LU501526B1

LU501526B1 - A small sample data life prediction method

Info

Publication number: LU501526B1
Application number: LU501526A
Authority: LU
Inventors: Yan Su; Hanying SANG; Zhenjun Zhang; Hongcai Chen; Tan Wen; Zhaojing Yang
Original assignee: Univ Nanjing Aeronautics & Astronautics; Nari Group Corp
Priority date: 2022-02-21
Filing date: 2022-02-21
Publication date: 2022-08-24

Abstract

The disclosure relates to a small sample data life prediction method, comprising the following steps: obtaining sample test data by an accelerated life test ,and augmenting the samples by a virtual augmentation method to obtain virtually augmented samples; performing bootstrap expansion for the virtually augmented samples to obtain first expanded samples; fitting the first expanded samples by a least square method to determine a Weibull distribution parameter; performing bootstrap expansion for the first expanded samples after fitting, to obtain second expanded samples; fitting the second expanded samples by the least square method to determine a first parameter and a second parameter of an Arrhenius model; determining mean time between failures according to the Weibull distribution parameter, the first parameter and the second parameter. The disclosure can achieve accelerated life prediction under insufficient sample size.

Description

A SMALL SAMPLE DATA LIFE PREDICTION METHOD LUS01526

TECHNICAL FIELD

[01] The disclosure relates to the life prediction field, and in particular, to a small sample data life prediction method.

BACKGROUND ART

[02] Aiming at life prediction algorithms for small sample accelerated life tests, at present, there is a classical statistical small sample method, a Bayes method for establishing a prediction model by prior information, a bootstrap method for expanding small sample data, a grey support vector machine which combines an artificial neural network and grey theory, etc. The existing algorithms have the following defects: (1) If only the classical statistical method is used, the sample size significantly influences the accuracy thereof. (2) The Bayes method greatly depends on the prior information, which easily causes subjective errors. (3) The bootstrap method is only applicable to single-direction expansion, so that the original small sample data range cannot be exceeded. (4) Under the condition that internal performance degradation research cannot be performed, such methods as the artificial neural network and the grey support vector machine are usually not considered.

[03] Therefore, it is required to provide an accelerated life prediction method that can address insufficient sample size.

SUMMARY

[04] The disclosure aims to provide a small sample data life prediction method to achieve the accelerated life prediction under insufficient sample size.

[05] In order to realize the purpose, the technical solution of the disclosure is as follows.

[06] A small sample data life prediction method comprises the following steps:

[07] obtaining sample test data by an accelerated life test;

[08] augmenting the sample test data by a virtual augmentation method to obtain virtually augmented samples;

[09] performing bootstrap expansion for the virtually augmented samples to obtain first expanded samples;

[10] fitting the first expanded samples by a least square method to determine a Weibull distribution parameter;

[11] performing bootstrap expansion for the first expanded samples after fitting, to obtain second expanded samples;

[12] fitting the second expanded samples by the least square method to determine a first parameter and a second parameter of an Arrhenius model; 1

[13] determining mean time between failures according to the Weibull distribution parameter, the first parameter and the second parameter. LU501526

[14] According to specific embodiments provided by the disclosure, the disclosure has the following technical effects.

[15] Inthe disclosure, the sample test data are obtained by the accelerated life test; then, the sample test data are augmented by the virtual augmentation method, the bootstrap expansion is subsequently performed twice, and fitting is performed twice by the least square method to determine the Weibull distribution parameter as well as the first parameter and the second parameter of the Arrhenius model; finally, the mean time between failures is determined according to the Weibull distribution parameter, the first parameter and the second parameter, so as to achieve the accelerated life prediction under insufficient sample size.

BRIEF DESCRIPTION OF THE DRAWINGS

[16] In order to describe the technical solutions in the embodiments of the disclosure or the relevant art more clearly, the drawings required to be used in descriptions about the embodiments will be simply introduced below, obviously, the drawings described below are only some embodiments of the disclosure, and other drawings can further be obtained by those of ordinary skill in the art according to the drawings without creative work.

[17] FIG. 1 is a flow diagram of a small sample data life prediction method according to the disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[18] The following clearly and completely describes the technical solutions in the embodiments of the disclosure with reference to the accompany drawings in the embodiments of the disclosure. The described embodiments are only part of embodiments of the disclosure rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the disclosure without creative efforts shall fall within the protection scope of the disclosure.

