A longitudinal wave-based method for evaluating a liquid-filled state of a pipeline

Abstract

The application discloses a kind of pipe liquid-filled state evaluation methods based on longitudinal guided wave, first outside pipe arrangement excitation sensor array and receiving sensor array, then carry out sweep-frequency experiment under different pipe liquid-filled state, obtain longitudinal guided wave response signal, according to the characteristic extraction of multiple characteristic parameters in advance to longitudinal guided wave response signal, obtain the characteristic vector set reflecting different excitation frequency and liquid-filled state, then the characteristic vector set under each excitation frequency is carried out information gain and statistical significance calculation, obtain optimal monitoring frequency, select the characteristic vector set under optimal monitoring frequency, carry out normalization processing, form the standardized characteristic vector set, input support vector machine classification model and train, using the support vector machine classification model of training completion new monitoring sample, can real-time evaluate the liquid-filled state of pipe.Compared with prior art, the method not only improves the accuracy of evaluation, but also enhances the applicability and reliability of the method.

Description

Technical Field

[0001] This invention relates to a method for assessing the liquid filling status of pipelines based on longitudinal guided waves, belonging to the field of pipeline non-destructive testing technology. Background Technology

[0002] Accurate assessment of the fluid level within industrial pipelines is crucial for their operation and maintenance. Traditional pipeline monitoring methods rely on physical contact sensors or indirect monitoring methods such as pressure sensors, flow meters, or visual inspection systems. While these methods perform well in some applications, they typically have limitations, such as complex installation, high maintenance costs, environmental sensitivity, and long response times.

[0003] In recent years, non-destructive testing technologies have attracted widespread attention because they can monitor the condition of pipelines without interfering with their normal operation. In particular, monitoring methods based on guided wave technology, which can penetrate pipeline materials and propagate over long distances, have shown great potential in industrial pipeline monitoring because they can effectively detect the integrity and internal condition of pipelines.

[0004] Guided wave detection technology, as an effective pipeline assessment tool, can evaluate the liquid filling state within a pipeline based on the propagation characteristics of guided waves. The propagation speed and attenuation characteristics of guided waves vary with the presence and distribution of liquid within the pipeline, enabling non-contact, remote monitoring. However, relying solely on traditional signal processing methods to analyze guided wave data often fails to achieve high-precision and high-reliability monitoring results, especially when liquid filling states are diverse. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of the existing technology by providing a pipeline filling status assessment method based on longitudinal guided waves. This method employs non-invasive measurement, can accurately assess the pipeline filling status in real time, and has high accuracy and reliability.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A method for assessing the liquid filling status of a pipeline based on longitudinal guided waves includes the following steps:

[0008] (1) An excitation sensor array and a receiving sensor array are arranged outside the pipeline to excite and receive longitudinal guided wave signals;

[0009] (2) Frequency sweeping experiments were conducted under different pipeline filling conditions to obtain longitudinal guided wave response signals at each excitation frequency;

[0010] (3) Set multiple characteristic parameters, extract features from the longitudinal guided wave response signal, and obtain a set of feature vectors reflecting different excitation frequencies and liquid filling states;

[0011] (4) Calculate the information gain and statistical significance of the feature vector set under each excitation frequency to obtain the optimal monitoring frequency that can most effectively distinguish different filling states;

[0012] (5) Select the feature vector set under the optimal monitoring frequency, normalize it to form a standardized feature vector set, and input it into the support vector machine classification model for training;

[0013] (6) Use the trained support vector machine classification model to process new monitoring samples, and evaluate the liquid filling status of the pipeline in real time through the classification results output by the model.

[0014] Further, in step (1), the arrangement method of the sensor array is as follows: 16 piezoelectric sheets are uniformly installed in the circumference of the first port of the pipe to form an excitation sensor array, and the same voltage signal is applied to each piezoelectric sheet to excite the longitudinal modal guided wave; 8 piezoelectric sheets are uniformly installed in the circumference of the second port of the pipe to form a receiving sensor array, which is used to receive the guided wave response signal at multiple angles.

