A method for simulating non-gaussian stochastic processes based on randomized eigenvalue decomposition

Abstract

The application discloses a non-Gaussian random process simulation method based on random eigenvalue decomposition, and relates to the technical field of structure engineering. The method first defines a target covariance function and discretizes the target covariance function into a covariance matrix; then a Gaussian random matrix is constructed to perform random projection on the covariance matrix, an orthogonal basis matrix is obtained through QR decomposition, and original matrix eigenvalue decomposition is compressed to a low-dimensional subspace and mapped back to the original space, so that main eigenvalue pairs required by Karhunen-Loeve expansion are efficiently extracted; subsequently, Gaussian random variables are generated, and initial samples are generated by using K-L expansion; the samples are iteratively distributed corrected by an inverse transformation method, so that edge cumulative distribution functions of the samples gradually approach target distributions, and finally, random process simulation samples are output. While the optimal convergence of the K-L expansion is maintained, the method greatly reduces the calculation time and memory requirement, and is suitable for efficient and high-precision simulation of large-scale non-stationary and non-Gaussian random processes.

Description

Technical Field

[0001] This invention relates to the field of structural engineering technology, and in particular to a method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition. Background Technology

[0002] Accurate simulation of non-stationary, non-Gaussian stochastic processes is a crucial foundation for structural dynamic reliability analysis, extreme response prediction, and fatigue damage assessment in fields such as structural engineering, wind engineering, marine engineering, and earthquake engineering. Dynamic excitations widely present in real-world engineering environments, such as pulsating wind speeds, strong earthquake ground motion, extreme wave loads, and random traffic loads, typically exhibit two significant non-classical statistical characteristics: first, non-stationarity, manifested as a significant shift in the process's mean, variance, and power spectral density over time; and second, non-Gaussianity, meaning that the probability density function of its instantaneous amplitude deviates significantly from a normal distribution. How to efficiently and accurately generate stochastic process samples that simultaneously match the time-varying structural characteristics and higher-order probability distribution features of the target system has become a key fundamental issue in numerical simulation research in these engineering fields.

[0003] Spectral Representation Method (SRM) is a classic approach for simulating non-stationary stochastic processes. Its core idea lies in constructing latent Gaussian process samples by superimposing simple harmonic wave components with random phases onto the evolved power spectral density (EPSD), and then achieving a non-Gaussian mapping through probability transformation. However, the engineering applicability of this method is constrained by two inherent problems: First, for a given non-stationary autocovariance function, there can be multiple compatible families of evolved spectral functions, the specific form of which is highly dependent on the selection of the modulation function, lacking a uniqueness criterion; second, the model parameters of the evolved spectrum usually need to be identified from finite-length measured samples. Due to sampling randomness and data length limitations, parameter estimation errors often significantly affect the spectral-preserving accuracy of subsequent simulations. These dual uncertainties pose a severe challenge to the robustness and accuracy of spectral representation in non-stationary, non-Gaussian scenarios.

[0004] The Karhunen-Loève (KL) expansion method provides another effective path to circumvent the difficulties of evolution spectrum estimation. This method does not rely on the spectral density function but directly decomposes the objective covariance function of the stochastic process using an integral operator. By solving the second kind of Fredholm integral equation, it obtains the orthogonal eigenfunctions and corresponding eigenvalues, thus achieving the optimal orthogonal expansion of the stochastic process in the mean-square sense. Compared to spectral representation, the KL expansion not only has explicit convergence optimality but also captures most of the process's energy with fewer expansion terms, resulting in a more direct and stable numerical implementation. However, the real bottleneck in its application lies in the fact that when the simulation time history is long, the sampling step size is dense, or when facing multivariate random field extensions, the covariance matrix formed after discretization becomes extremely large. Performing classical eigenvalue decomposition (EVD) on it will cause the computational complexity to increase cubically, accompanied by extremely high memory consumption, severely limiting the practical deployment efficiency of the KL expansion method in large-scale, high-dimensional engineering problems. Therefore, while fully preserving the theoretical advantages of the KL method, how to develop a fast, low-rank, and high-fidelity eigenvalue decomposition strategy for large covariance matrices has become a core technical bottleneck that urgently needs to be overcome in the current research on efficient simulation of non-Gaussian stochastic processes. Summary of the Invention

