Track Analysis Method and System Integrating Frequency Domain Features and Multi-Dimensional Evaluation Metrics

Abstract

The present invention discloses a track analysis method and system that integrates frequency domain features and multi-dimensional evaluation indicators, including: filtering abnormal flight numbers in the track data collected by a track data acquisition device; obtaining second track data through fast Fourier transform, and extracting high-frequency energy features and peak frequency features; screening out target track data to be used for analysis through a unified standardization processing method; adaptively determining the interval where the optimal number of clustering clusters is located; traversing each number of clustering clusters within this interval, and determining the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, so as to perform clustering analysis through the K-Means++ algorithm according to the optimal number of clustering clusters K; visually displaying the clustering analysis results of the target track data. It can provide a theoretical reference for the clustering analysis of the arrival and departure tracks of aircraft in the terminal area through an analysis method that integrates frequency domain features and multi-dimensional evaluation indicators, making the optimization results conform to the actual situation and having more practical application potential.

Description

Technical Field

[0001] The present disclosure relates to the technical field of air traffic management, and in particular, to a flight track analysis method and system that integrates frequency domain features and multi-dimensional evaluation indicators. Background Art

[0002] With the rapid development of China's civil aviation transportation industry, the flight density of route transport aircraft in the airspace has increased rapidly, and the continuous growth of flight flow has made the terminal area airspace an area prone to airspace congestion, flight delays, and flight accidents. Against this background, a variety of terminal area aircraft flight track analysis technologies have emerged, aiming to optimize the layout of the air traffic network and improve the fluency and safety of air traffic through data mining and analysis. Aircraft flight track analysis technology depicts the movement state of aircraft in the air, assisting air traffic controllers to monitor the flight trajectories of aircraft and timely discover and handle potential abnormal flight tracks and flight conflicts.

[0003] However, traditional clustering methods, such as HDBSCAN and spectral clustering, etc., mostly rely on single time domain feature information, which is time-consuming to calculate and the effect is not ideal; in addition, the flight trajectories collected in the terminal area have complex characteristics such as large data volume, strong time series, and high abnormal rate, so there are certain limitations in processing.

[0004] Therefore, those skilled in the art urgently need to find a new technical solution to solve the above problems. Summary of the Invention

[0005] To overcome the problems existing in the related art, the present disclosure provides a flight track analysis method and system that integrates frequency domain features and multi-dimensional evaluation indicators.

[0006] According to the first aspect of the embodiments of the present disclosure, a flight track analysis method that integrates frequency domain features and multi-dimensional evaluation indicators is provided. The method includes:

[0007] Filter abnormal flight numbers from the flight track data collected by a flight track data acquisition device to obtain first flight track data after filtering, and the first flight track data is arranged in time sequence;

[0008] Traverse the first flight track data through fast Fourier transform, perform frequency domain conversion on the first flight track data to obtain second flight track data, and extract high-frequency energy features and peak frequency features from the second flight track data;

[0009] Through a unified standardization processing method, screen out the target flight track data required for analysis from the high-frequency energy features and peak frequency features;

[0010] According to the number of label data in the target flight track data, adaptively determine the interval where the optimal number of clustering clusters is located;

[0011] By means of a fusion traversal algorithm, each number of clustering clusters within the said interval is traversed, and the optimal number of clustering clusters K is determined according to the silhouette coefficient and the Davies-Bouldin index, so as to perform clustering analysis on the target track data according to the optimal number of clustering clusters K by means of the K-Means++ algorithm;

[0012] Visualize the clustering analysis results of the target track data.

[0013] Optionally, the abnormal flight number filtering of the track data collected by the track data acquisition device to obtain the filtered first track data includes:

[0014] For each flight number corresponding to a track node in the track data, extract the flight number and the corresponding feature label data;

[0015] Fill the flight number and the corresponding feature label into an empty dictionary created;

[0016] After the empty dictionary is filled, detect the number of label data corresponding to the flight number in the filled dictionary by means of a dictionary search algorithm;

[0017] If the number of label data is greater than 1, determine the flight number as an abnormal flight number;

[0018] Filter the abnormal flight numbers in the track data to obtain the first track data.

