CN112036586A - Statistical distribution inspection method for aviation equipment maintenance equipment demand - Google Patents

Statistical distribution inspection method for aviation equipment maintenance equipment demand Download PDF

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CN112036586A
CN112036586A CN202010825768.8A CN202010825768A CN112036586A CN 112036586 A CN112036586 A CN 112036586A CN 202010825768 A CN202010825768 A CN 202010825768A CN 112036586 A CN112036586 A CN 112036586A
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郭峰
赵宏强
孙中华
徐凤磊
张素琴
郭星香
高伟
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Qingdao Campus of Naval Aviation University of PLA
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Abstract

The invention relates to a statistical distribution inspection method for aviation equipment maintenance equipment demand, which comprises the following steps: collecting historical fault data required by various typical equipment, and making a historical fault data sample set according to the historical fault data; drawing a histogram of the requirement of each typical equipment according to the historical fault data sample set; obtaining the statistical distribution of the requirements of each typical equipment according to the histogram fitting of the requirements of each typical equipment; the statistical distribution of the requirements of each typical equipment is tested by a chi-square test method; testing the statistical distribution of the requirements of each typical equipment by adopting a K-S testing method; and judging the statistical distribution of the requirements of each typical equipment according to the test result of the chi-square test method and the test result of the K-S test method. The invention adopts two detection methods of chi-square detection and K-S detection to detect the distribution of equipment requirements, so as to avoid false abandon or false extraction errors as much as possible, ensure the correctness of detection and improve the accuracy and reliability of detection results.

Description

Statistical distribution inspection method for aviation equipment maintenance equipment demand
Technical Field
The invention belongs to the technical field of aviation maintenance support, and particularly relates to a statistical distribution inspection method for aviation equipment maintenance equipment demand.
Background
At present, many application studies on the distribution of aviation equipment maintenance equipment needs are carried out at home and abroad, different documents adopt different statistical distributions for the same equipment, and the A.A. Syntetos is equal to that in 2012, statistical distribution test studies such as Poisson distribution, normal distribution, gamma distribution and the like are carried out on the needs of military equipment spare parts and electronic equipment spare parts in the United states, the United kingdom and the Europe by adopting a chi-square distribution test method. They set the observation period to 1 month, and the sample observation time was 84 months maximum and 48 months minimum. In practice, however, the observation period should not be too short, otherwise a large variation in the difference-to-average ratio may result, which may have a large effect on the distribution test results. However, they do not provide sample data and verification processes. Lengu D equals 2014, studies on the goodness of fit of various composite poisson distributions and proposes a distribution-based classification method, but does not specify statistical distributions to which various equipment is subjected. The prior literature assumes, without demonstration, statistical distributions of different classes of equipment, and does not verify and account for other distributions that are appropriate for the proposed class of equipment.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a statistical distribution testing method for aviation equipment maintenance equipment requirements, which can improve the accuracy and reliability of testing results.
In order to achieve the aim, the invention provides a statistical distribution testing method for aviation equipment maintenance equipment requirements, which comprises the following steps of:
collecting historical fault data required by various typical equipment, and making a historical fault data sample set according to the historical fault data;
drawing a histogram of the requirement of each typical equipment according to the historical fault data sample set;
obtaining the statistical distribution of the requirements of each typical equipment according to the histogram fitting of the requirements of each typical equipment;
the statistical distribution of the requirements of each typical equipment is tested by a chi-square test method;
testing the statistical distribution of the requirements of each typical equipment by adopting a K-S testing method;
and judging the statistical distribution of the requirements of each typical equipment according to the test result of the chi-square test method and the test result of the K-S test method.
Preferably, when the statistical distribution of each typical equipment is judged, if the test result of the chi-square test method is consistent with the test result of the K-S test method, the statistical distribution of the requirements of the typical equipment is determined to obey the statistical distribution obtained according to the histogram; and if the test result of the chi-square test method is inconsistent with the test result of the K-S test method, determining that the statistical distribution of the typical equipment requirements is not compliant with the statistical distribution obtained according to the histogram.
Preferably, the checking by the chi-square checking method comprises the following steps:
suppose that:
H0: distribution function of X is F0(x)(1)
Where X is the historical fault data sample set for typical equipment requirements, F0(x) Obtaining a statistical distribution function of typical equipment requirements for the histogram, wherein X is a sample in a historical fault data sample set X;
test statistic χ2Comprises the following steps:
Figure BDA0002636104010000021
wherein n is the total number of samples x, niFor observed values equal to xiThe frequency of actual measurement, xiIs the ith sample，piTo theoretical probability, npiK is the theoretical frequency, and k is the packet number;
H0the receiving domain is:
χ2≤χ2 1-α(k-1) (3)
in the formula, x2 1-α(k-1) as test statistic χ2A is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (3).
