A matrix transformation-based meta-heuristic test design optimization method

Abstract

The application discloses a kind of meta-heuristic test design optimization methods based on matrix transformation, belong to computer-aided test design and statistical modeling technical field, this method uses a two-stage optimization framework: first, in global exploration stage, with minimizing global uniformity index as target to carry out unconstrained optimization, to obtain well space coverage sample set;Subsequently in local development stage, with minimizing adjacent distance variance as target, optimization is carried out under the constraint of maintaining the good global coverage obtained, to fine adjustment sample point local spacing distribution, improve uniformity;The application encodes test design scheme into matrix form individual, is updated iteratively by meta-heuristic algorithm, and adopts global normalization strategy to ensure that sample is always located in design space;The application can systematically consider the global coverage of sample and local uniformity, significantly improve the sample distribution quality in high-dimensional space, generate comprehensive performance excellent test design scheme.

Description

Technical Field

[0001] This invention relates to the field of computer-aided experimental design and statistical modeling technology, and in particular to a metaheuristic experimental design optimization method based on matrix transformation. Background Technology

[0002] Experimental design aims to obtain sufficient and effective information at minimal cost, providing a reliable foundation for subsequent statistical inference and system modeling. Through the analysis of experimental data, researchers can conduct analysis of variance, range analysis, regression analysis, surrogate model construction, and global sensitivity analysis to reveal the mapping relationship between input variables and output responses and assess the influence of each factor. In high-dimensional complex engineering problems, the uniformity of sample point distribution in the design space directly determines the prediction accuracy and robustness of the surrogate model. Therefore, uniformly distributed samples can more comprehensively capture the input-output mapping relationship and avoid modeling bias caused by missing local information.

[0003] To improve the uniformity of sample distribution, researchers have proposed various space-filling criteria, such as the potential energy criterion and the maximum-minimum distance criterion, and developed a series of optimization methods based on these criteria. Among them, the Latin Hypercube Design (LHD) has been widely used because it can guarantee one-dimensional projection uniformity. To further improve the overall uniformity of LHD in multidimensional space, various optimization methods have been proposed, such as the Column Pair Exchange Algorithm (CP) and the Enhanced Stochastic Evolutionary Algorithm (ESE), which optimize the initial LHD through row / column exchange strategies.

[0004] Limitations of existing methods:

[0005] (1) Limited optimization space: Most methods use LHD as the initial sample, and the optimization process is limited to row / column swapping operations within the initial sample matrix. This combination of row and column swapping within the matrix essentially limits the algorithm's global exploration capability and makes it difficult to fully cover the unexplored areas in the high-dimensional design space.

[0006] (2) Insufficient overall uniformity: Existing methods often rely on a single spatial filling index, which often ignores the local uniformity between samples while optimizing global coverage. A sample set with good global coverage may have uneven spacing in local areas, resulting in some areas being too dense and others being too sparse. Summary of the Invention

[0007] The purpose of this invention is to solve the technical problems existing in the prior art and to provide a metaheuristic experimental design optimization method based on matrix transformation.

[0008] To achieve the above objectives, the technical solution provided by this invention is: a metaheuristic experimental design optimization method based on matrix transformation, wherein the optimization method specifically includes the following steps:

[0009] (1) Population initialization

[0010] Set the sample dimension P, sample size N, population size n, maximum number of generations Maxgen, and sample lower bound vector. With upper limit vector The population consists of an N×P dimensional matrix of experimental design schemes, as shown in the equation. As shown:

[0011]

[0012] The initial population is generated based on randomization methods (such as random sampling and low-difference sequences). ;

[0013] (2) Global Exploration

[0014] The global exploration phase is an unconstrained optimization process, expressed as: The mathematical form shown; will As an optimization objective, this metric focuses on evaluating the spatial coverage characteristics of the sample set, by minimizing... The value aims to explore the space to obtain a well-covered sample set. This lays the foundation for subsequent optimizations;

[0015]

[0016] in The global uniformity index of the sample is defined mathematically as follows:

[0017] (3)

[0018] in This represents the Euclidean distance between points i and j, where p is a constant; as shown in the formula, when the distance between any two points is... When decreasing, It will grow exponentially, significantly penalizing clustering; and by utilizing the minimum distance maximization mechanism, it effectively avoids sample gaps and ensures spatial coverage without dead zones. The smaller the value, the larger the overall distance between points, and the better the sample coverage of the space;

[0019] (3) Local development

[0020] The local development phase is a constrained optimization process, which can be expressed as shown in the formula. As shown, The criterion, as an optimization objective, aims to maintain good spatial coverage obtained during the exploration phase by optimizing the spatial coverage of the target area. Minimizing the value refines the local spacing distribution between sample points, thereby significantly improving spacing uniformity.

