Multi-point non-stationary non-Gaussian random process efficient simulation method based on spectral representation method

Abstract

The invention discloses a multi-point non-stationary non-Gaussian random process efficient simulation method based on a spectral representation method, and belongs to the technical field of random process simulation. Aiming at the problem of low efficiency of existing potential Gaussian random process power spectrum matrix solving and matrix decomposition, the method comprises the following steps: determining a power spectrum matrix and non-Gaussian features of a target process, establishing a transfer equation by using a Mehler formula, calculating time and frequency interpolation points based on the amplitude of the power spectrum matrix, and performing matrix decomposition and orthogonal decoupling at the interpolation points. And generating a potential Gaussian process sample through interpolation and fast Fourier transform, and finally converting to obtain a target process sample. According to the method, time-frequency domain resource allocation is optimized through a self-adaptive interpolation strategy, so that interpolation points are concentrated in an area with high energy, redundancy decomposition is avoided, and the calculation efficiency is remarkably improved on the premise of ensuring the simulation precision.

Description

Technical Field

[0001] The present invention relates to the technical field of random process simulation, in particular to a multi-point non-stationary non-Gaussian random process efficient simulation method based on spectral representation. Background Art

[0002] In engineering, accurate simulation of multi-point non-stationary non-Gaussian random processes is crucial for structural seismic and wind-resistant design. Existing simulation methods based on spectral representation usually rely on the construction of a latent Gaussian random process, but they suffer from significant efficiency bottlenecks. On the one hand, solving the power spectrum matrix of the latent Gaussian random process requires iterative spectral updates or solving the nonlinear equation of the Mehler formula at each time point and frequency point. The computational complexity grows exponentially with the number of discrete points, making it extremely inefficient. On the other hand, although the number of decompositions can be reduced by interpolation points during power spectrum matrix decomposition, the existing interpolation point distribution must be manually determined in advance, and it is impossible to adaptively adjust to the time-frequency energy distribution characteristics of the latent Gaussian process. This leads to redundant decomposition in energy-sparse regions and insufficient accuracy in key areas. In addition, traditional methods use uniform discretization in the time-frequency domain and do not consider the non-uniform distribution of non-stationary process energy. This results in wasted computational resources in energy-sparse regions and insufficient accuracy in concentrated areas, creating a dual contradiction of "low efficiency" and "insufficient accuracy", making it difficult to meet the needs of efficient and high-precision simulation in engineering practice. Summary of the Invention

[0003] In response to the problem of low efficiency in solving the power spectrum matrix and matrix decomposition of potential Gaussian random processes in the existing technology, the present invention provides an efficient simulation method for multi-point non-stationary non-Gaussian random processes based on spectral representation. It optimizes the allocation of time-frequency domain computing resources through adaptive interpolation strategy, and significantly improves computing efficiency while ensuring simulation accuracy.

[0004] To achieve the above objectives, this application provides the following solutions:

[0005] An efficient simulation method for multi-point non-stationary non-Gaussian random processes based on spectral representation includes the following steps:

[0006] Determine the target multi-point non-stationary non-Gaussian vector random process Y(t)={Y1(t),Y2(t),…,Y n (t)} power spectrum matrix and non-Gaussian probability characteristics;

[0007] Calculate Y j (t) and Y k The time-varying correlation coefficient ρ of (t) NG,j,k (t,s);

[0008] Establish ρ through Mehler formula NG,j,k (t,s) and the corresponding latent Gaussian process correlation coefficient ρ j,kThe transfer equation of (t,τ);

[0009] Based on the transfer equation, the time-varying correlation coefficient ρ of the potential Gaussian random process is calculated through the discrete data mapping relationship j,k (t,τ) and its power spectrum matrix;

[0010] Based on the amplitude information of the power spectrum matrix of the potential Gaussian random process, the time interpolation points and the frequency interpolation points are calculated;

[0011] Perform matrix decomposition on the potential Gaussian random process at the time interpolation point and the frequency interpolation point to obtain a decomposition matrix;

[0012] Perform orthogonal decoupling on the decomposition matrix to obtain time parameters and frequency parameters;

[0013] Obtain the decoupling parameters of all discrete points through interpolation;

[0014] Fast Fourier transform is used to generate samples of the latent Gaussian random process;

[0015] The target non-stationary non-Gaussian vector random process Y is obtained by nonlinear transfer function transformation j The corresponding time-course samples of (t).