[0015] Further, in step (2), the different pipe filling states include:

[0016] Unfilled state: There is no liquid in the pipeline;

[0017] 1 / 4 filling state: The liquid level in the pipe reaches 1 / 4 of the pipe diameter;

[0018] Half-filled state: The liquid level in the pipe reaches 1 / 2 of the pipe diameter;

[0019] 3 / 4 filling state: The liquid level in the pipe reaches 3 / 4 of the pipe diameter;

[0020] Fully filled: The pipe is completely filled with liquid.

[0021] Further, in step (2), the frequency sweep experiment method is as follows: A gradually increasing voltage signal from 60kHz to 100kHz is applied to each piezoelectric element of the excitation array, increasing by 10kHz each time, forming a frequency sweep sequence. Voltage signals of various frequencies are applied sequentially to the excitation array, and corresponding longitudinal guided wave response signal samples are received through a receiving array. This method can excite longitudinal guided waves of the corresponding frequency and capture the guided wave response signal from the pipe using a receiving array. The guided wave response signal obtained at each frequency is recorded, and this process is repeated for each filling state.

[0022] Furthermore, in step (3), the plurality of feature parameters are as follows:

[0023] Energy characteristic parameters include root mean square, maximum value, and peak-to-peak value, denoted as , ... , , ;

[0024] Complexity features include box dimension, sample entropy, spectral entropy, and zero-crossing rate, denoted as […]. , , , ;

[0025] Frequency characteristic parameters include power spectral density and frequency center, denoted as , respectively. , ;

[0026] Statistical characteristic parameters include skewness and kurtosis, denoted as _____, ... , .

[0027] Furthermore, in step (4), the method for obtaining the optimal monitoring frequency includes the following steps:

[0028] Step 1: Construct the feature vector matrix

[0029] For each frequency f, the eigenvector matrix Represented as:

[0030]

[0031] eigenvector matrix It contains multiple features extracted from the guided wave response signal. Each row represents a liquid filling state, and each column represents a specific feature. Here, n represents the total number of observations under different liquid filling states; m is the total number of features extracted at that frequency. Each liquid filling state has 8*11=88 feature parameters. This represents the calculated value of the i-th sample on the j-th feature, for frequency f;

[0032] Step 2: Define category labels

[0033] Define a category vector Y, which contains the category label of each sample, corresponding to the fluid filling category of each row in matrix X;

[0034] Step 3: Calculate information gain

[0035] First, calculate the total entropy of the sample set:

[0036]

[0037] in, is the probability that the sample belongs to class i in the sample set, and c is the total number of classes;

[0038] Next, for each feature (each column) in matrix X, calculate its conditional entropy:

[0039] ,

[0040] These are all possible values ​​of feature A. It is a subset when the value of feature A is v. and These are the sizes of the subset and the original set, respectively;

[0041] Finally, the information gain is calculated based on the total entropy and conditional entropy:

[0042]

[0043] The information gain of the eigenvector matrix X corresponding to each excitation frequency is calculated;

[0044] Step 4: Calculate the statistical significance index p.

[0045] First, calculate the sum of squares between groups:

[0046]

[0047] The average value of all eigenvalues ​​in the eigenvector matrix X;

[0048] The average value of the eigenvalues ​​in each row of the eigenvector matrix X, i.e., in each filled state;

[0049] Next, calculate the sum of squares within the group:

[0050]

[0051] in, It is the i-th feature value in the j-th group;

[0052] Finally, the F-statistic is calculated based on the between-group sum of squares and the within-group sum of squares:

[0053]

[0054] Where N is the total number of samples, which is the number of rows in the feature vector matrix X; k is the number of groups, representing the number of types of liquid filling states;

[0055] Based on the F-statistic and the corresponding degrees of freedom between groups and intragroup freedom The p-value can be found through the F-distribution table;

[0056] The p-value of the eigenvector matrix X corresponding to each excitation frequency is calculated;

[0057] Step 5: Optimal Frequency Evaluation

[0058] Define evaluation metrics:

[0059] E = Info(f) * (1-p(f)), where f represents the excitation frequency;

[0060] Based on the information gain value and p-value of the eigenvector matrix X corresponding to each excitation frequency, the frequency with the largest evaluation index is selected as the optimal monitoring frequency.

