Dynamical Bayesian Significance Testing for Information on Performance Variation of Rolling Bearing for Space Applications

A dynamical Bayesian significance testing method is proposed to examine information on performance variation of rolling bearings for space applications under the condition of an unknown probability distribution and trend in advance. Sub-series of time series of rolling bearing performance are obtained via a regularly sampling, probability density functions of sub-series are acquired with bootstrap and maximum entropy theory, a referenced sequence from sub-series is found by minimum variance principle, posterior probability density function is established according to Bayesian theory, and mutation probability is defined in the light of fuzzy set theory. At the given significance level, dynamical Bayesian significance testing for information on performance variation of rolling bearings is put into effect with the help of mutation probability. Experimental investigation presents that the method proposed can effectively detect variation information of rolling bearing performance with unknown probability distributions and trends.

data are obtained.Let X r stand for the rth time series that is given by   (1), (2),..., ( ),..., ( ) ; 1, 2,..., where x r (h) is the hth datum in X r ; h is a sequence number, h = 1, 2, …, H; and H is the number of data in X r .
The rth time series X r is divided into D sub-series and the dth sub-series is given by   (1), (2),..., ( ),..., ( ) ; 1, 2,..., where x rd (i) stands for the ith datum in X rd ; i for a sequence number, I = 1, 2, …, I; and I for the number of data, which is expressed as According to bootstrap, an equiprobable resampling with replacement from X rd is implemented by following steps: (1) Let the constant B be equal to 500000, and let the variable b take a value 1, where B is the number of the resampling samples and b is the bth equiprobable resampling.
(2) Let one datum be drawn by an equiprobable resampling with replacement from X rd .
(3) Let the step (2) be repeated I times, so that I data can be sampled.
(4) Calculate the mean y rd (b) of I data, which is considered as one of the data in the generated data series Y rd .
(6) If b>B, go to the step (7); otherwise go to the step (2).
(7) Let the generated data series be of size B = 500000, so that many generated data are obtained.
Via steps (1) to ( 7), the generated data series Y rd is gained, as follows: with where θ b (i) is the ith data obtained and y rd (b) is the mean of I data in the bth sampling.
The origin moment of X rd is as follows: where M rd is the highest order of the origin moments and M rdm is the mth order origin moment.
Assume x is a random variable for describing rolling bearing performance data.According to maximum entropy theory, a probability density function f rd (x) is obtained by where c rdk is the kth Lagrangian multiplier about X rd and k = 0,1,…, M rd .
In Equation ( 6), the Lagrangian multiplier c rdk (k = 1,2,…, M rd ) can be solved by The first Lagrangian multiplier c rd0 can be obtained by where R rd is the integrating range of x about X rd .
Let r = 1 in Equation ( 6), then the probability density function of the dth sub-series X 1d in the first time series X 1 is obtained as For the first time series X 1 , let X 1d be both a prior sample and a current sample and f 1d (x) be both a prior distribution and a current sample distribution.According to Bayesian statistics, the posterior probability density function of X 1d is obtained as According to statistics, the mathematical expectation E 1d of X 1d is defined as and the variance D 1d of X 1d is defined as According to the minimum variance principle, the minimum variance D 1min is given by ) ,..., ,..., , min( For the first data series, suppose the sub-series with the minimum variance D 1min is marked by X 1min and the posterior probability density function of X 1min is marked by φ 1min (x).Define X 1min and f 1min (x) as the referenced sequence and the referenced distribution, respectively.
For the rth time series (r = 2, 3, …, R), let X rd and f rd (x) be the current sample and current sample distribution, respectively, then according to Bayesian statistics the posterior probability density function φ rd (x) of X rd is as follows: where R 0 is the integrating range of x.
According to statistics, the mathematical expectation E rd of X rd is defined as and the variance D rd is defined as In the light of concept of intersection of fuzzy sets, a mutation probability α 1,rd is defined as where A(φ rd (x)∩φ 1min (x)) stands for the area of the intersection of φ rd (x) and φ 1min (x).
The mutation probability α 1,rd can take values in [0,1].Let significance level be α=0.1, then significance testing for performance variation of rolling bearings can be conducted. If then variation of X rd is of significance; otherwise, variation of X rd is of no significance.