Preferably, when the actual statistical distribution of the typical equipment demands is determined according to the formula (3), if the formula (3) is satisfied, the actual statistical distribution of the typical equipment demands is determined to comply with the statistical distribution of the typical equipment demands obtained by the histogram, and if the formula (3) is not satisfied, the actual statistical distribution of the typical equipment demands is determined not to comply with the statistical distribution of the typical equipment demands obtained by the histogram.
Preferably, the step of performing the test by using the K-S test method comprises:
suppose that:
H0：F(x)＝F0(x) (4)
where F (x) is the actual statistical distribution of typical equipment requirements;
test statistic D is:
Figure BDA0002636104010000022
in the formula (I), the compound is shown in the specification,
Figure BDA0002636104010000031
for the actual accumulated frequency, miIs observed to be less than or equal to xiThe total number of n samples x;
H0the receiving domain is:
D≤Dα (6)
in the formula, DαA distribution critical value of the test statistic D, wherein alpha is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (6).
Preferably, when the actual statistical distribution of the typical equipment demands is determined according to the formula (6), if the formula (6) is satisfied, the actual statistical distribution of the typical equipment demands is determined to comply with the statistical distribution of the typical equipment demands obtained by the histogram, and if the formula (6) is not satisfied, the actual statistical distribution of the typical equipment demands is determined not to comply with the statistical distribution of the typical equipment demands obtained by the histogram.
Preferably, when the statistical distribution of each typical equipment is two or more, if the test result of the chi-square test method and the test result of the K-S test method of each statistical distribution are consistent, the statistical distribution with the maximum P value is selected as the final statistical distribution of the typical equipment, wherein the P value is determined by the test statistic χ2Or the test statistic D.
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention comprehensively uses two testing methods of chi-square testing and K-S testing to test the distribution of equipment requirements, so as to avoid false abandon or false extraction errors as much as possible, ensure the correctness of the testing and improve the accuracy and reliability of the testing result. The method lays a scientific theoretical basis for researches such as demand prediction, formulation of measure supply standards, inventory optimization and the like of aviation equipment by using a probability statistics method, and has high scientificity and popularization and application values.
(2) The invention carries out the empirical test by judging whether the requirements of various typical equipments comply with the same distribution and whether the requirements of the same equipment comply with various distributions. According to the test results, the requirements of most typical equipment obey Poisson distribution, and the requirements of the same typical equipment can obey various distributions, so that the problem that the equipment requirement distribution assumptions in the existing literature are contradictory, lack of example verification and difficult to serve as scientific basis is solved.
Drawings
FIG. 1 is a histogram of a type of accumulator for a type A aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 2 is a histogram of a centrifugal pump of a type of aircraft maintenance equipment according to an embodiment of the present invention;
FIG. 3 is a histogram of a sensor of a type for a type A aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 4 is a histogram of a type of radio station for type A aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 5 is a histogram of a type of altimeter for a type A aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 6 is a top histogram of a type of aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 7 is a histogram of a servo valve of a type B aircraft maintenance equipment according to an embodiment of the present invention;
FIG. 8 is a histogram of a processor of a type B aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 9 is a top histogram of a type B aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 10 is a histogram of a type C aircraft maintenance equipment type conditioner according to an embodiment of the present invention;
FIG. 11 is a histogram of a receiver of a type C aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 12 is a histogram of a certain type of electromagnetic valve of a D-type aircraft maintenance equipment according to an embodiment of the present invention;
FIG. 13 is a histogram of a type D fuel pump of an aircraft maintenance rig in accordance with an embodiment of the present invention;
FIG. 14 is a histogram of a centrifugal pump of a type E aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 15 is a histogram of a control box of a type E aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 16 is a histogram of a compass of a type E for aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 17 is a histogram of a transceiver of a type E aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 18 is a histogram of a sensor of a type E aircraft maintenance equipment in accordance with an embodiment of the present invention;
FIG. 19 is a histogram of a type E aircraft maintenance equipment instrument of an embodiment of the present invention;
FIG. 20 is a block diagram of a starter according to an embodiment of the present invention;
FIG. 21 is a graph fitting a normal distribution, a Weibull distribution, and an exponential distribution according to an embodiment of the present invention;
FIG. 22 is a fitting curve of Poisson distribution and negative binomial distribution according to an embodiment of the present invention.