[0016] Optionally, the calculation Y j (t) and Y k The time-varying correlation coefficient ρ of (t) NG,j,k (t,s), which is calculated using the following formula:

[0017]

[0018] in, and Y j (t) and Y k The time-varying variance of (t); S NG,j,k (t,ω) is the power spectrum matrix S NG The element in the jth row and kth column of (t,ω).

[0019] Optionally, the Mehler formula is used to establish ρ NG,j,k (t,s) and the corresponding latent Gaussian process correlation coefficient ρ j,k The transfer equation of (t,τ) is:

[0020]

[0021] Where Q is the cutoff number of the Mehler formula, usually Q = 8;

[0022] y GH,l (t) and w GH,l(t) is the Gauss–Hermite integration point and weight, and the number of integration points d is usually 11;

[0023] H m (·) is a Hermite polynomial of order m, satisfying:

[0024]

[0025] Optionally, the time-varying correlation coefficient ρ of the potential Gaussian random process is calculated based on the transfer equation through the discrete data mapping relationship j,k (t,τ) and its power spectrum matrix, including:

[0026] The time-varying correlation coefficient ρ of the underlying Gaussian random process j,k The domain of (t,τ) [-1,1] is discretized into s points with equal spacing:

[0027] Will Substituting the discrete points in into the transfer equation, we can get the corresponding discrete set of non-Gaussian correlation coefficients:

[0028] Construct ρ based on discrete sets j,k (t,τ) and ρ NG,j,k The mapping function of (t,τ);

[0029] Backcalculate the target ρ by interpolation NG,j,k The corresponding ρ in (t,τ) j,k (t,τ).

[0030] Optionally, the calculating of the time interpolation points and the frequency interpolation points based on the amplitude information of the power spectrum matrix of the potential Gaussian random process includes:

[0031] For any discrete time point t p and frequency discrete points ω l , calculate the variance at that point and the proportion of the variance in the entire time domain or frequency domain respectively;

[0032] Determine the moment t at which the variance in the time domain is the largest max The frequency ω with the largest variance in the sum frequency domain max ;

[0033] t max The time interval is expanded symmetrically around the center until the variance ratio in the interval exceeds the first preset threshold, and the core time interval [t c ,t d ]; Similarly, with ω max The frequency interval is expanded symmetrically around the center until the variance ratio in the interval reaches a second preset threshold value or more, and the core frequency interval [ω c,ω d ];

[0034] Divide the entire time domain into core time intervals [t c ,t d ] and the edge time interval [t b ,t c ]∪[t d ,t u ], and divide the full frequency domain into core frequency intervals [ω c ,ω d ] and the edge frequency interval [ω b ,ω c ]∪[ω d ,ω u ];

[0035] Calculate the time interpolation step of the core time interval and the edge time interval, as well as the frequency interpolation step of the core frequency interval and the edge frequency interval respectively;

[0036] Generate time interpolation points according to the time interpolation step And generate frequency interpolation points according to the frequency interpolation step

[0037] Optionally, the matrix decomposition of the potential Gaussian random process at the time interpolation point and the frequency interpolation point is specifically:

[0038] By formula Decomposition, get the decomposition matrix

[0039] Optionally, the orthogonal decoupling of the decomposition matrix to obtain the time parameter and the frequency parameter is specifically:

[0040] By formula Decomposition matrix Decoupling, where:

[0041] is the time parameter, is the frequency parameter, and r is the cutoff order.

