A method for predicting the correlation between material-related parameters based on a stochastic degradation process

Abstract

The application discloses a kind of prediction method between material related parameters based on random degradation process, comprising: the parameter pair of sample data between related parameters is carried out sample distribution calculation, obtains the parameter array of distribution compliance and determines distribution type;So as to construct distribution model;New test sample data sequence is generated as training sample data, the input vector and output vector of training sample are extracted to carry out regression model training, obtain the decision function of training sample regression model, and the mean vector of training sample input vector is calculated, the mean prediction vector is obtained by importing regression model prediction;The input vector of training sample is imported into regression model prediction, the prediction output vector of related parameter is obtained, and the weighted mean square error of prediction output vector and the output vector of training sample is calculated;So as to calculate the confidence interval of mean prediction vector, the prediction value confidence interval is calculated by combining linear difference method, for evaluating the reliability of prediction value.The method of the application can accurately predict and evaluate the related parameter test data of related material applied to mechanical engineering in degradation process.

Description

Technical Field

[0001] This invention belongs to the field of reliability analysis of materials in engineering machinery products under random degradation, and specifically relates to a prediction method for material-related parameters based on the random degradation process. Background Technology

[0002] The physicochemical properties of materials used in engineering machinery are crucial for product reliability evaluation. However, in practical applications, these properties change non-linearly as the product degrades. Therefore, effective reliability assessment requires regular measurement and analysis of these materials, which is a significant workload. Furthermore, the large number of physicochemical parameters and the difficulty in measuring some parameters further complicates the measurement process, necessitating substantial human and material resources.

[0003] Since physicochemical parameters are intrinsic properties of materials, meaning that there is a strong correlation between certain parameters, such as the hardness and tensile strength of steel, and the shear modulus and toughness of rubber, and since the measurement of some of these parameters is relatively simple and convenient, such as hardness and shear modulus, this correlation can be used in conjunction with regression analysis to predict the difficult-to-measure related parameters in the product material during the random degradation process, using easily measurable parameter data as independent variables. Summary of the Invention

[0004] Purpose of the invention: In order to overcome the shortcomings of the prior art, the present invention provides a prediction method for material-related parameters based on random degradation process. Based on the SVM regression prediction method of the present invention, the experimental data of relevant parameters of materials that are difficult to measure during the degradation process of materials applied in mechanical engineering can be accurately predicted and evaluated, and regression prediction of physicochemical parameters with correlation in materials can be achieved.

[0005] Technical Solution: Firstly, this invention provides a method for predicting material-related parameters based on a random degradation process, comprising:

[0006] Obtain parameter pairs from sample data of relevant parameters;

[0007] The parameter pairs are used to calculate the sample distribution, thereby obtaining a parameter array that conforms to the distribution and determining the distribution type;

[0008] Construct a distribution model based on the parameter array of the distribution and the determined distribution type;

[0009] Generate new simulated data sequences based on the parameter pairs of the distribution model and relevant parameter sample data;

[0010] The new experimental sample data sequence is used as the training sample data for the SVM regression algorithm. The input vector of the training sample and the mean vector of the training sample are extracted based on the training sample data.

[0011] The regression model is trained using the training sample data to obtain the regression model; the input vector of the training sample is used as the input vector of the test sample, and the model and the mean vector of the training sample are imported into the regression model for prediction, respectively obtaining the relevant parameter prediction output vector and the mean prediction vector.

[0012] The weighted mean square error is calculated separately for the predicted output vector and the output vector of the training samples.

[0013] The confidence interval of the mean prediction vector is calculated based on the weighted mean square error and the mean prediction vector.

[0014] The confidence interval of the predicted mean vector and the mean vector of the training samples are substituted into the linear interpolation method to obtain the confidence interval of the predicted value.

[0015] The reliability of the predicted values ​​of relevant parameters is assessed based on the confidence interval of the predicted values.

[0016] In a further embodiment, the method for obtaining parameter pairs of sample data between relevant parameters is as follows:

[0017] During the random degradation experiment of product materials, multiple experimental samples are set for relevant parameters; among which the relevant parameters are two or more interrelated intrinsic material properties.

[0018] Measure the parameters of multiple experimental samples separately and record the sample data of relevant parameters at different times;

[0019] The sample data of multiple experimental samples are organized into a set of parameter pairs of sample data between relevant parameters in chronological order;

[0020] The expression for the parameter pairs in the relevant parameter sample data is: {(α ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n

[0021] In the formula, α and β represent different properties of the product material and are related to each other, n represents time, m represents the total number of batches of test samples, j represents any time, and i represents the i-th batch of test samples at time j.

[0022] In a further embodiment, the method for calculating the sample distribution of the parameter pairs to obtain a parameter array conforming to the distribution and determining the distribution type includes:

[0023] Extract the array of relevant parameters at the same time point from the set of parameter pairs in the relevant parameter sample data;

[0024] Substitute the relevant parameter array into the probability density function for testing to determine whether the relevant parameters are random distribution variables that conform to the distribution parameters;

[0025] The maximum likelihood ratio is then used to calculate the maximum likelihood ratio values ​​corresponding to the multiple experimental samples that have passed the test.

[0026] The magnitude of the maximum likelihood ratios corresponding to multiple parameter arrays is determined by comparing their values.

[0027] The distribution type of the relevant parameters and the parameter set that conforms to the distribution are determined based on the magnitude of the maximum likelihood ratio;

[0028] The probability density function is:

[0029] f(α|θ αj1 ,θ αj2 ,...,θ αjk (1)

[0030] The expression for testing by substituting the relevant parameter array into the probability density function is:

[0031]

[0032] In the formula, (θ αj1 ,θ αj2 ,...,θ αjk H0 and H1 are the parameter set of the distribution model of parameter α at time t; H0 and H1 are the hypotheses to be tested.

