A Successful Envelope Analysis Method for Satellite Products Based on Ground Data

Abstract

This invention proposes a successful envelope analysis method for satellite products based on ground data. It can determine the data type and remove outliers by testing and judging, and finally use control charts to solve for the optimal envelope result. The method includes the following steps: Step 1: Perform a normality test on the ground data of satellite products for each attribute and each operating condition to determine whether the data follows a normal distribution; Step 2: Determine whether the ground data of satellite products for each attribute and each operating condition contains outliers, and perform outlier testing and removal; Step 3: Under each attribute, apply a pairwise bipopulation test to the data under different operating conditions of that attribute to determine whether the data belong to the same population under these operating conditions; Step 4: Determine which control chart to use based on the data under the different tests described in Steps 1 to 3, and construct the control interval; Step 5: Determine whether the Hotelling statistic is within the control chart boundaries based on the constructed control interval.

Description

Technical Field

[0001] This invention relates to a method for successful envelope analysis of satellite products based on ground data, belonging to the field of satellite data processing technology, and is used to provide a reliable basis for correcting data boundary indicators. Background Technology

[0002] Reliability is the foundation of satellite product quality, and high inherent reliability is ensured through reliability design. However, during the manufacturing process, it is often necessary to establish performance indicators to guarantee that the manufactured satellites meet high reliability standards. These indicators often rely on human experience, which can deviate from real-world scenarios, requiring multiple iterations and revisions. Therefore, determining reasonable indicators to guarantee satellite reliability while reducing testing costs is a crucial problem that needs to be addressed.

[0003] Current research on satellite reliability mainly focuses on how to ensure the robust operation of individual components after the satellite is assembled, as it is a complex system. However, research on the successful envelope of ground data from satellite products is insufficient. For example:

[0004] In patent CN113326628A, a single-class support vector machine is used to perform envelope analysis on aerospace data. However, the resulting envelope cannot provide the probability that the correct data lies within the envelope, making it difficult to apply directly. But similar problems exist in industry, such as the successful envelope analysis of satellite products. This problem is known as Statistical Process Control (SPC), a modern quality management approach that uses mathematical statistics (such as control charts) to monitor the production process, promptly identify anomalies, and take corrective measures to eliminate their impact, thus controlling product quality. Since Shewhart proposed using the normal distribution to construct control charts, numerous well-known control charts have emerged, such as mean-standard deviation control charts, moving range control charts, and Hotelling control charts for multivariate analysis.

[0005] In recent studies applying control charts for quality control, patent CN111896279A uses envelope analysis to obtain the upper and lower envelopes of the air gap / slot gap curves of linear motor trains. Patent CN113139078A discloses a method for generating control charts, providing a series of processing steps from data to control charts. Patent CN112326248A constructs Hotelling control charts using the spectral kurtosis and envelope spectral kurtosis of bearings. However, the above studies did not consider the verification of the data itself. Since data may not necessarily follow a normal distribution, and there may be outliers within the data, and different groups of data may be quite similar, such complex internal data conditions need to be seriously considered when applying them.

[0006] The aforementioned studies merely applied the data to control chart processing, neglecting numerous data verification steps, potentially compromising the accuracy of control limits. Wu Yansheng's book, *Data Envelopment Analysis of Successful Flight Samples for Aerospace Products*, provides numerous recommended data envelopment methods for aerospace products, along with suggestions and verification methods for the applicability of certain methods. However, these verifications are insufficient and overlook the complex distributions that may occur in aerospace data. Furthermore, the book does not provide a complete operational procedure for actual engineering projects; therefore, direct application may lead to inaccurate envelope results.

[0007] In summary, current research both domestically and internationally indicates an urgent need for a complete workflow from data verification to the construction of control charts based on the verification results. This workflow should fully utilize the information within the data, be able to process different data distributions, be applicable to a wide range of data types, ensure the accuracy of the envelope results, and, given the data, directly obtain the most suitable control chart envelope results through software, thus automating the processing.

[0008] The control charts mentioned above are quality control charts. Their basic principle is to collect certain quality characteristic data after production, find the steady-state of the production process to create a control chart, and finally use this control chart to monitor the production process online. Currently, the control charts with generally recognized good performance are shown in Table 1.