[0019] Optionally, the traversing of the first track data by means of the fast Fourier transform, performing frequency domain conversion on the first track data, and obtaining the second track data, and extracting the high-frequency energy feature and the peak frequency feature in the second track data includes:

[0020] Traverse the first track data by means of the fast Fourier transform is the sequence length of the first track data;

[0021] Perform frequency domain conversion on the first track data to obtain the second track data It can be represented by the following formula:

[0022]

[0023]

[0024] where is the th frequency component of the sequence, is the imaginary unit, is the frequency index from 0 to ​ is the FFT result of the even-index vector, is the FFT result of the odd-index vector, is the rotation factor in the FFT;

[0025] Extract the high-frequency energy feature in the second track data, and the high-frequency energy feature ε high can be expressed by the following formula:

[0026]

[0027] where, and are the lower and upper index of the high-frequency cut-off frequency respectively;

[0028] Extract the peak frequency feature in the second track data, and the peak frequency feature can be expressed by the following formula:

[0029]

[0030] where, is when it is the maximum value index, is the sampling frequency.

[0031] Optionally, the method of screening the target track data to be analyzed from the high-frequency energy feature and the peak frequency feature through the unified standardization processing method includes:

[0032] Calculate the mean and standard deviation of the features in each combination based on the combination of the high-frequency energy feature and the combination of the peak frequency feature;

[0033] For each feature, subtract the mean corresponding to the feature from the feature to obtain the difference, and then divide the difference by the standard deviation corresponding to the feature to obtain the quotient value, so as to obtain the target track data screened from the high-frequency energy feature and the peak frequency feature according to the quotient value corresponding to each feature.

[0034] Optionally, the method of traversing each number of clustering clusters in the interval according to the silhouette coefficient and the Davies-Bouldin index to determine the optimal number of clustering clusters K through the fusion traversal algorithm includes:

[0035] Traverse each number of clustering clusters in the area, and calculate the silhouette coefficient and the Davies-Bouldin index of each number of clustering clusters. Among them, the silhouette coefficient SC is expressed by the following formula:

[0036]

[0037] where, SC(n) represents the silhouette coefficient of data point n, where SC(n) ∈ [-1, 1]. represents the average distance from data point n to other data points in the same cluster. represents the minimum value of the average distance from data point n to all data points in other clusters, and M is the number of clustering clusters.

[0038] The Davies-Bouldin index DBI is represented by the following formula:

[0039]

[0040] where represents the average distance from data points in each clustering cluster k to the cluster center. represents the distance between different clustering clusters k and m. represents the scatter degree of each clustering cluster k. represents the maximum value of the scatter degrees of clustering cluster k and all other clustering clusters m, and M represents the number of clustering clusters.

[0041] Calculate a weighted score based on the silhouette coefficient and the Davies-Bouldin index to determine the optimal number of clustering clusters K according to the weighted score. Wherein, the weighted score WS is represented by the following formula:

[0042] WS = ω1 * SC - ω2 * DBI

[0043] where ω1 + ω2 = 1, ω1, ω2 > 0.

[0044] Optionally, the method further includes:

[0045] Evaluate the effect of the clustering analysis through the adjusted Rand coefficient and the normalized mutual information.

[0046] Optionally, the visualization display of the clustering analysis result of the target track data includes:

[0047] Visualize the clustering analysis result of the target track data from the aspects of visualization change situation, the number of tracks in each cluster, and the planar and spatial distribution of the tracks in each cluster.

[0048] According to the second aspect of the disclosed embodiments of the present invention, there is provided a track analysis system integrating frequency domain features and multi-dimensional evaluation indicators. The system includes:

[0049] An abnormal flight number filtering module that filters the track data collected by the track data acquisition device to obtain the filtered first track data, and the first track data is arranged in time sequence.

[0050] A frequency domain conversion module, connected to the abnormal flight number filtering module, traverses the first track data through fast Fourier transform, performs frequency domain conversion on the first track data, obtains second track data, and extracts high-frequency energy features and peak frequency features in the second track data;

[0051] A track data screening module, connected to the frequency domain conversion module, screens out the target track data to be used for analysis from the high-frequency energy features and peak frequency features through a unified standardization processing method;

[0052] An optimal number of clustering clusters determination module, connected to the track data screening module, adaptively determines the interval where the optimal number of clustering clusters is located according to the number of label data in the target track data;

[0053] A clustering analysis module, connected to the optimal number of clustering clusters determination module, traverses each number of clustering clusters in the interval through a fusion traversal algorithm, determines the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, and performs clustering analysis on the target track data according to the optimal number of clustering clusters K through the K-Means++ algorithm;

[0054] A visualization display module, connected to the clustering analysis module, visually displays the clustering analysis results of the target track data.

