absolute difference through a genetic algorithm, and obtaining the chromosome corresponding to the maximum fitness as an optimal bandwidth parameter of an approximate probability density function and an approximate cumulative distribution function; and substituting the optimal bandwidth parameter into the approximate probability density function and the approximate cumulative distribution function to obtain a target approximate probability density function and a target approximate cumulative distribution function.

Description

Technical Field

[0001] The present invention relates to the technical field of geotechnical parameter analysis, and in particular to a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition. Background Art

[0002] Geotechnical parameters, including effective cohesion, effective friction angle, and undrained shear strength, are important parameters for assessing the safety and stability of geotechnical structures, such as soil slopes, retaining walls, excavations, and shallow foundations. However, these parameters are usually uncertain due to the combined influence of multiple sources of uncertainty (such as spatial variability, conversion uncertainty, statistical uncertainty, and measurement error), among which the variability of geotechnical parameters is one of the most important issues.

[0003] Since the variability of geotechnical parameters has a significant impact on geotechnical engineering analysis and design, how to correctly describe this variability plays a key role in geotechnical engineering. Data statistics (such as mean, standard deviation and quantile) and probability distribution (such as probability density function and cumulative distribution function) have been widely used to describe the variability of geotechnical parameters. Some classical distributions (such as normal distribution and lognormal distribution) are also often used to fit the data of geotechnical parameters. In this way, the corresponding statistical data and probability distribution of geotechnical parameters can be easily derived.

[0004] In actual geotechnical engineering, due to time, cost and site constraints, it is impossible to investigate a large amount of data, and the data usually comes from field measurements or experimental tests. Sometimes field data must be collected at a specific location, resulting in very sparse and limited data. Geotechnical parameters based on sparse data may not follow the above-mentioned normal or log-normal distribution. Instead, these parameters may follow other commonly used distributions (e.g., Weibull, exponential, Gamma, and extreme value type I distributions, etc.) or irregular distributions, which may exhibit multimodal characteristics. Insufficient data and multimodal characteristics make it challenging to reasonably estimate the probability distribution of geotechnical parameters. Summary of the invention

[0005] In view of this, an embodiment of the present invention provides a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition, so as to solve the problem in the prior art that due to insufficient data and multi-peak characteristics of geotechnical parameters in geotechnical engineering, it is necessary to first perform a complex calculation process on the variability description of the geotechnical parameters before solving the probability distribution model.

[0006] The embodiment of the present invention provides a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition, comprising:

[0007] Establish a sparse data set based on the collected sample data and sample quantity;

[0008] Based on sparse data sets, Gaussian function is used as kernel function, and the approximate probability density function and approximate cumulative distribution function are obtained through kernel density estimation algorithm;

[0009] Get the empirical cumulative distribution function of a sparse data set;

[0010] The maximum absolute difference is obtained by the difference between the approximate cumulative distribution function and the empirical cumulative distribution function;

[0011] Several random bandwidth parameters are used as the initial population of chromosomes, and the maximum absolute difference is optimized and solved by genetic algorithm, and the chromosome corresponding to the maximum fitness is obtained as the optimal bandwidth parameter of the approximate probability density function and the approximate cumulative distribution function;

[0012] Substitute the optimal bandwidth parameter into the approximate probability density function and the approximate cumulative distribution function to obtain the target approximate probability density function and the target approximate cumulative distribution function.

[0013] Optionally, the approximate probability density function is:

[0014] ;

[0015] The approximate cumulative distribution function is:

[0016] ;

[0017] Among them, x i ∈ sparse dataset {x1,x2,…,x n}, n is the number of samples in the sparse data set; h0 is the bandwidth parameter to be optimized; K(*) is the kernel function, x is the input value, represents the approximate probability density function at x; is the standard normal cumulative distribution function.

[0018] Optionally, based on the sparse data set, obtaining an empirical cumulative distribution function of the sparse data set includes:

[0019] ;

[0020] ;

[0021] Among them, x i ∈ sparse dataset {x1,x2,…,x n}; x is the input value; n is the number of samples in the sparse data set.

[0022] Optionally, the maximum absolute difference is obtained by the difference between the approximate cumulative distribution function and the empirical cumulative distribution function, including:

[0023] ;

[0024] Where D max is the objective function to be optimized.

[0025] Optionally, several random bandwidth parameters are used as the initial population of chromosomes, and the maximum absolute difference is optimized and solved by a genetic algorithm to obtain the chromosome corresponding to the maximum fitness as the optimal bandwidth parameter of the approximate probability density function and the approximate cumulative distribution function, including:

[0026] Randomly generate N chromosomes and obtain the initial population {h1,h2,…,h N};

[0027] Take the inverse of the objective function to be optimized and get the fitness function:

[0028] ;

[0029] The fitness function is used to calculate the initial population {h1,h2,…,h N The fitness value of {f1,f2,…,f N}.