Detailed Description
The invention is described in detail below by way of exemplary embodiments. It should be understood, however, that elements, structures and features of one embodiment may be beneficially incorporated in other embodiments without further recitation.
The invention provides a statistical distribution inspection method for aviation equipment maintenance equipment demand, which comprises the following steps:
and S1, collecting historical fault data of various typical equipment requirements, and making a historical fault data sample set according to the historical fault data.
And S2, drawing a histogram of the requirement of each typical equipment according to the historical fault data sample set.
And S3, obtaining the statistical distribution of the demand of each typical equipment according to the histogram fitting of the demand of each typical equipment.
And S4, testing the statistical distribution of the requirement of each typical equipment by adopting a chi-square test method. The method comprises the following specific steps:
suppose that:
H0: distribution function of X is F0(x) (1)
Where X is the historical fault data sample set for typical equipment requirements, F0(x) Obtaining a statistical distribution function of typical equipment requirements for the histogram, wherein X is a sample in a historical fault data sample set X;
test statistic χ2Comprises the following steps:
Figure BDA0002636104010000051
wherein n is the total number of samples x, niFor observed values equal to xiThe frequency of actual measurement, xiIs the ith sampleThis, piTo theoretical probability, npiK is the theoretical frequency, and k is the packet number;
H0the receiving domain is:
χ2≤χ2 1-α(k-1) (3)
in the formula, x2 1-α(k-1) as test statistic χ2A is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (3).
Specifically, when the actual statistical distribution of the typical equipment demands is determined according to the formula (3), if the formula (3) is satisfied, the actual statistical distribution of the typical equipment demands is determined to comply with the statistical distribution of the typical equipment demands obtained by the histogram, and if the formula (3) is not satisfied, the actual statistical distribution of the typical equipment demands is determined not to comply with the statistical distribution of the typical equipment demands obtained by the histogram.
Generally, the larger the theoretical frequency, the closer the distribution is to the chi-squared distribution, and the better the chi-squared distribution is when the theoretical frequency is greater than or equal to 5. Therefore, when the expected theoretical frequency is small, it is generally necessary to merge adjacent groups to try to make the theoretical frequency of each group not less than 5. In this example, the significance level α is 0.05.
S5, testing the statistical distribution of the requirement of each typical equipment by adopting a K-S test method. The method comprises the following specific steps:
suppose that:
H0：F(x)＝F0(x) (4)
where F (x) is the actual statistical distribution of typical equipment requirements;
test statistic D is:
Figure BDA0002636104010000061
in the formula (I), the compound is shown in the specification,
Figure BDA0002636104010000062
for the actual accumulated frequency, miIs observed to be less than or equal to xiThe total number of n samples x;
H0the receiving domain is:
D≤Dα (6)
in the formula, DαA distribution critical value of the test statistic D, wherein alpha is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (6).
Specifically, when the actual statistical distribution of the typical equipment demands is determined according to the formula (6), if the formula (6) is satisfied, the actual statistical distribution of the typical equipment demands is determined to comply with the statistical distribution of the typical equipment demands obtained by the histogram, and if the formula (6) is not satisfied, the actual statistical distribution of the typical equipment demands is determined not to comply with the statistical distribution of the typical equipment demands obtained by the histogram.
And S6, judging the statistical distribution of the requirements of each typical equipment according to the test result of the chi-square test method and the test result of the K-S test method.
Specifically, when the statistical distribution of each typical equipment is judged, if the chi-square test result is consistent with the K-S test result, the statistical distribution of the typical equipment requirements is determined to obey the statistical distribution obtained according to the histogram; if the results of the chi-squared test and the K-S test are inconsistent, then it is determined that the statistical distribution of such typical equipment requirements is not amenable to statistical distribution derived from the histogram.
It should be noted that, when the statistical distribution of each typical equipment is two or more, if the test result of the chi-square test method and the test result of the K-S test method of each statistical distribution are consistent, the statistical distribution with the largest P value is selected as the final statistical distribution of the typical equipment, wherein the P value is determined by the test statistic χ2Or the test statistic D. Specifically, when the statistical distribution is judged by the P value, if P is<Alpha, the type of equipment does not obey the statistical distribution obtained according to the histogram, and if P is larger than or equal to alpha, the type of equipment obeys the statistical distribution obtained according to the histogramAnd (5) counting distribution.
