Five-directional stressometer monitoring data anomaly identification method based on PCA confidence interval analysis

Abstract

The invention discloses a five-directional stressometer monitoring data anomaly identification method based on PCA confidence interval analysis, and relates to the technical field of hydropower engineering. According to the method, circle centers (zt1, zt2) and radiuses r1r2 of two-dimensional coordinate data points in each time period are obtained, all the circle centers are subjected to arithmetic average to obtain a final circle center, all the radiuses are subjected to arithmetic average to obtain a final radius, and therefore a global confidence circle is formed; and when new data appear, performing the same PCA transformation and coordinate mapping on the new data, and judging whether a new point is in the final confidence circle so as to realize anomaly detection. According to the method, data dimensionality reduction is performed on data of each monitoring point of the five stress meters by adopting a principal component analysis method, and a final confidence circle is formed by performing arithmetic averaging on two-dimensional circle center coordinates and radiuses of monitoring data in multiple periods; judging whether the data of each monitoring point is normal or not by judging whether the circle center coordinate of the new confidence circle is in the final confidence circle or not; the whole method is suitable for hydropower engineering dam monitoring, equipment diagnosis and data analysis scenes of rapid anomaly detection.

Description

Technical Field

[0001] The present invention relates to the technical field of hydropower engineering, and in particular to a method for identifying anomalies in five-way stress gauge monitoring data based on PCA confidence interval analysis, which is simple, efficient, and easy to deploy in an online system. Background Art

[0002] With the rapid development of sensor and data acquisition technologies, large amounts of multidimensional data are being acquired in real time. Five-axis strain gauges, a common monitoring instrument in hydropower projects, play a vital role in dam safety monitoring. However, traditional analysis methods only analyze component data from a single direction within a single time period, ignoring the interactions among the five directional components at the same point in time. Summary of the Invention

[0003] The present invention aims to solve the problems existing in the existing technical solutions and provide a simple, efficient and easy-to-deploy method for anomaly identification of five-axis stress gauge monitoring data based on PCA confidence interval analysis.

[0004] The present invention is based on the PCA confidence interval analysis of the five-axis stress gauge monitoring data anomaly recognition method, the characteristic of which is that the recognition method obtains the center (z t1 ,z t2 ) and radius r1r2, take the arithmetic average of all circle centers to get the final circle center, and take the arithmetic average of all radii to get the final radius, thus forming a global confidence circle; when new data appears, perform the same PCA transformation and coordinate mapping on it to determine whether the new point is within the final confidence circle to achieve anomaly detection; the specific steps are as follows:

[0005] 1. Data and PCA Dimensionality Reduction

[0006] Suppose there are n time slices (t=1,2,...,n), each of which has a 5-dimensional vector data:

[0007]

[0008] X t It is the 5-dimensional vector data in a single time slice, R 5 is the entire nx5 dataset, Represents the transposed matrix

[0009] Make all the data into a matrix:

[0010]

[0011] Centralization:

[0012] Calculate the mean vector

[0013]

[0014] Centralize X:

[0015]

[0016] Where 1 is an n-dimensional all-one column vector, is the mean vector The transpose of X represents the data matrix, X c is the centered matrix;

[0017] Covariance matrix:

[0018]

[0019] in, represents the transpose of the centered matrix;

[0020] Eigendecomposition:

[0021] Find the eigendecomposition of C:

[0022] ,j=1,…,5

[0023] Let λ1≥λ2≥λ3≥λ4≥λ5 be the eigenvalues, and v1,v2,…,v5 be the corresponding eigenvectors;

[0024] Select the first two principal components:

[0025] Take the first two eigenvalues ​​corresponding to the eigenvectors V1 and V2 to form a matrix

[0026]

[0027] Project the data into two-dimensional space:

[0028]

[0029] This way we get n two-dimensional points:

[0030]

[0031] 2. Generate n confidence circles

[0032] For each time point t, set a two-dimensional coordinate Z t , let the confidence circle of each time point t be Z t is the center of the circle; the radius is selected according to the degree of discreteness of the local data;

[0033] Let the radius of each circle be r t Determined by the variation of the local point in the time slice; suppose there are 5 sub-data points in time t , projecting it into two dimensions yields , then the radius of the circle can be defined as the radius of these points relative to its center, that is The average Euclidean distance of:

[0034]

[0035] 3. Generate the coordinates of the center of the confidence circle

[0036] Suppose there are n confidence circles, each with its center at z t ,but:

[0037]

[0038] By calculating the coordinates x and y respectively by the arithmetic mean method, the final coordinate z can be obtained mean final ;

[0039] 4. Final confidence circle radius

[0040] Calculate the standard deviation of the projected points within each time slice:

[0041] There are m five-dimensional original data points in each time slice t, which are projected into two-dimensional space after PCA dimensionality reduction to obtain m two-dimensional points {Z t,1 ,Z t,2 ,…,Z t,m}, the coordinates of each point are (x t,i ,y t,i ), calculate the two-dimensional coordinate mean μ t :

[0042]

[0043] Standard deviation σ t , which measures the degree of data dispersion:

[0044]

[0045] When there is only one projection point in a time slice, the historical window or adjacent time slice data is used for supplementary calculations;

[0046] Define the weight function:

[0047]

[0048] Among them, ϵ is a smoothing term, set ϵ=10 2 , avoid σ t =0 when the weight is infinite;

[0049] When σ t The smaller it is, the more concentrated the data is, and the weight w tThe larger it is, the higher the confidence level is;

[0050] When σ t The larger the value, the more dispersed the data is, and the weight w t The smaller it is, the more it suppresses the impact of low-quality data;

[0051] Calculate the final confidence circle radius

[0052] Input the radius of each time slice {r1,r2,…,r n} and weights {w1,w2,…,w n};

[0053] Weighted average formula:

[0054]

[0055] 5. Final Confidence Circle

[0056] The final confidence circle is defined as:

[0057] Center of circle: z final

[0058] Radius: R;

[0059] 6. Anomaly Detection Methods

[0060] Given new data x^(5-dimensional), first perform steps 1 to 5 above to obtain the coordinates of the center of the circle to be detected :

[0061] judge Is it within the final confidence circle?

[0062] ≤R normal

[0063] >R abnormal

[0064] At this point, the recognition is completed.

[0065] The present invention's five-axis stress gauge monitoring data anomaly identification method based on PCA confidence interval analysis uses principal component analysis to perform data dimensionality reduction on the data of each monitoring point of the five stress gauges, and forms a final confidence circle by taking the arithmetic average of the two-dimensional center coordinates and radius of the monitoring data of multiple time periods; then, whether the data of each monitoring point is normal is judged by whether the center coordinates of the new confidence circle are within the final confidence circle; the measurement point component data of each direction and different dimensions of the five-axis stress gauge are spatially aggregated and analyzed in the same dimension, thereby solving the problem of identifying abnormal data under the correlation of all directions of the five-axis stress gauge; the entire method is simple in calculation, highly real-time, and robust, and is suitable for hydropower project dam monitoring, equipment diagnosis, and data analysis scenarios for rapid anomaly detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 This is a screenshot of the test data 1 results.

