An optimal design method and system for ozone test based on differential evolution algorithm
By optimizing ozone experimental design using the differential evolution algorithm, the problem of optimal design in ozone experiments is solved, the number of experiments is reduced, the design efficiency and accuracy are improved, and the assumption dependence of quantile regression models is avoided.
Patent Information
- Application Number
- CN202310786395.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-06-29
- Publication Date
- 2026-02-06
- Estimated Expiration
- 2043-06-29
AI Technical Summary
Existing technologies struggle to find the optimal design for ozone experiments, resulting in numerous experiments, high costs, and the need for many assumptions and unresolved local optima issues when solving quantile regression models.
A differential evolution algorithm combined with a quantile regression model was used to establish the relationship between temperature and reaction time, the design variables of ozone experiments. The temperature design variable was then optimized through iterative updates using a nested differential evolution algorithm to obtain the optimal design.
It reduces the number of ozone experiments, avoids a large number of assumptions, improves the efficiency and accuracy of the design, and solves the problem of solving quantile regression models in ozone experiments.
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Figure CN116759003B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application belongs to the technical field of ozone reaction experiment test, and particularly relates to an ozone test optimal design method and system based on a differential evolution algorithm. BACKGROUND
[0002] The statements in this section merely provide background information related to the present application and do not necessarily constitute the prior art.
[0003] Koenker and Bassett proposed quantile regression in 1978, which has been widely used in fields including pharmacology, medicine and information science due to its excellent robustness. However, due to the complexity of quantile regression calculation, the optimal experimental design for constructing quantile regression has not been paid attention by researchers. Although Dette and Trampisch provided a limited local and standardized maximum and minimum quantile regression D-optimal design theorem in 2012, the theory has significant limitations. Researchers have also proposed robust optimal designs for quantile regression, however, their theory is based on a large number of assumptions about the model. In recent years, some researchers have proposed optimal designs for quantile regression of logical models in restricted spaces. However, this design has strict restrictions on the model and design space. So far, finding the optimal design of quantile regression model is still a challenging task. Although there are some theories trying to solve this problem, they have limitations, require a large number of assumptions, and lead to limited practical applications.
[0004] The design of quantile regression can meet the application of experimental design in heteroscedastic models, which is of great significance for the application of optimal experimental design in reality, because many data distributions in reality conform to heteroscedastic distribution, but many optimal criteria are not convex functions of quantile regression design, and these optimal criteria can only provide sufficient and unnecessary conditions for the optimal design of quantile regression.
[0005] The reaction of carbon monoxide and ozone captures the loss process of the tropospheric and stratospheric ozone, in order to study the time dependence on temperature of the reaction, the experimental design supported at two temperature points is sought, when the temperature dependence of the reaction is studied, if the optimal experimental design is not carried out, a plurality of designs in the range of design space X need to be carried out, such as: if the range of X is [1, 100], then the experimental design needs to be carried out in the range of 1-100, even several hundred times of experiments are needed to cover the temperature range, so as to know the time dependence on temperature of the reaction, therefore, the optimal design of the experiment can reduce the number of experiments or the number of samples, and help researchers save more funds and energy in the experiment. In the currently available methods, some optimal experimental design criteria can be used to design the experiment, but the method has great defects, and has many requirements for the experiment, and in the practical application, it is often difficult or impossible to solve. SUMMARY
[0006] In order to overcome the above-mentioned deficiencies of the prior art, the present application provides an ozone experiment optimal design method and system based on a differential evolution algorithm, a quantile regression model of a design variable temperature and a reaction time in an ozone experiment is established, the quantile regression model is solved by combining a differential evolution algorithm, and optimal design of the design variable temperature in the ozone experiment is obtained, and the problem that a large number of assumptions are needed when solving the quantile regression model in the ozone experiment is avoided.
[0007] To achieve the above object, the first aspect of the present application provides an ozone experiment optimal design method based on a differential evolution algorithm, comprising:
[0008] A quantile regression model of a design variable temperature and a reaction time in an ozone experiment is established;
[0009] Based on the established quantile regression model, quantile regression estimation of the relationship parameters of the design variable temperature and the reaction time is determined;
[0010] Based on the quantile regression estimation of the relationship parameters, an asymptotic covariance formula of the relationship parameters is determined;
[0011] Based on the local D-optimal design, the optimal function is determined according to the asymptotic covariance formula of the relationship parameters;
[0012] The asymptotic covariance formula is taken as an inner fitness function of the differential evolution algorithm, and the optimal function is taken as an outer fitness function of the differential evolution algorithm, and iterative updating is carried out, so as to obtain the optimal design of the design variable temperature.