[0005] The purpose of this invention is to propose a simulation method for non-Gaussian stochastic processes based on randomized eigenvalue decomposition. By compressing the eigenvalue decomposition of a large-scale covariance matrix into a low-dimensional subspace, this method significantly reduces computation time and memory overhead while accurately simulating non-stationary non-Gaussian stochastic processes, thereby improving its applicability in engineering.

[0006] To achieve the above objectives, this invention proposes a method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition, the specific steps of which are as follows: Step S1: Define the objective covariance function of the stochastic process and the corresponding Fredholm integral equation of the second kind. Use numerical integration to discretize the integral equation to obtain the covariance matrix B. Step S2: Construct a Gaussian random matrix and randomly project it onto the covariance matrix B to construct a random sampling matrix Z. Then, perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B. Step S3: Using the orthogonal basis matrix Q, compress the feature information of matrix B into a low-dimensional submatrix E, calculate the eigenvalues ​​and eigenvectors of matrix E, and map the decomposition results back to the original space to obtain the principal feature pairs required for the KL expansion of the stochastic process. Step S4: Simulate and generate M×N Gaussian random variables with zero mean; Step S5: Generate a stochastic process using the KL expansion.N Given an initial sample, estimate the marginal cumulative distribution function of the current sample, use the inverse transform method to correct the distribution of the initial sample to approximate the target probability distribution, and update the Gaussian random variable based on the corrected sample. Step S6: Repeat step S5 until the empirical distribution function of the simulated sample satisfies the preset convergence condition, and output the final random process simulation sample.

[0007] Preferably, in step S1, the specific steps are as follows: Step S11: Define the objective covariance function of the stochastic process. Construct the corresponding Fredholm integral equation of the second kind, as follows: ; in, To simulate the total duration, For the first i eigenfunction of order, For the first i eigenvalues ​​of order 1 , Let KL be the number of terms in the expansion. t , s Time is the independent variable; Step S12: Discretize both sides of the integral equation and then perform numerical integration to obtain the discrete integral equation, as shown in the following formula: ; in, For time step, For the first k A discrete point in time, For the first l A discrete point in time, The number of discrete points; Step S13: Transform the discrete integral equation into matrix form, as shown in the following formula: ; in, , for The result after discretization L × L A symmetric positive semidefinite covariance matrix, whose elements are: , For B's i indivual L 3D eigenvector, its 3rd eigenvector k The elements are .

[0008] Preferably, in step S2, the specific steps are as follows: Step S21: Construct a standard Gaussian random matrix Ω∈ L×(M+P) ,in, P The oversampling parameters are used; matrix B is subjected to exponentiation and random projection to construct the random sampling matrix Z, as shown in the following formula: ; Where T is the matrix transpose operator; Step S22: Perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B, as shown in the following formula: ; Where, R∈ (M+P)×(M+P) Let Q be an upper triangular matrix, Q∈ L×(M+P) It is an orthogonal basis matrix.

[0009] Preferably, in step S3, the specific steps are as follows: Step S31: Use an orthogonal matrix Q The feature information of matrix B is compressed into a low-dimensional submatrix E, as shown in the following formula: ; Step S32: Decompose the eigenvalues ​​of matrix E, as shown in the following formula: ; Where, U∈ (M+P)×(M+P) , Σ∈ (M+P)×(M+P) Let these represent the eigenvalue matrix and eigenfunction matrix of matrix E, respectively. Step S33: Map the eigenvalue decomposition result of matrix E back to the original space to obtain the truncated eigenvalue matrix of matrix B. The formula for the truncated eigenvector matrix V is as follows: ; Among them, the truncated eigenvalue matrix of matrix B The matrix E is the same, while the truncated eigenfunction matrix V of matrix B is calculated as follows: ; In the KL simulation, retain The former M The diagonal elements and the corresponding V's first... M The column contains the eigenvalues ​​required for the original eigenvalue problem. and characteristic function .