[0061] The beneficial effects of this invention are as follows: This invention provides a method for assessing the liquid filling status of pipelines using longitudinal guided waves. This method employs an external excitation and receiving sensor array to generate and receive longitudinal guided wave signals. Data is recorded under different liquid filling states through frequency sweep experiments. By setting various characteristic parameters and extracting feature vector sets, information gain and statistical significance analysis are used to determine the optimal monitoring frequency, improving the accuracy of feature selection. Finally, a support vector machine (SVM) classification model is used with a normalized feature vector set as input to achieve real-time and accurate assessment of the pipeline's liquid filling status. Compared to existing technologies, this invention not only improves the accuracy of the assessment but also enhances the applicability and reliability of the method. Furthermore, this invention uses non-invasive measurement, which is beneficial for protecting the integrity of the pipeline system. Attached Figure Description

[0062] Figure 1 This is a flowchart of the pipeline liquid filling state assessment method based on longitudinal guided waves in Example 1;

[0063] Figure 2 This is a diagram showing the arrangement of the sensor array in Example 1;

[0064] Figure 3 This is a flowchart of obtaining the optimal monitoring frequency in Example 1;

[0065] Figure 4 The result of information gain calculation in Example 1;

[0066] Figure 5 The statistical significance calculation results are from Example 1;

[0067] Figure 6 The results are the calculation results of the evaluation indicators in Example 1. Detailed Implementation

[0068] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and should not be used to limit the scope of protection of the present invention.

[0069] Example 1

[0070] like Figure 1 A method for evaluating the liquid filling state of a pipeline based on longitudinal guided waves includes the following steps:

[0071] (1) An excitation sensor array and a receiving sensor array are arranged outside the pipeline to excite and receive longitudinal guided wave signals;

[0072] In this embodiment, the experimental object used is a steel pipe of a specific specification, with an outer diameter of 113 mm, an inner diameter of 110 mm, a length of 3 meters, a Young's modulus of 210 GPa, a Poisson's ratio of 0.3, and a density of 7850 kg / m³. Excitation sensor arrays and receiving sensor arrays are designed at both ends of the pipe, respectively.

[0073] The sensor array is arranged as follows: Figure 2 As shown, 16 piezoelectric elements are uniformly installed circumferentially at the first port of the pipe to form an excitation sensor array. Applying the same voltage signal to each piezoelectric element simultaneously can excite the longitudinal modal guided wave. At the second port of the pipe, 8 piezoelectric elements are uniformly installed circumferentially to form a receiving sensor array, which is used to receive the guided wave response signal at multiple angles.

[0074] (2) Frequency sweeping experiments were conducted under different pipeline filling conditions to obtain longitudinal guided wave response signals at each excitation frequency;

[0075] In this embodiment, the different pipe filling states include 5 categories:

[0076] Unfilled state: There is no liquid in the pipeline;

[0077] 1 / 4 filling state: The liquid level in the pipe reaches 1 / 4 of the pipe diameter;

[0078] Half-filled state: The liquid level in the pipe reaches 1 / 2 of the pipe diameter;

[0079] 3 / 4 filling state: The liquid level in the pipe reaches 3 / 4 of the pipe diameter;

[0080] Fully filled: The pipe is completely filled with liquid.

[0081] The frequency sweeping experiment method is as follows: Apply a gradually increasing voltage signal from 60kHz to 100kHz to each piezoelectric element of the excitation array, increasing by 10kHz each time, f1=60kHz, f2=70kHz, f3=80kHz, f4=90kHz, f5=100kHz, forming a frequency sweeping sequence. Apply voltage signals of each frequency to the excitation array in sequence, and receive the corresponding longitudinal guided wave response signal samples through the receiving array.