[0013] The second aspect of the present application provides an ozone experiment optimal design system based on a differential evolution algorithm, comprising:
[0014] Model establishing module: establish quantile regression model of design variable temperature and reaction time in ozone test;
[0015] Estimation module: determine quantile regression estimation of relationship parameter of design variable temperature and reaction time based on the established quantile regression model;
[0016] Asymptotic covariance determining module: determine asymptotic covariance formula of relationship parameter based on quantile regression estimation of relationship parameter;
[0017] Optimal function determining module: determine optimal function according to asymptotic covariance formula of relationship parameter based on local D-optimal design;
[0018] Optimal design module: take asymptotic covariance formula as inner layer fitness function of differential evolution algorithm, take optimal function as outer layer fitness function of differential evolution algorithm, and perform iterative update to obtain optimal design of design variable temperature.
[0019] The third aspect of the present application provides a computer device, comprising: a processor, a memory and a bus, the memory stores machine readable instructions executable by the processor, when the computer device is running, the processor and the memory communicate through the bus, the machine readable instructions are executed by the processor to perform an optimal design method of ozone test based on differential evolution algorithm.
[0020] The fourth aspect of the present application provides a computer readable storage medium, the computer readable storage medium stores a computer program, the computer program is executed by the processor to perform an optimal design method of ozone test based on differential evolution algorithm.
[0021] The above one or more technical solutions have the following beneficial effects:
[0022] In the present application, quantile regression model of design variable temperature and reaction time in ozone test is established, and the quantile regression model is solved by combining differential evolution algorithm to obtain optimal design of design variable temperature in ozone test, so that the problems that a large number of assumptions are needed when solving quantile regression model in ozone test at present, and the problems that local optimum is trapped and even it is difficult to solve when solving optimal design are solved.
[0023] The advantages of the additional aspects of the present application will be partially given in the following description, partially will become obvious from the following description, or will be understood by the practice of the present application. BRIEF DESCRIPTION OF DRAWINGS
[0024] The drawings accompanying the specification of the present application form a part thereof, serve to provide further understanding of the present application, and together with the description of the exemplary embodiments of the present application and their description serve to explain the present application, and do not constitute improper limitations on the present application.
[0025] Figure 1 Flow chart of nested differential evolution algorithm in embodiment one of the present application;
[0026] Figure 2 Flow chart of general differential evolution algorithm in embodiment one of the present application. DETAILED DESCRIPTION
[0027] It should be noted that the following detailed description is merely exemplary in nature and is intended to provide further description of the present application. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the application belongs.
[0028] It is to be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application.
[0029] In the case of no conflict, the embodiments in the present application and the features in the embodiments can be combined with each other.
[0030] Embodiment one
[0031] The present embodiment discloses an optimal design method of ozone test based on differential evolution algorithm, comprising:
[0032] Establishing quantile regression model of design variable temperature and reaction time in ozone test;
[0033] Based on the established quantile regression model, determining quantile regression estimation of the relationship parameters of design variable temperature and reaction time;
[0034] Based on the quantile regression estimation, determining the asymptotic covariance formula of the quantile regression estimation;
[0035] Based on the local D-optimal design, determining the optimal function according to the asymptotic covariance formula of the quantile regression estimation;
[0036] Taking the asymptotic covariance formula as the inner fitness function of the differential evolution algorithm and taking the optimal function as the outer fitness function of the differential evolution algorithm, iterative updating is performed to obtain the optimal design of the design variable temperature.
[0037] The reaction capture of NO+O3→NO2+O2 captures the loss process of the ozone in the troposphere and the stratosphere, and the correlation formula of the reaction time and the temperature is defined as follows:
[0038]
[0039] In the formula, Er represents the reaction time. Er is an exponential function of the activation temperature θ2, and the parameter θ1 is a multiplication constant. The design variable is the temperature x, and a good selection made for it can produce effective estimation of θ1 and θ2.
[0040] The equation is nonlinear with respect to θ2, so any optimal design will depend on a set of initial parameter values θ 0 For this example, θ 0 = (3.0 x 10 -12 , 1500) was chosen based on preliminary results from NASA Jet Propulsion Laboratory. x The acceptable temperature range, i.e., the design space X, was fixed at [212, 422].