[0055] Optionally, the abnormal flight number filtering module includes:

[0056] A label data extraction unit, which extracts the flight number and the corresponding feature label data for each flight number corresponding to the track nodes in the track data;

[0057] An empty dictionary filling unit, connected to the label data extraction unit, fills the flight number and the corresponding feature label into the created empty dictionary;

[0058] A dictionary search unit, connected to the empty dictionary filling unit, after the empty dictionary is filled, detects the number of label data corresponding to the flight number in the filled dictionary through a dictionary search algorithm;

[0059] An abnormal flight number determination unit, connected to the dictionary search unit, if the number of label data is greater than 1, determines the flight number as an abnormal flight number;

[0060] An abnormal flight number filtering unit, connected to the abnormal flight number determination unit, filters the abnormal flight numbers in the track data to obtain the first track data.

[0061] Optionally, the track data screening module includes:

[0062] A mean and standard deviation calculation unit calculates the mean and standard deviation of the features within each combination based on the combination of the high-frequency energy features and the combination of the peak frequency features;

[0063] A track data screening unit is connected to the mean and standard deviation calculation unit. For each feature, the difference is obtained by subtracting the mean corresponding to the feature from the feature, and then the quotient is obtained by dividing the difference by the standard deviation corresponding to the feature, so as to obtain the target track data screened from the high-frequency energy features and the peak frequency features according to the quotient corresponding to each feature.

[0064] In summary, through the technical solution disclosed in the present invention, the following beneficial effects can be brought:

[0065] (1) The approach for analyzing terminal area tracks by integrating frequency domain features and multi-dimensional evaluation metrics can provide a theoretical reference for the clustering analysis of the arrival and departure tracks of aircraft in the terminal area. By considering the frequency domain features of the aircraft flight trajectories, the optimization results are in line with the actual situation and have greater potential for practical applications;

[0066] (2) Through frequency domain analysis, the periodic characteristics contained in the tracks can be more accurately reflected, providing richer information for track feature analysis, thereby optimizing the accuracy and efficiency of the analysis;

[0067] (3) It can clearly distinguish and represent the flight trajectories in each direction, and can also efficiently identify abnormal tracks, showing a high clustering effect and reliability.

[0068] Other features and advantages of the present invention will be described in detail in the subsequent specific implementation section. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] The drawings are used to provide a further understanding of the present disclosure and constitute a part of the specification. Together with the following specific implementation, they are used to explain the present disclosure, but do not constitute a limitation to the present disclosure. In the drawings:

[0070] Figure 1 is a schematic flowchart of a track analysis method that integrates frequency domain features and multi-dimensional evaluation metrics;

[0071] Figure 2 is a schematic diagram of the principle for obtaining and collecting track data;

[0072] Figure 3 is a schematic diagram of track data label search mapping and abnormal flight number filtering;

[0073] Figure 4 is a schematic diagram of adaptive re-clustering analysis for multi-dimensional attribute tracks;

[0074] Figure 5It is a structural diagram for the multi-level visualization display of the track clustering results in the terminal area;

[0075] Figure 6 It is a graph showing the variation of the silhouette coefficient and the Davies-Bouldin index with the number of clusters in the said example;

[0076] Figure 7 It is a distribution graph of the number of tracks included in each cluster in the said example;

[0077] Figure 8 It is a two-dimensional plane display diagram of some track clusters in the said example;

[0078] Figure 9 It is a three-dimensional solid display diagram of some track clusters in the said example;

[0079] Figure 10 It is a structural block diagram of a track analysis system that fuses frequency domain features and multi-dimensional evaluation indexes shown according to an exemplary embodiment;

[0080] Figure 11 It is according to Figure 10 shown is a structural block diagram of an abnormal flight number filtering module;

[0081] Figure 12 It is according to Figure 10 shown is a structural block diagram of a track data screening module. Detailed implementation manners

[0082] The following will explain in detail the specific implementation manners disclosed in the present invention with reference to the accompanying drawings. It should be understood that the specific implementation manners described herein are only for the purpose of illustration and explanation of the present disclosure, and are not used to limit the present disclosure.