[0009] Table 1 Typical Control Charts

[0010]

[0011]

[0012] For satellite product ground data that lacks batch concept, the most suitable control chart format is the single-value control chart. Also known as an X-chart, X-single-value control chart, or X-control chart, it is a measurement-based control chart used in Total Quality Management (TQM) to control data based on a single acquired data point. It is suitable for situations where data is scarce and inconvenient to break down, or where measurement is typically difficult or expensive. Therefore, when using subgroups to create a control chart is unsuitable, the subgroup size is set to 1, and a single-value control chart is created.

[0013] Individual value control charts do not provide the average value and variation of product quality, offering limited information, but they eliminate the need for tedious calculations, making them a simple and convenient way to determine process stability. Currently, the main data situations suitable for using individual value control charts are as follows:

[0014] (1) The cost of obtaining sample data is very high or the time interval between obtaining sample data is very long:

[0015] (2) There is no basis for reasonable grouping (e.g., monitoring environmental conditions).

[0016] (3) Dividing the sample into subgroups may result in homogeneity within the subgroups. Summary of the Invention

[0017] To overcome the above-mentioned shortcomings, this invention proposes a successful envelope analysis method for satellite products based on ground data. This method can identify data types and eliminate outliers by checking and judging them, and finally use control charts to solve for the optimal envelope result.

[0018] The present invention is achieved through the following technical solution.

[0019] A method for successful envelope analysis of satellite products based on ground data includes the following steps:

[0020] Step 1: Perform a normality test on the ground data of satellite products for each attribute and each operating condition to determine whether the data follows a normal distribution;

[0021] Step 2: Determine whether the satellite product ground data for each attribute and operating condition contains outliers, and perform outlier detection and removal.

[0022] Step 3: Under each attribute, perform a pairwise bipopulation test on the data for different working conditions to determine whether the data belong to the same population under these multiple working conditions;

[0023] Step 4: Based on the data described in Steps 1 to 3 under several tests, determine which control chart should be used for the data, and then construct the control interval;

[0024] Step 5: Determine whether the Hotelling statistic is within the control chart limits based on the constructed control interval.

[0025] The beneficial effects of this invention are:

[0026] 1. This invention classifies, cleans, and merges satellite product ground data under different attributes and operating conditions using normality tests, outlier tests, and dual-population tests. Finally, it analyzes the data using several different statistical methods to obtain relatively reliable boundaries. This method can be applied not only to satellite product ground data but also to on-orbit data and other satellite indicators with independent randomness.

[0027] 2. This invention uses the Epps-Pulley test to test normality, which is applicable to samples with n≥8 and can remain sensitive to heavy-tailed samples;

[0028] 3. This invention uses the Bayesian method to construct the control interval. This method requires less sample data, and the estimation is more accurate when the prior data is relatively reliable. As the number of samples gradually increases, the resulting envelope also gradually stabilizes.

[0029] 4. This invention uses the Grubbs test to determine whether there are outliers in the data, which improves the accuracy of the test.

[0030] 5. This invention constructs control spaces for large and small sample data respectively, achieving high accuracy and wide applicability;

[0031] 6. Since the amount of satellite test data available under normal circumstances is relatively small, and in most cases, non-human-controlled random data will approximately follow a normal distribution, it is recommended to use the EP test with a confidence level of 99% when conducting the test. Therefore, in this invention, α is taken as 0.01 as the test standard. Attached Figure Description

[0032] Figure 1 This is a flowchart of the successful envelope analysis method for satellite products based on ground data, as described in this invention. Detailed Implementation

[0033] Exemplary embodiments of the present invention will now be described in detail with reference to the accompanying drawings. It should be understood that the embodiments shown and described in the drawings are merely exemplary and are intended to illustrate the principles and spirit of the present invention, and are not intended to limit the scope of the present invention.

[0034] like Figure 1 As shown, the successful envelope analysis method for satellite products based on ground data of the present invention specifically includes the following steps:

[0035] Step 1: Perform a normality test on the ground data of satellite products for each attribute and each operating condition to determine whether the data follows a normal distribution.