The chi-square test has a wide application range. However, when the expected frequency is small, the chi-square test needs to merge adjacent groups and then calculate, which results in some information loss of the sample. The K-S check does not require grouping and therefore retains more information. In fact, in either method of verification, it is possible to obtain the opposite result from the previous one if the confidence level or the number of samples is changed, and chi-squared verification is also affected by the grouping. In order to ensure the correctness of the detection, the method of the invention adopts two methods of chi-square detection and K-S detection to carry out nonparametric hypothesis detection on various statistical distributions, thereby reducing the incidence rate of false abandon or false extraction errors.
The above-mentioned method of the present invention is described in detail below with reference to specific embodiments.
Example 1: take A type airplane maintenance equipment as an example
The airplane has small changes in equipment strength and flight mission quantity, the statistical age of fault data samples is 2003-2017, the sample capacity counted in half a year is 30, and the inspection requirements are met.
(1) Certain type of pressure accumulator
The equipment is mechanical equipment, and the fault data sample of the equipment is shown in table 1.
TABLE 1
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 1 6 3 2 1 6 4 0 4 1 2 4 2 7 1 2 4 0 5 1 0 2 0 2 2 0 1 5 2
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 1.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 2.3333. The results of the chi-square test are shown in table 2.
TABLE 2
Figure BDA0002636104010000071
Statistic χ of chi-square test2＝3.7503，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(2) Centrifugal pump of certain type
The equipment was electromechanical equipment and the fault data samples are shown in table 3.
TABLE 3
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 1 0 0 3 4 1 5 6 0 2 6 9 4 1 8 3 5 6 7 2 6 11 1 2 5 8 8 0 6 7
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 2.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 4.2333. The results of the chi-square test are shown in table 4.
TABLE 4
Figure BDA0002636104010000081
Statistic χ of chi-square test2＝3.4757，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(3) Certain type of sensor
The equipment is an ad hoc equipment and the fault data samples are shown in table 5.
TABLE 5
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 1 1 1 6 2 14 14 11 8 10 8 14 9 9 4 6 5 8 1 7 4 5 5 7 12 17 6 21 2
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 3.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 4 groups. The desired average is 7.2667. The results of the chi-square test are shown in table 6.
TABLE 6
Figure BDA0002636104010000082
Figure BDA0002636104010000091
Statistic χ of chi-square test2＝4.9993，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(3) 7.8147. Obviously, χ2<χ2 0.95(3) The equipment requirements are subject to a poisson distribution.
(4) Certain type radio station
The equipment was ad hoc and the fault data samples are shown in table 7.
TABLE 7
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 1 6 6 4 6 7 4 11 5 11 9 5 8 10 5 14 2 10 8 8 5 8 10 14 15 7 9 9 5
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 4.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The expected mean value is 7.4. The results of the chi-square test are shown in table 8.
TABLE 8
Figure BDA0002636104010000092
Statistic χ of chi-square test2＝2.2177，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(5) Certain type altimeter
The equipment is an ad hoc equipment and the fault data samples are shown in table 9.
TABLE 9
i 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 1 3 6 5 11 8 11 9 2 1 4 11 9 9 6 2 6 0 4 10 7 12 7 5 17 6 8 2
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 5.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 6.1333. The results of the chi-square test are shown in table 10.
Watch 10
Figure BDA0002636104010000101
Statistic χ of chi-square test2＝4.4212，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2)＝5.9915。Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(6) Certain type top
The equipment is an ad hoc equipment and the fault data samples are shown in table 11.
TABLE 11
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 2 6 1 3 4 2 6 6 9 11 7 10 8 8 9 6 4 5 5 7 8 8 10 9 8 5 8 10 0
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in figure 6.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 4 groups. The desired average is 6.1667. The results of the chi-square test are shown in table 12.
TABLE 12
Figure BDA0002636104010000102
Statistic χ of chi-square test2＝1.3821，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(3) 7.8147. Obviously, χ2<χ2 0.95(3) The equipment requirements are subject to a poisson distribution.
Example 2: take B type airplane maintenance equipment as an example
The airplane has the advantages that the equipment strength and the flight mission volume of the airplane do not change greatly, the statistical age of fault data samples is 2005-2017, the sample capacity counted by half year is 26, and the inspection requirements are basically met.