[19] In order to make the purposes, features and advantages of the disclosure become more apparent and easier to understand, the disclosure will be further described in detail below in combination with the drawings and the specific implementation modes.

[20] See FIG. 1. The small sample data life prediction method of the disclosure comprises the following steps:

[21] Step 101: obtaining sample test data by an accelerated life test.

[22] Step 102: augmenting the sample test data by a virtual augmentation method to obtain virtually augmented samples.

[23] Step 103: performing bootstrap expansion for the virtually augmented samples to obtain first expanded samples.

[24] Step 104: fitting the first expanded samples by a least square method to determine a 2

Weibull distribution parameter. LU501526

[25] Step 105: performing bootstrap expansion for the first expanded samples after fitting, to obtain second expanded samples.

[26] Step 106: fitting the second expanded samples by the least square method to determine a first parameter and a second parameter of an Arrhenius model.

[27] Step 107: determining mean time between failures according to the Weibull distribution parameter, the first parameter and the second parameter.

[28] The disclosure is applicable to an accelerated life test under high-temperature constant stress by minimum samples.

[29] For the solution of the accelerated life test under high-temperature constant stress in the disclosure, (D) the minimum sample test data under various temperature stress levels are augmented to more than 10 by semi-empirical virtual augmentation; @) the small samples are expanded into large samples by an improved bootstrap method; ©) the large sample data are subsequently subjected to least square linear fitting to obtain an unknown parameter of a Weibull distribution model required for life prediction, @ the test data under various stress levels are expanded again by the improved bootstrap method and are subjected to the least square linear fitting for the second time, to obtain unknown parameters of an Arrhenius model required for life prediction. When all parameters of the mathematical model are known, MTBF value of a product under normal stress can be directly solved through an MTBF solution formula (the mean time between failures is namely a product life measurement index). In the cases of small samples/minimum subsamples, the estimation accuracy of the disclosure is superior to that of traditional algorithms.

[30] Specific algorithm process steps are as follows:

[31] Step 1: semi-empirical virtual augmentation of minimum subsamples

[32] Minimum sample data with size less than 5, which are obtained directly by a test, are augmented to obtain virtually augmented samples.

[33] Existing data: i groups of minimum sample test data obtained by an accelerated life test under high-temperature constant stress (i is the number of constant stress set in the test.)

[34] Under stress 1: t11, t12, ... (minimum samples with size less than 5, similarly hereinafter) Under stress 2: t21, t22, ... Under stress 1: tir, tio, ...

[35] The only 1 test datum under each stress level is augmented by the semi-empirical virtual augmentation to more than 10 subsamples (13 subsamples are used in the disclosure). The virtually augmented subsamples are required to have a certain randomness, and the difference between the randomness of the virtually augmented subsamples and that of original samples is 3 required to be controlled in a certain range. Therefore, the virtually augmented subsamples are required to meet the following conditions: (1) the mean value of the virtually augmented LU501526 subsamples is equal to that of the original subsamples; (2) the standard deviation of the virtually augmented subsamples is equal to that of the subsamples of similar pieces. The distribution form of historical data of similar pieces can be taken as a reference.

[36] Step 2: bootstrap expansion for virtually augmented samples

[37] The step aims to fully explore the unknown general distribution information carried by existing small samples through bootstrap re-sampling and expand the small samples into large samples, thus avoiding accidental errors caused by statistical analysis under insufficient sample size.

[38] The virtually augmented samples are expanded by the bootstrap method.

[39] Existing data: i groups of virtually augmented samples:

[40] Under stress 1: ti, t12, ..., t113 Under stress 2: t21, to, ..., t213 Under stress 1: tir, ti2, ..., ti13

[41] The i groups of data are expanded by an improved bootstrap method respectively, and the steps are as follows:

[42] Among the i groups of data, one group is taken as an example: under stress 1, ti1, tio, … ti13, in order to distinguish from the expanded data, all to-be-expanded data are marked with asterisk: t11*, t12*, …, ti13*.

[43] (1) A random number 6 uniformly distributed in an interval of [0, 1] is generated by matlab.

[44] (2) 0 =(n-16 and J H0|+1 are set, wherein |0] is a round-down integer of 6. The above aims to generate a natural number that is not greater than the number of original test samples, that is, the two steps determines which data of the original sample data are taken as the benchmark for the randomly generated samples. Therein, 6 is an intermediate parameter, and n is expansion frequency.