[0033] The formula for calculating the maximum likelihood ratio is:

[0034]

[0035] In the formula, For (θ) αj1 ,θ αj2 ,...,θ αjk The maximum likelihood estimate of ) is given by λ, where λ is the maximum likelihood ratio; the judgment expression for determining the magnitude of multiple maximum likelihood ratios is:

[0036]

[0037] In the formula, K is the critical value.

[0038] In a further embodiment, the distribution parameter set determined according to the maximum likelihood method Given a defined distribution function type, the expression for constructing the distribution model of the material parameters is as follows:

[0039]

[0040] In a further embodiment, the method for generating a new experimental sample data sequence based on the parameters of the distribution model and the relevant parameter sample data includes:

[0041] For the relevant parameters, α and β are established as distribution models L of the parameters at time t, and the Monte Carlo (MC) method is used to generate a set of parameters that conform to the distribution. and Simulated data sequence and Composition of sample simulation data Where m0 represents the number of simulated data at time j.

[0042] The sample data {(α) ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n and simulation data To form a new experimental sample data sequence {(α i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n This effectively includes the statistical information of the relevant parameters in the α and β distribution models. Among them:

[0043] m' = m + m0, representing the total number of batches of new experimental sample data over n time points;

[0044] i' represents the i' batch of experimental samples at time j;

[0045] The constraints on m0 are:

[0046] 40<m+m0≤50 (6)

[0047] In equation (6), if the number of measured sample data m exceeds 50, the Monte Carlo method is not used to generate simulated data; if the number of measured sample data m does not exceed 40, numerical data simulation should be performed appropriately according to the MC method mentioned above so that the sum of the measured sample data and the simulated sample data satisfies equation (6).

[0048] In further implementation, the new experimental sample data sequence is used as the training sample data for the SVM regression algorithm. The method for extracting the input vector of the training samples and calculating the mean vector of the training samples based on the training sample data is as follows:

[0049] From the new experimental sample data sequence {(α i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n Choose one of the easily measurable parameters α as the input vector {α} for the training samples. i',j} i'＝1,2,...m'；j＝1,2,...nAnd calculate the mean vector of the training samples at n time points.

[0050] In further implementation, to avoid loss of generality, the training process for the regression model in n-dimensional space is as follows:

[0051] Let W be the hyperplane in n-dimensional space. T X+b=0, where W=(w1,w2,...w n ) and b are the eigenvectors and constant term of the hyperplane, respectively, X = (x1, x2, ... x2) n () represents an n-dimensional training sample point vector;

[0052] Let the normal vector of the eigenvector W of the n-dimensional hyperplane be the positive direction distance, and the opposite direction be the negative direction distance; y k As a scalar, when the distance from the training sample point vector to the hyperplane is in the positive direction, y k =1, when y is the distance in the negative direction. k =-1;

[0053] Where the total number of training sample point vectors is r, (r = m'·n), any training sample point X k ∈{X1,X2,...X r The distance d to the hyperplane can be expressed as:

[0054]

[0055] In the formula X k' ∈{X1,X2,...X r} represents the training sample point vector farthest from the hyperplane;

[0056] The purpose of regression training is to obtain the hyperplane W with feature vector W. T X+b=0, which minimizes the distance d from the training sample points to the hyperplane. This is equivalent to finding the optimal solution for the feature vector W in formula (7), which is also the constrained optimization problem of formula (8) as follows:

[0057]

[0058] In equation (8), d' is a constant, determined by the training sample data. It is treated as a constant term in subsequent calculations and does not affect the training results of the regression model.

[0059] For optimal solution problems with constraints, the Lagrange method is usually used, which introduces the Lagrange multiplier vector α = (α1, α2, ... α) r ) T As an undetermined coefficient of the constraint condition in formula (8), its Langran function equation is defined as:

[0060]

[0061] Where α k | k＝1,2,...,r The kth component of the Lagrange multiplier;

[0062] To solve an optimization problem where the constraints in formula (8) are inequalities, the optimal solution must satisfy the following conditions, namely the KKT (Karush-Kuhn-Tucker) conditions:

[0063]

[0064] Combining the Lagrange function (9) and the necessary KKT conditions (10), the above optimal solution problem formula (8) is equivalent to:

[0065]

[0066] Among them, (X) i ·X j () represents the inner product between the vectors of the training sample points;

[0067] By combining the training sample point vector data, the optimal solution of the above formula (11) is solved, and the optimal solution of the Lagrange multipliers is obtained as follows:

[0068] α * =(α1) * ,α2 * ,...,α r * ) T (12)

[0069] The decision function for the regression problem is:

[0070]

[0071] Where α k * b is the k-th component of the optimal solution of the Lagrange multipliers. * For the fitting constant term;

[0072] Select an appropriate radial basis kernel function K(X) i ·X j ) replace the inner product (X) between training sample point vectors i ·X j The optimal solution problem of the above formula (8) can be subjected to nonlinear regression analysis. The radial basis kernel function can be a Gaussian kernel function, a polynomial kernel function, etc.; and a slack variable ξ is introduced. k |k=1,2,...,r and penalty parameter C>0, where slack variable ξ k|k=1,2,...,r represents the degree to which the k-th sample point does not meet the regression conditions, and the penalty parameter C represents the tolerable error of the decision function, making the optimal solution problem of the above formula (8) a nonlinear soft regression problem, that is:

[0073]

[0074] By solving formulas (9) to (12) above, the optimal solution of the Lagrange multipliers is calculated sequentially, and the regression decision function is obtained as follows:

[0075]

[0076] In a further embodiment, the input vector of the training samples is imported into the regression model for training to obtain the output vector of the training samples; and the input vector and mean vector of the training samples are imported into the regression model for prediction to obtain the relevant parameter prediction output vector and mean prediction vector, respectively.