[0041] In this implementation, the optimal design for quantile regression model is solved by the nested differential evolution algorithm, which first determines the relationship model between the response reaction time and the design temperature x. A univariate nonlinear quantile regression model is introduced, which is defined as follows:
[0042]
[0043] where g and σ are the location function and scale function, respectively, and ∈ is an independent and identically distributed variable following the distribution function F, and its τ-quantile is 0, i.e., F -1 (τ) = 0. x represents the explanatory variable in the design space X = (xl, xu), is the vector of model location parameters, in this embodiment,
[0044] where the scale function is used to represent the error link. In order to illustrate that the current error is consistent with quantile regression, y(x) is shown in the actual time, because compared with the ideal reaction formula (1), formula (2) increases the error in the experiment.
[0045] Assuming that the location function g is differentiable with respect to the parameter , and the vector of partial derivatives of g with respect to is denoted by formula (3) and the gradient of the θ prediction value:
[0046]
[0047] The quantile regression estimate of the parameter in the model can be defined as:
[0048]
[0049] where the value range of τ is from 0 to 1, N is the number of support points, i.e., how many test points are given in practice, and the optimal design is represented as follows: ξ, which is the total number of points to be given is specified. If the approximate design on X is ξ = {x i , w i}(i = 1, 2,..., n), where wi is the weight corresponding to support point x i , and Then y(x i ) represents the corresponding observation value, which is obtained according to formula (2). Θ i represents the value range of θ i . ρ τ represents the test function, also represents the partition loss of the quantile regression. It can be proved that, under certain assumptions, is asymptotically normally distributed, that is:
[0050]
[0051] wherein the matrices D0and D1are defined as follows:
[0052]
[0053]
[0054] The asymptotic covariance matrix of the quantile regression estimate is defined as:
[0055]
[0056] In this embodiment, the local optimal design is used as the basis of the optimality criterion of the maximization principle, and is defined as:
[0057]
[0058] is the D-optimality efficiency of the given design ξ, wherein is the local D-optimal design of the quantile regression, and p is the number of location parameters in the model. According to the standard maximum-minimum quantile regression optimization experimental design, it is required to maximize the function:
[0059]
[0060] In order to achieve this, firstly, the local D-optimal design needs to be found, then the function Φ Θ (ξ) is maximized, and finally the maximum-minimum optimal design of the quantile regression is obtained. In constructing the nested DE algorithm for the optimization design of the quantile model, the difference evolution optimization process of the formula is incorporated into the fitness function of the external differential evolution algorithm. In the given value range X=(xl,xu), θ2=(θ 2l , θ 2u ), the fitness function of the outer layer differential evolution is set as obj, that is -Φ Θ(ξ), set the inner layer differential evolution fitness function as |H|, H is formula (8). In the differential evolution optimization process in the inner layer, search for the minimum D efficiency value in the range of θ2, and find the corresponding optimal design in X in the outer layer algorithm.
[0061] The flow chart of the general differential evolution algorithm is shown in Figure 2 , and the specific steps are as follows:
[0062] (1) Determine the differential evolution algorithm control parameters, and determine the fitness function. The differential evolution algorithm control parameters include: population size NP, scaling factor F, and crossover probability CR;
[0063] (2) Randomly generate an initial population;
[0064] (3) Evaluate the initial population, that is, calculate the fitness value of each individual in the initial population.
[0065] (4) Determine whether the termination condition is reached or the evolution number reaches the maximum. If yes, terminate the evolution and output the best individual as the optimal solution; if no, continue;
[0066] (5) Perform mutation and crossover operations to obtain an intermediate population;
[0067] (6) Select individuals from the original population and the intermediate population to obtain a new generation population;
[0068] (7) Evolution number g = g + 1, go to step (4);
[0069] As shown in Figure 1 , in this embodiment, the specific steps of solving the optimal design in X using the differential evolution algorithm include:
[0070] S1: Determine the differential evolution algorithm control parameters NP, F, and CR, and use the asymptotic covariance formula as the inner layer fitness function of the differential evolution algorithm, and use the optimal function as the outer layer fitness function of the differential evolution algorithm;
[0071] S2: Randomly initialize the population, and calculate the outer layer fitness function obj of each individual in the initial population;
[0072] S3: Determine whether the termination condition of the outer layer fitness function obj is reached or the evolution iteration number reaches the maximum, if yes, terminate the evolution and output the best individual as the optimal solution; if no, continue S4;
[0073] S4: Calculate the inner layer fitness function |H| of each individual in the initial population;
[0074] S5: Determine whether the inner fitness function |H| reaches the termination condition or the number of iterations reaches the maximum, if yes, terminate the evolution, output the best individual as the optimal solution, and go to S3 to continue; if not, go to S6.