[0082] The above excitation and reception process was repeated in the states of no liquid filling, 1 / 4 liquid filling, half liquid filling, 3 / 4 liquid filling and full liquid filling to obtain guided wave response signal samples of 5 frequencies in 5 liquid filling states, for a total of 25 guided wave response signal samples in 25 states, with each sample containing 8 signals.

[0083] (3) Set multiple characteristic parameters, extract features from the longitudinal guided wave response signal, and obtain a set of feature vectors reflecting different excitation frequencies and liquid filling states;

[0084] The plurality of feature parameters include:

[0085] Energy characteristic parameters: root mean square, maximum value, peak-to-peak value, reflecting the all-round distribution of energy;

[0086] Complexity feature parameters: box dimension, sample entropy, spectral entropy, zero crossover rate, used to analyze the complexity and irregularity of signals;

[0087] Frequency characteristic parameters: power spectral density, frequency center, used to identify frequency variations;

[0088] Statistical characteristic parameters: skewness and kurtosis. Through statistical analysis of data from different perspectives, a more comprehensive set of state characteristics can be obtained.

[0089] (4) Calculate the information gain and statistical significance of the feature vector set under each excitation frequency to obtain the optimal monitoring frequency that can most effectively distinguish different filling states;

[0090] like Figure 3 The optimal monitoring frequency acquisition process is as follows:

[0091] Step 1: Construct the feature vector matrix

[0092] Load the characteristic parameters of the frequency sweep signal; for each frequency f, the eigenvector matrix... Represented as:

[0093]

[0094] eigenvector matrix It contains multiple features extracted from the guided wave response signal, with each row representing a liquid filling state and each column representing a specific feature;

[0095] Where n represents the total number of observations under different liquid filling conditions;

[0096] m is the total number of features extracted at this frequency. Each filling state has 8*11=88 feature parameters.

[0097] This represents the calculated value of the i-th sample on the j-th feature, for frequency f;

[0098] Step 2: Define category labels

[0099] Define a category vector Y, which contains the category label of each sample, corresponding to the fluid filling category of each row in matrix X;

[0100] Step 3: Calculate information gain

[0101] First, calculate the total entropy of the sample set:

[0102]

[0103] in, is the probability that the sample belongs to class i in the sample set, and c is the total number of classes;

[0104] Next, for each feature (each column) in matrix X, calculate its conditional entropy:

[0105] ,

[0106] These are all possible values ​​of feature A. It is a subset when the value of feature A is v. and These are the sizes of the subset and the original set, respectively;

[0107] Finally, the information gain is calculated based on the total entropy and conditional entropy:

[0108]

[0109] The information gain of the eigenvector matrix X corresponding to each excitation frequency is calculated;

[0110] Step 4: Calculate the statistical significance index p.

[0111] First, calculate the sum of squares between groups:

[0112]

[0113] The average value of all eigenvalues ​​in the eigenvector matrix X;

[0114] The average value of the eigenvalues ​​in each row of the eigenvector matrix X, i.e., in each filled state;

[0115] Next, calculate the sum of squares within the group:

[0116]

[0117] in, It is the i-th feature value in the j-th group;

[0118] Finally, the F-statistic is calculated based on the between-group sum of squares and the within-group sum of squares:

[0119]

[0120] Where N is the total number of samples, which is the number of rows in the feature vector matrix X;

[0121] k is the number of groups, representing the number of different liquid filling states;

[0122] Based on the F-statistic and the corresponding degrees of freedom between groups and intragroup freedom The p-value can be found through the F-distribution table;

[0123] The p-value of the eigenvector matrix X corresponding to each excitation frequency is calculated;

[0124] Step 5: Optimal Frequency Evaluation

[0125] Define evaluation metrics:

[0126] E = Info(f) * (1-p(f)), where f represents the excitation frequency;

[0127] Based on the information gain value and p-value of the eigenvector matrix X corresponding to each excitation frequency, the frequency with the largest evaluation index is selected as the optimal monitoring frequency.