[0083] Figure 1 It is a schematic flowchart of a track analysis method that fuses frequency domain features and multi-dimensional evaluation indexes, as Figure 1 shown, the method includes:

[0084] In step 101, the track data collected by the track data acquisition device is filtered for abnormal flight numbers to obtain the first track data after filtering.

[0085] Among them, the first track data is arranged in chronological order.

[0086] Exemplarily, as Figure 2 shown, through the data acquisition device, the first track data arranged in chronological order is collected and input. The track data consists of a large number of chronological track nodes, and each track node contains multi-dimensional attributes such as "flight number", "x coordinate", "y coordinate", "z coordinate", "timestamp", etc. The multi-dimensional attributes of the track node constitute the track vector of this track node.

[0087] Specifically, for each flight number corresponding to a track node in the track data, extract the flight number and the corresponding feature label data; fill the flight number and the corresponding feature label into an empty dictionary created; after the empty dictionary is filled, detect the number of label data corresponding to the flight number in the filled dictionary through a dictionary search algorithm; if the number of label data is greater than 1, determine the flight number as an abnormal flight number; filter the abnormal flight numbers in the track data to obtain the first track data.

[0088] Exemplarily, as Figure 3 shown, perform label search mapping on the track data. In the disclosed embodiments of the present invention, a dictionary search algorithm is used to filter abnormal flight numbers. The multi-dimensional attributes of track nodes constitute the track vector of this track node, and each flight number has several feature label data. Check whether there is a label in the first track data. If so, create an "empty dictionary" to store the label list corresponding to each flight number. Perform label search mapping on the first track data, that is, for each track data, extract the current flight number and its label data, and at the same time fill the label corresponding to this flight number into the empty dictionary and output the corresponding result (it should be noted that when a label is detected, the data is verification set data; when no label is detected, the data is training set data, that is, through the disclosed embodiments of the present invention, it can be automatically determined whether the input data is training set data or verification set data). Use a dictionary search algorithm to filter abnormal flight numbers, that is, after the empty dictionary is filled, through the dictionary search algorithm, check the number of labels held by all track data in the dictionary. If the number of labels held is greater than 1, determine the track data as the abnormal flight number to be filtered. Through the above method, all non-abnormal flight numbers can be extracted from the original data for filtering.

[0089] In step 102, traverse the first track data through a fast Fourier transform, perform frequency domain conversion on the first track data to obtain the second track data, and extract the high-frequency energy feature and peak frequency feature in the second track data.

[0090] Specifically, traverse the first track data through a fast Fourier transform is the sequence length of the first track data; perform frequency domain conversion on the first track data to obtain the second track data It can be represented by the following formula:

[0091]

[0092] where is the th frequency component of the sequence, is the imaginary unit, is the frequency index from 0 to The frequency index, is the FFT result of the even-index vector, is the FFT result of the odd-index vector, is the rotation factor in the FFT; extract the high-frequency energy feature in the second track data, and the high-frequency energy feature ε high can be expressed by the following formula:

[0093]

[0094] wherein, and are the lower and upper limits of the high-frequency cut-off frequency index respectively;

[0095] Extract the peak frequency feature in the second track data, and the peak frequency feature can be expressed by the following formula:

[0096]

[0097] wherein, is When it is the largest Value index, is the sampling frequency.

[0098] Exemplarily, the second track data after filtering abnormal flight numbers is processed by the Fast Fourier Transform (FFT) method, the time-domain data is converted into frequency-domain data, and the frequency-domain features in each direction are extracted. In the disclosed embodiments of the present invention, all the filtered track data is automatically traversed, and the FFT method is applied to perform unified frequency-domain conversion for the multi-dimensional attributes of each second track data, and then the high-frequency energy features and peak frequency features of the data points in the three spatial dimensions of "x, y, z" are respectively extracted to construct a feature vector. The fast Fourier transform technology is introduced, and its unique advantages in frequency-domain analysis are utilized, such as reducing noise interference, capturing subtle differences, reducing computational complexity, and enhancing the robustness of the algorithm, in order to break through the limitations of the existing track analysis methods.