[0042] Optionally, the decoupling parameters of all discrete points are obtained by interpolation, specifically:

[0043] Based on time parameters and frequency parameters Interpolate all discrete time points and frequency discrete points to obtain the time parameters at discrete time points and frequency parameters

[0044] Optionally, the generating of the latent Gaussian random process sample by using fast Fourier transform is specifically as follows:

[0045] By formula Perform FFT operation to generate sample u j (t), where:

[0046] i is an imaginary number, i 2 =-1; Δω is the frequency interval; Re{·} represents the real number in the brackets; is a random phase angle uniformly distributed in [0,2π].

[0047] Optionally, the target non-stationary non-Gaussian vector random process Y is obtained by nonlinear transfer function conversion j The corresponding time-course samples of (t) are:

[0048] By formula Perform the conversion, where:

[0049] F j The inverse function of (·,t), Φ j (·,t) is u j Cumulative distribution function of (t).

[0050] Through the above technical solutions, the beneficial effects of the present invention are as follows: This application optimizes the allocation of computing resources in the time-frequency domain through an adaptive interpolation strategy, significantly improving computing efficiency while ensuring simulation accuracy. Specifically, based on the amplitude information of the power spectrum matrix of the potential Gaussian random process, the core time-frequency interval is determined and the interpolation step size is dynamically adjusted, so that the interpolation points are concentrated in the high-energy area, avoiding redundant decomposition in the energy-sparse area, and solving the dual contradictions of "low efficiency" and "insufficient accuracy" in the traditional method. Experiments show that the time-varying statistical moments of the non-stationary non-Gaussian random process generated by this method are highly consistent with the target value, the probability density function is well matched, and the simulation time is greatly shortened, which fully verifies its advantages of efficient and high-precision simulation in engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention, and the embodiments in the drawings do not constitute any limitation to the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0052] Figure 1 A schematic diagram of a flow chart of an efficient simulation method for a multi-point non-stationary non-Gaussian random process based on spectral representation provided in one embodiment of the present application;

[0053] Figure 2A functional relationship diagram of the time-varying correlation coefficient and the corresponding potential Gaussian process correlation coefficient provided in an embodiment of the present application;

[0054] Figure 3 A scatter plot of the time-varying statistical moments of Y1(t) provided in one embodiment of the present application;

[0055] Figure 4 A time-varying power spectrum diagram of a potential Gaussian random process provided in an embodiment of the present application;

[0056] Figure 5 A time-frequency interpolation point distribution diagram provided in an embodiment of the present application;

[0057] Figure 6 A probability density function curve diagram at different times provided in an embodiment of the present application;

[0058] Figure 7 A schematic diagram of the structure of a computer device provided in one embodiment of the present application;

[0059] The realization of the objectives, functional features and advantages of this application will be further explained in conjunction with embodiments and with reference to the accompanying drawings. DETAILED DESCRIPTION

[0060] It should be understood that the specific embodiments described herein are merely for explaining the present application and are not intended to limit the present application. Rather, these embodiments are provided to make the present disclosure more thorough and complete and to fully convey the scope of the present disclosure to those skilled in the art.

[0061] The above and other technical contents, features and effects of the present invention are described below with reference to the attached Figure 1-7 The detailed description of the embodiments will clearly show that the structural contents mentioned in the following embodiments are all based on the accompanying drawings.

[0062] Various exemplary embodiments of the present invention will be described below with reference to the accompanying drawings.

[0063] In an exemplary embodiment, Figure 1 As shown, a method for efficiently simulating multi-point non-stationary non-Gaussian random processes based on spectral representation is provided. In an embodiment of the present invention, the method includes the following steps A to J. Among them:

[0064] Step A, determine the target multi-point non-stationary non-Gaussian vector random process Y(t) = {Y1(t), Y2(t), ..., Y n (t)} power spectrum matrix and non-Gaussian probability characteristics.

[0065] Step B, calculate Y j (t) and Y k The time-varying correlation coefficient ρ of (t)NG,j,k (t,s), which is calculated using the following formula:

[0066]

[0067]

[0068] in, and Y j (t) and Y k The time-varying variance of (t); S NG,j,k (t,ω) is the power spectrum matrix S NG The element in the jth row and kth column of (t,ω).