[0036] The rationale behind this step is that satellite-to-ground data often lacks a clear directional trend. The national standard GB / T4882-2001, "Statistical Processing and Interpretation of Data: Normality Test," recommends two normality tests: the Shapiro-Wilks test and the Epps-Pulley test. The Shapiro-Wilks test is suitable for samples with 8 ≤ n ≤ 50, while the Epps-Pulley test is suitable for samples with n ≥ 8. Although the sample size usually meets the requirements of both tests, the Shapiro-Wilks test is insensitive to heavy-tailed samples. Therefore, this embodiment uses the Epps-Pulley test, or EP test.

[0037] The Epps-Pulley test is used to perform the normality test, and the specific method is as follows:

[0038] Let x1, x2, ..., x n For a normal population N(μ,σ) 2 For a sample of ), the EP test statistic is defined as:

[0039]

[0040] in Let {T} be the sample mean and sample second moment, and its rejection region be {T}. EP ≥T 1-α,EP (n)},T 1-α,EP (n) is the 1-α quantile of the EP statistic (distribution under the null hypothesis) for a sample size of n, where α is usually considered to be 0.05 or 0.01. Since the amount of satellite test data available is generally small, and in most cases, non-human-controlled random data will approximately follow a normal distribution, it is recommended to be conservative when performing the test and use the EP test with a confidence level of 99%. Therefore, in this embodiment, α is taken as 0.01 as the test standard; the steps are as follows:

[0041] Step 1.1: Store the sample size n and sample x. i (i = 1, ..., n);

[0042] Step 1.2: Calculate and store the mean based on the sample size and sample number. and sample second central moments

[0043] Step 1.3: Calculate and store the mean and sample second central moments.

[0044] Step 1.4: Calculate and store the second-order central moments of the samples.

[0045] Step 1.5: Calculate the test statistic based on the results of steps 1.3 and 1.4.

[0046] Step 2: Determine whether the ground data of satellite products under each attribute and operating condition contains outliers, and perform outlier detection and removal.

[0047] The rationale behind this step is that testing environments are often subject to environmental noise interference, and certain randomly occurring large noises can cause outliers within the data. GBT4883-2008, "Statistical Processing and Interpretation of Data: Judgment and Handling of Outliers in Normal Samples," recommends using the Grubbs test and the Dixon test, with a detection level of 0.05 and a rejection level of 0.01. However, the Dixon test uses a different calculation formula depending on the sample size, which can affect the accuracy of the test.

[0048] Therefore, this embodiment uses the Grubbs test to determine whether the data contains outliers, as follows:

[0049] Let the detection level be α, and the critical value be G. n (1-α) and remember The Grubbs test statistic and judgment criteria are as follows:

[0050] Upper side inspection: If G n >G n (1-α), determine x (n) If it is an outlier, then it is not;

[0051] Lower side inspection: If G′ n >G n (1-α), determine x (1) If it is an outlier, then it is not;

[0052] Two-tailed test: If G n >G′ n and Determine x (n) It is an outlier; if G′ n >G n and Determine x (1) If G is an outlier, then no outlier was found; otherwise, no outlier was found. n =G n In such cases, the limit on the number of outliers to be detected should be reconsidered.

[0053] Step 3: For each attribute, there may be test data under multiple operating conditions. For the data of these attributes under different operating conditions (such as low temperature, normal temperature, and high temperature), use a two-population test in pairs to determine whether the data belong to the same population under multiple operating conditions.

[0054] The main purpose of this step is to verify whether the data of different working conditions under each attribute come from the same population, that is, whether they can be put together for joint processing.