[0128] In this embodiment, the calculation results of information gain and statistical significance are as follows: Figure 3 , Figure 4 As shown, the calculation results of the evaluation indicators are as follows: Figure 6 ,according to Figure 6 The optimal monitoring frequency was selected as the frequency corresponding to the maximum value of the evaluation index, which is 60kHz.

[0129] (5) Select the feature vector set under the optimal monitoring frequency, normalize it to form a standardized feature vector set, and input it into the support vector machine classification model for training;

[0130] (6) Use the trained support vector machine classification model to process new monitoring samples, and evaluate the liquid filling status of the pipeline in real time through the classification results output by the model.

[0131] In this embodiment, the classification labels of the SVM training model (i.e., the support vector machine classification model) are shown in Table 1. 50% of the data is used for training the model, and the remaining 50% is used to test the model's accuracy. Furthermore, different machine learning classification algorithms are used for training and comparison, and the prediction accuracy test results are shown in Table 2.

[0132] Table 1

[0133] 1. No filling fluid 10 2.1 / 4 filling 10 3.1 / 2 Filling 10 4.3 / 4 filling 10 5. Full fill fluid 10

[0134] Table 2

[0135] KNN 80% TREE 90% Linear Discriminant 90% SVM 100%

[0136] As shown in Table 2, the support vector machine has a 100% accuracy in predicting the liquid filling state of the pipeline, which is better than the other three algorithms.

[0137] To verify the effectiveness of the optimal monitoring frequency used in this invention, other excitation frequencies were compared with the optimal frequency. The training and testing process of the optimal monitoring frequency dataset was repeated, and the accuracy results are shown in Table 3.

[0138] Table 3

[0139] 70kHz 95% 80kHz 95% 90kHz 90% 100kHz 85%

[0140] As can be seen from the test data in Table 3, the guided wave excited by the optimal monitoring frequency used in this invention is more sensitive to changes in the liquid filling state of the pipeline and can be better used to identify the liquid filling state of the pipeline.

[0141] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0142] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0143] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0144] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0145] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for evaluating the liquid filling state of a pipeline based on longitudinal guided waves, characterized in that, Includes the following steps: (1) An excitation sensor array and a receiving sensor array are arranged outside the pipeline to excite and receive longitudinal guided wave signals; the arrangement method of the sensor array is as follows: 16 piezoelectric plates are uniformly installed around the first port of the pipeline to form an excitation sensor array, and the same voltage signal is applied to each piezoelectric plate to excite the longitudinal modal guided wave; 8 piezoelectric plates are uniformly installed around the second port of the pipeline to form a receiving sensor array to receive guided wave response signals at multiple angles; (2) Frequency sweeping experiments were conducted under different pipeline filling conditions to obtain longitudinal guided wave response signals at each excitation frequency; (3) Set multiple characteristic parameters, extract features from the longitudinal guided wave response signal, and obtain a set of feature vectors reflecting different excitation frequencies and liquid filling states; (4) Calculate the information gain and statistical significance of the feature vector set under each excitation frequency to obtain the optimal monitoring frequency that can most effectively distinguish different filling states; The method for obtaining the optimal monitoring frequency includes the following steps: Step 1: Construct the feature vector matrix Step 2: Define category labels Define a category vector Y, which contains the category label of each sample, corresponding to the fluid filling category of each row in matrix X; Step 3: Calculate information gain First, calculate the total entropy of the sample set; Next, for each feature (each column) in matrix X, calculate its conditional entropy; Finally, the information gain is calculated based on the total entropy and conditional entropy; Step 4: Calculate the statistical significance index p. First, calculate the sum of squares between groups; Second, calculate the sum of squares within each group; Finally, the F-statistic is calculated based on the between-group sum of squares and the within-group sum of squares. Based on the F-statistic and the corresponding degrees of freedom between groups and intragroup freedom The p-value can be found through the F-distribution table; The p-value of the eigenvector matrix X corresponding to each excitation frequency is calculated; Step 5: Optimal Frequency Evaluation Define evaluation metrics: E = Info(f) * (1-p(f)), where f represents the excitation frequency; Based on the information gain value and p value of the eigenvector matrix X corresponding to each excitation frequency, the frequency with the largest evaluation index is selected as the optimal monitoring frequency. (5) Select the feature vector set under the optimal monitoring frequency, normalize it to form a standardized feature vector set, and input it into the support vector machine classification model for training; (6) Use the trained support vector machine classification model to process new monitoring samples, and evaluate the liquid filling status of the pipeline in real time through the classification results output by the model.