[0069] Step C, establish ρ by Mehler formula NG,j,k (t,s) and the corresponding latent Gaussian process correlation coefficient ρ j,k The transfer equation of (t,τ) is:

[0070]

[0071] Where Q is the cutoff number of the Mehler formula, usually Q = 8;

[0072] y GH,l (t) and w GH,l (t) is the Gauss–Hermite integration point and weight, and the number of integration points d is usually 11;

[0073] H m (·) is a Hermite polynomial of order m, satisfying:

[0074]

[0075] Step D, based on the transfer equation, calculate the time-varying correlation coefficient ρ of the potential Gaussian random process through the discrete data mapping relationship j,k (t,τ) and its power spectrum matrix, including:

[0076] D1. The time-varying correlation coefficient ρ of the potential Gaussian random process j,k The domain of (t,τ) [-1,1] is discretized into s points with equal spacing:

[0077] D2. Substituting the discrete points in into the transfer equation, we can get the corresponding discrete set of non-Gaussian correlation coefficients:

[0078] D3. Construct ρ based on discrete sets j,k (t,τ) and ρ NG,j,kThe mapping function of (t,τ);

[0079] D4. Inversely calculate the target ρ by interpolation NG,j,k The corresponding ρ in (t,τ) j,k (t,τ).

[0080] In a specific embodiment, the interval length of the domain [-1, 1] defined in step D1 is 2. When it is divided into s points with equal spacing, the interval between adjacent points is:

[0081]

[0082] Will Substituting the discrete values ​​in into the equation in formula (4), we can get the corresponding values Based on ρ s G,j,k and ρ s NG,j,k The value of ρ can be established j,k (t,τ) and ρ NG,j,k The functional relationship between (t,τ), such as Figure 2 As shown. Figure 2 The functional relationship constructed in NG,j,k The corresponding ρ in (t,τ) j,k (t,τ) can be quickly calculated by interpolation method.

[0083] Step E, calculating the time interpolation points and the frequency interpolation points based on the amplitude information of the power spectrum matrix of the potential Gaussian random process, includes:

[0084] E1. For any discrete time point t p and frequency discrete points ω l , calculate the variance at that point and the proportion of the variance in the entire time domain or frequency domain respectively;

[0085] E2. Determine the moment t when the variance in the time domain accounts for the largest proportion max The frequency ω with the largest variance in the sum frequency domain max ;

[0086] E3, with t max The time interval is expanded symmetrically around the center until the variance ratio in the interval exceeds the first preset threshold, and the core time interval [t c ,t d ]; Similarly, with ω max The frequency interval is expanded symmetrically around the center until the variance ratio in the interval reaches a second preset threshold value or more, and the core frequency interval [ω c ,ω d ];

[0087] E4. Divide the entire time domain into core time intervals [t c ,t d ] and the edge time interval [t b ,t c ]∪[t d ,t u ], and divide the full frequency domain into core frequency intervals [ω c ,ω d ] and the edge frequency interval [ω b ,ω c ]∪[ω d ,ω u ];

[0088] E5. Calculate the time interpolation step size of the core time interval and the edge time interval, as well as the frequency interpolation step size of the core frequency interval and the edge frequency interval respectively;

[0089] E6. Generate time interpolation points based on time interpolation step And generate frequency interpolation points according to the frequency interpolation step

[0090] In a specific embodiment, for any discrete time point t p , and its corresponding variance is:

[0091]

[0092] Then t p The ratio of the variance at one moment to the variance of all moments is for:

[0093]

[0094] in,

[0095] By using formula (7), we can determine the moment when the prescription difference accounts for the largest proportion among all discrete time points, which is recorded as t max .