[0067] Figure 2 Screenshot of the test data 2 results.

[0068] Figure 3 This is a screenshot of the test data 3 results. DETAILED DESCRIPTION

[0069] Example 1: A method for identifying abnormalities in monitoring data of a five-axis stress gauge based on PCA confidence interval analysis, characterized in that the method obtains the center (z t1 ,z t2 ) and radius r1r2, take the arithmetic average of all circle centers to get the final circle center, and take the arithmetic average of all radii to get the final radius, thus forming a global confidence circle; when new data appears, perform the same PCA transformation and coordinate mapping on it to determine whether the new point is within the final confidence circle to achieve anomaly detection; the specific steps are as follows:

[0070] 1. Data and PCA Dimensionality Reduction

[0071] Data acquisition:

[0072] In the monitoring system of the five-axis strain gauge, the data of the measuring points in the five directions are collected at a fixed time, forming an n×5 data matrix. The following is the monitoring data of the five-axis strain gauge of a power station dam:

[0073] Serial number Measurement point number Time Time of entry Strain 1 Strain 2 Strain 3 Strain 4 Strain 5 1 T0-S501 2025-03-24 07:00:00 2025-03-24 07:56:47 51.97332995 -22.92551473 -74.84747024 45.18082797 -38.68818269 2 T0-S501 2025-03-23 07:00:00 2025-03-23 07:56:46 58.82634333 -22.92551473 -75.52479403 36.91489955 -46.95027611 3 T0-S501 2025-03-22 07:00:00 2025-03-22 07:56:38 58.20667759 -23.46841905 -75.52479403 36.20138664 -48.41298917 4 T0-S501 2025-03-21 07:00:00 2025-03-21 07:56:36 50.73024459 -23.46841905 -68.6613255 43.04577862 -63.48509601 5 T0-S501 2025-03-20 07:00:00 2025-03-20 07:56:34 57.58715047 -16.4607947 -77.55609896 43.04577862 -56.68209144 6 T0-S501 2025-03-19 07:00:00 2025-03-19 07:56:38 56.9677621 -24.01114384 -77.55609896 35.48796704 -49.87537539 7 T0-S501 2025-03-18 07:00:00 2025-03-18 07:56:34 49.48771 -24.55368892 -85.77914617 49.18435968 -58.14794651 8 T0-S501 2025-03-17 07:00:00 2025-03-17 07:56:47 56.34851254 -24.55368892 -85.77914617 41.62288385 -58.14794651 9 T0-S501 2025-03-16 07:00:00 2025-03-16 07:56:40 56.34851254 -25.09605417 -86.45774481 34.06140801 -59.61347771 10 T0-S501 2025-03-15 07:00:00 2025-03-15 07:56:37 55.72940194 -25.09605417 -94.00574358 34.06140801 -59.61347771 11 T0-S501 2025-03-14 07:00:00 2025-03-14 07:56:46 55.72940194 -32.65009077 -80.26294739 33.34826874 -60.3461217 12 T0-S501 2025-03-13 07:00:00 2025-03-13 07:56:30 62.59591811 -32.65009077 -94.68606345 40.20036692 -60.3461217 13 T0-S501 2025-03-12 07:00:00 2025-03-12 07:56:32 62.59591811 -18.08235762 -87.81461055 40.20036692 -67.15861465 14 T0-S501 2025-03-11 07:00:00 2025-03-11 07:56:33 62.59591811 -32.65009077 -88.49287742 40.20036692 -60.3461217 15 T0-S501 2025-03-10 07:00:00 2025-03-10 07:56:35 55.1104304 -25.63823942 -88.49287742 40.20036692 -61.07868456 16 T0-S501 2025-03-09 07:00:00 2025-03-09 07:56:32 54.49159799 -26.72206951 -82.29191203 40.20036692 -54.26056795 17 T0-S501 2025-03-08 07:00:00 2025-03-08 07:56:36 54.49159799 -33.73797254 -74.73474516 31.92227105 -54.99114599 18 T0-S501 2025-03-07 07:00:00 2025-03-07 07:56:46 53.87290487 -19.70229093 -75.40900662 31.92227105 -55.72164177 19 T0-S501 2025-03-06 07:00:00 2025-03-06 07:56:43 67.62143346 -34.28164458 -83.64399588 45.63795433 -56.45205519 20 T0-S501 2025-03-05 07:00:00 2025-03-05 07:56:29 67.62143346 -19.70229093 -83.64399588 38.06730132 -57.1823862 21 T0-S501 2025-03-04 07:00:00 2025-03-04 07:56:37 67.62143346 -19.70229093 -84.31986977 51.79439264 -57.1823862 22 T0-S501 2025-03-03 07:00:00 2025-03-03 08:53:48 60.74528287 -19.70229093 -76.7571907 44.92896032 -57.91263475 23 T0-S501 2025-03-02 07:00:00 2025-03-02 08:44:14 53.87290487 -12.67863375 -84.31986977 51.79439264 -57.91263475 24 T0-S501 2025-03-01 07:00:00 2025-03-01 09:07:48 60.12868515 -27.26371403 -84.31986977 44.92896032 -57.91263475 25 T0-S501 2025-02-28 07:00:00 2025-02-28 08:55:44 60.12868515 -27.26371403 -77.43111316 51.08743055 -57.91263475 26 T0-S501 2025-02-27 07:00:00 2025-02-27 07:56:42 60.12868515 -20.24190588 -84.99563155 51.79439264 -57.91263475 27 T0-S501 2025-02-26 07:00:00 2025-02-26 07:56:35 53.25435111 -13.21621709 -84.99563155 51.79439264 -51.08125472 28 T0-S501 2025-02-25 07:00:00 2025-02-25 07:56:44 67.62143346 -12.67863375 -84.99563155 44.92896032 -58.64280074 29 T0-S501 2025-02-24 07:00:00 2025-02-24 07:56:42 60.12868515 -12.67863375 -84.99563155 51.79439264 -51.08125472 30 T0-S501 2025-02-23 07:00:00 2025-02-23 07:56:35 53.25435111 -26.72206951 -84.31986977 44.92896032 -51.80951022 31 T0-S501 2025-02-23 12:00:00 2025-02-23 07:37:45 53.06 3.34 -83.61 41.4 -60.16 32 T0-S501 2025-02-22 07:00:00 2025-02-22 07:56:46 67.62143346 -26.72206951 -77.43111316 44.92896032 -51.08125472 33 T0-S501 2025-02-21 07:00:00 2025-02-21 07:56:44 60.74528287 -26.72206951 -91.88254885 45.63795433 -57.91263475 34 T0-S501 2025-02-20 07:00:00 2025-02-20 07:56:47 60.74528287 -27.26371403 -91.88254885 31.20941277 -51.08125472 35 T0-S501 2025-02-19 07:00:00 2025-02-19 07:56:36 53.87290487 -34.28164458 -84.31986977 38.77822848 -50.35291569 36 T0-S501 2025-02-18 07:00:00 2025-02-18 07:56:38 53.87290487 -26.72206951 -84.31986977 31.20941277 -50.35291569 37 T0-S501 2025-02-17 07:00:00 2025-02-17 07:56:38 67.62143346 -19.70229093 -76.7571907 52.5014516 -57.1823862 38 T0-S501 2025-02-16 07:00:00 2025-02-16 07:56:47 67.62143346 -27.26371403 -91.20483658 45.63795433 -64.74027919 39 T0-S501 2025-02-15 07:00:00 2025-02-15 07:56:44 60.74528287 -19.70229093 -76.7571907 38.06730132 -49.62449318 40 T0-S501 2025-02-14 07:00:00 2025-02-14 07:56:42 67.62143346 -27.26371403 -77.43111316 51.79439264 -57.91263475 41 T0-S501 2025-02-13 03:00:00 