[0077] The input vector of the training samples is imported into the regression model for training, and the output vector of the training samples is obtained. The output vector of the training samples is a parameter array {β} of the difficult-to-measure relevant parameter β. i',j} i'＝1,2,...,m'；j＝1,2,...n ;

[0078] The input vector of the training samples {α i',j} i'＝1,2,...m'；j＝1,2,...n The input vector used as the test sample is imported into the regression model to predict the relevant parameter β and the predicted output vector.

[0079] At the same time, the mean vector of the training samples The new test sample input vector is imported into the regression model for prediction, yielding the mean prediction vector. in,

[0080] In a further embodiment, the weighted mean square error is calculated by separately calculating the predicted output vector and the output vector of the training samples.

[0081] The output vectors {β} of the training samples at different time points are respectively... i',j} i'＝1,2,...,m'；j＝1,2,...n and predicted output vector The weighted mean square error σ j 2 The calculation is as follows:

[0082]

[0083] In the formula, f(·) is the weight function, which has the following properties: f(0) = 0, f(1) = 1, and is monotonically and continuously increasing on [0,1].

[0084] In a further embodiment, the method of substituting the confidence interval of the mean prediction vector and the mean vector of the training samples into the linear interpolation method to obtain the confidence interval of the predicted value includes:

[0085] The weighted root mean square error σ at different times t j The confidence interval for the mean prediction vector is defined as follows:

[0086] Furthermore, based on the confidence level requirements of the actual problem, multiple weighted root mean square error widths are set: the confidence interval can be... The value of n' is determined based on the confidence level requirements of the actual problem.

[0087] In a further embodiment, the method for assessing the reliability of the predicted values ​​of relevant parameters based on the confidence interval of the predicted values ​​is as follows:

[0088] Using any measured parameter α' as the input vector of the regression prediction model, the confidence of its predicted value β' is evaluated; if the predicted output value β' is within the prediction confidence interval... Within this range, the predicted value of the relevant parameter β, which is difficult to measure, is assessed as reliable.

[0089] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0090] Based on the SVM regression prediction method of this invention, we can accurately predict and evaluate the experimental data of relevant parameters that are difficult to measure during the degradation process of materials used in mechanical engineering. This method enables regression prediction between physicochemical parameters that are related in the materials. For nonlinear regression analysis problems with random perturbations in experimental data and with small amounts of data containing random perturbations, the Support Vector Machine (SVM) regression algorithm has the advantage of being highly flexible in its applicability, so that it can be applied to the reliability assessment and analysis of engineering machinery products.

[0091] Compared with the multinomial regression method, the SVM regression method can control the predicted values ​​of the test samples within a high confidence range and avoid the problem of overfitting, achieving a good balance between prediction accuracy and practicality.

[0092] By comparing deep learning regression methods, the SVM regression prediction method still performs well in nonlinear regression problems even with a small number of training samples, demonstrating greater flexibility and wider applicability. Furthermore, the SVM method eliminates the need for a two-step mapping process; its transformation is completed in one step using a kernel function, and the predicted representation is a linear combination of radial basis function kernels, significantly reducing computational cost compared to deep learning.

[0093] Compared with Bayesian linear regression, SVM regression prediction method has less dependence on the number of training samples. It still has a good regression model when the number of training samples is small, and its prediction results are decision results rather than posterior probability distribution intervals, which have high certainty. Attached Figure Description

[0094] Figure 1 This is a flowchart of the prediction method between relevant parameters of the present invention;

[0095] Figure 2 This is a graph showing the prediction results when the predicted boundary width is ±1σ.

[0096] Figure 3 This is a graph showing the prediction results when the predicted boundary width is ±2σ. Detailed Implementation

[0097] To better understand the technical content of the present invention, the technical solution of the present invention will be further introduced and explained below with reference to specific embodiments, but is not limited thereto.

[0098] Example 1:

[0099] Combination Figure 1 This embodiment further illustrates a method for predicting material-related parameters based on a random degradation process, including:

[0100] Obtain parameter pairs from sample data of relevant parameters;

[0101] The parameter pairs are used to calculate the sample distribution, thereby obtaining a parameter array that conforms to the distribution and determining the distribution type;

[0102] Construct a distribution model based on the parameter array of the distribution and the determined distribution type;

[0103] Generate new simulated data sequences based on the parameter pairs of the distribution model and relevant parameter sample data;

[0104] The new experimental sample data sequence is used as the training sample data for the SVM regression algorithm. The input vector of the training sample and the mean vector of the training sample are extracted based on the training sample data.

[0105] The regression model is trained using the training sample data to obtain the regression model; the input vector of the training sample is used as the input vector of the test sample, and the model and the mean vector of the training sample are imported into the regression model for prediction, respectively obtaining the relevant parameter prediction output vector and the mean prediction vector.

[0106] The weighted mean square error is calculated separately for the predicted output vector and the output vector of the training samples.

[0107] The confidence interval of the mean prediction vector is calculated based on the weighted mean square error and the mean prediction vector.

[0108] The confidence interval of the predicted mean vector and the mean vector of the training samples are substituted into the linear interpolation method to obtain the confidence interval of the predicted value.

[0109] The reliability of the predicted values ​​of relevant parameters is assessed based on the confidence interval of the predicted values.