[0030] Optionally, it also includes:

[0031] Select M chromosomes from the initial population, and the probability of the jth chromosome being selected is:

[0032] ;

[0033] The remaining NM chromosomes are generated through crossover operation, with the crossover probability set to c1. The chromosomes obtained through crossover operation are mutated, with the mutation probability set to c2, to obtain a new generation of N chromosomes.

[0034] After a preset number of iterations, the current optimal solution is output and the optimal bandwidth parameters are obtained.

[0035] Optionally, the crossover probability c1 is set to 0.5; the mutation probability c2 is set to 0.5; and the preset number of iterations is set to 100.

[0036] Optionally, it also includes:

[0037] Before reaching the preset number of iterations, if the fitness function output result has reached convergence, the iteration is stopped.

[0038] Beneficial effects of the present invention:

[0039] 1. This embodiment provides a sparse data probability distribution modeling method for geotechnical parameter data acquisition, which transforms the bandwidth parameter selection problem of kernel density estimation into an optimization problem. The goal is to minimize the maximum absolute difference between the approximate cumulative distribution function and the empirical cumulative distribution function, and effectively improve the modeling accuracy of the probability distribution by adaptively solving the optimal bandwidth.

[0040] 2. This embodiment provides a sparse data probability distribution modeling method for geotechnical parameter data acquisition, which does not require subjective assumptions about bandwidth and parameter distribution type and has higher accuracy than traditional modeling methods.

[0041] 3. The sparse data probability distribution modeling method for geotechnical parameter data acquisition provided in this embodiment can reasonably estimate the probability distribution under insufficient data and multi-peak characteristics, and has higher applicability than traditional methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The features and advantages of the present invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the present invention in any way. In the accompanying drawings:

[0043] Figure 1 A flow chart of a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition in an embodiment of the present invention is shown;

[0044] Figure 2 A schematic diagram showing the absolute difference between an approximate cumulative distribution function and an empirical cumulative distribution function in an embodiment of the present invention is shown;

[0045] Figure 3 An optimization iterative process diagram of the absolute difference between an approximate cumulative distribution function and an empirical cumulative distribution function in an embodiment of the present invention is shown;

[0046] Figure 4 A probability density function comparison diagram of a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition in an embodiment of the present invention and other methods is shown;

[0047] Figure 5 A comparison diagram of cumulative distribution functions of a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition in an embodiment of the present invention and other methods is shown;

[0048] Figure 6 An optimization iterative process diagram of the absolute difference between an approximate cumulative distribution function and an empirical cumulative distribution function in an embodiment of the present invention is shown;

[0049] Figure 7A probability density function comparison diagram of a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition in an embodiment of the present invention and other methods is shown;

[0050] Figure 8 A comparison diagram of the cumulative distribution functions of a sparse data probability distribution modeling method applied to geotechnical parameter data acquisition in an embodiment of the present invention and other methods is shown. DETAILED DESCRIPTION

[0051] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work are within the scope of protection of the present invention.

[0052] The embodiment of the present invention provides a sparse data probability distribution modeling method for geotechnical parameter data acquisition, such as Figure 1 As shown, including:

[0053] Step S10, establishing a sparse data set according to the collected sample data and the number of samples.

[0054] In this embodiment, the sample data set {x1, x2, ..., x n} is a sparse data set, the data is obtained through field measurement or experimental testing, and the sample size is n.

[0055] Step S20, based on the sparse data set, using a Gaussian function as a kernel function, and obtaining an approximate probability density function and an approximate cumulative distribution function through a kernel density estimation algorithm.

[0056] In this embodiment, the approximate probability density function is:

[0057] .

[0058] .

[0059] Among them, x i ∈ sparse dataset {x1,x2,…,x n}; h0 is the bandwidth parameter to be optimized, which is an unknown value; K(*) is the kernel function, x is the input value, represents the approximate probability density function at x; is the standard normal cumulative distribution function.

[0060] The approximate probability density function can be rewritten as:

[0061] ;

[0062] In the formula, is the standard normal probability density function.

[0063] The approximate probability density function can be rewritten as:

[0064] .

[0065] In the formula, .

[0066] Step S30, obtaining the empirical cumulative distribution function of the sparse data set.

[0067] In this embodiment, the empirical cumulative distribution function of the sparse data set is:

[0068] .

[0069] .

[0070] Among them, x i ∈ sparse dataset {x1,x2,…,x n}; x is the input value; n is the number of samples in the sparse data set. From the above formula, we can see that T(0)=0, T(x n )=1.