2025-02-13 05:56:03 70.18004 -29.82313999 -81.61199999 40.54071 -63.06619 42 T0-S501 2025-02-13 12:00:00 2025-02-13 10:10:46 53.06 3.34 -83.61 41.4 -60.16 43 T0-S501 2025-02-12 12:00:00 2025-02-12 03:56:22 53.06 3.34 -83.61 41.4 -60.16 44 T0-S501 2025-02-11 12:00:00 2025-02-11 08:53:07 53.06 3.34 -83.61 41.4 -60.16 45 T0-S501 2025-02-09 12:00:00 2025-02-09 10:50:16 73.89 -28.35 -88.37 41.39 -54.72 46 T0-S501 2025-02-08 12:00:00 2025-02-08 11:04:04 73.89 -28.35 -88.37 41.39 -54.72 47 T0-S501 2025-02-07 12:00:00 2025-02-07 04:52:00 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 48 T0-S501 2025-02-06 12:00:00 2025-02-06 05:03:20 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 49 T0-S501 2025-02-05 12:00:00 2025-02-05 04:45:55 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 50 T0-S501 2025-02-04 12:00:00 2025-02-04 02:55:42 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 51 T0-S501 2025-02-03 12:00:00 2025-02-03 09:03:16 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 52 T0-S501 2025-02-02 12:00:00 2025-02-02 08:57:59 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 53 T0-S501 2025-02-01 12:00:00 2025-02-01 05:43:58 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 54 T0-S501 2025-01-31 12:00:00 2025-01-31 09:29:36 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 55 T0-S501 2025-01-30 12:00:00 2025-01-30 09:30:16 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 56 T0-S501 2025-01-29 12:00:00 2025-01-29 08:02:40 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 57 T0-S501 2025-01-28 12:00:00 2025-01-28 05:23:16 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 58 T0-S501 2025-01-27 12:00:00 2025-01-27 05:54:51 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 59 T0-S501 2025-01-26 10:00:00 2025-01-26 10:56:42 73.88867983 -28.34646134 -88.37275529 41.38542941 -54.72169559 60 T0-S501 2025-01-25 10:00:00 2025-01-25 10:56:44 66.3922956 -20.78133922 -95.94648448 27.64652999 -56.17728538 61 T0-S501 2025-01-24 10:00:00 2025-01-24 10:56:39 73.88867983 -13.21621709 -88.37275529 42.80255361 -63.02206006 62 T0-S501 2025-01-23 10:00:00 2025-01-23 10:56:38 67.00679375 -28.34646134 -88.37275529 42.80255361 -55.44953242 63 T0-S501 2025-01-22 10:00:00 2025-01-22 10:56:38 67.00679375 -5.65109497 -95.26944064 43.51125987 -55.44953242 64 T0-S501 2025-01-21 10:00:00 2025-01-21 10:56:37 67.00679375 -12.67863375 -94.59228517 57.95857726 -40.3044772 65 T0-S501 2025-01-20 10:00:00 2025-01-20 05:17:04 60.12868515 -34.28164458 -94.59228517 58.66360126 -54.72169559 66 T0-S501 2025-01-19 10:00:00 2025-01-20 05:24:11 67.62143346 -33.73797254 -87.02224317 51.79439264 -53.99377488 67 T0-S501 2025-01-19 12:00:00 2025-01-19 04:25:24 57.58715047 -9.4493041 -87.13623297 27.93015544 -31.60877334 68 T0-S501 2025-01-18 10:00:00 2025-01-20 05:24:11 60.74528287 -33.19412125 -86.34681832 38.06730132 -54.72169559 69 T0-S501 2025-01-18 12:00:00 2025-01-18 05:08:07 69.46620093 -25.09605417 -85.67128107 39.48925036 -45.69873779 70 T0-S501 2025-01-17 10:00:00 2025-01-20 05:24:10 61.36202101 -18.08235762 -86.34681832 38.77822848 -53.99377488 71 T0-S501 2025-01-17 12:00:00 2025-01-17 10:12:13 53.06 3.34 -83.61 41.4 -60.16 72 T0-S501 2025-01-16 10:00:00 2025-01-16 10:56:36 69.46620093 -25.09605417 -85.67128107 39.48925036 -45.69873779 73 T0-S501 2025-01-16 12:00:00 2025-01-16 03:33:19 53.06 3.34 -83.61 41.4 -60.16 74 T0-S501 2025-01-15 10:00:00 2025-01-15 10:56:49 62.59591811 -24.55368892 -77.43111316 32.63522307 -44.97247935 75 T0-S501 2025-01-15 12:00:00 2025-01-15 02:57:15 53.06 3.34 -83.61 41.4 -60.16 76 T0-S501 2025-01-14 10:00:00 2025-01-14 10:56:50 63.21307685 -24.01114384 -83.64399588 40.91157813 -57.91263475 77 T0-S501 2025-01-14 12:00:00 2025-01-14 03:33:16 53.06 3.34 -83.61 41.4 -60.16 78 T0-S501 2025-01-13 10:00:00 2025-01-13 10:49:22 56.9677621 -16.4607947 -74.73474516 34.7746408 -42.06660019 79 T0-S501 2025-01-12 10:00:00 2025-01-12 10:56:43 57.58715047 -9.4493041 -87.13623297 27.93015544 -31.60877334 80 T0-S501 2025-01-11 10:00:00 2025-01-11 10:56:43 57.58715047 -2.43394427 -100.8714822 13.53344401 -30.15829604 81 T0-S501 2025-01-11 12:00:00 2025-01-11 03:13:58 53.06 3.34 -83.61 41.4 -60.16 82 T0-S501 2025-01-10 10:00:00 2025-01-10 10:56:42 56.9677621 -9.9879809 -87.81461055 34.7746408 -31.60877334 83 T0-S501 2025-01-10 12:00:00 2025-01-10 09:13:26 53.06 3.34 -83.61 41.4 -60.16 84 T0-S501 2025-01-09 10:00:00 2025-01-09 10:49:08 63.21307685 -18.08235762 -81.61570223 19.65535433 -33.05891058 85 T0-S501 2025-01-09 12:00:00 2025-01-09 10:24:58 53.06 3.34 -83.61 41.4 -60.16 86 T0-S501 2025-01-08 10:00:00 2025-01-08 10:54:49 62.59591811 -18.62251665 -82.96800993 25.78495936 -26.95081418 87 T0-S501 2025-01-08 12:00:00 2025-01-08 10:18:22 53.06 3.34 -83.61 41.4 -60.16 88 T0-S501 2025-01-07 10:00:00 2025-01-07 10:56:50 55.1104304 -19.16249446 -82.96800993 25.07007919 -35.23347762 89 T0-S501 2025-01-07 12:00:00 2025-01-07 10:33:49 53.06 3.34 -83.61 41.4 -60.16 90 T0-S501 2025-01-06 10:00:00 2025-01-06 10:47:26 61.36202101 -5.11536161 -92.56014991 24.35529174 -44.2461363 91 T0-S501 2025-01-06 12:00:00 2025-01-06 09:00:40 53.06 3.34 -83.61 41.4 -60.16 92 T0-S501 2025-01-05 10:00:00 2025-01-05 10:56:48 60.74528287 -19.70229093 -86.34681832 23.64059709 -38.85604823 93 T0-S501 2025-01-05 12:00:00 2025-01-05 04:17:05 53.06 3.34 -83.61 41.4 -60.16 94 T0-S501 2025-01-04 10:00:00 2025-01-04 10:56:40 60.74528287 -19.16249446 -87.02224317 30.49664831 -47.15100058 95 T0-S501 2025-01-03 10:00:00 2025-01-03 10:56:43 53.87290487 -40.75000306 -87.69755552 38.06730132 -53.99377488 96 T0-S501 2025-01-03 12:00:00 2025-01-03 09:06:34 53.06 3.34 -83.61 41.4 -60.16 97 T0-S501 2025-01-02 10:00:00 2025-01-02 10:56:49 60.74528287 -25.63823942 -79.45220114 38.06730132 -54.72169559