[0110] In a further embodiment, the method for obtaining parameter pairs of sample data between relevant parameters is as follows:

[0111] During the random degradation experiment of product materials, multiple experimental samples are set for relevant parameters; among which the relevant parameters are two or more interrelated intrinsic material properties.

[0112] Measure the parameters of multiple experimental samples separately and record the sample data of relevant parameters at different times;

[0113] The sample data of multiple experimental samples are organized into a set of parameter pairs of sample data between relevant parameters in chronological order;

[0114] The expression for the parameter pairs in the relevant parameter sample data is: {(α ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n

[0115] In the formula, α and β represent different properties of the product material and are related to each other, n represents time, m represents the total number of batches of test samples, j represents any time, and i represents the i-th batch of test samples at time j.

[0116] In a further embodiment, the method for calculating the sample distribution of the parameter pairs to obtain a parameter array conforming to the distribution and determining the distribution type includes:

[0117] Extract the array of relevant parameters at the same time point from the set of parameter pairs in the relevant parameter sample data;

[0118] Substitute the relevant parameter array into the probability density function for testing to determine whether the relevant parameters are random distribution variables that conform to the distribution parameters;

[0119] The maximum likelihood ratio is then used to calculate the maximum likelihood ratio values ​​corresponding to the multiple experimental samples that have passed the test.

[0120] The magnitude of the maximum likelihood ratios corresponding to multiple parameter arrays is determined by comparing their values.

[0121] The distribution type of the relevant parameters and the parameter set that conforms to the distribution are determined based on the magnitude of the maximum likelihood ratio;

[0122] The probability density function is:

[0123] f(α|θ αj1 ,θ αj2 ,...,θ αjk (1)

[0124] The expression for testing by substituting the relevant parameter array into the probability density function is:

[0125]

[0126] In the formula, (θ αj1 ,θ αj2 ,...,θ αjk H0 and H1 are the parameter set of the distribution model of parameter α at time t; H0 and H1 are the hypotheses to be tested.

[0127] The formula for calculating the maximum likelihood ratio is:

[0128]

[0129] In the formula, For (θ) αj1 ,θ αj2 ,...,θ αjk The maximum likelihood estimate of ) is given by λ, where λ is the maximum likelihood ratio; the judgment expression for determining the magnitude of multiple maximum likelihood ratios is:

[0130]

[0131] In the formula, K is the critical value.

[0132] In a further embodiment, the distribution parameter set determined according to the maximum likelihood method Given a defined distribution function type, the expression for constructing the distribution model of the material parameters is as follows:

[0133]

[0134] In a further embodiment, the method for generating a new experimental sample data sequence based on the parameters of the distribution model and the relevant parameter sample data includes:

[0135] For the relevant parameters, α and β are established as distribution models L of the parameters at time t, and the Monte Carlo (MC) method is used to generate a set of parameters that conform to the distribution. and Simulated data sequence and Composition of sample simulation data Where m0 represents the number of simulated data at time j.

[0136] The sample data {(α) ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n and simulation data To form a new experimental sample data sequence {(α i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n This effectively includes the statistical information of the relevant parameters in the α and β distribution models. Among them:

[0137] m' = m + m0, representing the total number of batches of new experimental sample data over n time points;

[0138] i' represents the i' batch of experimental samples at time j;

[0139] The constraints on m0 are:

[0140] 40<m+m0≤50 (6)

[0141] In equation (6), if the number of measured sample data m exceeds 50, the Monte Carlo method is not used to generate simulated data; if the number of measured sample data m does not exceed 40, numerical data simulation should be performed appropriately according to the MC method mentioned above so that the sum of the measured sample data and the simulated sample data satisfies equation (6).

[0142] In further implementation, the new experimental sample data sequence is used as the training sample data for the SVM regression algorithm. The method for extracting the input vector of the training samples and calculating the mean vector of the training samples based on the training sample data is as follows:

[0143] From the new experimental sample data sequence {(α i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n Choose one of the easily measurable parameters α as the input vector {α} for the training samples. i',j} i'＝1,2,...m'；j＝1,2,...n And calculate the mean vector of the training samples at n time points.

[0144] In further implementation, to avoid loss of generality, the training process for the regression model in n-dimensional space is as follows:

[0145] Let W be the hyperplane in n-dimensional space. T X+b=0, where W=(w1,w2,...w n ) and b are the eigenvectors and constant term of the hyperplane, respectively, X = (x1, x2, ... x2) n () represents an n-dimensional training sample point vector;

[0146] Let the normal vector of the eigenvector W of the n-dimensional hyperplane be the positive direction distance, and the opposite direction be the negative direction distance; y k As a scalar, when the distance from the training sample point vector to the hyperplane is in the positive direction, y k =1, when y is the distance in the negative direction. k =-1;

[0147] Where the total number of training sample point vectors is r, (r = m'·n), any training sample point X k ∈{X1,X2,...X r The distance d to the hyperplane can be expressed as:

[0148]

[0149] In the formula X k' ∈{X1,X2,...X r} represents the training sample point vector farthest from the hyperplane;

[0150] The purpose of regression training is to obtain the hyperplane W with feature vector W. T X+b=0, which minimizes the distance d from the training sample points to the hyperplane. This is equivalent to finding the optimal solution for the feature vector W in formula (7), which is also the constrained optimization problem of formula (8) as follows:

[0151]

[0152] In equation (8), d' is a constant, determined by the training sample data. It is treated as a constant term in subsequent calculations and does not affect the training results of the regression model.