[0096] According to the following algorithm 1, the core time interval [t c ,t d ], ensuring that the power spectrum of the underlying Gaussian random process in this area accounts for more than a first preset threshold. In this embodiment, both the first and second preset thresholds are set to 95%. The specific description of Algorithm 1 is as follows:

[0097]

[0098] For any frequency discrete point ω l , and its corresponding variance is:

[0099]

[0100] For ω l The proportion of variance of all frequency discrete points Calculated by the following formula:

[0101]

[0102] The core frequency range [ω c ,ω d ], ensuring that the variance in this area accounts for more than 95% of the total variance.

[0103] For the time interval [t c ,t d ], the total time T is divided into two parts TT1 and TT2, where TT1 = [t c ,t d ], TT2={[t b ,t c ]∪[t d ,t u For TT1 and TT2, their corresponding time interpolation steps are recorded as Δt1 and Δt2. The steps for determining Δt1 and Δt2 are the same, so this embodiment mainly takes Δt1 as an example for explanation.

[0104] For interval TT1, at frequency ω max The variance m of the time interpolation step Δt k,TT1 for:

[0105]

[0106] Similarly, at frequency ω max The variance m of the time interpolation step Δt1 k,AT for:

[0107]

[0108] The convergence criterion is determined as:

[0109]

[0110] Where, ε1=0.01.

[0111] Based on the above convergence criteria, the time interpolation step Δt1 can be determined using the following Algorithm 2. The specific description of Algorithm 2 is as follows:

[0112]

[0113]

[0114] Using the above algorithm 2, the time interpolation step Δt2 can also be determined.

[0115] For the frequency region [ω c ,ω d ], the total frequency W is divided into two parts WW1 and WW2, where WW1 = [ω c ,ω d ],WW2={[ω b ,ω c ]∪[ω d ,ω u For WW1 and WW2, their corresponding frequency interpolation steps are recorded as Δω1 and Δω2. The steps for determining Δω1 and Δω2 are the same, so the present invention mainly takes Δω1 as an example for explanation.

[0116] For interval WW1, at time t max The variance m of the frequency interval Δω k,WW1 for:

[0117]

[0118] Similarly, at time t max The variance m of the frequency interval Δω1 kAW for:

[0119]

[0120] The convergence criterion is determined as:

[0121]

[0122] Where, ε2 = 0.01.

[0123] Accordingly, based on Algorithm 2, the specific value of Δω1 can be determined by using the convergence criterion of Equation (15). Furthermore, the value of Δω2 can also be determined.

[0124] According to TT1, TT2, WW1, WW2, Δt1, Δt2, Δω1 and Δω2, the corresponding time and frequency interpolation points are:

[0125]

[0126] Where, M1=(t c -t b ) / Δt2,M2=(t d -t c ) / Δt1,M3=(t u -t d ) / Δt2,M1+M2+M3 <M。

[0127]

[0128] Where, N1=(ω c -ω b ) / Δω2,N2=(ω d -ω c ) / Δω1,N3=(ω u -ω d ) / Δω2,N1+N2+N3 <N。

[0129] Step F, at the time interpolation point and frequency interpolation points The matrix decomposition of the potential Gaussian random process is performed at:

[0130] By formula Decomposition, get the decomposition matrix

[0131] Step G, performing orthogonal decoupling on the decomposition matrix to obtain time parameters and frequency parameters, specifically:

[0132] By formula Decomposition matrix Decoupling, where:

[0133] is the time parameter, is the frequency parameter, r is the truncation order, and usually 6 is selected to obtain satisfactory accuracy.

[0134] Step H, obtain the decoupling parameters of all discrete points by interpolation, specifically:

[0135] Based on time parameters and frequency parameters Interpolate all discrete time points and frequency discrete points to obtain the time parameters at discrete time points and frequency parameters

[0136] Step I: Generate a latent Gaussian random process sample using fast Fourier transform, specifically:

[0137] By formula Perform FFT operation to generate sample u j (t), where:

[0138] i is an imaginary number, i 2 =-1, Δω is the frequency interval, Re{·} represents the real number in the brackets, is a random phase angle uniformly distributed in [0,2π].