[0074] The above data is constructed into a matrix

[0075]

[0076] Centralization:

[0077] Calculate the mean vector

[0078]

[0079] Centralize X:

[0080]

[0081] Where 1 is an n-dimensional all-one column vector, X c is the centralized matrix, X represents the data matrix

[0082] Covariance matrix:

[0083]

[0084] in, Represents the transpose of the centered matrix

[0085] Eigendecomposition:

[0086] Find the eigendecomposition of C:

[0087] ,j=1,…,5

[0088] Let λ1≥λ2≥λ3≥λ4≥λ5 be the eigenvalues, and v1,v2,…,v5 be the corresponding eigenvectors;

[0089] Select the first two principal components:

[0090] Take the first two eigenvalues ​​corresponding to the eigenvectors V1 and V2 to form a matrix

[0091]

[0092] Project the data into two-dimensional space:

[0093]

[0094] This way we get n two-dimensional points:

[0095]

[0096] The data after standardization is as follows:

[0097] [[-9.49192980e+00 -3.78125598e+00 1.00931991e+01 5.07188547e+001.49947048e+01]

[0098] [-2.63891642e+00 -3.78125598e+00 9.41587534e+00 -3.19404295e+00 6.73261134e+00]

[0099] [-3.25858216e+00 -4.32416030e+00 9.41587534e+00 -3.90755586e+005.26989828e+00]

[0100] [-1.07350152e+01 -4.32416030e+00 1.62793439e+01 2.93683612e+00 -9.80220856e+00]

[0101] [-3.87810928e+00 2.68346405e+00 7.38457041e+00 2.93683612e+00 -2.99920399e+00]

[0102] [-4.49749765e+00 -4.86688509e+00 7.38457041e+00 -4.62097546e+003.80751206e+00]

[0103] [-1.19775498e+01 -5.40943017e+00 -8.38476804e-01 9.07541718e+00 -4.46505906e+00]

[0104] [-5.11674721e+00 -5.40943017e+00 -8.38476804e-01 1.51394135e+00 -4.46505906e+00]

[0105] [-5.11674721e+00 -5.95179542e+00 -1.51707544e+00 -6.04753449e+00 -5.93059026e+00]

[0106] [-5.73585781e+00 -5.95179542e+00 -9.06507421e+00 -6.04753449e+00 -5.93059026e+00]

[0107] [-5.73585781e+00 -1.35058320e+01 4.67772198e+00 -6.76067376e+00 -6.66323425e+00]

[0108] [ 1.13065836e+00 -1.35058320e+01 -9.74539408e+00 9.14244156e-02 -6.66323425e+00]

[0109] [ 1.13065836e+00 1.06190113e+00 -2.87394118e+00 9.14244156e-02 -1.34757272e+01]

[0110] [ 1.13065836e+00 -1.35058320e+01 -3.55220805e+00 9.14244156e-02 -6.66323425e+00]

[0111] [-6.35482935e+00 -6.49398067e+00 -3.55220805e+00 9.14244156e-02 -7.39579711e+00]

[0112] [-6.97366176e+00 -7.57781076e+00 2.64875734e+00 9.14244156e-02 -5.77680503e-01]

[0113] [-6.97366176e+00 -1.45937138e+01 1.02059242e+01 -8.18667145e+00 -1.30825854e+00]

[0114] [-7.59235488e+00 -5.58032181e-01 9.53166275e+00 -8.18667145e+00 -2.03875432e+00]

[0115] [ 6.15617371e+00 -1.51373858e+01 1.29667349e+00 5.52901183e+00 -2.76916774e+00]

[0116] [ 6.15617371e+00 -5.58032181e-01 1.29667349e+00 -2.04164118e+00 -3.49949875e+00]

[0117] [ 6.15617371e+00 -5.58032181e-01 6.20799596e-01 1.16854501e+01 -3.49949875e+00]

[0118] [-7.19976881e-01 -5.58032181e-01 8.18347867e+00 4.82001782e+00 -4.22974730e+00]

[0119] [-7.59235488e+00 6.46562500e+00 6.20799596e-01 1.16854501e+01 -4.22974730e+00]