[0153] For optimal solution problems with constraints, the Lagrange method is usually used, which introduces the Lagrange multiplier vector α = (α1, α2, ... α) r ) T As an undetermined coefficient of the constraint condition in formula (8), its Langran function equation is defined as:

[0154]

[0155] Where α k | k＝1,2,...,r The kth component of the Lagrange multiplier;

[0156] To solve an optimization problem where the constraints in formula (8) are inequalities, the optimal solution must satisfy the following conditions, namely the KKT (Karush-Kuhn-Tucker) conditions:

[0157]

[0158] Combining the Lagrange function (9) and the necessary KKT conditions (10), the above optimal solution problem formula (8) is equivalent to:

[0159]

[0160] Among them, (X) i ·X j () represents the inner product between the vectors of the training sample points;

[0161] By combining the training sample point vector data, the optimal solution of the above formula (11) is solved, and the optimal solution of the Lagrange multipliers is obtained as follows:

[0162] α * =(α1) * ,α2 * ,...,α r * ) T (12)

[0163] The decision function for the regression problem is:

[0164]

[0165] Where α k * b is the k-th component of the optimal solution of the Lagrange multipliers. * For the fitting constant term;

[0166] Select an appropriate radial basis kernel function K(X) i ·X j ) replace the inner product (X) between training sample point vectors i ·X j The optimal solution problem of the above formula (8) can be subjected to nonlinear regression analysis. The radial basis kernel function can be a Gaussian kernel function, a polynomial kernel function, etc.; and a slack variable ξ is introduced. k |k=1,2,...,r and penalty parameter C>0, where slack variable ξ k |k=1,2,...,r represents the degree to which the k-th sample point does not meet the regression conditions, and the penalty parameter C represents the tolerable error of the decision function, making the optimal solution problem of the above formula (8) a nonlinear soft regression problem, that is:

[0167]

[0168] By solving formulas (9) to (12) above, the optimal solution of the Lagrange multipliers is calculated sequentially, and the regression decision function is obtained as follows:

[0169]

[0170] In a further embodiment, the input vector of the training samples is imported into the regression model for training to obtain the output vector of the training samples; and the input vector and mean vector of the training samples are imported into the regression model for prediction to obtain the relevant parameter prediction output vector and mean prediction vector, respectively.

[0171] The input vector of the training samples is imported into the regression model for training, and the output vector of the training samples is obtained. The output vector of the training samples is a parameter array {β} of the difficult-to-measure relevant parameter β. i',j} i'＝1,2,...,m'；j＝1,2,...n ;

[0172] The input vector of the training samples {α i',j} i'＝1,2,...m'；j＝1,2,...n The input vector used as the test sample is imported into the regression model to predict the relevant parameter β and the predicted output vector.

[0173] At the same time, the mean vector of the training samples The new test sample input vector is imported into the regression model for prediction, yielding the mean prediction vector. in,

[0174] In a further embodiment, the weighted mean square error is calculated by separately calculating the predicted output vector and the output vector of the training samples.

[0175] The output vectors {β} of the training samples at different time points are respectively... i',j} i'＝1,2,...,m'；j＝1,2,...n and predicted output vector The weighted mean square error σ j 2 The calculation is as follows:

[0176]

[0177] In the formula, f(·) is the weight function, which has the following properties: f(0) = 0, f(1) = 1, and is monotonically and continuously increasing on [0,1].

[0178] In a further embodiment, the method of substituting the confidence interval of the mean prediction vector and the mean vector of the training samples into the linear interpolation method to obtain the confidence interval of the predicted value includes:

[0179] The weighted root mean square error σ at different times t j The confidence interval for the mean prediction vector is defined as follows:

[0180] Furthermore, based on the confidence level requirements of the actual problem, multiple weighted root mean square error widths are set: the confidence interval can be... The value of n' is determined based on the confidence level requirements of the actual problem.

[0181] In a further embodiment, the method for assessing the reliability of the predicted values ​​of relevant parameters based on the confidence interval of the predicted values ​​is as follows:

[0182] Using any measured parameter α' as the input vector of the regression prediction model, the confidence of its predicted value β' is evaluated; if the predicted output value β' is within the prediction confidence interval... Within this range, the predicted value of the relevant parameter β, which is difficult to measure, is assessed as reliable.

[0183] Example 2:

[0184] Combination Figures 2 to 3 This paper further illustrates the implementation of the "hardness-strength" related parameters of steel, and realizes regression prediction between related physicochemical parameters in the material, so as to be applied to the reliability assessment and analysis of engineering machinery products.

[0185] Step 1: During the random degradation process of engineering machinery products, at n different times, the "hardness-strength" related parameters of m test samples are compared with {(α... ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n , to conduct measurements and collect and organize data.

[0186] Step 2: Apply the "hardness-strength" related parameters obtained in Step 1 to {(α) ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n Using the likelihood ratio test of the distribution, the parameters {α} of m experimental samples at a certain time t are determined respectively. ij | i＝1,2,...,m} and {β ij | i＝1,2,...,m The distribution types of} are identified, such as Gaussian distribution, Weibull distribution, Gamma distribution, etc.; and distribution models L(α) are established for each. j |θ αj1 ,θ αj2 ,...,θ αjk ) and L(β) j |θ βj1 ,θ βj2 ,...,θβjk ), where (θ αj1 ,θ αj2 ,...,θ αjk ) and (θ βj1 ,θ βj2 ,...,θ βjk Let be the model parameter set for the distribution models of parameters α and β at time t. The distribution type and parameters, determined by the likelihood ratio test, are independently determined at different times. Taking parameter α as an example, the specific process of the likelihood ratio test for the distribution is as follows:

[0187] At a certain time t, assume that the parameter α conforms to the distribution of the unknown parameter set (θ). αj1 ,θ αj2 ,...,θ αjk The random distribution variable is given by the probability density function f(α|θ). αj1 ,θ αj2 ,...,θ αjk Consider the verification problem:

[0188]

[0189] In the formula, (θ αj1 ,θ αj2 ,...,θ αjk H0 and H1 are the parameter set of the distribution model of parameter α at time t; H0 and H1 are the hypotheses to be tested.