[0139] Step J, obtain the target non-stationary non-Gaussian vector random process Y through nonlinear transfer function conversion jThe corresponding time-course samples of (t) are:

[0140] By formula j∈(1,2,...,n) to perform the transformation, where:

[0141] F j The inverse function of (·,t), Φ j (·,t) is u j Cumulative distribution function of (t).

[0142] This example uses the simulation of a three-point non-stationary non-Gaussian random process as an example to elaborate on the specific execution flow, parameter setting, and verification process of the efficient simulation method based on spectral representation:

[0143] In this embodiment, the non-Gaussian probability characteristic of the three-point non-stationary non-Gaussian random vector process Y(t) = {Y1(t), Y2(t), Y3(t)} is a log-normal distribution, and its corresponding time-varying statistical moment is as follows: Figure 3 As shown, Figure 3 (a) is the time-varying mean m 1,1 ; (b) is the time-varying second-order statistical moment m 1,2 ; (c) is the time-varying third-order statistical moment m 1,3 ; (d) is the time-varying fourth-order statistical moment m 14 .

[0144] Y i (t) and Y j The cross-evolution power spectrum of (t) is expressed as:

[0145]

[0146] Among them, A l (t,ω) is a non-uniform modulation function and is expressed as

[0147]

[0148] Where a = 0.55s -1 , b=a+0.001s -1 , c = 0.005;

[0149]

[0150] Among them, S 0,j,j (ω) is the Clough-Penzien spectrum, that is:

[0151]

[0152] ρ i,j (ω)=Aexp[-2d i,j(1-A+αA) / αθ(ω)]+(1-A)exp[2d i,j (1-A+αA) / θ(ω)] (23)

[0153] Among them, ω g =15.7rad / s,ω f =1.57rad / s,ξ g =0.60,ξ f =0.60, C j =0.08m 2 ·s -3 ;

[0154] θ(ω)=K[1+(ω / 2πf0) b ] -1 / 2 (twenty four)

[0155] Where A = 0.636, α = 0.0186, K = 31200, f0 = 1.51, b = 2.95;

[0156] d i,j =20|ij| (25)

[0157] In this example, the number of frequency discretes is 512, the frequency interval is 0.3927 rad / s, the cutoff frequency is 201.0619 rad / s, the number of time discretes is 512, and the time interval is 0.0156 s.

[0158] As mentioned above, we first compute the time-varying power spectrum matrix of the underlying Gaussian random process, such as Figure 4 As shown, Figure 4 (a) is the time-varying power spectrum S 1,1 The standard solution of (t,ω) is given by (b), and the proposed method is given by (c). The standard solution is obtained by comparing the analytical expression of the reference (Liu PL, Kiureghian AD. Multivariate distribution models with prescribed marginals and covariances. Probabilist Eng Mech. 1986; 1(2): 105-112.). The results show that the power spectrum of the potential Gaussian random process evolution obtained by the method of the present invention is highly consistent with the standard solution.

[0159] Based on the algorithm of the present invention, the key parameters are determined as follows: max =1.7344s,ω max=12.5664rad / s, Δt1=0.8736s, Δt2=0.1092s, Δω1=2.5362rad / s, Δω2=20.4204rad / s. The corresponding time-frequency interpolation point distribution is as follows Figure 5 As shown in the figure, the interpolation points are significantly concentrated in the time-frequency region with higher amplitude of the evolving power spectrum, which verifies that the adaptive interpolation strategy can accurately capture the energy distribution characteristics.

[0160] Based on the time-frequency interpolation points, the power spectrum matrix of the potential Gaussian random process is decomposed to obtain The time and frequency parameters are obtained through orthogonal decomposition. After interpolation and expansion to all discrete points, the fast Fourier transform is used to generate the potential Gaussian process time history samples, and finally the target non-stationary non-Gaussian random process time history samples are obtained through nonlinear transfer relations.