[0010] Preferably, in step S4, the Gaussian random variable The following orthogonality conditions must be met: ; in, j =1,2,…, N ; For mathematical expectation, Let Kronecker function be used.

[0011] Preferably, in step S5, the specific steps are as follows: Step S51: Generate a stochastic process using the KL decomposition method. N For each sample, the formula is as follows: ; in, For sample functions; Step S52: Based on the generated sample function The marginal cumulative distribution is calculated using the following formula: ; in, Let be the empirical distribution function. x is the independent variable of the distribution function; This is an indicator function; it equals 1 if the event occurs, and 0 otherwise. Step S53, due to the empirical distribution function Initially deviating from the target, each sample is transformed to match the target distribution function. The formula is as follows: ; in, The transformed number j One sample function, Let be the target marginal cumulative distribution function of the stochastic process, and for The inverse function of the random variable, and thus the random variable Updated to: ; in, It is a transformed sample Z ( t The overall average; Step S54: For random variables Standardize and orthogonalize the material to meet the orthogonality conditions specified in step S4.

[0012] Preferably, in step S6, step S5 is repeated until the empirical distribution function of the simulated sample is obtained. Satisfy the target distribution function Output the final simulated random process sample: .

[0013] Therefore, this invention proposes a method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition, which has the following advantages: (1) The method of the present invention transforms the eigenvalue decomposition problem of a large covariance matrix into the eigenvalue decomposition problem of a low-dimensional matrix by randomizing eigenvalue decomposition, which significantly reduces computation time and memory requirements while maintaining the accuracy of the main KL mode solution.

[0014] (2) The method of the present invention can enhance the ability of the random subspace to capture the dominant feature space by introducing the power iteration technique, and improve the approximate accuracy of eigenvalues ​​and eigenfunctions. It is especially suitable for large-scale covariance matrices with slow eigenvalue decay.

[0015] (3) The method of the present invention combines KL expansion and iterative distribution matching, which can not only maintain the covariance structure of the target random process, but also accurately reproduce its non-Gaussian marginal distribution characteristics, and is suitable for efficient simulation of non-stationary non-Gaussian random processes.

[0016] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0017] Figure 1 This is a flowchart of a non-Gaussian stochastic process simulation method based on randomized eigenvalue decomposition according to the present invention; Figure 2 This is a comparison chart of the first 650 eigenvalues ​​obtained by the method of the present invention and the direct eigenvalue decomposition method in the embodiments; Figure 3 The diagram shows a comparison of the first and 650th eigenfunctions obtained by the method of the present invention and the direct eigenvalue decomposition method in the embodiment; wherein, (a) is a comparison diagram of the first-order eigenfunction, (b) is a comparison diagram of the 650th-order eigenfunction, and (c) is a comparison diagram of the 650th-order eigenfunction at t∈[24,26] s; Figure 4 The above are comparison charts of the sample covariance function and the target covariance function in the embodiments of the present invention; wherein, (a) is the comparison chart at t=5s, (b) is the comparison chart at t=25s, and (c) is the comparison chart at t=45s. Figure 5 The above are comparison diagrams of the cumulative distribution function of the sample edge and the cumulative distribution function of the target edge in an embodiment of the present invention; wherein, (a) is the comparison diagram at t=5s, (b) is the comparison diagram at t=25s, and (c) is the comparison diagram at t=45s. Detailed Implementation

[0018] To make the technical solutions, advantages, and objectives of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below. The described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0019] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0020] Example 1 This embodiment uses a non-stationary, non-Gaussian random process with an analytical covariance function and a specified marginal distribution function as an example to illustrate the implementation process and performance of the method proposed in this invention. This random process is defined within a time interval t∈[0,50] seconds, and its target covariance function is... and marginal cumulative distribution function They are respectively: ; ; in, and This represents the distribution parameters that change over time, used to ensure that the random process meets the requirements of zero mean and target variance.