[0071] Step S40: obtaining the maximum absolute difference through the difference between the approximate cumulative distribution function and the empirical cumulative distribution function.

[0072] In this embodiment, the maximum absolute difference The choice of bandwidth parameter is a key parameter affecting the maximum absolute difference and an important factor affecting the accuracy of kernel density estimation. Therefore, the maximum absolute difference D max As the objective function, it is optimized and solved:

[0073] .

[0074] In a specific embodiment, for a given point x0, the absolute difference between the approximate cumulative distribution function and the empirical cumulative distribution function is like Figure 2 shown.

[0075] Step S50, taking several random bandwidth parameters as the initial population of chromosomes, optimizing and solving the maximum absolute difference through a genetic algorithm, and obtaining the chromosome corresponding to the maximum fitness as the optimal bandwidth parameter of the approximate probability density function and the approximate cumulative distribution function.

[0076] In this embodiment, the genetic algorithm optimization solution process is as follows:

[0077] Randomly generate N chromosomes and obtain the initial population {h1,h2,…,h N}.

[0078] Since the genetic algorithm is to find the optimal solution with the maximum fitness value, the inverse of the objective function to be optimized is taken to obtain the fitness function:

[0079] .

[0080] The fitness function is used to calculate the initial population {h1,h2,…,h N The fitness value of {f1,f2,…,f N}.

[0081] Select M chromosomes from the initial population, and the probability of the jth chromosome being selected is:

[0082] .

[0083] Fitness value f j The larger the jth chromosome h j That is to say, as the algorithm proceeds, chromosomes with small fitness values ​​will be eliminated, and the overall quality of chromosomes will be improved.

[0084] The remaining NM chromosomes are generated by crossover operation, the crossover probability is set to c1, and the chromosomes obtained by the crossover operation are mutated, the mutation probability is set to c2, and a new generation of N chromosomes is obtained. In a specific embodiment, the crossover probability c1 is set to 0.5; the mutation probability c2 is set to 0.5.

[0085] After a preset number of iterations, the current optimal solution is outputted to obtain the optimal bandwidth parameter. In a specific embodiment, the preset number of iterations is set to 100 times.

[0086] In a specific embodiment, if the output result of the fitness function has reached convergence, the iteration is stopped. In a specific implementation, when the calculation result tends to be stable, it is judged that convergence has been reached.

[0087] Step S60: Substitute the optimal bandwidth parameter into the approximate probability density function and the approximate cumulative distribution function to obtain a target approximate probability density function and a target approximate cumulative distribution function.

[0088] The present embodiment provides a sparse data probability distribution modeling method for geotechnical parameter data acquisition, which transforms the bandwidth parameter selection problem of kernel density estimation into an optimization problem, with the goal of minimizing the maximum absolute difference between the approximate cumulative distribution function and the empirical cumulative distribution function, and effectively improves the modeling accuracy of the probability distribution by adaptively solving the optimal bandwidth. There is no need to make subjective assumptions about the bandwidth and parameter distribution type, and it has higher accuracy than traditional modeling methods. In addition, the present embodiment provides a sparse data probability distribution modeling method for geotechnical parameter data acquisition, which can reasonably estimate the probability distribution under insufficient data and multi-peak characteristics, and has higher applicability than traditional methods.

[0089] Table 1 Data of effective cohesion

[0090] ;

[0091] Taking "Efficient sampling of the irregular probability distributions of geotechnical parameters for reliability analysis" (Structural Safety Volume 101, March 2023, 102309) and "Site-specific probability distribution of geotechnical properties" (Computers and Geotechnics Volume 70, October 2015, Pages 159-168) as references, the histogram and the methods proposed by Wang et al. and Jiang et al. were compared and analyzed with the method given in the present invention.

[0092] Taking the effective cohesion of 63 real data as an example, the data is shown in Table 1. The sparse data probability distribution modeling method provided by this embodiment is used to solve the optimization iteration process using the genetic algorithm. Figure 3 As shown, the optimal bandwidth parameter can be effectively found, and finally the probability distribution of effective cohesion is obtained.

[0093] The comparison results of probability density functions of different methods are as follows: Figure 4 In addition, in order to fully compare and illustrate, Figure 5 As shown in Figure 2, the cumulative distribution function comparison results of each method are also given. Figure 4 and Figure 5 It can be seen that the results of the method given in this embodiment are very close to those of the methods developed by Wang et al. and Jiang et al., which verifies the effectiveness of the proposed method.

[0094] Depend on Figure 3 It can be seen that the iterative process converges very quickly. After 10 iterations, the calculation results tend to be stable. The results of the first 12 iterations are detailed in Table 2.