[0120] [-1.33657460e+00 -8.11945528e+00 6.20799596e-01 4.82001782e+00 -4.22974730e+00]

[0121] [-1.33657460e+00 -8.11945528e+00 7.50955621e+00 1.09784880e+01 -4.22974730e+00]

[0122] [-1.33657460e+00 -1.09764713e+00 -5.49621843e-02 1.16854501e+01 -4.22974730e+00]

[0123] [-8.21090864e+00 5.92804166e+00 -5.49621843e-02 1.16854501e+012.60163273e+00]

[0124] [ 6.15617371e+00 6.46562500e+00 -5.49621843e-02 4.82001782e+00 -4.95991329e+00]

[0125] [-1.33657460e+00 6.46562500e+00 -5.49621843e-02 1.16854501e+012.60163273e+00]

[0126] [-8.21090864e+00 -7.57781076e+00 6.20799596e-01 4.82001782e+001.87337723e+00]

[0127] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0128] [ 6.15617371e+00 -7.57781076e+00 7.50955621e+00 4.82001782e+00 2.60163273e+00]

[0129] [-7.19976881e-01 -7.57781076e+00 -6.94187948e+00 5.52901183e+00 -4.22974730e+00]

[0130] [-7.19976881e-01 -8.11945528e+00 -6.94187948e+00 -8.89952973e+002.60163273e+00]

[0131] [-7.59235488e+00 -1.51373858e+01 6.20799596e-01 -1.33071402e+003.32997176e+00]

[0132] [-7.59235488e+00 -7.57781076e+00 6.20799596e-01 -8.89952973e+003.32997176e+00]

[0133] [ 6.15617371e+00 -5.58032181e-01 8.18347867e+00 1.23925091e+01 -3.49949875e+00]

[0134] [6.15617371e+00 -8.11945528e+00 -6.26416721e+00 5.52901183e+00 -1.10573917e+01]

[0135] [-7.19976881e-01 -5.58032181e-01 8.18347867e+00 -2.04164118e+004.05839427e+00]

[0136] [ 6.15617371e+00 -8.11945528e+00 7.50955621e+00 1.16854501e+01 -4.22974730e+00]

[0137] [ 8.71478025e+00 -1.06788812e+01 3.32866938e+00 4.31767496e-01 -9.38330255e+00]

[0138] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0139] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0140] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0141] [ 1.24247402e+01 -9.20574125e+00 -3.42933063e+00 1.28105750e+00 -1.03711255e+00]

[0142] [ 1.24247402e+01 -9.20574125e+00 -3.42933063e+00 1.28105750e+00 -1.03711255e+00]

[0143] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0144] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0145] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0146] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0147] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0148] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0149] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0150] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0151] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0152] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0153] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0154] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0155] [ 1.24234201e+01 -9.20220259e+00 -3.43208592e+00 1.27648691e+00 -1.03880814e+00]

[0156] [ 4.92703585e+00 -1.63708047e+00 -1.10058151e+01 -1.24624125e+01 -2.49439793e+00]

[0157] [ 1.24234201e+01 5.92804166e+00 -3.43208592e+00 2.69361111e+00 -9.33917261e+00]

[0158] [ 5.54153400e+00 -9.20220259e+00 -3.43208592e+00 2.69361111e+00 -1.76664497e+00]

[0159] [ 5.54153400e+00 1.34931638e+01 -1.03287713e+01 3.40231737e+00 -1.76664497e+00]

[0160] [ 5.54153400e+00 6.46562500e+00 -9.65161580e+00 1.78496348e+011.33784102e+01]

[0161] [-1.33657460e+00 -1.51373858e+01 -9.65161580e+00 1.85546588e+01 -1.03880814e+00]

[0162] [ 6.15617371e+00 -1.45937138e+01 -2.08157380e+00 1.16854501e+01 -3.10887433e-01]

[0163] [-3.87810928e+00 9.69495465e+00 -2.19556360e+00 -1.21787871e+012.20741141e+01]

[0164] [-7.19976881e-01 -1.40498625e+01 -1.40614895e+00 -2.04164118e+00 -1.03880814e+00]

[0165] [ 8.00094118e+00 -5.95179542e+00 -7.30611704e-01 -6.19692144e-017.98414966e+00]

[0166] [-1.03238741e-01 1.06190113e+00 -1.40614895e+00 -1.33071402e+00 -3.10887433e-01]

[0167] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0168] [ 8.00094118e+00 -5.95179542e+00 -7.30611704e-01 -6.19692144e-017.98414966e+00]

[0169] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0170] [ 1.13065836e+00 -5.40943017e+00 7.50955621e+00 -7.47371943e+008.71040810e+00]

[0171] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0172] [ 1.74781710e+00 -4.86688509e+00 1.29667349e+00 8.02635626e-01 -4.22974730e+00]

[0173] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0174] [-4.49749765e+00 2.68346405e+00 1.02059242e+01 -5.33430170e+001.16162873e+01]

[0175] [-3.87810928e+00 9.69495465e+00 -2.19556360e+00 -1.21787871e+012.20741141e+01]

[0176] [-3.87810928e+00 1.67103145e+01 -1.59308128e+01 -2.65754985e+01 2.35245914e+01]

[0177] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0178] [-4.49749765e+00 9.15627785e+00 -2.87394118e+00 -5.33430170e+002.20741141e+01]

[0179] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0180] [ 1.74781710e+00 1.06190113e+00 3.32496714e+00 -2.04535882e+012.06239769e+01]

[0181] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0182] [ 1.13065836e+00 5.21742099e-01 1.97265944e+00 -1.43239831e+012.67320733e+01]

[0183] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0184] [-6.35482935e+00 -1.82357110e-02 1.97265944e+00 -1.50388633e+011.84494098e+01]

[0185] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0186] [-1.03238741e-01 1.40288971e+01 -7.61948054e+00 -1.57536508e+019.43675115e+00]

[0187] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0188] [-7.19976881e-01 -5.58032181e-01 -1.40614895e+00 -1.64683454e+011.48268392e+01]

[0189] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0190] [-7.19976881e-01 -1.82357110e-02 -2.08157380e+00 -9.61229419e+006.53188687e+00]

[0191] [-7.59235488e+00 -2.16057443e+01 -2.75688615e+00 -2.04164118e+00 -3.10887433e-01]

[0192] [-8.40525975e+00 2.24842587e+01 1.33066937e+00 1.29105750e+00 -6.47711255e+00]

[0193] [-7.19976881e-01 -6.49398067e+00 5.48846823e+00 -2.04164118e+00 -1.03880814e+00]

[0194] [7.38587740e+00 -6.49398067e+00 -2.08157380e+00 -2.04164118e+00 -3.10887433e-01]]

