Sensitivity analysis method of computer numerical calculation software considering input parameter distribution rule

Abstract

The invention discloses a sensitivity analysis method for computer numerical calculation software considering an input parameter distribution rule, belongs to the technical field of computer numerical calculation and system sensitivity analysis, and is suitable for scenes such as engineering simulation. The method aims to overcome the defect that an existing gradient method neglects input parameter distribution imbalance, and input-output parameter relevance is quantized through entropy in the information theory. The method comprises the specific steps of determining a parameter combination according to input parameter use frequency, substituting the parameter combination into software to obtain output parameters, counting state frequency distribution of the output parameters, calculating unconditional entropies and conditional entropies of the output parameters so as to obtain uncertainty when the input parameters are removed, and obtaining the input parameters through drawing an uncertainty change trend chart. And judging the input parameter sensitivity according to the slope. The method highlights the influence of high-frequency value points, is accurate in quantification uncertainty, is suitable for a multi-model scene, is visual and easy to understand in result, and can provide technical support for output parameter regulation and control.

Description

Technical Field

[0001] This invention relates to the field of computer numerical calculation and system sensitivity analysis technology, and is specifically applicable to scenarios such as engineering simulation, industrial data analysis, and system optimization. It aims to accurately identify the input parameters with the highest sensitivity to output parameters by quantifying the distribution law of input parameters on output parameters, and to provide technical support for the control of output parameters. Background Technology

[0002] Computer numerical calculation software, given certain input parameters, can calculate corresponding output parameters to meet the data processing needs of various industries. However, these software programs typically have multiple input parameters, and the degree of influence of these input parameters on the output parameters varies. It is necessary to identify the few input parameters that have the greatest impact on the output parameters in order to adjust and control the output parameters.

[0003] Studying the influence of input parameters on output parameters is essentially a sensitivity analysis problem. The gradient method is commonly used in the calculation, and mathematically it can be written as:

[0004]

[0005] However, this method struggles to reflect the impact of the uneven distribution of input parameters on the output parameters. Since input parameters exhibit a certain distribution pattern within their range, some values ​​are used more frequently than others. Relatively speaking, the more frequently used values ​​have a greater impact on the output parameters. The gradient method, on the other hand, considers the impact of all input parameter values ​​on the output parameters evenly across the range, neglecting the influence of the uneven distribution of input parameters on the sensitivity of the output parameters. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of existing gradient methods that "ignore the distribution patterns of input parameters" and to provide a sensitivity analysis method that considers the frequency of use of input parameters. This method quantifies the correlation between input and output parameters by using "unconditional entropy" and "conditional entropy" from information theory and identifies highly sensitive input parameters by combining the trend of uncertainty changes.

[0007] The present invention adopts the following technical solution:

[0008] This invention treats input and output parameters as "information sources" and uses "entropy" to describe the uncertainty of the output parameters:

[0009] Unconditional entropy H(Y): The uncertainty of the output parameters when the distribution of input parameters is unknown;

[0010] Conditional entropy H(Y|X): The uncertainty of the output parameters when the distribution of input parameters is known;

[0011] Uncertainty U: Normalized by conditional entropy, used to compare the impact of input parameters on uncertainty;

[0012] The sensitivity of the input parameters can be determined by the slope of the change in uncertainty after removing the input parameters—the larger the slope, the higher the sensitivity.

[0013] The specific steps are as follows:

[0014] Step 1: Determine multiple combinations of input parameters based on their usage frequency.

[0015] Step 2: Substitute the input parameters into the software to obtain the output parameters.

[0016] Step 3: Statistically analyze the frequency distribution of each state of the output parameters.

[0017] Step 4: Calculate the unconditional entropy H(Y) of the output parameters.

[0018] Step 5: Calculate the conditional entropy H(Y|X) of the output parameters.

[0019] - Input parameter states and probabilities are determined.

[0020] -- Calculation of joint probability and conditional probability.

[0021] --Conditional entropy calculation.

[0022] - Calculation of conditional entropy for multiple models.

[0023] Step 6: Calculate the uncertainty U when input parameters are removed.

[0024] Step 7: Plot the trend of uncertainty and determine the sensitivity.