[0190] For m parameter data, the formula for calculating the maximum likelihood ratio is:

[0191]

[0192] In the formula, For (θ) αj1 ,θ αj2 ,...,θ αjk The maximum likelihood estimate of ) is given by λ, where λ is the maximum likelihood ratio; the judgment expression for determining the magnitude of multiple maximum likelihood ratios is:

[0193]

[0194] In the formula, K is the critical value.

[0195] After determining the distribution type and parameter set of parameter α by the magnitude of the likelihood ratio λ, its distribution model at time t can be expressed as:

[0196]

[0197] Step 3: Based on the parameter distribution model L determined in Step 2 at time t, generate a set of parameters that conforms to the distribution using the Monte Carlo method. and Simulated data sequence and Composition of sample simulation data Where m0 represents the number of simulated data at time j.

[0198] The sample data {(α) ij ,β ij )}| i＝1,2,...,m；j＝1,2,...n and simulation data To form a new experimental sample data sequence {(α i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n This effectively includes the statistical information of the relevant parameters in the α and β distribution models. Among them:

[0199] m' = m + m0, representing the total number of batches of new experimental sample data over n time points;

[0200] i' represents the i' batch of experimental samples at time j;

[0201] The constraints on m0 are:

[0202] 40<m+m0≤50 (5)

[0203] In equation (5), if the number of measured sample data m exceeds 50, the Monte Carlo method is not used to generate simulated data; if the number of measured sample data m does not exceed 40, numerical data simulation should be performed appropriately according to the MC method mentioned above so that the sum of the measured sample data and the simulated sample data satisfies equation (5).

[0204] Step 4: The new simulation data sequence {(α) generated in Step 3 is used to generate the new simulation data sequence {(α)}. i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n As training sample data in the SVM regression algorithm, since the hardness of steel is relatively easy to measure, {α} is used. i',j} i'＝1,2,...m'；j＝1,2,...n For the input vector of the training samples, the intensity parameter {β}, which is difficult to measure, is... i',j} i'＝1,2,...,m'；j＝1,2,...n The output vector, used as the training sample, is used to train the regression model; the specific training process is as follows:

[0205] Let the equation of the two-dimensional regression line be W T X + b = 0, where W = (w1, w2) and b are the eigenvector and constant term of the regression curve, respectively, and X = (x1, x2) = {(α...} i'j ,βi'j )}| i'＝1,2,...,m'；j＝1,2,...n The training sample point vector.

[0206] Let the normal vector of the eigenvector W of the two-dimensional regression line be the positive direction distance, and the opposite direction be the negative direction distance; y k As a scalar, y represents the distance from the training sample point vector to the two-dimensional regression line in the positive direction. k =1, when the distance is in the negative direction.

[0207] y k =-1;

[0208] Where the total number of training sample point vectors is r, (r = m'·n), any training sample point X k ∈{X1,X2,...X r The distance d to the hyperplane can be expressed as:

[0209]

[0210] In the formula X k' ∈{X1,X2,...X r} represents the training sample point vector farthest from the hyperplane;

[0211] The purpose of regression training is to obtain the hyperplane W with feature vector W. T X+b=0, which minimizes the distance d from the training sample points to the hyperplane. This is equivalent to finding the optimal solution for the feature vector W in formula (6), which is also the constrained optimization problem of formula (7) as follows:

[0212]

[0213] In equation (7), d' is a constant determined by the training sample data. It is treated as a constant term in subsequent calculations and does not affect the training results of the regression model.

[0214] For optimal solution problems with constraints, the Lagrange method is usually used, which introduces the Lagrange multiplier vector α = (α1, α2, ... α) r ) T As an undetermined coefficient of the constraint condition in formula (7), its Langran function equation is defined as:

[0215]

[0216] Where α k | k＝1,2,...,r The kth component of the Lagrange multiplier;

[0217] To solve an optimization problem where the constraints in formula (7) are inequalities, the optimal solution must satisfy the following conditions, namely the KKT (Karush-Kuhn-Tucker) conditions:

[0218]

[0219] Combining the Lagrange function (8) and the necessary KKT conditions (9), the above optimal solution problem formula (7) is equivalent to:

[0220]

[0221] Among them, (X) i ·X j () represents the inner product between the vectors of the training sample points;

[0222] By combining the training sample point vector data, the optimal solution of the above formula (10) is solved, and the optimal solution of the Lagrange multipliers is obtained as follows:

[0223] α * =(α1) * ,α2 * ,...,α r * ) T (11)

[0224] The decision function for the regression problem is:

[0225]

[0226] Where α k * b is the k-th component of the optimal solution of the Lagrange multipliers. * For the fitting constant term;

[0227] Select an appropriate radial basis kernel function K(X) i ·X j ) replace the inner product (X) between training sample point vectors i ·X j The optimal solution problem of the above formula (7) can be subjected to nonlinear regression analysis. The radial basis kernel function can be a Gaussian kernel function, a polynomial kernel function, etc.; and a slack variable ξ is introduced. k |k=1,2,...,r and penalty parameter C>0, where slack variable ξ k |k=1,2,...,r represents the degree to which the k-th sample point does not meet the regression conditions, and the penalty parameter C represents the tolerable error of the decision function, making the optimal solution problem of the above formula (7) a nonlinear soft regression problem, that is:

[0228]

[0229] By solving formulas (8) to (11) above, the optimal solution of the Lagrange multipliers is calculated sequentially, and the regression decision function is obtained as follows:

[0230]

[0231] Step 5: Based on the training regression model from Step 4, the training samples {(α)} are processed. i'j ,β i'j )}| i'＝1,2,...,m'；j＝1,2,...n The parameter vector {α i',j} i'＝1,2,...m'；j＝1,2,...n The input vector, used as the test sample, is used to make predictions through a regression model to obtain the predicted output vector. On the other hand, the mean vector of training samples at n time points As the new test sample input vector, obtain the new prediction vector for the test sample. These are referred to as the mean test vector and the mean prediction vector, respectively.