[0021]

[0022] in The standard deviation of the nearest neighbor distance is defined as follows, and its core idea is to ensure that the spacing between samples is uniform.

[0023] (5)

[0024] (6)

[0025] Record the Euclidean distance from point i to the nearest point among all sample points. The standard deviation of the Euclidean distance of all nearest points; the standard deviation of the nearest neighbor distance. This reflects the uniformity among the sample points: The smaller the value, the more uniform the distance between sample points, and the better the uniformity between samples;

[0026] (4) Result output

[0027] After the global exploration and local development phases, a high-quality sample set that balances global coverage and local uniformity is output.

[0028] Beneficial effects of this invention:

[0029] 1. This invention is not limited to swapping operations within the design matrix. Instead, it utilizes direct matrix operations and a global normalization strategy based on metaheuristic algorithms to enable sample points to be freely updated throughout the entire continuous design space. This fundamentally overcomes the problem of limited exploration space caused by the fixed matrix structure in existing methods, significantly improving the algorithm's global search and coverage capabilities in high-dimensional spaces.

[0030] 2. This invention decomposes the optimization of overall uniformity into two sequential stages: "global exploration" and "local development." In the global exploration stage, the goal is to minimize... The guiding principle is to prioritize ensuring broad coverage of the design space by the sample; during the local development phase, the goal is to minimize the standard deviation of the proximity distance. ) as the target, and in Under the constraint of non-deterioration, the uniformity of local spacing between sample points is finely adjusted. This phased, multi-objective optimization strategy effectively solves the problem of insufficient overall uniformity caused by existing methods relying on a single index. Attached Figure Description

[0031] The accompanying drawings, which are provided to further illustrate the invention and constitute a part of this invention, are illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention.

[0032] Figure 1 This is a schematic diagram of the optimized process in this invention;

[0033] Figure 2 This is a schematic diagram comparing the optimization results in this invention. Detailed Implementation

[0034] This section will describe in detail specific embodiments of the present invention. Preferred embodiments of the present invention are shown in the accompanying drawings. The purpose of the drawings is to supplement the textual description with graphics, so that people can intuitively and vividly understand each technical feature and overall technical solution of the present invention, but they should not be construed as limiting the scope of protection of the present invention.

[0035] This invention discloses a metaheuristic experimental design optimization method based on matrix transformation. The following detailed explanation, using the Particle Swarm Optimization (PSO) algorithm and initialization with a Sobol sequence, illustrates this method. The flowchart is shown below. Figure 1 As shown, the specific algorithm steps are as follows:

[0036] (1) Population initialization

[0037] Set the sample dimension P, sample size N, population size n, and maximum number of generations Maxgen. Each particle position... , representing a set of experimental design schemes, whose speed It is also an N×P matrix. The initial position and velocity of the particles are initially generated based on the Sobol sequence, as shown in the formula. As shown:

[0038]

[0039] based on Criteria for assessing the fitness of the current particle And set the current particle position as the particle's best historical position. And among all particles, the position of the particle with the lowest fitness is selected as the global optimal position. .

[0040] (2) Global Exploration

[0041] During the global exploration phase, with The criterion is to perform unconstrained optimization for a single objective. The velocity and position of each particle in the population are updated as follows:

[0042] 1) Speed ​​update formula:

[0043]

[0044] in For inertial weights, , As a learning factor, It is a random number. This represents the current iteration number.

[0045] 2) Position update formula:

[0046]

[0047] For the updated sample matrix To avoid the bias that may be introduced by the boundary absorption processing strategy for samples, this invention adopts a global normalization method: after each particle position update, the entire sample matrix is ​​linearly mapped to the interval [0,1], ensuring that all sample points are located within the defined design space while strictly maintaining the relative spatial position relationship of all sample points after iterative update.

[0048] After updating the particle positions, assess the fitness of the current individual. If the fitness of the new position is better than the particle's previous best performance. Then update If the fitness of the new position is better than the current global optimum Then update .

[0049] The algorithm terminates after reaching the maximum number of generations and outputs the globally optimal solution. and its corresponding sample matrix Repeat the global exploration process until n optimal sample matrices are obtained.

[0050] (3) Local development

[0051] Partial development phase To optimize the objective, based on the high-quality population obtained in the global exploration phase, we continue to perform single-objective constraint optimization using the PSO algorithm to finely adjust the sample distribution and improve the local uniformity among samples. The specific process is as follows:

[0052] 1) Population initialization

[0053] The n optimal sample matrices obtained during the global exploration phase ( This serves as the initial population for the local development phase. The values ​​of each particle are calculated. Value, and calculate the average. The value serves as a constraint.