(1) Certain type of servo valve
The fixture was an electromechanical fixture and its fault data samples are shown in table 13.
Watch 13
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 6 2 6 5 5 3 8 8 1 6 5 1 2 9 9 0 7 4 1 1 0 4 2 2 4 5 9 2
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in FIG. 7.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 5.1923. The results of the chi-square test are shown in table 14.
TABLE 14
Figure BDA0002636104010000111
Statistic χ of chi-square test2＝1.078，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(2) Type processor
The equipment was ad hoc and the fault data samples are shown in table 15.
Watch 15
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 0 0 1 0 1 1 8 3 6 4 8 7 2 5 1 2 5 7 3 3 3 3 3 2 2 3 4
And drawing a histogram according to the actual measurement frequency of the equipment faults, and obtaining the statistical distribution of the equipment requirements as Poisson distribution as shown in FIG. 8.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 3.6923. The results of the chi-square test are shown in table 16.
TABLE 16
Figure BDA0002636104010000121
Statistic χ of chi-square test2＝0.0829，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(3) Certain type top
The equipment was ad hoc and the fault data samples are shown in table 17.
TABLE 17
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 0 1 1 4 0 3 4 3 5 3 7 7 5 6 4 2 2 4 4 6 5 7 7 3 2 3
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 9.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 3.7692. The results of the chi-square test are shown in table 18.
Watch 18
Figure BDA0002636104010000122
Statistic χ of chi-square test2＝0.0491，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
Example 3: case of C-shaped airplane maintenance equipment
The airplane has the advantages that the equipment strength and the flight mission volume of the airplane do not change greatly, the statistical age of fault data samples is 2005-2017, the sample capacity counted by half year is 26, and the inspection requirements are basically met.
(1) Certain type regulator
The fixture was an electromechanical fixture and its fault data samples are shown in table 19.
Watch 19
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 1 1 2 0 0 2 9 4 2 2 4 0 0 3 2 2 1 0 6 2 4 4 9 2 2 2
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 10.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 2.5385. The results of the chi-square test are shown in table 20.
Watch 20
Figure BDA0002636104010000131
Statistic χ of chi-square test2＝0.2306，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(2) Receiver of a certain type
The equipment is ad hoc and the fault data samples are shown in table 21.
TABLE 21
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 0 2 0 0 0 2 0 1 3 2 1 1 0 4 4 0 2 4 4 6 1 3 4 1 1 8 9 4
A histogram is drawn according to the measured frequency of the equipment fault, and as shown in fig. 11, the statistical distribution of the equipment demand is obtained as poisson distribution.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 3.2692. The results of the chi-square test are shown in table 22.
TABLE 22
Figure BDA0002636104010000141
Statistic χ of chi-square test2＝1.8388，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
Example 4: take D type airplane maintenance equipment as an example
The airplane has the advantages that the equipment strength and the flight mission volume of the airplane do not change greatly, the statistical age of fault data samples is 2005-2017, the sample capacity counted by half year is 26, and the inspection requirements are basically met.
(1) Electromagnetic valve
The fixture was an electromechanical fixture and its fault data samples are shown in table 23.
TABLE 23
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 1 0 6 3 2 2 1 0 6 4 3 4 5 3 5 6 2 1 4 3 3 2 2 2 3 1
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 12.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 2.8462. The results of the chi-square test are shown in table 24.
Watch 24
Figure BDA0002636104010000142
Figure BDA0002636104010000151
Statistic χ of chi-square test2＝0.0734，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(2) Certain type fuel pump
The fixture was an electromechanical fixture and its fault data samples are shown in table 25.
TABLE 25
i 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 26
X i 1 2 2 1 1 5 6 5 1 1 4 4 1 4 7 2 2 9 4 2 7 4 3 3 7 4 3
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 13.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average value is 4. The results of the chi-square test are shown in table 26.
Watch 26
Figure BDA0002636104010000152
Statistic χ of chi-square test2＝2.992，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
Example 5: take E type airplane maintenance equipment as an example
The capacity and the flight mission volume of the airplane equipment do not change greatly, the statistical year of fault data samples is 2005-2018, the sample capacity counted according to the half year is 27, and the specific equipment of the airplane is subjected to statistical distribution inspection.
(1) Centrifugal pump of certain type
The fixture was an electromechanical fixture and its fault data samples are shown in table 27.