[0055] Suppose that for any one of these attributes, the variance σ of its data under different operating conditions follows a normal distribution. 2 If (unknown) are equal, then first select the data of a certain attribute under the two working conditions, and let their data be x respectively. i ,i=1,…,n;y j If j = 1, ..., m, then test whether the normal population means μ1 and μ2 under the two working conditions are equal; the specific steps are as follows:

[0056] Step 3.1: Establish the hypothesis (H0: μ1=μ2 vs H1: μ1≠μ2) and determine the significance level α;

[0057] Step 3.2: Calculate the test statistic based on the sample size under both operating conditions; the specific formula is as follows:

[0058]

[0059] Where n and m are the number of samples belonging to the two working conditions, respectively;

[0060] Step 3.3: Determine the P-value according to the aforementioned test level and t-critical value table. If in For a t-distribution with n+m-2 degrees of freedom If the quantiles are used, the null hypothesis is accepted, meaning that the data from the two scenarios can be processed together; otherwise, the null hypothesis is rejected, meaning that the data from different attributes do not come from the same population and cannot be processed together.

[0061] Step 4: Based on the data described in Steps 1 to 3 under several tests, determine which control chart to use for the data, and thus construct the control interval. Specifically:

[0062] If the data passes the Epps-Pulley test, a 3σ control chart or a Bayesian control chart is used, and the choice between a one-sided or two-sided control chart to construct the control interval is determined based on engineering requirements; specifically, the following scenarios are included:

[0063] A. Based on the assumption of a large sample size, a one-sided control chart is used to construct the control interval;

[0064] The traditional 3σ method in the prior art uses standard deviation to construct control limits, as shown in the following formula:

[0065]

[0066] in Represents the obtained samples x1, x2, ..., x k The mean, i.e. s is the sample standard deviation of the k samples obtained, and its calculation formula is: With a large sample size, the sample mean and standard deviation will tend to the population mean and standard deviation, and all points will fall within this control interval with a 99.7% probability. However, when the sample size is small, using s to estimate the population standard deviation σ is biased.

[0067] Therefore, in this embodiment, c4(k) is introduced to correct the sample standard deviation s, making it an unbiased estimate of the population standard deviation σ. The formula for calculating c4(k) is:

[0068]

[0069] Then the 3σ control limit based on c4(k) is:

[0070]

[0071] In practical implementation, since some indicators only require control of the upper or lower limit, it is also necessary to provide a one-sided control chart under a given confidence level. Unlike a two-sided control chart, a one-sided control chart only has one-sided control limits. To ensure equal confidence levels, this embodiment provides the control limits for the upper and lower limit control charts under both biased and unbiased conditions. The specific formulas are as follows:

[0072] (1) The control limits for the upper limit control charts under biased and unbiased conditions are as follows:

[0073]

[0074]

[0075] (2) The control limits for the lower limit control charts under biased and unbiased conditions are as follows:

[0076]

[0077]

[0078] Among them, z 0.997 It is the 0.997 quantile of the standard normal distribution.

[0079] B. Constructing control intervals based on small samples;

[0080] If the total amount of data is small, the estimated mean and variance cannot be highly accurate. Therefore, using the control limits under the general large sample scenario would lead to a higher-than-expected probability of false positives, necessitating small sample corrections. This results in small sample corrections for mean-standard deviation, mean-range, median-standard deviation, and median-range plots. However, after testing, the latter three methods are found to be very complex in their calculations, and the interval lengths given for a given probability are longer than those for mean-standard deviation. Therefore, the final decision was to use the control limits based on the mean-standard deviation for small samples. The formula for the control limits based on the mean-standard deviation is shown below.

[0081]

[0082] The explanation for 's' remains the same as above, and for 't'... 1-α / 2 (k-1) represents the upper 1-α / 2 quantile of the t-distribution with k-1 degrees of freedom. In this paper, α is chosen as 0.003, i.e., the 99.7% confidence interval is still selected. Similar to the case under the large sample assumption above, the one-sided control limits for the small sample case are also given here:

[0083] (1) The control limits for the upper limit control chart are:

[0084]

[0085] (2) The control limits for the lower limit control chart are:

[0086]

[0087] C. Construct control intervals based on Bayesian methods;

[0088] The core idea behind this step is that the Bayesian method is an effective way to make reasonable estimates of the control interval when the sample size is small. Because aerospace data often shows a certain degree of inheritance between indicators of different generations of products, when the sample size is small, data from previous generations can be used as prior data to generate a prior distribution of parameters. Then, the sample data is used as posterior data to correct this prior distribution, resulting in the posterior distribution of the sample parameters, and finally, the envelope curve of the data. This method requires less sample data, and the estimation is more accurate when the prior data is relatively reliable. As the sample size gradually increases, the resulting envelope curve also gradually stabilizes. However, this method assumes that the original data distribution is normally distributed, which may have limited effectiveness in non-normal cases. Furthermore, the Bayesian method is very sensitive to prior data; if the prior is not chosen appropriately, it can cause estimation bias that is even greater than that of traditional estimation methods.