[0075] S6: Mutate, cross, and select the population of the inner fitness function |H| to obtain a new generation of population, and go to S5.
[0076] As shown in Figure 2 , the specific steps of mutation, crossover and selection operations are as follows:
[0077] Mutation: In the differential evolution algorithm, mutation is regarded as the disturbance of random elements. The target vector in G generation is represented as The mutation operation will make two target vectors produce a weighted difference, and then add a third target vector to generate a new individual, which is usually called donor vector Please note that each target vector is different from the donor vector, and each target vector is randomly selected. According to the formula, the donor vector can be obtained, where r0, r1, r2 are different:
[0078]
[0079] Crossover: The crossover operation mixes the donor vector and the target vector to form a candidate for the next generation of individual vectors, which is usually called test vector. The specific formula is as follows:
[0080]
[0081] Where, and represent their j-th dimensional components, and rand(0,1) represents a random number between 0 and 1. We can see that CR here guarantees the proportion of the test vector and the target vector. rand is a randomly selected index that guarantees at least one component from .
[0082] Selection: The selection operation compares each test vector with the target vector, and uses the fitness function h to select the most suitable one as the next generation vector The formula is as follows:
[0083]
[0084] Implementation column two
[0085] The embodiment discloses an ozone test optimal design system based on a differential evolution algorithm, comprising:
[0086] Model establishing module: establishing quantile regression model of design variable temperature and reaction time in ozone test;
[0087] Estimating module: determining quantile regression estimation of relationship parameter of design variable temperature and reaction time based on the established quantile regression model;
[0088] Asymptotic covariance determining module: determining asymptotic covariance formula of relationship parameter based on quantile regression estimation of relationship parameter;
[0089] Optimal function determining module: determining optimal function based on asymptotic covariance formula of relationship parameter according to local D-optimal design;
[0090] Optimal design module: taking asymptotic covariance formula as inner fitness function of differential evolution algorithm and optimal function as outer fitness function of differential evolution algorithm to perform iterative updating to obtain optimal design of design variable temperature.
[0091] Embodiment three
[0092] The purpose of this embodiment is to provide a computing device, comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the above method when executing the program.
[0093] Embodiment four
[0094] The purpose of this embodiment is to provide a computer-readable storage medium.
[0095] A computer-readable storage medium having stored thereon a computer program, the program being executable by a processor to perform the steps of the above method.
[0096] The steps and methods involved in the above embodiments two, three and four correspond to embodiment one, and the specific embodiments can be seen from the relevant description part of embodiment one. The term "computer-readable storage medium" should be understood as including a single medium or multiple media of one or more instruction sets; it should also be understood as including any medium capable of storing, encoding or carrying instruction sets for execution by a processor and causing the processor to perform any method in the present application.
[0097] Those skilled in the art should understand that the above modules or steps of the present application can be realized by a general computer device, alternatively, they can be realized by program codes executable by a computing device, so that they can be stored in a storage device for execution by a computing device, or they can be respectively made into individual integrated circuit modules, or a plurality of modules or steps among them can be made into a single integrated circuit module to realize. The present application is not limited to any specific combination of hardware and software.
[0098] The above describes the specific embodiments of the present application in combination with the drawings, but is not a limitation on the protection scope of the present application. Those skilled in the art should understand that various modifications or variations made by those skilled in the art on the basis of the technical solutions of the present application without creative labor are still within the protection scope of the present application.
Claims
1. An optimal design method for ozone experiments based on differential evolution algorithm, characterized in that, include: Establish a quantile regression model for the design variables temperature and reaction time in ozone experiments; Based on the established quantile regression model, the quantile regression estimate of the relationship parameter between the design variable temperature and the response time is determined; Based on the quantile regression estimation of the relation parameters, the asymptotic covariance formula of the relation parameters is determined; Based on local D-optimal design, the optimal function is determined according to the asymptotic covariance formula of the relation parameters; The asymptotic covariance formula is used as the inner fitness function of the differential evolution algorithm, and the optimal function is used as the outer fitness function of the differential evolution algorithm. The algorithm is iteratively updated to obtain the optimal design of the design variable temperature.