[0021] like Figure 1 As shown, this invention provides a method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition, with the following specific steps: Step S1: Define the objective covariance function of the stochastic process and the corresponding Fredholm integral equation of the second kind. Discretize the integral equation numerically using numerical integration methods to obtain the covariance matrix B. The specific steps are as follows: Step S11: Define the objective covariance function of the stochastic process. Construct the corresponding Fredholm integral equation of the second kind, as follows: ; in, To simulate the total duration, For the first i eigenfunction of order, For the first i eigenvalues ​​of order 1 , Let KL be the number of terms in the expansion. t , s Time is the independent variable; In this embodiment, =50s, =650; Step S12: Discretize both sides of the integral equation and then perform numerical integration to obtain the discrete integral equation, as shown in the following formula: ; in, For time step, For the first k A discrete point in time, For the first l A discrete point in time, The number of discrete points; In this embodiment, , The time step Δt is uniformly discretized from the time domain [0,50]s. =0.001s, =50001; Step S13: Transform the discrete integral equation into matrix form, as shown in the following formula: ; in, , for The result after discretization L × L A symmetric positive semidefinite covariance matrix, whose elements are: , For B's i indivual L 3D eigenvector, its 3rd eigenvector k The elements are .

[0022] Step S2: Construct a Gaussian random matrix and randomly project it onto the covariance matrix B to construct a random sampling matrix Z. Then, perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B. The specific steps are as follows: Step S21: Construct a standard Gaussian random matrix Ω∈ L×(M+P) ,in, P The oversampling parameters are used; matrix B is subjected to exponentiation and random projection to construct the random sampling matrix Z, as shown in the following formula: ; Where T is the matrix transpose operator; in this embodiment, P= 1.5 =975; Step S22: Perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B, as shown in the following formula: ; Where, R∈ (M+P)×(M+P) Let Q be an upper triangular matrix, Q∈ L×(M+P) It is an orthogonal basis matrix.

[0023] Step S3: Using the orthogonal basis matrix Q, compress the feature information of matrix B into a low-dimensional submatrix E, calculate the eigenvalues ​​and eigenvectors of matrix E, and map the decomposition result back to the original space to obtain the principal feature pairs required for the KL expansion of the stochastic process. The specific steps are as follows: Step S31: Use an orthogonal matrix Q The feature information of matrix B is compressed into a low-dimensional submatrix E, as shown in the following formula: ; Step S32: Decompose the eigenvalues ​​of matrix E, as shown in the following formula: ; Where, U∈ (M+P)×(M+P) , Σ∈ (M+P)×(M+P) Let these represent the eigenvalue matrix and eigenfunction matrix of matrix E, respectively. Step S33: Map the eigenvalue decomposition result of matrix E back to the original space to obtain the truncated eigenvalue matrix of matrix B. The formula for the truncated eigenvector matrix V is as follows: ; Among them, the truncated eigenvalue matrix of matrix B The matrix E is the same, while the truncated eigenfunction matrix V of matrix B is calculated as follows: ; In the KL simulation, retain The former M The diagonal elements and the corresponding V's first... M The column contains the eigenvalues ​​required for the original eigenvalue problem. and characteristic function .

[0024] Step S4: Simulate and generate M×N Gaussian random variables with a mean of zero; where, the Gaussian random variables... The following orthogonality conditions must be met: ; in, j =1,2,…, N ; For mathematical expectation, This is the Kronecker function. In this embodiment, N=10000.