[0089] To address this issue, the control interval is constructed in this embodiment using the following method:

[0090] Let v = n + 2α0 + 3, then:

[0091] Bilateral control chart:

[0092] Upper limit control chart:

[0093] Lower limit control chart:

[0094] Among them, t v (α) is the α quantile of the t-distribution of degrees of freedom v.

[0095]

[0096]

[0097] In the formula, λ0=m, m represents the mean, sample variance, and sample size of the prior data, respectively. α0 is a constant related to m, which is a solution to the following equation:

[0098]

[0099] D. Constructing control intervals based on the bootstrap method

[0100] The basic idea behind this step is to replace the characteristics of the original population with the statistical characteristics of the experimental observation data. This eliminates the dependence on distribution assumptions in traditional statistical methods, making it suitable for estimating any distribution and any parameter of interest. Furthermore, it considers corrections for estimation bias, thus the lower confidence limit of the small sample population percentile values ​​obtained from this step is closer to the true value. Specifically:

[0101] Assume the obtained sample is x i For i = 1, ..., n, estimate the empirical distribution function of the sample using the sample data. in The expression is:

[0102]

[0103] in,

[0104] Then, sampling is performed within the empirical distribution, with n samples drawn each time, denoted as the Bootstrap sample, and represented as X. (i) =x i,1 ,x i,2 ,…,x n,1 If i = 1, ..., n, then the mean of this Bootstrap sample is Perform such sampling N times continuously to obtain

[0105] Then, under the given significance level of α, the control limits obtained by the Bootstrap method are:

[0106] The two-sided control chart is:

[0107] The upper control chart is:

[0108] The lower control chart is:

[0109] Where is the k-th largest number after sorting all Step Five: Judge whether the Hotelling statistic is within the control chart boundaries according to the constructed control interval; where the Hotelling statistic is:

[0110]

[0111]

[0112] Where is the sample mean of the multi-dimensional data, is the estimate of the sample covariance matrix. For the Hotelling control chart, there is only an upper control limit and no lower control limit;

[0113] The formula for the upper control limit is: where p is the data dimension, k is the number of samples, and F α,p,k-p is an F-statistic quantile with degrees of freedom p and k - p and a significance level of α; when T 2 <UCL, it can be determined that the sample is normal.

[0114] Here, in order to make a comparison with the single-value control chart, the confidence level used in this test is also 99.7%.

[0115] Example 1: [[ID=5--4]]

[0116] The satellite product success envelope analysis method based on ground data proposed by the present invention envelopes the satellite under various attributes and working conditions to obtain reliable delimiting indicators. Assume that the ground test data of a certain satellite and the technical indicators set artificially are shown in Table 2. At the same time, under each attribute, there are three sets of data for working conditions: low temperature, normal temperature, and high temperature.

[0117] Table 2 Satellite Ground Test Project Names and Technical Indicators

[0118] Project Name Technical indicators Project 1 ≥0dBm Project 2 >70dBC Project 3 >60dB Project 4 >80dB Project 5 ≤4dB Project 6 <-112dBm Project 7 <-118dBm Project 8 (-35,35)kHz Project 9 Better than ±115kHz (i.e., absolute value | item 9 | > 115kHz) Project 10 >32kHz / s Project 11 >49dBHz Project 12 (500,1000)mW Project 13 Better than ±10 ns (i.e., absolute value |Item 13| < 10 ns)

[0119] (1) Results of the Epps-Pulley test

[0120] Engineering analysis shows that the test results and test methods for projects 2, 4 (which overlap with project 3), 7 (which overlaps with project 6), and 9 / 10 / 13 are closely related; therefore, envelope analysis is not required for these projects. Envelope analysis will be performed on projects 1, 3, 5, 6, 8, 11, and 12. Regarding the issue of some data being in dB, we believe that using the transformed dB unit is more appropriate than the original unit. The original unit results in data that differs by many orders of magnitude, leading to calculation errors, and the data distribution is more complex. The transformed data, however, more closely approximates a normal distribution. Therefore, choosing dB as the unit for this data is more suitable.