[0195] PCA principal component variance ratio [ 0.46765468 0.27017025 ]

[0197] The data after PCA dimensionality reduction is as follows:

[0198] [[ 2.09635517e-01 8.06973963e+00]

[0199] [-1.75115971e+00 6.96113260e+00]

[0200] [-1.91650907e+00 6.35587409e+00]

[0201] [ 1.25642697e+00 -9.62460031e+00]

[0202] [ 4.26560979e+00 -4.54834783e+00]

[0203] [-1.95617602e+00 5.87164545e+00]

[0204] [-8.27021821e-01 -8.59529840e+00]

[0205] [-2.97833472e+00 -3.91997880e+00]

[0206] [-2.91383605e+00 -8.58743637e-02]

[0207] [-3.11419030e+00 3.11041899e-01]

[0208] [-9.12882582e+00 -1.25050189e-01]

[0209] [-1.32183463e+01 -4.04561009e+00]

[0210] [ 6.28974298e-01 -1.02312889e+01]

[0211] [-1.28525593e+01 -4.35553391e+00]

[0212] [-3.46888789e+00 -5.00173796e+00]

[0213] [-3.99059697e+00 -6.60594540e-02]

[0214] [-9.31042619e+00 4.67451924e+00]

[0215] [ 3.72242644e+00 3.52815512e+00]

[0216] [-1.65390789e+01 -5.23733236e+00]

[0217] [-2.64601761e+00 -1.56750979e+00]

[0218] [-3.71837738e+00 -1.04073933e+01]

[0219] [-1.38121865e-02 -6.68908143e+00]

[0220] [8.15905970e+00 -1.08582127e+01]

[0221] [-7.11820220e+00 -5.94423429e+00]

[0222] [-7.17452308e+00 -1.02700322e+01]

[0223] [-1.26487247e+00 -1.06709812e+01]

[0224] [ 7.72942281e+00 -5.59342933e+00]

[0225] [ 3.20020082e+00 -7.36697970e+00]

[0226] [5.49474576e+00 -5.83124011e+00]

[0227] [-4.02725611e+00 -1.12185712e+00]

[0228] [ 2.39745366e+01 -6.59133428e+00]

[0229] [-9.33172057e+00 -1.35903588e+00]

[0230] [-7.36823514e+00 -6.06808981e+00]

[0231] [-6.92169969e+00 8.47103626e+00]

[0232] [-1.07411199e+01 4.28821827e+00]

[0233] [-3.27136931e+00 8.83379502e+00]

[0234] [-3.32488344e+00 -1.12429197e+01]

[0235] [-1.04045758e+01 -1.14745694e+01]

[0236] [ 3.27265273e-01 4.03977825e+00]

[0237] [-1.01982615e+01 -1.09593325e+01]

[0238] [-1.28406320e+01 -7.35020322e+00]

[0239] [ 2.39745366e+01 -6.59133428e+00]

[0240] [ 2.39745366e+01 -6.59133428e+00]

[0241] [ 2.39745366e+01 -6.59133428e+00]

[0242] [-1.36060258e+01 -1.40639958e+00]

[0243] [-1.36060258e+01 -1.40639958e+00]

[0244] [-1.36020554e+01 -1.40471616e+00]

[0245] [-1.36020554e+01 -1.40471616e+00]

[0246] [-1.36020554e+01 -1.40471616e+00]

[0247] [-1.36020554e+01 -1.40471616e+00]

[0248] [-1.36020554e+01 -1.40471616e+00]

[0249] [-1.36020554e+01 -1.40471616e+00]

[0250] [-1.36020554e+01 -1.40471616e+00]

[0251] [-1.36020554e+01 -1.40471616e+00]

[0252] [-1.36020554e+01 -1.40471616e+00]

[0253] [-1.36020554e+01 -1.40471616e+00]

[0254] [-1.36020554e+01 -1.40471616e+00]

[0255] [-1.36020554e+01 -1.40471616e+00]

[0256] [-1.36020554e+01 -1.40471616e+00]

[0257] [-3.10776129e+00 6.63535883e+00]

[0258] [ 2.77716989e-01 -9.31819356e+00]

[0259] [-1.09648874e+01 -2.66009310e+00]

[0260] [9.29111868e+00 -3.81543063e+00]

[0261] [1.50988548e+00 -1.36613591e+00]

[0262] [-1.52313690e+01 -1.15635524e+01]

[0263] [-1.67572794e+01 -7.20641312e+00]

[0264] [ 1.07074275e+01 2.44186483e+01]

[0265] [-1.24470306e+01 1.26897908e+00]

[0266] [-8.77005196e+00 6.52485764e+00]

[0267] [ 1.03384676e+00 6.48963186e-01]

[0268] [ 2.39745366e+01 -6.59133428e+00]

[0269] [-8.77005196e+00 6.52485764e+00]

[0270] [ 2.39745366e+01 -6.59133428e+00]

[0271] [-4.56433152e+00 1.12827352e+01]

[0272] [ 2.39745366e+01 -6.59133428e+00]

[0273] [-5.02997117e+00 -3.62607526e+00]

[0274] [ 2.39745366e+01 -6.59133428e+00]

[0275] [ 4.99129634e+00 1.17740536e+01]

[0276] [ 1.07074275e+01 2.44186483e+01]

[0277] [ 1.73520811e+01 3.51917305e+01]

[0278] [ 2.39745366e+01 -6.59133428e+00]

[0279] [ 9.90642414e+00 2.00720069e+01]

[0280] [ 2.39745366e+01 -6.59133428e+00]

[0281] [ 1.57564839e+00 2.86125047e+01]

[0282] [ 2.39745366e+01 -6.59133428e+00]

[0283] [ 6.57401156e-01 2.93996202e+01]

[0284] [ 2.39745366e+01 -6.59133428e+00]

[0285] [ 3.36084263e+00 2.38295469e+01]

[0286] [2.39745366e+01 -6.59133428e+00]

[0287] [1.33822835e+01 1.70893148e+01]

[0288] [2.39745366e+01 -6.59133428e+00]

[0289] [6.18563053e-01 2.20221640e+01]

[0290] [2.39745366e+01 -6.59133428e+00]

[0291] [7.30892418e-01 1.13007667e+01]

[0292] [-1.67146778e+01 2.44936752e+00]

[0293] [2.39745366e+01 -6.59133428e+00]

[0294] [-5.14270069e+00 5.76934094e-01]

[0295] [-8.81881625e+00 1.25717198e+00]];

[0296] 2. Generate n confidence circles

[0297] For each time point t, set a two-dimensional coordinate Z t , let the confidence circle of each time point t be Z t is the center of the circle; the radius is selected according to the degree of discreteness of the local data;