[45] (3) A random number x uniformly distributed in an interval of [0,1 /(1-e)] is 1, =| Crx(d, 0). . generated, and / | 7 x( JH ;) | is set. The distance from the random datum to the benchmark sample 1s greater than the distance from the sample data at the adjacent right of the benchmark sample to the benchmark sample by a certain proportion, that is, the newly generated samples are out of the original sample data range.

[46] (4) A random number y uniformly distributed in an interval of [0,1 /(1-0)] is 4 generated, and a I " —>(e ah ) is set. After the benchmark is determined, a random LU501526 datum 1s generated at the right of the sample datum.

[47] | (5)tj and t;+1 are generated new sample data. The above steps are required to be repeated according to specific expansion.

[48] After m cycles, 2m groups of expanded data are generated, that is, there are 13+2m data under each stress level.

[49] Under stress 1: tii, t12, …, t113+2m Under stress 2: t1, t22, …, t213+2m Under stress 1: tir, ti, …, ti13+2m

[50] Step 3: the first least square fitting for samples augmented by semi-empirical augmentation and expanded by improved bootstrap method

[51] The output data of the Step 2 are taken as the input data of the Step 3:

[52] Under stress 1: tii, t12, … tin Under stress 2: t21, t22, …, ton Under stress 1: tir, ti2, …, tin (n=13+2m)

[53] Empirical distribution functions F(t) are respectively constructed for i groups of data according to a general formula, F1(ta)=q/n(q=1,2, ..., n) F2(ta)=q/n(q=1,2, ..., n) Fi(tq)=q/n(q=1,2, ..., n)

[54] Therein, tq is the qth discrete time point, and q represents a sequence from 1 to n.

[55] According to a two-parameter Weibull distribution function form, a linear form can be obtained through taking the logarithm twice:

Flt)=1—exp[-(t/n)"],t>0 (r)=1-expl[~(t/7)"], LU501526 Inlnf1 > F()]" =mlint-mln7

[56] For F(t) data under various stress levels: —1 (577 2 =ato=FOO] 35 set x=Int

[58] The least square linear fitting is performed for n (x, y) data points under various stress levels to fit them into a straight line Y 7 7" +b , and then the following formulae are obtained: a=m b'=-mlnn . That is, the estimated values of the unknown parameters m and n of the 1 p groups of Weibull distribution under i temperature stress levels can be obtained.

[59] (The i temperature stress levels in the test are T1, T2, ..., Ti) T1: M1, M T2: M, M Ti: mi, Ni

[60] The estimated value of parameter m of Weibull distribution is as follows: m=(mi+my+...+m;)/i.

[61] Step 4: the second least square fitting for data under different stress levels after another bootstrap expansion

[62] The step aims to fit the linear relation between 1/T; and Inni; therein, ; is a temperature stress level, and ni is an estimated value obtained through the least square fitting of the data under this stress level in the Step 3. For example, if 4 temperature stress levels are set in the test, there are 4 groups of (1/Ti, Inmi) points; the bootstrap expansion is performed for the 4 groups of points, and then the least square linear fitting is performed to obtain a slope and an intercept of a fitted straight, which are parameters a and b in the Arrhenius model respectively.

[63] Existing data: the estimated value of the Weibull distribution parameter under each stress level, which is obtained in the Step 3: T1: M1, M T2: M, M Ti: mi, Ni 6

[64] According to the Arrhenius model: Inn =a+b/T , 1/T and Inn have linear relation. LU501526 Therefore, the i (1/T, Inn) data points (less than ten stress levels are usually set in the test, and there are less data points) under 1 test stress levels are still expanded by the improved bootstrap method, so as to obtain z data points (more data points).

[65] The least square linear fitting is again used for z data points, the slope and the intercept obtained by fitting are namely the unknown parameters a and b of the Arrhenius model. Through the above 4 steps, the mathematical model of the accelerated life test aims to determine the unknown parameters (that is, m, a and b) of the Arrhenius model and the Weibull distribution, and the MTBF value can be solved directly according to the MTBF solution formula.

[66] A predicted value of the reliability index MTBF under a normal temperature stress level is determined according to the following formula.