[0054]

[0055] Local homogeneity index As a fitness function, it evaluates the fitness of the current particle. The penalty function method is then used to calculate the fitness value of the penalty function. :

[0056]

[0057] Set the current particle position as the particle's best historical position. And among all particles, the position of the particle with the smallest penalty function fitness value is selected as the global optimal position. .

[0058] 2) Iterative update

[0059] Speed ​​update formula:

[0060]

[0061] in For inertial weights, , As a learning factor, It is a random number. This represents the current iteration number.

[0062] Position update formula:

[0063]

[0064] The updated particles are subjected to the same normalization strategy as those used in the exploration phase.

[0065] After updating the particle position, evaluate the current particle fitness value. And update the penalty function fitness value for each particle. If the penalty function fitness value at the new position is better than the particle's previous best value... Then update If the fitness value of the penalty function at the new position is better than the current global optimum... Then update .

[0066] 3) Result Output

[0067] The algorithm terminates after reaching the maximum number of generations and outputs the globally optimal solution. and its corresponding sample matrix and will The results are mapped to the upper and lower limits of the input samples to obtain the final experimental design scheme.

[0068] Taking a sample size of 20×2 as an example, the output of the sample point distribution is shown in the figure. The Sobol sequence before optimization is compared with the existing mainstream methods: column pair exchange algorithm (CP) and augmented random evolution algorithm (ESE).

[0069] Based on the optimization results, the method of this invention is significantly superior to Sobol sequences and existing optimization methods (CP, ESE). Figure 2 As shown, the Sobol sequence, as an unoptimized initial space-filling design, exhibits strong randomness in the distribution of sample points. The value (18.214) and The value (0.0656) is the highest. After optimization by the method of this invention, the distribution of sample points becomes extremely uniform and regular, with complete spatial coverage and no obvious clusters or blank areas; its quantitative indicators Value (3.859) and The values ​​(0.00445) all decreased to the minimum. In comparison, although the two existing optimization methods, CP and ESE, are better than Sobol sequences, they still exhibit obvious local sample point clustering and incomplete spatial coverage; The values ​​(5.780 and 6.237, respectively) are approximately 50% higher than those of the method of this invention. The values ​​(0.0190 and 0.0485, respectively) are several times higher. The results fully demonstrate that the method proposed in this invention significantly improves the global space-filling performance of the design. ) and local point spacing uniformity ( It has significant advantages in all aspects and can generate higher quality experimental design schemes.

[0070] Without causing conflict, those skilled in the art can freely combine and use the above-mentioned additional technical features.

[0071] The above description is only a preferred embodiment of the present invention. Any technical solution that achieves the purpose of the present invention by essentially the same means is within the protection scope of the present invention.

Claims

1. A metaheuristic experimental design optimization method based on matrix transformation, characterized in that: The optimization method specifically includes the following steps: (1) Population initialization Set the sample dimension P, sample size N, population size n, maximum number of generations Maxgen, and sample lower bound vector. With upper limit vector The population consists of an N×P dimensional matrix of experimental design schemes, as shown in the equation. As shown: ； The initial population is generated based on a randomization method. ; (2) Global Exploration The global exploration phase is an unconstrained optimization process, expressed as: The mathematical form shown; will As an optimization objective, this metric focuses on evaluating the spatial coverage characteristics of the sample set, by minimizing... The value aims to explore the space to obtain a well-covered sample set. This lays the foundation for subsequent optimizations; ； in The global uniformity index of the sample is defined mathematically as follows: (3) in This represents the Euclidean distance between points i and j, where p is a constant; as shown in the formula, when the distance between any two points is... When decreasing, It will grow exponentially, significantly penalizing clustering; and by utilizing the minimum distance maximization mechanism, it effectively avoids sample gaps and ensures spatial coverage without dead zones. The smaller the value, the larger the overall distance between points, and the better the sample coverage of the space; (3) Local development The local development phase is a constrained optimization process, which can be expressed as shown in the formula. As shown, The criterion, as an optimization objective, aims to maintain good spatial coverage obtained during the exploration phase by optimizing the spatial coverage of the target area. Minimizing the value refines the local spacing distribution between sample points, thereby significantly improving spacing uniformity. ； in The standard deviation of the nearest neighbor distance is defined as follows, and its core idea is to ensure that the spacing between samples is uniform. (5) (6) Record the Euclidean distance from point i to the nearest point among all sample points. The standard deviation of the Euclidean distance of all nearest points; the standard deviation of the nearest neighbor distance. This reflects the uniformity among the sample points: The smaller the value, the more uniform the distance between sample points, and the better the uniformity between samples; (4) Result output After the global exploration and local development phases, a high-quality sample set that balances global coverage and local uniformity is output.

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