[0099] It should be noted that the DFT (Discrete Fourier Transform) is a mathematical tool that converts a signal from the time domain to the frequency domain, and it can analyze the frequency components of the signal. The FFT is an algorithm for efficiently calculating the discrete Fourier transform and its inverse transform. The emergence of the FFT algorithm has greatly improved the calculation efficiency of the DFT, enabling fast frequency-domain analysis in practical applications.

[0100] The basic principle of the DFT algorithm is as follows: For a sequence of length its DFT is:

[0101]

[0102] where is the th frequency component of the sequence, is the imaginary unit, is the frequency index (ranging from 0 to ).

[0103] The basic principle of the FFT algorithm is to reduce the computational complexity of the DFT from to The specific implementation of the track vector FFT divide-and-conquer algorithm based on Cooley-Tukey is shown in Table 1:

[0104] Table 1 Flow of the FT divide-and-conquer algorithm for track vectors

[0105]

[0106] The FFT reduces the number of multiplications and additions through the butterfly operation. The butterfly operation is a specific complex multiplication and addition operation used to combine the results of two smaller DFTs to construct a larger DFT. For a sequence of length its FFT result can be obtained by combining the FFT results of two subsequences of length :

[0107]

[0108] where is the FFT result of the even-index vector, is the FFT result of the odd-index vector, is the rotation factor in the FFT.

[0109] Exemplarily, in the flight trajectory of an aircraft, high-frequency components are often related to rapidly changing motion characteristics, such as sudden turns, accelerations, or decelerations. These features are crucial for understanding the dynamic behavior of the aircraft; the peak frequency can indicate the main vibration or motion mode, helping to identify specific flight patterns or abnormal behaviors.

[0110] High-frequency energy generally refers to the total energy of the signal in the high-frequency part. In the frequency domain, the energy of the signal can be obtained by calculating the square of the modulus of the FFT result. If is the signal ​the FFT result, then the energy of the signal at the th frequency component where is the th modulus of the frequency component. Among them, and are the lower and upper limits of the high-frequency cut-off frequency index respectively, and in the actual calculation process, only the sum of the frequency energies with the absolute value of the frequency greater than 0.1 is calculated.

[0111] The peak frequency refers to the frequency component with the maximum signal energy. In the FFT result, the peak frequency corresponds to the index of the frequency component with the maximum energy. For its peak frequency means finding the that makes maximum. The actual value of the peak frequency can be obtained by converting the index to the actual frequency. If is the sampling frequency, is the number of points of the FFT, then the actual frequency of the th frequency component

[0112] In step 103, through a unified standardization processing method, the target track data for analysis is screened out from the high-frequency energy features and peak frequency features.

[0113] Specifically, based on the combination of the high-frequency energy features and the combination of the peak frequency features, calculate the mean and standard deviation of the features within each combination; for each feature, subtract the mean corresponding to the feature from the feature to obtain the difference, and then divide the difference by the standard deviation corresponding to the feature to obtain the quotient value, so as to obtain the target track data screened out from the high-frequency energy features and peak frequency features according to the quotient value corresponding to each feature.

[0114] Input and select the feature data for cluster analysis. First, select the multi-dimensional frequency domain features for cluster analysis, screen out the frequency domain feature quantities for analysis from the peak frequency and high-frequency energy, so as to further analyze the track features in the terminal area under different problem scenarios. Subsequently, it is necessary to standardize the selected multi-dimensional frequency domain features, that is, based on the combination of the above-selected feature quantities, calculate the mean and standard deviation of each feature quantity. For each value in each feature quantity, subtract the mean of the feature quantity from the value, and then divide by the standard deviation of the feature quantity. The data processed in this way will have a mean of 0 and a unit standard deviation, realizing the standardization of the features. The standardization of the multi-dimensional frequency domain feature quantities is convenient for the calculation and comparison of subsequent multi-dimensional evaluation indicators.

[0115] In step 104, according to the number of label data in the target track data, adaptively determine the interval where the optimal number of clustering clusters is located.

[0116] Exemplarily, if it is necessary to perform clustering analysis on a large amount of track data, it is necessary to determine the optimal number of track clustering clusters. According to the number of labels of the track data (validation set data), adaptively determine the interval where the optimal number of clustering clusters K is located, which is convenient for subsequent track clustering analysis.