[0161] The verification results show that: Figure 3 The four time-varying statistical moments are in good agreement with the target values; Figure 6 The probability density function at a specific moment in [(a) Y1(t); (b) Y2(t); (c) Y3(t)] fully matches the target distribution. In terms of computational efficiency, the total simulation time for the non-stationary non-Gaussian random process Y3(t) is 7.44 seconds, of which the calculation of the underlying Gaussian process EPSD matrix takes 3.82 seconds, the calculation of the time-frequency interpolation points takes only 0.09 seconds, the matrix decomposition takes 0.02 seconds, the interpolation takes 0.01 seconds, and the fast Fourier transform takes 3.50 seconds. This fully demonstrates that the proposed method significantly improves simulation efficiency while ensuring accuracy.

[0162] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Figure 7 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O) and a communication interface. The processor, the memory and the input / output interface are connected via a system bus, and the communication interface is connected to the system bus via the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store random process simulation data. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal via a network connection. When the computer program is executed by the processor, an efficient simulation method for multi-point non-stationary non-Gaussian random processes based on spectral representation is implemented.

[0163] Those skilled in the art will understand that Figure 7 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than shown in the figure, or combine certain components, or have a different component arrangement.

[0164] In an exemplary embodiment, a computer device is further provided, including a memory and a processor. The memory stores a computer program, and the processor implements the steps in the above method embodiments when executing the computer program.

[0165] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program. When the computer program is executed by a processor, the steps in the above-mentioned method embodiments are implemented.

[0166] In an exemplary embodiment, a computer program product is provided, including a computer program. When the computer program is executed by a processor, the steps in the above method embodiments are implemented.

[0167] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0168] Those skilled in the art will appreciate that all or part of the processes in the above-mentioned embodiment methods can be implemented by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to memory, database or other media used in the embodiments provided in this application may include at least one of non-volatile and volatile memory. Non-volatile memory may include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory may include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0169] The databases involved in the various embodiments provided herein may include at least one of a relational database and a non-relational database. Non-relational databases may include, but are not limited to, distributed databases based on blockchains. The processors involved in the various embodiments provided herein may include, but are not limited to, general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic units, data processing logic units based on quantum computing, and the like.

[0170] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0171] This document uses specific examples to illustrate the principles and implementation methods of this application. The description of the above examples is only intended to help understand the method and core concept of this application. At the same time, for those skilled in the art, based on the concept of this application, there may be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. An efficient simulation method for multi-point non-stationary non-Gaussian random processes based on spectral representation, characterized by: The steps include: Determine the target multi-point non-stationary non-Gaussian vector random process Y(t)={Y1(t),Y2(t),…,Y n (t)} power spectrum matrix and non-Gaussian probability characteristics; Calculate Y j (t) and Y k The time-varying correlation coefficient ρ of (t) NG,j,k (t,s); Establish ρ through Mehler formula NG,j,k (t,s) and the corresponding latent Gaussian process correlation coefficient ρ j,k The transfer equation of (t,τ); Based on the transfer equation, the time-varying correlation coefficient ρ of the potential Gaussian random process is calculated through the discrete data mapping relationship j,k (t,τ) and its power spectrum matrix; Based on the amplitude information of the power spectrum matrix of the potential Gaussian random process, the time interpolation points and the frequency interpolation points are calculated; Perform matrix decomposition on the potential Gaussian random process at the time interpolation point and the frequency interpolation point to obtain a decomposition matrix; Perform orthogonal decoupling on the decomposition matrix to obtain time parameters and frequency parameters; Obtain the decoupling parameters of all discrete points through interpolation; Fast Fourier transform is used to generate samples of the latent Gaussian random process; The target non-stationary non-Gaussian vector random process Y is obtained by nonlinear transfer function transformation j The corresponding time-course samples of (t).

2. The method according to claim 1, characterized in that The calculation Y j (t) and Y k The time-varying correlation coefficient ρ of (t) NG,j,k (t,s), which is calculated using the following formula: in, and Y j (t) and Y k The time-varying variance of (t); S NG,j,k (t,ω) is the power spectrum matrix S NG The element in the jth row and kth column of (t,ω).