Case Studies
This case involves with experiment on vibration acceleration of a rolling bearing for space applications.The rolling bearing that was installed on a specialized performance rig worked for 46 days (time interval: 8 November 2010 to 23 December 2010, running conditions of axial load of 49N and of rotational speed of 1000 r/min) and test data, in dB, were sampled 10 times (viz., R = 10), one time every 5 days and 4000 data (viz., H = 4000) every time, as shown in Figures 1 and 2. It is easy to see from Figures 1 and 2 that as time series, information of rolling bearing vibration acceleration presents a complex and variational status, with an unknown probability distribution and trend.
From Figures 1 and 2, every 400 data are considered as a sub-series, viz., I = 400, and 4000 data in the first sub-series X 1 are regarded as prior information that includes ten sub-series, X 1,1 , X 1,2 , …, X 1d , …, X 1,10 (including 400 data in every sub-series).
Using Equations ( 9) to ( 13), the mathematical expectation E 1d and the variance D 1d of X 1d are calculated for selection of the referenced sequence X 1min and results are listed in Table 1.The first sub-series The first sub-series -0.003 5.8993 The second sub-series The second sub-series -0.0045 16.294 The third sub-series The third sub-series -0.0056 3.7922 The fourth sub-series The fourth sub-series -0.0054 1.4836 The fifth sub-series The fifth sub-series -0.0048 2.4824 The sixth sub-series The sixth sub-series -0.0012 8.7588 The seventh sub-series The seventh sub-series -0.0070 9.4744 The eighth sub-series The eighth sub-series -0.0024 43.029 The ninth sub-series The ninth sub-series 0.0005 31.533 The tenth sub-series The tenth sub-series -0.0035 22.149 According to Table 1, the fourth sub-series, viz., X 1min = X 1,4 , is selected as the referenced sequence due to its minimum variance D 1,4 = D 1min .Based on this, with the help of Equations ( 15), (17), and (18), the mathematical expectation, the variance ratio, and the mutation probability are obtained as shown in Figures (3), (4), and (5), respectively.
-   From Figure 5, overall, from beginning of the second sub-series (corresponding to abscissa values 1 to 10 in Figure 5) to end of the third sub-series (corresponding to abscissa values 11 to 20 in Figure 5), the mutation probability is in a rising trend; from beginning of the fourth sub-series (corresponding to abscissa values 21 to 30 in Figure 5) to end of the ninth sub-series (corresponding to abscissa values 71 to 80 in Figure 5), the mutation probability that takes values in the range from 0.03 to 0.32 is in a large fluctuation; and from beginning of the tenth sub-series (corresponding to abscissa values 80 to 90 in Figure 5), the mutation probability that takes values about 0.1 is in a new stability.As a result, variation information of rolling bearing vibration acceleration is tested as follows: (1) From 8 November to 18 November, vibration performance variation becomes gradually significant, showing an early degradation phase; (2) From 23 November to 18 December, vibration performance variation is complex and variable, alternating significance and no significance and revealing a transitional period from early degradation phase to gradual degradation phase; (3) On 23 December, vibration performance variation is not significant, meaning a start of gradual degradation phase.
It can be seen from the above that the method proposed is able to test information on rolling bearing performance variation.

Conclusions
The dynamical Bayesian significance testing method, under the condition of unknown probability distributions and trends in advance, can examine information on rolling bearing performance variation for the early detection of the hidden danger of failure of rolling bearing performance, thus avoiding serious accident.Experimental investigation on vibration acceleration of the rolling bearing for space applications shows correctness of the method.

Figure 1 .Figure 2 .
Figure 1.Experimental data of time series from X 1 to X 5

Figure
Figure 3. Mathematical expectation

Figure
Figure 5. Mutation probability

Table 1 .
Selection of referenced sequence