[0121] The following is a summary of the results of whether each group of data passed the test.

[0122] Table 3. Epps-Pulley Test Results for High Temperature Data

[0123]

[0124] Table 4. Results of Epps-Pulley Test for Low Temperature Data

[0125]

[0126] Table 5. Epps-Pulley Test Results for Room Temperature Data

[0127]

[0128] Methods 1, 2, 3, and 4 all assume the data follows a normal distribution. Attribute 11 of the LOW property fails the normality assumption (0.573 > 0.566) because it contains an outlier. This outlier can be detected and removed using an outlier test. After removing the outlier, the EP test value is 0.183 (0.183 < 0.566), which is much smaller than the critical EP test value, thus passing the normality test.

[0129] (2) Outlier test results

[0130] During envelope analysis, some indicators only require control over the upper bound, lower bound, or both bounds. Therefore, we focus on outliers that affect or exceed the control bounds. In such cases, an upper-tailed test is used when only the upper bound needs to be controlled, and a lower-tailed or two-tailed test is used when controlling the lower bound or both bounds. Thus, attributes 1, 3, and 11 are tested using a lower-tailed test, 5 and 6 using an upper-tailed test, and 8 and 12 using a two-tailed test. Based on this, the test results are as follows:

[0131] Table 6 Outlier Test Results for High Temperature Data

[0132]

[0133]

[0134] Table 7 Outlier Test Results for Low Temperature Data

[0135] project <![CDATA[G′ n ]]> <![CDATA[G n ]]> Do outliers exist? 5 / 1.857557 no 6 / 1.856903 no 8 3.120485 1.910148 It is (the minimum value) 11 2.993233 / It is (the minimum value) 12 2.120744 1.497088 no

[0136] Table 8 Outlier Test Results for Room Temperature Data

[0137] project <![CDATA[G′ n ]]> <![CDATA[G n ]]> Do outliers exist? 1 2.228894 / no 3 1.422396 / no 5 / 2.060678 no 6 / 2.278311 no 8 1.941414 1.833922 no Project 11 1.810827 no Project 12 2.187764 1.586748 no

[0138] After a removal operation with a level of 0.01, two points were removed: the minimum value of attribute 8 (LOW) and the minimum value of attribute 11 (LOW). After removing outliers, the EP test value for attribute 11 (LOW) was 0.1830, which is much lower than the critical EP value, thus passing the normality test. The EP test value for attribute 8 (LOW) was 0.1134, indicating that removing outliers did not affect its normality. Therefore, these two attributes will also be processed using methods for normal data.

[0139] (3) Results of the dual population test

[0140] Based on the above algorithms, we grouped the three operating conditions into three groups: high temperature-low temperature, low temperature-normal temperature, and high temperature-normal temperature. Within each group, we used the above algorithms to calculate the t-test value of the attribute under both operating conditions to determine whether they are similar at a 95% confidence level. The test results are as follows:

[0141] Table 9. Results of the dual population test for the high-temperature and low-temperature group data.

[0142] property p-value Are they similar? 5 0.18131527 yes 6 5.19E-09 no 8 0.01402139 no 11 0.01931675 no 12 0.00222475 no

[0143] Table 10 Results of the double population test for the low-temperature-normal-temperature group data.

[0144] property p-value Are they similar? 5 0.88750742 yes 6 5.05E-07 no 8 2.81E-07 no 11 0.84530876 yes 12 0.06689199 yes

[0145] Table 11 Results of the double population test for the high temperature-normal temperature group data

[0146] property p-value Are they similar? 5 0.21341034 yes 6 0.17942383 yes 8 0.0009701 no 11 0.02150699 no 12 0.11254546 yes

[0147] The above tests show that only attribute 5 is completely similar. Other attributes differ under different working conditions. Therefore, in the following analysis, we will still process each attribute separately. If there are other requirements for a certain attribute, they can be handled specially.