[0025] The positive effects of this invention are as follows:

[0026] 1) Consider the distribution pattern of input parameters: By mapping "usage frequency-probability", the influence of high-frequency value points is highlighted, avoiding the bias of the "uniform distribution assumption" of the gradient method;

[0027] 2) Quantifying uncertainty: The uncertainty of the output parameters is accurately described using entropy theory, rather than the "local derivative" of the gradient method, and the results are more in line with the actual scenario;

[0028] 3) Applicable to multi-model scenarios: By extending to combinations of multiple sub-models through the link theorem, it overcomes the limitation of existing methods that are only applicable to a single model;

[0029] 4) Intuitive results: Sensitivity can be directly judged by the slope of the uncertainty trend graph without the need for complicated derivative calculations, making it easy for engineers to understand and apply. Attached Figure Description

[0030] Fig. 1 Schematic diagram of the calculation process

[0031] Fig. 2 Schematic diagram of the combined model

[0032] Fig. 3 A schematic diagram illustrating the trend of uncertainty changes when input parameters are removed. (Detailed implementation method) Detailed Implementation

[0033] As attached Figs. 1-3 As shown, Embodiment 1 of the present invention is as follows:

[0034] 1. Determine the input and output parameters.

[0035] Input parameters (3):

[0036] X_1: Material elastic modulus (GPa), historical statistics show that its value distribution is "200±5GPa (frequency 60%)", "215±10GPa (30%)", "other (10%)";

[0037] X_2: Cross-sectional area (m²) 2 The value distribution is "0.01±0.001m". 2 (70%)" "0.013±0.002m" 2 (30%)”;

[0038] X_3: Load magnitude (kN), with values ​​distributed as “100kN (65%)”, “150kN (25%)”, and “200kN (10%)”;

[0039] Output parameter Y: Maximum stress of the simply supported beam (MPa), a continuous value, with an analysis accuracy requirement of ±5MPa.

[0040] 2. Implementation Steps

[0041] 2.1 Step 1: Generate input parameter combinations

[0042] 1) Adjust sampling weights according to usage frequency: For X_1, the "200±5GPa" interval was sampled 60 times, "215±10GPa" 30 times, and "other" 10 times; for X_2, the "0.01±0.001m" interval was sampled 10 times. 2 "70 times,"0.013±0.002m 2 30 times; X_3's "100kN" 65 times, "150kN" 25 times, "200kN" 10 times;

[0043] 2) Latin hypercube sampling is used to generate 100 sets of input parameter combinations (covering the high-frequency range of each parameter).

[0044] 2.2 Step Two: Substitute the parameters into the software to obtain the output parameters

[0045] 1) Convert 100 sets of input parameters into the format of the structural mechanics analysis software (convert the unit of elastic modulus to Pa, i.e., 200GPa=2×10^11Pa);

[0046] 2) Calculate in batches using the APDL software script to obtain 100 output parameters Y (maximum stress value, range 50-350MPa);

[0047] 3) Remove one set of abnormal data (Y = 450MPa, because the input parameter exceeds the software's calculation range), and retain 99 sets of valid data.

[0048] 2.3 Step 3: Statistical analysis of output parameter state frequency

[0049] Divide the ranges into ±5MPa intervals to obtain the output parameter states: 50-55MPa (b_1), 55-60MPa (b_2), ..., 345-350MPa (b_60); count the number of occurrences in each interval, for example, b_10 (95-100MPa) occurs 12 times, b_20 (145-150MPa) occurs 18 times, and calculate the frequency (e.g., the frequency of b_10 = 12 / 99 ≈ 0.121).

[0050] Step 4: Calculate the unconditional entropy H(Y)

[0051] 1) State symbol table B = (b_1, b_2, ..., b_60), probability vector Q = (q_1, q_2, ..., q_60) (q_j is the frequency of each interval);

[0052] 2) Calculate H(Y) = -∑(j=1 to 60)q_jlogq_j≈5.8 bits according to formula (2);

[0053] 3) H_max = log60 ≈ 5.9 bits (because when the 60 states are uniformly distributed, q_j = 1 / 60).

[0054] Step 5: Calculate the conditional entropy H(Y|X_1,X_2,X_3)

[0055] 1) Input parameter state division:

[0056] X_1: a_1(200±5GPa), a_2(215±10GPa), a_3(other), probability P=(0.6,0.3,0.1);

[0057] X_2:a_4(0.01±0.001m 2), a_5(0.013±0.002m 2 The probability P = (0.7, 0.3) is given.

[0058] X_3: a_6 (100kN), a_7 (150kN), a_8 (200kN), probability P = (0.65, 0.25, 0.1);

[0059] 2) Calculation of conditional probability: Count the number of times "input parameter state combination → output parameter state". For example, the number of times a_1(X_1)+a_4(X_2)+a_6(X_3) corresponds to b_10(95-100MPa) is 8. The total number of times this input combination is 15. Then P(b_10|a_1,a_4,a_6)=8 / 15≈0.533;

[0060] 3) Calculate H(Y|X_1,X_2,X_3)≈2.1 bits according to formula (3).