Watch 27
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 3 1 5 6 0 2 6 9 4 1 8 3 4 6 5 2 6 10 1 2 4 8 8 0 6 8 5
A histogram is drawn according to the measured frequency of the equipment fault, and as shown in fig. 14, the statistical distribution of the equipment demand is obtained as poisson distribution.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 4.5556. The results of the chi-square test are shown in table 28.
Watch 28
Figure BDA0002636104010000161
Statistic χ of chi-square test2＝2.4388，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(2) Control box of certain type
The equipment is ad hoc and the fault data samples are shown in table 29.
Watch 29
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 0 0 2 5 4 5 4 6 3 5 1 2 6 4 5 4 8 12 4 7 5 11 3 14 12 8 3
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 15.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 5.2963. The results of the chi-square test are shown in table 30.
Watch 30
Figure BDA0002636104010000162
Figure BDA0002636104010000171
Statistic χ of chi-square test2＝0.767，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(3) Certain type compass
The equipment is ad hoc and the fault data samples are shown in table 31.
Watch 31
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 4 4 4 5 8 2 3 8 8 4 0 1 4 5 4 0 1 1 3 2 4 2 8 4 1 3 0
A histogram is drawn according to the measured frequency of the equipment fault, and as shown in fig. 16, the statistical distribution of the equipment demand is obtained as poisson distribution.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 3.4444. The results of the chi-square test are shown in table 32.
Watch 32
Figure BDA0002636104010000172
Statistic χ of chi-square test2＝0.3046，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(4) Transceiver of a certain type
The equipment is ad hoc and the fault data samples are shown in table 33.
Watch 33
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 3 6 6 3 5 8 5 3 3 6 3 2 1 3 1 1 2 1 5 2 7 5 4 5 4 5 5
Drawing a histogram according to the actual measurement frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 17.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 3.8519. The results of the chi-square test are shown in table 34.
Watch 34
Figure BDA0002636104010000181
Statistic χ of chi-square test2＝1.5154，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(5) Certain type of sensor
The equipment is ad hoc and the fault data samples are shown in table 35.
Watch 35
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 6 2 15 14 14 9 10 8 14 8 9 3 6 5 7 1 7 4 5 5 7 12 18 6 7 4 2
A histogram is drawn according to the measured frequency of the equipment fault, and as shown in fig. 18, the statistical distribution of the equipment demand is obtained as poisson distribution.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The desired average is 7.7037. The results of the chi-square test are shown in table 36.
Watch 36
Figure BDA0002636104010000182
Statistic χ of chi-square test2＝2.391，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
(6) Certain type instrument
The equipment was ad hoc and the fault data samples are shown in table 37.
Watch 37
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
X i 8 11 12 13 10 8 17 11 8 8 11 14 13 12 6 7 4 5 6 6 3 13 10 8 8 17 8
Drawing a histogram according to the measured frequency of the equipment fault, and obtaining the statistical distribution of the equipment demand as poisson distribution as shown in fig. 19.
Under the condition that the theoretical frequency of each group is not less than 5, the samples are divided into 3 groups. The expected mean value is 9.5185 and the results of the chi-square test are shown in table 38.
Watch 38
Figure BDA0002636104010000191
Statistic χ of chi-square test2＝0.3215，χ2Critical value of distribution χ2 0.95(k-1)＝χ2 0.95(2) 5.9915. Obviously, χ2<χ2 0.95(2) The equipment requirements are subject to a poisson distribution.
To further verify the results of the chi-square test, a K-S test was performed using statistical analysis software SPSS Statistics on typical equipment for type five aircraft in the 5 examples described above, with the results shown in table 39.
It can be seen from the table that the P values of all the appliances are greater than 0.05, and the requirements of the appliances are subject to poisson distribution, thereby further verifying the results of the chi-square distribution.
Watch 39
Figure BDA0002636104010000192
Figure BDA0002636104010000201
Example 6: taking a certain starter as an example, a non-parametric hypothesis test is performed on several more common statistical distributions by using statistical analysis software easy fit.
This type of starter is an electromechanical device and its fault data samples are shown in table 40.
Watch 40
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
X i 0 0 3 3 5 2 4 4 9 4 6 8 7 8 6 11 8 8 17 14 9 11 9 13 12 12 9 8 4 3
A histogram is drawn according to the actual measurement frequency of the equipment fault, and as shown in fig. 20, the obtained demand statistical distribution of the equipment may be continuous distribution such as normal distribution, weibull distribution, exponential distribution and the like, and discrete distribution such as poisson distribution, negative binomial distribution and the like.