[0148] (4) Control chart envelope results

[0149] Although the modified 3σ control chart and small sample control chart have better theoretical properties, since their final calculated results are very similar to those of the ordinary 3σ control chart, only the results of the ordinary 3σ control chart, Bayesian control chart, and bootstrap control chart are plotted in the graph.

[0150] For Bayesian control charts, due to the lack of data from previous satellites, when applying the Bayesian method to the data, we used ambient temperature data as prior data for high and low temperature data, and high temperature data as prior data for ambient temperature data, to obtain the final Bayesian control limits for each attribute. However, there are some differences in the data under different operating conditions, which may limit the effectiveness of the Bayesian method. More data is needed to select appropriate priors. The control limits given by this method are plotted in the final result graph. The results show that when data with strong correlation to the target data is used as priors, the interval length is smaller and the results are more accurate. Otherwise, the interval length will be wider, giving a relatively conservative result.

[0151] For bootstrap control charts, the control limits provided by this method are also shown in the final result chart. However, the effectiveness of this method is very limited, with many points falling outside the control limits because it provides confidence intervals for the mean rather than confidence intervals for the overall population distribution. Further processing is required if the bootstrap method is used.

[0152] (5) Results of multivariate Hotelling control chart

[0153] The data input to the Hotelling control chart consists of all variables that passed the normality test. For data under normal temperature conditions, since some data is missing, only variables with complete data are used to construct the control chart. The final result is that all points are within the control limits.

[0154] (6) Conclusions regarding the case study

[0155] First, this envelope analysis project is an interactive process between domain experts and data analysts. Domain experts can provide basic descriptions and qualitative judgments for each measurement, such as the relationships between different parameters and the specific measurement methods, so that the most representative variables can be extracted as the analysis variables in the final discussion.

[0156] Secondly, after obtaining the data for the analytical variables, considering that the product was measured under high, low, and normal temperature conditions, it was necessary to examine whether the data from these three conditions could be aggregated. In this project, we used a double population test to quantify the correlation of data under different operating conditions. When the data are completely similar, they can be aggregated. Among the several attributes, only attribute number five is completely similar across all operating conditions. However, to consider comparison with other attributes and the application of methods such as Bayesian analysis, each operating condition was processed separately.

[0157] After applying different envelope methods to the preprocessed data, we observed that the bootstrap method provides the narrowest bound, followed by the large-sample envelope method, while the Bayesian method generally provides the widest control limit. However, depending on the data, the Bayesian control limit may be narrower than that of the large-sample envelope method. Since the bootstrap method provides such a narrow bound that it is essentially ineffective for control, we do not recommend its use. The large-sample envelope method provides a more traditional control result, and previous experimental comparisons have shown that this method is not significantly different from other correction results under small sample conditions, making its results relatively reliable. The Bayesian method utilizes the most data and can achieve better results when the prior data is appropriately selected, but it requires a certain degree of similarity between the prior data and the experimental data. Therefore, the large-sample envelope method and the Bayesian method are generally recommended.

[0158] During this project, we found that when the prior data and sample data are not significantly different—for example, attribute 5 under all operating conditions and attribute 12 under high-temperature conditions—the control limits given by the large-sample envelope method and the Bayesian method are very similar. In this case, either method is feasible. The similarity between the data can be determined by a two-population test. When there are differences between the prior data and sample data, but the prior data is still reasonable—for example, attribute 8 under all operating conditions—considering the gain from more data, the more conservative Bayesian method may be more appropriate. If subsequent experiments can obtain more data related to a certain attribute, then the Bayesian method may be the optimal choice.

[0159] When determining the final control limits, the initial EP test value can be used as a reference. The lower the EP test value for this attribute, the more it conforms to a normal distribution, and the more reliable the result is. The control limits determined by this result can be used as the final control limits. Otherwise, empirical judgment is still needed as the basis for determining the control limits.

[0160] Finally, for cases where some data may not be normally distributed, further research is needed under the special distribution of these data to obtain their envelope analysis results.