CY sb / 4) MTBF =n T|1+—|=e T-T|1+— \ Mn ; \ MA ;

[67] Therein, I” is a gamma function (Euler’s second integral), T is a temperature stress value normally borne by a product .

[68] (1) Introduction of semi-empirical virtual augmentation

[69] Inthe disclosure, the minimum subsamples are augmented to more than 10 by the semi- empirical virtual augmentation, and the historical data information of the similar pieces is taken as the reference to a certain extent (that is, semi-empirical augmentation), so that the augmented samples are relatively approximate to actual data. Meanwhile, the samples are not virtually augmented by an unique specific formula, and the formula can be correspondingly adjusted according to actual situations, so as to be applicable to various actual situations and achieve high flexibility.

[70] (2) Improvement of bootstrap expansion algorithm

[71] The traditional bootstrap expansion algorithm is very simple, and the process is as follows:

[72] Arandom number uniformly distributed in an interval of [0, 1] is generated by matlab;

[73] 0 =(n-16 and | AO] + are set, wherein |0| is a round-down integer of 6; t=t +@—-i+)(t,-1) .

[74] 5% (0 ME ) is set;

[75] According to the specific expansion requirements, the above steps are repeated for n times, so as to obtain the bootstrap re-sampling samples ti=(t1, t2, t3, …, tn).

[76] Therein, ti* is the to-be-expanded original test sample data, t1*, t2*, ..., tn* are the original test sample data which are sorted from small to large, and ti is the new sample data generated by this method. 7

[77] The disclosure widens the expanded sample interval through introducing a significance factor a and adding a coefficient 1/(1-a) in the traditional bootstrap sample generation method, LU501526 so that the limitation of the original data range is avoided. Moreover, the relative deviation can be used as a measurement index to prove that the accuracy of the improved algorithm is superior to that of the original algorithm.

[78] (3) Integration of bootstrap method based on semi-empirical virtual augmentation and least square method

[79] After expansion, the large samples are used for the least square fitting to solve the problem that a single data point cannot be directly linearly fitted under the condition of only 1 minimum subsample, and the relative deviation is significantly reduced.

[80] The disclosure has the following advantages:

[81] 1. High algorithm efficiency

[82] (1) Compared with a traditional maximum likelihood parameter estimation method, the integration of the improved bootstrap method based on semi-empirical virtual augmentation and the least square method can avoid the problems of complex solution and multiple iterations in the maximum likelihood parameter estimation method, and aims to perform simple transformation and linear fitting for the expanded data, thus achieving high algorithm efficiency and fast operation speed.

[83] (2) The improved bootstrap expansion algorithm can perform data expansion toward left and right directions at the same time, and the data generation efficiency is double of that of traditional algorithms; double data can be generated in the same loop nesting or the cycle number can be reduced under the condition of meeting the data generation requirement; therefore, the algorithm program has faster speed.

[84] 2. Cost saving

[85] Small subsamples and even minimum subsample data only having 1 sample can be also used for relatively accurate life prediction; under the same accuracy requirement, the sample size can be reduced to save test cost. Meanwhile, under limited test conditions, the disclosure can also maximally improve the life prediction accuracy.

[86] The embodiments in the specification are described in a progressive manner, each embodiment is focused on differing from the other embodiments, and the same and similar parts of the embodiments are referred to with respect to each other.

[87] The principle and implementation mode of the disclosure have been described herein using specific examples, and the above description of the embodiments is only intended to help understand the method of the disclosure and its core idea. Meanwhile, those of ordinary skill in the art may make variations in terms of specific implementation mode and scope of application in accordance with the idea of the disclosure. In conclusion, the content of the specification shall not be understood as any limitation of the disclosure.

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Claims

WHAT IS CLAIMED IS: LUS01526

1. À small sample data life prediction method, comprising the following steps: obtaining sample test data by an accelerated life test: augmenting the sample test data by a virtual augmentation method to obtain virtually augmented samples; performing bootstrap expansion for the virtually augmented samples to obtain first expanded samples; fitting the first expanded samples by a least square method to determine a Weibull distribution parameter; performing bootstrap expansion for the first expanded samples after fitting, to obtain second expanded samples; fitting the second expanded samples by the least square method to determine a first parameter and a second parameter of an Arrhenius model; determining mean time between failures according to the Weibull distribution parameter, the first parameter and the second parameter.

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