2. The pipeline filling state assessment method based on longitudinal guided waves as described in claim 1, characterized in that, In step (2), the different pipe filling states include: Unfilled state: There is no liquid in the pipeline; 1 / 4 filling state: The liquid level in the pipe reaches 1 / 4 of the pipe diameter; Half-filled state: The liquid level in the pipe reaches 1 / 2 of the pipe diameter; 3 / 4 filling state: The liquid level in the pipe reaches 3 / 4 of the pipe diameter; Fully filled: The pipe is completely filled with liquid.

3. The pipeline filling state assessment method based on longitudinal guided waves as described in claim 1, characterized in that, In step (2), the frequency sweep experiment method is as follows: apply a gradually increasing voltage signal from 60kHz to 100kHz to each piezoelectric element of the excitation array, increasing by 10kHz each time to form a frequency sweep sequence, apply voltage signals of each frequency to the excitation array in sequence, and receive the corresponding longitudinal guided wave response signal samples through the receiving array.

4. The pipeline filling state assessment method based on longitudinal guided waves as described in claim 1, characterized in that, In step (3), the multiple feature parameters are as follows: Energy characteristic parameters include root mean square, maximum value, and peak-to-peak value, denoted as , ... , , ; Complexity features include box dimension, sample entropy, spectral entropy, and zero-crossing rate, denoted as […]. , , , ; Frequency characteristic parameters include power spectral density and frequency center, denoted as , respectively. , ; Statistical characteristic parameters include skewness and kurtosis, denoted as _____, ... , .

5. The pipeline filling state assessment method based on longitudinal guided waves as described in claim 1, characterized in that, In the method for obtaining the optimal monitoring frequency: Step 1: Construct the feature vector matrix For each frequency f, the eigenvector matrix Represented as: ； eigenvector matrix It contains multiple features extracted from the guided wave response signal, with each row representing a liquid filling state and each column representing a specific feature; Where n represents the total number of observations under different liquid filling conditions; m is the total number of features extracted at this frequency. Each filling state has 8*11=88 feature parameters. This represents the calculated value of the i-th sample on the j-th feature, for frequency f; Step 3: Calculate information gain First, calculate the total entropy of the sample set: ； in, is the probability that the sample belongs to class i in the sample set, and c is the total number of classes; Next, for each feature (each column) in matrix X, calculate its conditional entropy: ； These are all possible values ​​of feature A. It is a subset when the value of feature A is v. and These are the sizes of the subset and the original set, respectively; Finally, the information gain is calculated based on the total entropy and conditional entropy: ； The information gain of the eigenvector matrix X corresponding to each excitation frequency is calculated; Step 4: Calculate the statistical significance index p. First, calculate the sum of squares between groups: ； The average value of all eigenvalues ​​in the eigenvector matrix X; The average value of the eigenvalues ​​in each row of the eigenvector matrix X, i.e., in each filled state; Next, calculate the sum of squares within each group: ； in, It is the i-th feature value in the j-th group; Finally, the F-statistic is calculated based on the between-group sum of squares and the within-group sum of squares: ； Where N is the total number of samples, which is the number of rows in the feature vector matrix X; k is the number of groups, representing the number of different liquid filling states.

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