[0095] Table 2 Results of the first 12 iterations based on the dataset in Table 1

[0096] ;

[0097] The optimal bandwidth is determined to be h = 0.8369. Substituting n = 63 and h = 0.8369 into the approximate probability density function and the approximate cumulative distribution function in step S20 of this embodiment, we obtain:

[0098] .

[0099] .

[0100] The scarce data of 10 effective friction angles in this embodiment come from the target sand layer on the shore of Lake Mildred, and the specific values ​​are shown in Table 3.

[0101] Table 3 Data of effective friction angle

[0102] ;

[0103] Through the sparse data probability distribution modeling method provided in this embodiment, the optimization iteration process is as follows: Figure 6 And as shown in Table 4.

[0104] Table 4 Results of the first 35 iterations based on the dataset in Table 3

[0105] ;

[0106] Depend on Figure 6 As shown in Table 4, the iterative process converges after 33 iterations, and the optimal bandwidth is determined to be h = 0.3345. Substituting n = 10 and h = 0.3345 into the approximate probability density function and the approximate cumulative distribution function in step S20 of this embodiment, the corresponding approximate probability density function and the approximate cumulative distribution function can be obtained.

[0107] The comparison results of probability density function and cumulative distribution function of different methods are as follows: Figure 7 and Figure 8 It can be seen that the sparse data probability distribution modeling method for geotechnical parameter data acquisition proposed in this embodiment can effectively approximate the probability distribution of sparse data.

[0108] Although the embodiments of the present invention have been described in conjunction with the accompanying drawings, those skilled in the art may make various modifications and variations without departing from the spirit and scope of the present invention, and such modifications and variations are all within the scope defined by the appended claims.

Claims

1. A sparse data probability distribution modeling method for geotechnical parameter data acquisition, characterized in that: include: Establish a sparse data set based on the collected sample data and sample quantity; Based on the sparse data set, a Gaussian function is used as a kernel function, and an approximate probability density function and an approximate cumulative distribution function are obtained by a kernel density estimation algorithm; Obtaining an empirical cumulative distribution function of the sparse data set; Obtaining a maximum absolute difference through a difference between the approximate cumulative distribution function and the empirical cumulative distribution function; Using several random bandwidth parameters as the initial population of chromosomes, optimizing and solving the maximum absolute difference through a genetic algorithm, and obtaining the chromosome corresponding to the maximum fitness as the optimal bandwidth parameter of the approximate probability density function and the approximate cumulative distribution function; Substituting the optimal bandwidth parameter into the approximate probability density function and the approximate cumulative distribution function, a target approximate probability density function and a target approximate cumulative distribution function are obtained.

2. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 1 is characterized in that: The approximate probability density function is: ； The approximate cumulative distribution function is: ； Among them, x i ∈ sparse dataset {x1,x2,…,x n }, n is the number of samples in the sparse data set; h0 is the bandwidth parameter to be optimized; K(*) is the kernel function, x is the input value, represents the approximate probability density function at x; is the standard normal cumulative distribution function.

3. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 2 is characterized in that: Based on the sparse data set, obtaining an empirical cumulative distribution function of the sparse data set includes: ； ； Among them, x i ∈ sparse dataset {x1,x2,…,x n }; x is the input value; n is the number of samples in the sparse data set.

4. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 3 is characterized in that: Obtaining a maximum absolute difference through a difference between the approximate cumulative distribution function and the empirical cumulative distribution function, comprising: ； Where D max is the objective function to be optimized.

5. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 4 is characterized in that: Using several random bandwidth parameters as the initial population of chromosomes, optimizing and solving the maximum absolute difference through a genetic algorithm, and obtaining the chromosome corresponding to the maximum fitness as the optimal bandwidth parameter of the approximate probability density function and the approximate cumulative distribution function, including: Randomly generate N chromosomes to obtain the initial population {h1,h2,…,h N }; The inverse of the objective function to be optimized is taken to obtain the fitness function: ； The fitness function is used to calculate the initial population {h1,h2,…,h N The fitness value of {f1,f2,…,f N }.

6. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 5 is characterized in that: Also includes: M chromosomes are selected from the initial population, and the probability of the jth chromosome being selected is: ； The remaining NM chromosomes are generated through crossover operation, with the crossover probability set to c1. The chromosomes obtained through crossover operation are mutated, with the mutation probability set to c2, to obtain a new generation of N chromosomes. After a preset number of iterations, the current optimal solution is output and the optimal bandwidth parameter is obtained.

7. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 6 is characterized in that: The crossover probability c1 is set to 0.5; the mutation probability c2 is set to 0.5; and the preset number of iterations is set to 100.

8. The sparse data probability distribution modeling method for geotechnical parameter data acquisition according to claim 6 is characterized in that: Also includes: Before reaching the preset number of iterations, if the output result of the fitness function has reached convergence, the iteration is stopped.

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