3. The method according to claim 2, characterized in that The Mehler formula is used to establish ρ NG,j,k (t,s) and the corresponding latent Gaussian process correlation coefficient ρ j,k The transfer equation of (t,τ) is: Where Q is the cutoff number of the Mehler formula, usually Q = 8; y GH,l (t) and w GH,l (t) is the Gauss–Hermite integration point and weight, and the number of integration points d is usually 11; H m (·) is a Hermite polynomial of order m, satisfying:

4. The method according to claim 3, characterized in that Based on the transfer equation, the time-varying correlation coefficient ρ of the potential Gaussian random process is calculated through the discrete data mapping relationship j,k (t,τ) and its power spectrum matrix, including: The time-varying correlation coefficient ρ of the underlying Gaussian random process j,k The domain of (t,τ) [-1,1] is discretized into s points with equal spacing: Will Substituting the discrete points in into the transfer equation, we can get the corresponding discrete set of non-Gaussian correlation coefficients: Construct ρ based on discrete sets j,k (t,τ) and ρ NG,j,k The mapping function of (t,τ); Backcalculate the target ρ by interpolation NG,j,k The corresponding ρ in (t,τ) j,k (t,τ).

5. The method according to claim 4, characterized in that The calculation of the time interpolation points and the frequency interpolation points based on the amplitude information of the power spectrum matrix of the potential Gaussian random process includes: For any discrete time point t p and frequency discrete points ω l , calculate the variance at that point and the proportion of the variance in the entire time domain or frequency domain respectively; Determine the moment t at which the variance in the time domain is the largest max The frequency ω with the largest variance in the sum frequency domain max ; t max The time interval is expanded symmetrically around the center until the variance ratio in the interval exceeds the first preset threshold, and the core time interval [t c ,t d ]; Similarly, with ω max The frequency interval is expanded symmetrically around the center until the variance ratio in the interval reaches a second preset threshold value or more, and the core frequency interval [ω c ,ω d ]; Divide the entire time domain into core time intervals [t c ,t d ] and the edge time interval [t b ,t c ]∪[t d ,t u ], and divide the full frequency domain into core frequency intervals [ω c ,ω d ] and the edge frequency interval [ω b ,ω c ]∪[ω d ,ω u ]; Calculate the time interpolation step of the core time interval and the edge time interval, as well as the frequency interpolation step of the core frequency interval and the edge frequency interval respectively; Generate time interpolation points according to the time interpolation step And generate frequency interpolation points according to the frequency interpolation step 6. The method according to claim 5, characterized in that The matrix decomposition of the potential Gaussian random process at the time interpolation point and the frequency interpolation point is specifically: By formula Decomposition, get the decomposition matrix 7. The method according to claim 6, characterized in that The time parameters and frequency parameters are obtained by performing orthogonal decoupling on the decomposition matrix, specifically: By formula Decomposition matrix Decoupling, where: is the time parameter, is the frequency parameter, and r is the cutoff order.

8. The method according to claim 7, characterized in that The decoupling parameters of all discrete points are obtained by interpolation, specifically: Based on time parameters and frequency parameters Interpolate all discrete time points and frequency discrete points to obtain the time parameters at discrete time points and frequency parameters 9. The method according to claim 8, characterized in that The fast Fourier transform is used to generate the potential Gaussian random process sample, specifically: By formula Perform FFT operation to generate sample u j (t), where: i is an imaginary number, i 2 =-1; Δω is the frequency interval; Re{·} represents the real number in the brackets; is a random phase angle uniformly distributed in [0,2π].

10. The method according to claim 9, characterized in that The target non-stationary non-Gaussian vector random process Y is obtained by nonlinear transfer function conversion. j The corresponding time-course samples of (t) are: By formula Perform the conversion, where: F j The inverse function of (·,t), Φ j (·,t) is u j Cumulative distribution function of (t).

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