[0025] Step S5: Generate a stochastic process using the KL expansion. N Given an initial sample, estimate the marginal cumulative distribution function of the current sample. Use the inverse transform method to correct the distribution of the initial sample to approximate the target probability distribution. Update the Gaussian random variable based on the corrected sample. The specific steps are as follows: Step S51: Generate a stochastic process using the KL decomposition method. N For each sample, the formula is as follows: ; in, For sample functions; Step S52: Based on the generated sample function The marginal cumulative distribution is calculated using the following formula: ; in, Let be the empirical distribution function. x is the independent variable of the distribution function; This is an indicator function; it equals 1 if the event occurs, and 0 otherwise. Step S53, due to the empirical distribution function Initially deviating from the target, each sample is transformed to match the target distribution function. The formula is as follows: ; in, The transformed number j One sample function, Let be the target marginal cumulative distribution function of the stochastic process, and for The inverse function of the random variable, and thus the random variable Updated to: ; in, It is a transformed sample Z ( t The overall average; Step S54: For random variables Standardize and orthogonalize the material to meet the orthogonality conditions specified in step S4.

[0026] Step S6: Repeat step S5 until the empirical distribution function of the simulated sample is obtained. Satisfy the target distribution function Output the final simulated random process sample: .

[0027] To verify the accuracy of the method of the present invention in eigenvalue decomposition, the randomized eigenvalue decomposition method used in the present invention is compared with the traditional direct eigenvalue decomposition algorithm, namely the implicit restart Lanczos method.

[0028] like Figures 2-3 As shown, the eigenvalues ​​and eigenfunctions obtained by the method of this invention are in good agreement with the results obtained by the direct eigenvalue decomposition method, indicating that the randomized eigenvalue decomposition method can effectively capture the dominant feature space of the covariance matrix and meet the accuracy requirements for the extraction of the main feature modes in KL expansion.

[0029] Furthermore, to demonstrate the simulation effect of the method of the present invention, statistical analysis was performed on N=10000 samples of non-stationary non-Gaussian random processes obtained from the simulation. For example... Figures 4-5 As shown, the covariance function and marginal cumulative distribution function of the simulated sample both agree well with the target result, indicating that the random process sample generated by the method of the present invention can accurately reflect the second-order correlation structure and non-Gaussian probability distribution characteristics of the target random process. Regarding computational efficiency, in this embodiment, the time step is Δt = 0.001s, and the number of KL expansion terms is M = 650. Under the same computational conditions, the computer time for the direct eigenvalue decomposition method and the randomized eigenvalue decomposition method of the present invention to complete the eigenvalue decomposition is 2054.7s and 418.8s, respectively. It can be concluded that the method of the present invention significantly shortens the computation time for eigenvalue decomposition of large covariance matrices while ensuring the accuracy of eigenvalue and eigenfunction calculations, demonstrating a clear advantage in computational efficiency.

[0030] It is worth noting that all contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.

[0031] Therefore, this invention provides a method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition. By using randomized eigenvalue decomposition, the large-scale covariance matrix is ​​compressed into a low-dimensional subspace for solution, significantly reducing computation time and memory requirements while maintaining the accuracy of the principal modes in KL expansion. The power iteration technique further enhances the ability to capture the dominant eigenspace, making it suitable for large-scale matrices with slow eigenvalue decay. Simultaneously, combined with the iterative distribution matching method, the simulation samples accurately match the target covariance structure and the non-Gaussian marginal distribution, providing a complete solution for efficient and high-precision simulation of non-stationary non-Gaussian stochastic processes, and has significant engineering practical value.

[0032] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition, characterized in that, Includes the following steps: Step S1: Define the objective covariance function of the stochastic process and the corresponding Fredholm integral equation of the second kind. Use numerical integration to discretize the integral equation to obtain the covariance matrix B. Step S2: Construct a Gaussian random matrix and randomly project it onto the covariance matrix B to construct a random sampling matrix Z. Then, perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B. Step S3: Using the orthogonal basis matrix Q, compress the feature information of matrix B into a low-dimensional submatrix E, calculate the eigenvalues ​​and eigenvectors of matrix E, and map the decomposition results back to the original space to obtain the principal feature pairs required for the KL expansion of the stochastic process. Step S4: Simulate and generate M×N Gaussian random variables with zero mean; Step S5: Generate a stochastic process using the KL expansion. N Given an initial sample, estimate the marginal cumulative distribution function of the current sample, use the inverse transform method to correct the distribution of the initial sample to approximate the target probability distribution, and update the Gaussian random variable based on the corrected sample. Step S6: Repeat step S5 until the empirical distribution function of the simulated sample satisfies the preset convergence condition, and output the final random process simulation sample.

2. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 1, characterized in that, In step S1, the specific steps are as follows: Step S11: Define the objective covariance function of the stochastic process. Construct the corresponding Fredholm integral equation of the second kind, as follows: ； in, To simulate the total duration, For the first i eigenfunction of order, For the first i eigenvalues ​​of order 1 , Let KL be the number of terms in the expansion. t , s Time is the independent variable; Step S12: Discretize both sides of the integral equation and then perform numerical integration to obtain the discrete integral equation, as shown in the following formula: ； in, For time step, For the first k A discrete point in time, For the first l A discrete point in time, The number of discrete points; Step S13: Transform the discrete integral equation into matrix form, as shown in the following formula: ； in, , for The result after discretization L × L A symmetric positive semidefinite covariance matrix, whose elements are: , For B's i indivual L 3D eigenvector, its 3rd eigenvector k The elements are .

3. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 2, characterized in that, In step S2, the specific steps are as follows: Step S21: Construct a standard Gaussian random matrix Ω∈ L×(M+P) ,in, P The oversampling parameters are used; matrix B is subjected to exponentiation and random projection to construct the random sampling matrix Z, as shown in the following formula: ； Where T is the matrix transpose operator; Step S22: Perform QR decomposition on the random sampling matrix Z to obtain the orthogonal basis matrix Q corresponding to the principal eigenspace of matrix B, as shown in the following formula: ； Where, R∈ (M+P)×(M+P) Let Q be an upper triangular matrix, Q∈ L×(M+P) It is an orthogonal basis matrix.

4. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 3, characterized in that, In step S3, the specific steps are as follows: Step S31: Use an orthogonal matrix Q The feature information of matrix B is compressed into a low-dimensional submatrix E, as shown in the following formula: ； Step S32: Decompose the eigenvalues ​​of matrix E, as shown in the following formula: ； Where, U∈ (M+P)×(M+P) , Σ∈ (M+P)×(M+P) Let these represent the eigenvalue matrix and eigenfunction matrix of matrix E, respectively. Step S33: Map the eigenvalue decomposition result of matrix E back to the original space to obtain the truncated eigenvalue matrix of matrix B. The formula for the truncated eigenvector matrix V is as follows: ； Among them, the truncated eigenvalue matrix of matrix B The matrix E is the same, while the truncated eigenfunction matrix V of matrix B is calculated as follows: ； In the KL simulation, retain The former M The diagonal elements and the corresponding V's first... M The column contains the eigenvalues ​​required for the original eigenvalue problem. and characteristic function .

5. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 4, characterized in that, In step S4, the Gaussian random variable The following orthogonality conditions must be met: ； in, j =1,2,…, N ; For mathematical expectation, Let Kronecker function be used.

6. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 5, characterized in that, In step S5, the specific steps are as follows: Step S51: Generate a stochastic process using the KL decomposition method. N For each sample, the formula is as follows: ； in, For sample functions; Step S52: Based on the generated sample function The marginal cumulative distribution is calculated using the following formula: ； in, Let be the empirical distribution function. x is the independent variable of the distribution function; This is an indicator function; it equals 1 if the event occurs, and 0 otherwise. Step S53, due to the empirical distribution function Initially deviating from the target, each sample is transformed to match the target distribution function. The formula is as follows: ； in, The transformed number j One sample function, Let be the target marginal cumulative distribution function of the stochastic process, and for The inverse function of the random variable, and thus the random variable Updated to: ； in, It is a transformed sample Z ( t The overall average; Step S54: For random variables Standardize and orthogonalize the material to meet the orthogonality conditions specified in step S4.

7. The method for simulating non-Gaussian stochastic processes based on randomized eigenvalue decomposition according to claim 6, characterized in that, In step S6, step S5 is repeated until the empirical distribution function of the simulated sample is obtained. Satisfy the target distribution function Output the final simulated random process sample: 。

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