[0161] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for successful envelope analysis of satellite products based on ground data, characterized in that, Includes the following steps: Step 1: Perform a normality test on the ground data of satellite products for each attribute and each operating condition to determine whether the data follows a normal distribution; Step 2: Determine whether the satellite product ground data for each attribute and operating condition contains outliers, and perform outlier detection and removal. Step 3: Under each attribute, perform a pairwise bipopulation test on the data for different operating conditions to determine whether the data belong to the same population under these operating conditions; Step 4: Based on the data described in Steps 1 to 3 under several tests, determine which control chart should be used for the data, and then construct the control interval; Step 5: Determine whether the Hotelling statistic is within the control chart limits based on the constructed control interval; The Epps-Pulley test was used to examine normality, as detailed below: set up For a normal population For a sample of , the EP test statistic is: in , Let be the sample mean and sample second moment, and its rejection region be . , The sample size is EP statistic at time Quantiles As a testing standard; The Grubbs test is used to determine whether the data contains outliers. The specific method is as follows: Let the detection level be The critical value is and ,remember The Grubbs test statistic and judgment criteria are as follows: Upper side inspection: ,like ,determination If it is an outlier, then it is not; Lower side inspection: ,like ,determination If it is an outlier, then it is not; Two-tailed test: if and ,determination It is an outlier; if and ,determination If it is an outlier; otherwise, no outlier was found; when When this happens, reconsider limiting the number of outliers to be detected; The specific steps to determine whether data belongs to the same population under multiple operating conditions are as follows: Assuming that for any one of these attributes, the variance of its data follows a normal distribution with equal variances under different operating conditions, we first select the data of a certain attribute under two operating conditions, let their data be respectively... ; Then test the mean of the normal population under these two working conditions. Are they equal? Step 3.1: Establishing Hypotheses And establish testing standards ; Step 3.2: Calculate the test statistic based on the sample size under the two operating conditions. ; Step 3.3: Determine the P-value based on the aforementioned test level and t-critical value table; like ,in For degrees of freedom is t-distribution If the quantile is used, the null hypothesis is accepted, meaning that the data from the two working conditions can be processed together; otherwise, the null hypothesis is rejected, meaning that the data from different attributes do not come from the same population and cannot be processed together. The determination of which control chart is actually used for the data is specifically as follows: If the data passes the Epps-Pulley test, then use Use control charts or Bayesian control charts, and determine whether to use unilateral or bilateral control charts to construct the control interval based on engineering requirements.

2. The method for successful envelope analysis of satellite products based on ground data as described in claim 1, characterized in that, The Take 0.01 as the test standard.

3. The method for successful envelope analysis of satellite products based on ground data as described in claim 1, characterized in that, The use The control chart constructs control intervals as follows: A. Based on the assumption of a large sample size, a one-sided control chart is used to construct the control interval; (1) The control limits of the upper limit control charts under biased and unbiased conditions are as follows: (2) The control limits of the lower limit control charts under biased and unbiased conditions are as follows: in, It is the 0.997 quantile of the standard normal distribution; It is obtained The sample standard deviation of each sample. Used to correct sample standard deviation Make it the population standard deviation Unbiased estimation, The calculation formula is: ； B. Constructing control intervals based on small samples; (1) The control limits for the upper limit control chart are: (2) The control limits for the lower limit control chart are: 。 4. The method for successful envelope analysis of satellite products based on ground data as described in claim 1, characterized in that, The control interval is constructed using the Bayesian method, specifically as follows: make ,but: Bilateral control chart: Upper limit control chart: Lower limit control chart: in, For degrees of freedom t-distribution Quantiles In the formula, , , , , These are the mean, sample variance, and sample size of the prior data, respectively. , For a with The relevant constants are solutions to the following equation: 。 5. The successful envelope analysis method for satellite products based on ground data as described in claim 3 or 4, characterized in that, The Hotelling statistic is: ； in, The sample mean of the multidimensional data. For estimating the sample covariance matrix, the Hotelling control chart only has an upper control limit and no lower control limit; The formula for the upper limit of control is: ,in It refers to the data dimension. For the number of samples, It is a degree of freedom and The significance level is of Statistical quantiles; when The sample can be determined to be normal immediately.

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