1) Continuously distributed
(1) Normal distribution
Assuming that the starter demand follows a normal distribution, the results of the K-S test and the chi-squared test are shown in table 41.
It can be seen from the table that the P-value is greater than significance levels of 0.01, 0.02, 0.05, 0.1, 0.2, either by the K-S test or the chi-square test, which indicates that the material obeys a normal distribution regardless of the significance level.
Table 41
Figure BDA0002636104010000211
(2) Weibull distribution
Assuming that the starter requirements are subject to a Weibull distribution, the results of the K-S test and the Chi-Square test are shown in Table 42.
As can be seen from the table, the K-S test obeys the weibull distribution under the condition of significance levels of 0.01, 0.02, 0.05, 0.1, 0.2, and the chi-square test does not obey the weibull distribution only under the condition of significance level of 0.2.
Watch 42
Figure BDA0002636104010000221
(3) Distribution of index
Assuming that the starter requirements follow an exponential distribution, the results of the K-S test and the Chi-squared test are shown in Table 43.
As can be seen from the table, the K-S test obeys the exponential distribution under the condition of significance level of 0.01, 0.02, 0.05, and the chi-square test obeys the exponential distribution only under the condition of significance level of 0.01.
Watch 43
Figure BDA0002636104010000222
Figure BDA0002636104010000231
2) Discrete distribution
(1) Poisson distribution
Assuming that the starter requirements are subject to a poisson distribution, a K-S test is used and the results are shown in table 44.
As can be seen from the table, the starter obeys a poisson distribution at significance levels 0.01, 0.02, 0.05.
Watch 44
Figure BDA0002636104010000232
Figure BDA0002636104010000241
(2) Distribution of negative binomial
Assuming that the starter requirements follow a negative binomial distribution, a K-S test was used and the results are shown in Table 45.
As can be seen from the table, the starter follows a binomial distribution only at significance levels 0.01, 0.02.
TABLE 45
Figure BDA0002636104010000242
In addition, the difference average ratio of the negative binomial distribution is greater than 1, the difference average ratio of the binomial distribution is smaller than 1, and only one of the negative binomial distribution and the binomial distribution can accept the original assumption, so that the starter requirement is not in compliance with the binomial distribution.
It should be noted that the sample observation period of the starter is half a year, and if the observation period is one year, the standard deviation thereof is larger, that is, the fluctuation of the sample data of the fault thereof is larger or the dispersion degree of the number of the faults thereof is higher. As can be seen from Table 40, a certain number of the failures in serial number 19 in the equipment failure comparison set are cleared on site, and no spare parts are replaced, i.e., the failures do not require spare parts. Because the faults of the equipment are complex and various and the faults which can be eliminated on site are difficult to predict, the standards made by adopting fault data are larger. The equipment support department considers that the standards are properly larger, the requirements of the equipment can be better met, and the standards made by actually adopting fault data are reasonable and basically acceptable. However, there are also equipment with fewer replacement spare parts and more field failures, and the standard and actual deviation made by the number of failures are larger. Therefore, when the standard of supply of the equipment is formulated, the fault data of the quality control room of the crew is firstly adopted, and the repair and payment data of the shipping stock is secondly adopted.
Comparing K-S test statistics of various distributions, it can be seen that in three continuous distributions of normal distribution, Weibull distribution and exponential distribution, the test statistics of normal distribution is the smallest, and the test statistics of exponential distribution is the largest, which indicates that the fitness of normal distribution is the highest and the fitness of exponential distribution is the lowest; it can also be seen that in the two discrete distributions, Poisson distribution and negative binomial distribution, the fitting degree of Poisson distribution is higher, and the fitting degree of negative binomial distribution is lower. For the starter, under the condition that the significance level is 0.05, the normal distribution, the Weibull distribution, the Poisson distribution and the exponential distribution all accept the original hypothesis, so the method can be used for measuring and calculating equipment requirements.
The fitting curves of different distributions are shown in fig. 21 and 22, and it can be seen that the difference between the exponential distribution and the actual distribution is large, the poisson distribution, the normal distribution and the weibull distribution are relatively close to the actual distribution, and the poisson distribution, the normal distribution and the weibull distribution curves are closest.
In addition, as can be seen from fig. 21, the exponential distribution curve is much different from the actual, but at a significance level of 0.05, the chi-square test accepts the assumption that the exponential distribution is obeyed, but the K-S assumption does not. It should be noted that if the P value is greater than 0.05, it cannot indicate that the equipment failure does not follow the exponential distribution, but only indicates that the probability is low.