[0061] 2.6 Step Six: Calculate the uncertainty after removing input parameters

[0062] 1)U_ful l=H(Y|X_1,X_2,X_3) / H_max=2.1 / 5.9≈0.356;

[0063] 2) Remove X_1: Calculate H(Y|X_2,X_3)≈3.8 bits, U_1=3.8 / 5.9≈0.644;

[0064] Remove X_2: Calculate H(Y|X_1,X_3)≈2.9 bits, U_2=2.9 / 5.9≈0.492;

[0065] Remove X_3: Calculate H(Y|X_1,X_2)≈2.4 bits, U_3=2.4 / 5.9≈0.407.

[0066] 2.7 Step Seven: Determine Sensitivity

[0067] 1) Draw a trend chart: the horizontal axis is X_1, X_2, X_3, and the vertical axis is U(0.356→0.644→0.492→0.407);

[0068] 2) Calculate the slopes: k_1 = 0.644 - 0.356 = 0.288, k_2 = 0.492 - 0.356 = 0.136, k_3 = 0.407 - 0.356 = 0.051;

[0069] 3) Conclusion: X_1 (material elastic modulus) has the largest slope and the highest sensitivity to the output parameter Y (maximum stress); X_3 (load magnitude) has the lowest sensitivity.

Claims

1. A sensitivity analysis method for computer numerical calculation software considering the distribution of input parameters, characterized by Step one, determining multiple sets of input parameter combinations according to the frequency of use of input parameters; Step two, substituting each set of input parameters into the computer numerical calculation software to obtain the corresponding output parameters; Step three, statistically analyzing each state of the output parameters to obtain the frequency distribution of each state; Step four, calculating the unconditional entropy of the output parameters; Regarding the output parameters as a signal source Y, the numerous states of the output parameters can be regarded as an event set, and the probability of occurrence of a certain state y in the set is P(y). The information amount of this state is calculated as: I(y) = -logP(y) (1) Considering all states of the output parameters, the state symbol table and the probability of occurrence of these states are used, that is: B = (b1, b2,..., b n ) is the state symbol table, Q = (q1, q2,..., q n ) is the probability vector in the probability space; there are 3 cases: Case 1: The computer numerical calculation software outputs a limited number of discrete states. These discrete states can be directly converted into states that satisfy the unconditional entropy calculation, that is, these discrete states are directly mapped to the state symbol table B; Case 2: The computer numerical calculation software outputs an unlimited number of discrete states. Several states that do not have analytical value are uniformly listed as a class, and several states that have analytical value are listed separately. The unlimited number of discrete states is converted into a limited number of discrete states, and on this basis, the state symbol table B is mapped; Case 3: The computer numerical calculation software outputs continuous numerical values. According to the accuracy requirements of the analysis, the continuous numerical values are discretized to form several discrete states, and on this basis, the state symbol table B is mapped; After the above processing, the output parameters of the computer numerical calculation software can be mapped to multiple different discrete states, and the average information amount of the output parameters can be calculated as: That is, the average information amount of the output parameters is represented by the unconditional entropy H(Y); According to the unconditional entropy calculation method, the output parameter conditions of the computer numerical calculation software under various input parameter combinations can be calculated, and then statistical analysis of the output parameters is performed to obtain the frequency of each value state. Finally, formula (2) is used to calculate the unconditional entropy of the output parameters, and the calculation principle is shown in the following formula: (Y1, Y2, …) = f(X1, X2, X3, X4, …) Suppose the computer numerical calculation software has 3 input parameters, X1, X2, X3, whose values are a ij , and 1 output parameter Y, whose value is b k ; a ij According to the value distribution law of the input parameters X i , sampling is determined, and then it is substituted into the computer numerical calculation software, so that the value b k of the output parameter Y can be obtained; finally, b k is counted, that is, the number of values (b1, b2,..., b n ) is counted, on this basis, Q=(q1, q2,..., q n ) is calculated, and the unconditional entropy of the output parameter is calculated by formula (2); The calculated unconditional entropy has two extreme cases: Case 1: If Q obeys uniform distribution, I takes the maximum value, the average information of the output parameter is the minimum, because at this time q j =1 / n, it is impossible to determine which state occurs more, that is, the greater the unconditional entropy, the smaller the average information of the output parameter, and at this time the output parameter has lower available value; the unconditional entropy in this case can be used for uncertainty calculation; Case 2: If there is q j = 1, and I = 0, then the output parameter has no uncertainty because it is completely determined to be b j State occurs, at which time the output parameter has the highest value of usability, i.e., the conclusion corresponding to the minimum value of