2. The optimal design method for ozone experiments based on differential evolution algorithm as described in claim 1, characterized in that, Based on the quantile regression estimation of the relational parameters, the asymptotic covariance formula for the quantile regression estimation is determined as follows: In the ozone experiment, the partial derivative of the function of the design variable temperature and reaction time with respect to the relationship parameter of the design variable temperature and reaction time is obtained, and the first matrix is determined by integration based on the partial derivative and the transpose of the partial derivative. The second matrix is determined by integrating the partial derivatives, the transpose of the partial derivatives, and the proportional function of the design variables temperature and reaction time. The asymptotic covariance formula for quantile regression estimation is determined based on the first and second matrices.
3. The optimal design method for ozone experiments based on differential evolution algorithm as described in claim 1, characterized in that, Based on local D-optimal design, the optimal function is determined according to the asymptotic covariance formula of the relation parameters, specifically: The optimal function is determined based on the asymptotic covariance formula corresponding to the local D-optimal design obtained by quantile regression estimation, the asymptotic covariance formula corresponding to the quantile regression estimation, and the number of parameters relating the design variables temperature and reaction time.
4. The optimal design method for ozone experiments based on differential evolution algorithm as described in claim 1, characterized in that, Using the asymptotic covariance formula as the inner fitness function of the differential evolution algorithm and the optimal function as the outer fitness function, iterative updates are performed to obtain the optimal design for the design variable temperature. Specifically: S1: Determine the control parameters of the differential evolution algorithm, use the asymptotic covariance formula as the inner fitness function of the differential evolution algorithm, and use the optimal function as the outer fitness function of the differential evolution algorithm. S2: Randomly initialize the population and calculate the outer fitness function of each individual in the initial population; S3: Determine whether the outer fitness function's transition termination condition has been met or the number of iterations has reached its maximum. If yes, terminate and output the best individual obtained as the optimal solution; otherwise, continue to S4. S4: Calculate the inner fitness function for each individual in the initial population; S5: Determine whether the transition termination condition of the inner fitness function has been met or the number of iterations has reached its maximum. If yes, terminate, output the best individual obtained as the optimal solution, and go to S3 to continue; otherwise, continue to S6. S6: Perform mutation, crossover, and selection on the population with the inner fitness function to obtain a new generation population, and then transfer it to S5.
5. The optimal design method for ozone experiments based on differential evolution algorithm as described in claim 4, characterized in that, The local D-optimal design for quantile regression is obtained through the inner differential evolution optimization process.
6. An optimal design system for ozone experiments based on differential evolution algorithm, characterized in that, include: Model building module: Establishes a quantile regression model for the design variables temperature and reaction time in ozone experiments; Estimation module: Based on the established quantile regression model, determine the quantile regression estimate of the relationship parameter between the design variable temperature and the response time; The asymptotic covariance determination module: Based on the quantile regression estimation of the relationship parameters, it determines the formula for the asymptotic covariance of the relationship parameters; Optimal function determination module: Based on local D-optimal design, the optimal function is determined according to the asymptotic covariance formula of the relation parameters; Optimal Design Module: The asymptotic covariance formula is used as the inner fitness function of the differential evolution algorithm, and the optimal function is used as the outer fitness function of the differential evolution algorithm. The algorithm is iteratively updated to obtain the optimal design of the design variable temperature.
7. The optimal design system for ozone experiments based on differential evolution algorithm as described in claim 6, characterized in that, The asymptotic covariance determination module specifically includes: In the ozone experiment, the partial derivative of the function of the design variable temperature and reaction time with respect to the relationship parameter of the design variable temperature and reaction time is obtained, and the first matrix is determined by integration based on the partial derivative and the transpose of the partial derivative. The second matrix is determined by integrating the partial derivatives, the transpose of the partial derivatives, and the proportional function of the design variables temperature and reaction time. The asymptotic covariance formula for quantile regression estimation is determined based on the first and second matrices.
8. The optimal design system for ozone experiments based on differential evolution algorithm as described in claim 6, characterized in that, The optimal function determination module specifically includes: The asymptotic covariance formula corresponding to the local D-optimal design based on quantile regression estimation; The optimal function is determined by the asymptotic covariance formula corresponding to quantile regression estimation and the number of parameters relating the design variable temperature to reaction time.
9. A computer device, characterized in that, include: The computer device includes a processor, a memory, and a bus. The memory stores machine-readable instructions executable by the processor. When the computer device is running, the processor communicates with the memory via the bus. When the machine-readable instructions are executed by the processor, they perform an optimal ozone test design method based on differential evolution algorithm as described in any one of claims 1 to 5.
10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, which, when executed by a processor, performs an optimal ozone experiment design method based on differential evolution algorithm as described in any one of claims 1 to 5.