The above-described embodiments are intended to illustrate rather than to limit the invention, and any modifications and variations of the present invention are possible within the spirit and scope of the claims.

Claims (7)

1. A statistical distribution inspection method for aviation equipment maintenance equipment demand is characterized by comprising the following steps: collecting historical fault data required by various typical equipment, and making a historical fault data sample set according to the historical fault data;
drawing a histogram of the requirement of each typical equipment according to the historical fault data sample set;
obtaining the statistical distribution of the requirements of each typical equipment according to the histogram fitting of the requirements of each typical equipment;
the statistical distribution of the requirements of each typical equipment is tested by a chi-square test method;
testing the statistical distribution of the requirements of each typical equipment by adopting a K-S testing method;
and judging the statistical distribution of the requirements of each typical equipment according to the test result of the chi-square test method and the test result of the K-S test method.
2. The statistical distribution testing method for the requirements of the aviation equipment maintenance equipment according to claim 1, wherein when the statistical distribution of each typical equipment is judged, if the testing result of the chi-square testing method is consistent with the testing result of the K-S testing method, the statistical distribution of the requirements of the typical equipment is determined to be subjected to the statistical distribution obtained according to the histogram; and if the test result of the chi-square test method is inconsistent with the test result of the K-S test method, determining that the statistical distribution of the typical equipment requirements is not compliant with the statistical distribution obtained according to the histogram.
3. The method for statistical distribution of aircraft equipment maintenance equipment requirements as defined in claim 2, wherein the checking by the chi-square method comprises the steps of:
suppose that:
H0: distribution function of X is F0(x) (1)
Where X is the historical fault data sample set for typical equipment requirements, F0(x) Obtaining a statistical distribution function of typical equipment requirements for the histogram, wherein X is a sample in a historical fault data sample set X;
test statistic χ2Comprises the following steps:
Figure FDA0002636101000000011
wherein n is the total number of samples x, niFor observed values equal to xiThe frequency of actual measurement, xiFor the ith sample, piTo theoretical probability, npiK is the theoretical frequency, and k is the packet number;
H0the receiving domain is:
χ2≤χ2 1-α(k-1) (3)
in the formula, x2 1-α(k-1) as test statistic χ2A is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (3).
4. The method according to claim 3, wherein when the actual statistical distribution of the typical equipment requirements is determined according to the formula (3), if the formula (3) is satisfied, the actual statistical distribution of the typical equipment requirements is determined to comply with the statistical distribution of the typical equipment requirements obtained by the histogram, and if the formula (3) is not satisfied, the actual statistical distribution of the typical equipment requirements is determined not to comply with the statistical distribution of the typical equipment requirements obtained by the histogram.
5. The method for statistical distribution of aircraft equipment maintenance equipment requirements as defined in claim 3, wherein the step of performing the verification using the K-S verification method comprises:
suppose that:
H0：F(x)＝F0(x) (4)
where F (x) is the actual statistical distribution of typical equipment requirements;
test statistic D is:
Figure FDA0002636101000000021
in the formula (I), the compound is shown in the specification,
Figure FDA0002636101000000022
for the actual accumulated frequency, miIs observed to be less than or equal to xiThe total number of n samples x;
H0the receiving domain is:
D≤Dα (6)
in the formula, DαA distribution critical value of the test statistic D, wherein alpha is a significance level;
and (4) judging whether the actual statistical distribution of the typical equipment demands complies with the statistical distribution of the typical equipment demands obtained by the histogram according to the formula (6).
6. The method according to claim 5, wherein when the actual statistical distribution of the typical equipment requirements is determined according to the formula (6), if the formula (6) is satisfied, the actual statistical distribution of the typical equipment requirements is determined to comply with the statistical distribution of the typical equipment requirements obtained by the histogram, and if the formula (6) is not satisfied, the actual statistical distribution of the typical equipment requirements is determined not to comply with the statistical distribution of the typical equipment requirements obtained by the histogram.
7. The method according to claim 6, wherein when the statistical distribution of each typical equipment is two or more, if the chi-square test result of each statistical distribution and the K-S test result of each statistical distribution are consistent, the statistical distribution with the maximum P value is selected as the final statistical distribution of the typical equipment, wherein the P value is determined by the test statistic χ2Or the test statistic D.
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