[0232] Step 6: Based on Steps 4 and 5, calculate the output vector {β} of the training samples at different times t. i',j} i'＝1,2,...,m'；j＝1,2,...n and the output vector of the test sample The weighted mean square error σ j 2 The calculation process is as follows:

[0233]

[0234] Where f(·) is the weight function, its properties are: f(0)=0, f(1)=1, and it is monotonically and continuously increasing on [0,1], as shown below. wait.

[0235] Step 7: Based on steps 5 and 6, calculate the mean prediction vector. Confidence interval: based on the weighted mean square error σ in step 6 j 2 The weighted root mean square error σ at different times t j This serves as the boundary width of the confidence interval for the mean prediction vector, i.e., the confidence interval is... Multiple weighted root mean square error widths can also be set according to the confidence level requirements of the actual problem, that is, the confidence interval can be... The value of n' is determined based on the confidence level requirement of the actual problem, such as technical... Figure 2 and Figure 3 As shown in the image.

[0236] Step 8: Based on Step 7, for any measured hardness parameter The confidence interval for its associated intensity parameter β' can be determined by the following two steps using the linear interpolation method:

[0237] like The confidence interval width σ' of the corresponding prediction parameter β0' is calculated as follows:

[0238]

[0239] Where σ j and σ j+1 They are respectively and The width of the corresponding confidence interval;

[0240] The confidence interval for the predicted value β0' is expressed as follows: Its feature prediction value The calculation method is as follows:

[0241]

[0242] in and They are respectively and The corresponding predicted value.

[0243] Step 9: Based on Step 8, use any measured parameter α' as the input vector of the regression prediction model and evaluate the confidence of its predicted value β'; if the predicted output value β' is within the confidence interval calculated in Step 8...

[0244] If the predicted values ​​of the relevant parameters are within the acceptable range, then the predicted values ​​are considered acceptable. Figure 2 As shown in the image.

[0245] Compared with the multinomial regression method, the SVM regression method can control the predicted values ​​of the test samples within a high confidence range and avoid the problem of overfitting, achieving a good balance between prediction accuracy and practicality.

[0246] Compared to deep learning regression methods, SVM regression prediction methods still perform well in nonlinear regression problems even with limited training samples, demonstrating greater flexibility and wider applicability. Furthermore, SVM eliminates the need for a two-step mapping process; its transformation is completed in a single step using a kernel function, and the predicted representation is a linear combination of radial basis function kernels, significantly reducing computational cost compared to deep learning.

[0247] Compared with Bayesian linear regression, SVM regression prediction method has less dependence on the number of training samples. It still has a good regression model when the number of training samples is small. Moreover, its prediction results are decision results rather than posterior probability distribution intervals, which have high certainty.

[0248] In summary, the SVM regression prediction method of this invention can accurately predict and evaluate experimental data of relevant parameters that are difficult to measure during the degradation process of materials used in mechanical engineering. It can realize regression prediction between physicochemical parameters with correlation in materials. For nonlinear regression analysis problems with random perturbations in experimental data and small amounts of data with random perturbations, the Support Vector Machine (SVM) regression algorithm has the advantage of being highly flexible in its applicability, so that it can be applied to the reliability assessment and analysis of engineering machinery products.

[0249] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0250] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0251] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0252] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0253] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the technical principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for predicting between material-related parameters based on a stochastic degradation process, characterized in that, The method comprises the following steps: obtaining a parameter pair of sample data between hardness and strength related parameters of steel materials; performing sample distribution calculation on the parameter pair to obtain a parameter array conforming to a distribution and determine a distribution type; constructing a distribution model according to the parameter array conforming to the distribution and the determined distribution type; generating a simulation data sequence according to the distribution model and the parameter pair of sample data between hardness and strength related parameters, and combining the simulation data sequence with measured sample data to form a new test sample data sequence; using the new test sample data sequence as SVM regression algorithm training sample data, extracting an input vector of the training sample data and calculating a mean vector of the training sample data according to the training sample data; inputting the input vector of the training sample data into a regression model to obtain an output vector of the training sample data, and inputting the input vector of the training sample data and the mean vector of the training sample data into the regression model to obtain a prediction output vector and a mean prediction vector of the hardness and strength related parameters; performing calculation on the prediction output vector and the output vector of the training sample data to obtain a weighted mean square error; performing calculation on the weighted mean square error and the mean prediction vector to obtain a confidence interval of the mean prediction vector; performing calculation on the confidence interval of the mean prediction vector and the mean vector of the training sample data by using a linear difference method to obtain a prediction value confidence interval; According to the prediction value confidence interval, the reliability of the prediction value of the hardness and strength related parameters of the steel material is evaluated, which is used for reliability evaluation analysis of engineering machinery products; wherein the hardness parameter is ; and the strength parameter is .

2. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 1, characterized in that, The method for obtaining the parameter pair of sample data between hardness and strength related parameters comprises the following steps: during a random degradation experiment of product materials, a plurality of experimental samples are set for hardness and strength related parameters; the hardness and strength related parameters of the plurality of experimental samples are measured and recorded at different time points; the sample data of the plurality of experimental samples are sorted in time sequence to form a parameter pair set of sample data between hardness and strength related parameters; The expression of the parameter pair of the sample data of the hardness and the strength-related parameter is: ; wherein, , respectively represent the hardness and the strength of the product material and are correlated parameters, n represents the time, m represents the total batch of the test sample, represents an arbitrary time, represents the batch test sample at the time, time.

3. The method for prediction between material related parameters based on stochastic degradation process according to claim 1, wherein, the method for performing sample distribution calculation on the parameter pair to obtain a parameter array conforming to a distribution and determine a distribution type comprises the following steps: extracting a hardness and strength related parameter array at the same time point from the parameter pair set of sample data between hardness and strength related parameters; inputting the hardness and strength related parameter array into a probability density function to perform test and determine whether the hardness and strength related parameters are random distribution variables conforming to distribution parameters; calculating the parameter arrays of the plurality of experimental samples of the hardness and strength related parameters that pass the test by using maximum likelihood ratio in sequence to obtain maximum likelihood ratio values corresponding to the plurality of parameter arrays; determining the sizes of the maximum likelihood ratio values by comparing the maximum likelihood ratio values; determining the distribution type of the hardness and strength related parameters and the parameter group conforming to the distribution according to the sizes of the maximum likelihood ratio values; wherein the hardness and strength related parameters The expression of the array into the probability density function is: (1) Hardness and strength related parameters The expression for the array substitution probability density function test is: (2) wherein is a parameter a set of parameters of the distribution model at time t; and is the hypothesis problem to be tested; The calculation formula of the maximum likelihood ratio is: (3) In the formula, is the maximum likelihood estimate, is the maximum likelihood ratio; wherein the judgment expression for determining the size of the plurality of maximum likelihood ratios is: (4) In the formula, is a critical value.

4. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 3, characterized in that, The method for constructing a distribution model based on the parameter array of the distribution and the determined distribution type is as follows: The distribution parameter array is determined using the maximum likelihood method. Given a defined distribution function type, the expression for constructing the distribution model of material parameters is as follows: (5)。 5. The method for prediction between material related parameters based on stochastic degradation process according to claim 1, wherein, The method for generating the new test sample data sequence according to the distribution model and the parameter pair of sample data between hardness and strength related parameters comprises the following steps: For the hardness and strength related parameters, , The distribution models of the parameters at time t are established respectively And the Monte Carlo (MC) method is used to generate simulated data sequences and that conform to the distribution parameter groups and to form sample simulated data , wherein, represents the number of simulated data at time. The sample data and simulation data are combined to form a new test sample data sequence which effectively contains the hardness and strength correlation parameter pairs and statistical information in the distribution model; wherein: , indicates the total batch of new test sample data in one time. representing at the time point batch experimental samples; The constraints are: (6) In formula (6), if the measured sample data amount is more than 50, the Monte Carlo method is not used to generate simulation data; if the measured sample data amount is not more than 40, the numerical data simulation is appropriately performed according to the above MC method, so that the sum of the measured sample and the simulation sample data satisfies formula (6).

6. The method for prediction between material related parameters based on stochastic degradation process according to claim 1, wherein, using the new test sample data sequence as SVM regression algorithm training sample data, extracting an input vector of the training sample data and calculating a mean vector of the training sample data according to the training sample data. from a new test sample data sequence selecting one of the parameters which is easy to measure as the input vector of the training sample and calculating the mean vector of the training sample at n time points .

7. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 6, characterized in that, The input vector of the training sample is introduced into the regression model training to obtain an output vector of the training sample; and the input vector of the training sample and the mean vector of the training sample are introduced into the regression model for prediction to obtain a hardness and strength correlation parameter prediction output vector and a mean prediction vector, respectively. The training sample data sequence is imported into a two-dimensional space nonlinear equation, trained by a SVM regression algorithm, and a regression model is obtained The training sample data sequence is imported into a two-dimensional space nonlinear equation, trained by a SVM regression algorithm, and a regression model is obtained ; and the input vector of the training sample is imported into the regression model to predict the hardness and strength related parameters as the input vector of the test sample to obtain the hardness and strength related parameters predicted by the regression model the prediction output vector ; At the same time, the training sample mean vector As a new test sample input vector, the regression model prediction is imported to obtain the mean prediction vector ; wherein .

8. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 7, characterized in that, The prediction output vector and the output vector of the training sample are calculated respectively to obtain a weighted mean square error, and the method comprises the following steps: The output vectors of the training samples at different times are respectively and the predicted output vectors The weighted mean square error is calculated, and the calculation formula is as follows: (7) wherein is a weight function with the property that: , and is monotonically continuously increasing on [0,1].

9. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 8, characterized in that, The confidence interval of the mean prediction vector and the mean vector of the training sample are substituted into a linear difference method to obtain a prediction value confidence interval, and the method comprises the following steps: The weighted root mean square error at different t moments The confidence interval as the boundary width of the confidence interval of the mean prediction vector is ; Meanwhile, according to the confidence requirement of the actual problem, the width of the weighted root mean square error is set to be multiple, and the confidence interval can be wherein The value of the number is determined according to the confidence requirement of the actual problem.

10. The method of prediction between material-related parameters based on a stochastic degradation process according to claim 9, characterized in that, The reliability of the hardness and strength correlation parameter prediction value is evaluated according to the prediction value confidence interval, and the method comprises the following steps: any measured parameter as an input vector to a regression prediction model, whose predicted value is evaluated for its reliability; If the predicted value output quantity is located within the prediction value confidence interval , the predicted value of the hardness and strength related parameter that is difficult to measure is assessed as reliable.

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