[0298] Let the radius of each circle be r t Determined by the variation of the local point in the time slice; suppose there are 5 sub-data points in time t , projecting it into two dimensions yields , then the radius of the circle can be defined as the radius of these points relative to its center, that is The average Euclidean distance of:

[0299]

[0300] 3. Generate the coordinates of the center of the confidence circle

[0301] Suppose there are n confidence circles, each with its center at zt ,but:

[0302]

[0303] By calculating the coordinates x and y respectively by the arithmetic mean method, the final coordinate z can be obtained mean final ;

[0304] The center coordinates and radii of the n confidence circles are as follows:

[0305] Time slice 1: Center = [0.21 8.07], Radius = 3.087

[0306] Time slice 2: Center = [-1.751 6.961], Radius = 19.068

[0307] Time slice 3: Center = [-1.917 6.356], Radius = 19.366

[0308] Time slice 4: Center = [1.256 -9.625], Radius = 18.646

[0309] Time slice 5: Center = [4.266 -4.548], Radius = 18.848

[0310] Time slice 6: Center = [-1.956 5.872], Radius = 15.606

[0311] Time slice 7: Center = [-0.827 -8.595], Radius = 14.35

[0312] Time slice 8: Center = [-2.978 -3.92 ], Radius = 12.586

[0313] Time slice 9: Center = [-2.914 -0.086], Radius = 11.387

[0314] Time slice 10: Center = [-3.114 0.311], Radius = 11.972

[0315] Time slice 11: Center = [-9.129 -0.125], Radius = 16.44

[0316] Time slice 12: Center = [-13.218 -4.046], Radius = 17.141

[0317] Time slice 13: Center = [0.629 -10.231], Radius = 16.228

[0318] Slice 14: center = [-12.853 -4.356], radius = 16.473

[0319] Slice 15: center = [-3.469 -5.002], radius = 17.763

[0320] Slice 16: center = [-3.991 -0.066], radius = 17.755

[0321] Slice 17: center = [-9.314 4.675], radius = 20.376

[0322] Slice 18: center = [3.722 3.528], radius = 19.799

[0323] Slice 19: center = [-16.539 -5.237], radius = 22.716

[0324] Slice 20: center = [-2.646 -1.568], radius = 21.457

[0325] Slice 21: center = [-3.718 -10.407], radius = 22.356

[0326] Slice 22: center = [-0.014 -6.689], radius = 15.861

[0327] Slice 23: center = [8.159 -10.858], radius = 15.655

[0328] Slice 24: center = [-7.118 -5.944], radius = 15.533

[0329] Slice 25: center = [-7.175 -10.27], radius = 18.514

[0330] Slice 26: center = [-1.265 -10.671], radius = 15.969

[0331] Slice 27: center = [7.729 -5.593], radius = 14.738

[0332] Slice 28: center = [3.2 -7.367], radius = 13.671

[0333] Slice 29: center = [5.495 -5.831], radius = 24.448

[0334] Time slice 30: Center = [-4.027 -1.122], Radius = 30.031

[0335] Time slice 31: Center = [23.975 -6.591], Radius = 32.106

[0336] Time slice 32: Center = [-9.332 -1.359], Radius = 35.015

[0337] Time slice 33: Center = [-7.368 -6.068], Radius = 36.969

[0338] Time slice 34: Center = [-6.922 8.471], Radius = 16.243

[0339] Time slice 35: Center = [-10.741 4.288], Radius = 22.07

[0340] Time slice 36: Center = [-3.271 8.834], Radius = 25.213

[0341] Time slice 37: Center = [-3.325 -11.243], Radius = 24.703

[0342] Time slice 38: Center = [-10.405 -11.475], Radius = 25.126

[0343] Time slice 39: Center = [0.327 4.04 ], Radius = 19.874

[0344] Time slice 40: Center = [-10.198 -10.959], Radius = 38.065

[0345] Time slice 41: Center = [-12.841 -7.35 ], Radius = 43.378

[0346] Time slice 42: Center = [23.975 -6.591], Radius = 45.057

[0347] Time slice 43: Center = [23.975 -6.591], Radius = 47.279

[0348] Time slice 44: Center = [23.975 -6.591], Radius = 47.875

[0349] Time slice 45: Center = [-13.606 -1.406], Radius = 47.874

[0350] Time slice 46: Center = [-13.606 -1.406], Radius = 39.088

[0351] Time slice 47: Center = [-13.602 -1.405], Radius = 0.005

[0352] Time slice 48: Center = [-13.602 -1.405], Radius = 0.004

[0353] Time slice 49: Center = [-13.602 -1.405], Radius = 13.622

[0354] Time slice 50: Center = [-13.602 -1.405], Radius = 20.323

[0355] Time slice 51: Center = [-3.108 6.635], Radius = 19.699

[0356] Time slice 52: Center = [0.278 -9.318], Radius = 24.919

[0357] Time slice 53: Center = [-10.965 -2.66 ], Radius = 21.521

[0358] Time slice 54: Center = [9.291 -3.815], Radius = 25.088

[0359] Time slice 55: Center = [1.51 -1.366], Radius = 27.801

[0360] Time slice 56: Center = [-15.231 -11.564], Radius = 44.504

[0361] Time slice 57: Center = [-16.757 -7.206], Radius = 42.426

[0362] Time slice 58: Center = [10.707 24.419], Radius = 41.345

[0363] Time slice 59: Center = [-12.447 1.269], Radius = 37.364

[0364] Time slice 60: Center = [-8.77 6.525], Radius = 43.512

[0365] Time slice 61: Center = [1.034 0.649], Radius = 36.362

[0366] Time slice 62: Center = [23.975 -6.591], Radius = 41.202

[0367] Time slice 63: Center = [-8.77 6.525], Radius = 40.662

[0368] Time slice 64: Center = [23.975 -6.591], Radius = 43.64

[0369] Time slice 65: Center = [-4.564 11.283], Radius = 42.598

[0370] Time slice 66: Center = [23.975 -6.591], Radius = 40.483

[0371] Time slice 67: Center = [-5.03 -3.626], Radius = 39.957

[0372] Time slice 68: Center = [23.975 -6.591], Radius = 42.904

[0373] Time slice 69: Center = [ 4.991 11.774], Radius = 48.627

[0374] Time slice 70: Center = [10.707 24.419], Radius = 46.948

[0375] Time slice 71: Center = [17.352 35.192], Radius = 39.855

[0376] Time slice 72: Center = [23.975 -6.591], Radius = 46.471

[0377] Time slice 73: Center = [ 9.906 20.072], Radius = 50.473

[0378] Time slice 74: Center = [23.975 -6.591], Radius = 46.345

[0379] Time slice 75: Center = [ 1.576 28.613], Radius = 49.668

[0380] Time slice 76: Center = [23.975 -6.591], Radius = 53.396

[0381] Time slice 77: Center = [ 0.657 29.4 ], Radius = 51.331

[0382] Slice 78: center = [23.975 -6.591], radius = 50.496

[0383] Slice 79: center = [3.361 23.83], radius = 46.717

[0384] Slice 80: center = [23.975 -6.591], radius = 40.619

[0385] Slice 81: center = [13.382 17.089], radius = 43.47

[0386] Slice 82: center = [23.975 -6.591], radius = 40.889

[0387] Slice 83: center = [0.619 22.022], radius = 40.863

[0388] Slice 84: center = [23.975 -6.591], radius = 49.161

[0389] Slice 85: center = [0.731 11.301], radius = 49.161

[0390] Slice 86: center = [-16.715 2.449], radius = 45.101

[0391] Slice 87: center = [23.975 -6.591], radius = 38.462

[0392] Slice 88: center = [-5.143 0.577], radius = 40.673

[0393] Slice 89: center = [-8.819 1.257], radius = 38.879

[0394] IV. Final confidence circle radius