I, because at this time the analyst is more likely to determine which state is more likely to occur; Step five, calculating the conditional entropy of the output parameters; The input parameter is regarded as a signal source X and the output parameter is regarded as a signal source Y. Since the value of the input parameter conforms to a certain distribution rule, X is a random variable. Y, which changes with X, is also a random variable. That is, the random variables X and Y take values in two discrete sets A=(a1, a2,..., a n ), B=(b1, b2,..., b m ) respectively, and P=(p1, p2,..., p n ) is the probability vector of the signal source X. The output parameter state determination method in the unconditional entropy calculation method is also applicable to the determination of the output parameter state in the conditional entropy calculation method, and the discrete set B can be obtained; In the conditional entropy calculation method, the state determination method of the input parameters also needs to be considered, that is, which states are divided into each input parameter; According to the analysis needs, the input parameters are divided into several available states, at this time the discrete set A can be obtained, and P=(p1, p2,..., pn) is determined according to the use frequency of each state. n The basic principle for determining P is as follows: 1) If the frequency of use of each state is clear, P can be directly determined; 2) If the frequency of use of each state is not clear, and the input parameter is a variable related to the number of events, the Poisson distribution is used; 3) If the frequency of use of each state is not clear, and the input parameter is a variable related to time, such as equipment failure time, the exponential distribution or Rayleigh distribution is used; 4) If the frequency of each state is not clear, and the input parameter is a variable related to the static properties of things, the normal distribution is used; 5) If the frequency of each state is not clear, and it cannot be matched with the above principles, the uniform distribution is used; Obviously, X and Y are related, and they constitute a composite event set, which constitutes a two-dimensional product probability space, that is: where C ij is a i composite event consisting of the intersection of a j and b ij ; r i is the probability of a j and b ij occurring together, i.e., the joint probability of a i and b j : r ij = p(a i ) = p j ) = p i P ji , P ji = P(b j | a i ) Conditional probability P ji = P(b j | a i ) The calculation principle shown in Table 1 is adopted, that is, the function relationship Y = f(X1, X2, X3) described by computer numerical calculation software is expressed in conditional norm (Y|X1, X2, X3), and on this basis, the number of (Y|X1, X2, X3) under various value combinations is counted, so that the conditional probability P(Y|X1, X2, X3) can be calculated, that is, P ji = P(b j | a i ); The known a can be obtained according to the following formula i under the condition b j the conditional entropy, i.e.: where H m (P 1i , P 2i ,..., P mi ) is the entropy of the conditional probabilities of b i given a j , computed using equation (2); The evaluation criteria of conditional entropy and unconditional entropy are the same, that is, to obtain the output parameter with smaller conditional entropy; Step six, calculate the uncertainty of each input parameter when it is removed For a certain computer numerical calculation software, the minimum value of conditional entropy in theory is 0, at this time the output result of the model is deterministic; the maximum value in theory is H max , at this time the probability of each state of the output parameter of the model is equal; the conditional entropy of the output parameter under the support of the input parameter (X1, X2,...) is H(Y|X1, X2,...), and the uncertainty U is: Uncertainty has the following basic properties: ① 0 ≤ U ≤ 1, normalize the conditional entropy, which is convenient for comparing the degree of influence of each input parameter on the uncertainty of the system model; ② When H(Y|X1,X2,...) = 0, there is no uncertainty relationship in the model, at this time U = 0, which indicates that the relationship between the input parameter and the output parameter is deterministic; ③ When H(Y|X1,X2,...) = H max U = 1, which indicates that the relationship between the input parameters and the output parameters is the worst in certainty, and the relationship between the input parameters and the output parameters is described in the most conservative way. Step seven, draw the uncertainty trend chart Based on the uncertainty calculation method, draw the uncertainty trend chart. For computer numerical calculation supported by multiple input parameters, with the decrease of input parameters, the conditional entropy is non-decreasing, that is: H(Y|X1,X2,...) ≤ H(Y|X2,...), therefore the uncertainty trend chart gradually rises with the increase of the number of removed input parameters; According to the slope of each line segment, the influence degree of each input parameter on the output parameter can be judged, the greater the slope, the greater the influence degree of the input parameter.

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