[0395] Calculate the standard deviation of the projected points in each slice:

[0396] There are m five-dimensional original data points in each slice t, which are projected into two-dimensional space after PCA dimension reduction, and m two-dimensional points {Z t,1 ,Z t,2 ,…,Z t,m} are obtained, and the coordinates of each point are (x t,i , y t,i ). Calculate the two-dimensional coordinate mean μ t :

[0397]

[0398] Standard deviation σ t , which measures the degree of data dispersion:

[0399]

[0400] When there is only one projection point in a time slice, the historical window or adjacent time slice data is used for supplementary calculations;

[0401] Define the weight function:

[0402]

[0403] Among them, ϵ is the smoothing term, set ϵ=10 2 , avoid σ t =0, the weight is infinite;

[0404] When σ t The smaller it is, the more concentrated the data is, and the weight w t The larger it is, the higher the confidence level is;

[0405] When σ t The larger the value, the more dispersed the data is, and the weight w t The smaller it is, the more it suppresses the impact of low-quality data;

[0406] Calculate the final confidence circle radius

[0407] Input the radius of each time slice {r1,r2,…,r n} and weights {w1,w2,…,w n};

[0408] Weighted average formula:

[0409]

[0410] The final confidence circle radius is obtained: center = [1.37549 0.14205], radius = 28.95096;

[0411] 6. Anomaly Detection

[0412] After performing a dimensionality reduction operation on the new data consistent with the historical data, the Euclidean distance between the new data point and the center of the clustering circle is calculated to determine whether it exceeds the radius of the confidence circle and whether it is abnormal data.

[0413] Take a set of test data and the results are as follows:

[0414] T0-S501 T0-S501 T0-S501 T0-S501 T0-S501 T0-S501 T0-S501 T0-S501 1 T0-S501 68.85113715 -25.63823942 -87.02224317 38.06730132 -53.99377488 T0-S501 2 T0-S501 23.8421123 18.5313256 -87.25238437 8.75231232 -33.23278558 T0-S501 3 T0-S501 68.85113715 -15.63823942 -87.02224317 38.06730132 -32.99377488 Serial number Measurement point number Strain 1 Strain 2 Strain 3 Strain 4 Strain 5 Abnormality T0-S501 No T0-S501 Yes T0-S501 Yes

Claims

1. A method for identifying abnormalities in five-axis stress gauge monitoring data based on PCA confidence interval analysis, characterized in that The recognition method obtains the center (z t1 ,z t2 ) and radius r1r2, take the arithmetic average of all circle centers to get the final circle center, and take the arithmetic average of all radii to get the final radius, thus forming a global confidence circle; when new data appears, perform the same PCA transformation and coordinate mapping on it to determine whether the new point is within the final confidence circle to achieve anomaly detection; the specific steps are as follows:

1. Data and PCA Dimensionality Reduction Suppose there are n time slices (t=1,2,...,n), each of which has a 5-dimensional vector data: ， Among them, X t It is the 5-dimensional vector data in a single time slice, R 5 is the entire nx5 dataset, Represents the transposed matrix Make all the data into a matrix: ， Centralization: Calculate the mean vector 2. ， Centralize X: ， Where 1 is an n-dimensional all-one column vector, X c is the centralized matrix, X represents the data matrix Covariance matrix: ， in, Represents the transpose of the centered matrix Eigendecomposition: Find the eigendecomposition of C: ,j=1,…,5； Let λ1≥λ2≥λ3≥λ4≥λ5 be the eigenvalues, and v1,v2,…,v5 be the corresponding eigenvectors; Select the first two principal components: Take the first two eigenvalues ​​corresponding to the eigenvectors V1 and V2 to form a matrix ， Project the data into two-dimensional space: ， This way we get n two-dimensional points: ； 2. Generate n confidence circles For each time point t, set a two-dimensional coordinate Z t , let the confidence circle of each time point t be Z t is the center of the circle; the radius is selected according to the degree of discreteness of the local data; Let the radius of each circle be r t Determined by the variation of the local point in the time slice; suppose there are 5 sub-data points in time t , projecting it into two dimensions yields , then the radius of the circle can be defined as the radius of these points relative to its center, that is The average Euclidean distance of: ， 3. Generate the coordinates of the center of the confidence circle Suppose there are n confidence circles, each with its center at z t ,but: ， By calculating the coordinates x and y respectively by the arithmetic mean method, the final coordinate z can be obtained mean final ; 4. Final confidence circle radius Calculate the standard deviation of the projected points within each time slice: There are m five-dimensional original data points in each time slice t, which are projected into two-dimensional space after PCA dimensionality reduction to obtain m two-dimensional points {Z t,1 ,Z t,2 ,…,Z t,m }, the coordinates of each point are (x t,i ,y t,i ), calculate the two-dimensional coordinate mean μ t : ， Standard deviation σ t , which measures the degree of data dispersion: ， When there is only one projection point in a time slice, the historical window or adjacent time slice data is used for supplementary calculations; Define the weight function: ， Among them, ϵ is a smoothing term, set ϵ=10 2 , avoid σ t =0 when the weight is infinite; When σ t The smaller it is, the more concentrated the data is, and the weight w t The larger it is, the higher the confidence level is. When σ t The larger the value, the more dispersed the data is, and the weight w t The smaller it is, the more it suppresses the impact of low-quality data; Calculate the final confidence circle radius Input the radius of each time slice {r1,r2,…,r n } and weights {w1,w2,…,w n }; Weighted average formula: ， 5. Final Confidence Circle The final confidence circle is defined as: Center of circle: z final Radius: R; 6. Anomaly Detection Methods Given new data x^(5-dimensional), first go through steps 1 to 5 above to get the coordinates of the center of the circle to be detected ; judge Is it within the final confidence circle? ≤R normal, >R abnormal, At this point, the recognition is completed.

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