[0117] As Figure 4 shown, adaptively determine the interval where the number of clusters for track clustering is located. By fusing the traversal algorithm and multi-dimensional evaluation metrics, find the optimal number of clusters for clustering. Subsequently, use the K-Means++ algorithm to perform clustering analysis on the track data. Select an appropriate type of frequency domain feature according to the scenario of the analysis problem, and perform a standardization operation on it to unify the dimension; then determine the selection range of the number of clustering clusters k, that is, the interval where the k value is located, according to the number of labels of the track data set (validation set data); subsequently, apply the traversal search method, combined with the comprehensive evaluation system for the track clustering effect based on multi-dimensional evaluation metrics, to find the optimal number of clusters K for clustering within the above k value interval, and perform K-Means++ clustering analysis on the selected track frequency domain features with this K value.

[0118] In step 105, by fusing the traversal algorithm, traverse each number of clustering clusters in this interval, and determine the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, so as to perform clustering analysis on the target track data through the K-Means++ algorithm according to this optimal number of clustering clusters K.

[0119] Exemplarily, after obtaining the interval where the number of clustering clusters is located, the method for determining the optimal number of clustering clusters based on traversal search can traverse each number of clusters in the interval, and evaluate the track clustering effect under each number of clusters by applying the comprehensive evaluation system for the track clustering effect based on multi-dimensional evaluation metrics, output the optimal number of clustering clusters, and perform clustering analysis on the track accordingly.

[0120] Specifically, traverse each number of clustering clusters in this area, calculate the silhouette coefficient and the Davies-Bouldin index of each number of clustering clusters. Among them, the silhouette coefficient SC is expressed by the following formula:

[0121]

[0122] Among them, SC(n) represents the silhouette coefficient of data point n, SC(n) ∈ [-1, 1], represents the average distance from data point n to other data points in the same cluster, represents the minimum value of the average distance from data point n to all data points in other clusters, M is the number of clustering clusters,

[0123] The Davis-Bouldin Index (DBI) is represented by the following formula:

[0124]

[0125] where represents the average distance from the data points in each cluster k to the cluster center, represents the distance between different clusters k and m, represents the scatter of each cluster k, represents the maximum value of the scatter of cluster k and the scatters of all other clusters m, and M represents the number of clusters;

[0126] According to the silhouette coefficient and the Davis-Bouldin Index, a weighted score is calculated to determine the optimal number of clusters K based on this weighted score. Among them, the weighted score WS is represented by the following formula:

[0127] WS = ω1 * SC - ω2 * DBI,

[0128] where ω1 + ω2 = 1, ω1, ω2 > 0.

[0129] The comprehensive evaluation system for the track clustering effect based on multi-dimensional evaluation indicators includes the silhouette coefficient (Silhoutte Coefficient, abbreviated as SC), the Davis-Bouldin Index (Davis-Bouldin Index, abbreviated as DBI), and the comprehensive weighted index (Weighted Score, abbreviated as WS) of the internal evaluation indicators; as well as the calculation and evaluation of the external evaluation indicators of the adjusted Rand index (Adjusted Rand Index, abbreviated as ARI) and the normalized mutual information (Normalized Mutual Information, abbreviated as NMI).

[0130] SC is a widely recognized metric in clustering analysis, used to evaluate the similarity and separation of samples in different clusters. The value range of this coefficient is from -1 to 1, where 1 indicates that the sample is closely connected to the samples in its cluster and is significantly separated from the samples in other clusters, thus indicating an ideal clustering effect. On the contrary, a value of -1 indicates that the sample may be misassigned to the current cluster and the clustering effect is poor. An SC value close to 0 means that the sample is on the decision boundary between two clusters and the clustering effect is average. Assume that for each data point n, its average distance to all other points in the same cluster is the cohesion within the cluster; the minimum value of the average distance from data point n to all points in other clusters is the separation between clusters.

[0131] DBI is an important tool for measuring the performance of clustering algorithms. It evaluates the clustering effect by calculating the ratio of the within-cluster distance to the between-cluster distance. The lower the DBI value, the closer the samples within the cluster and the higher the separation between clusters, indicating a better clustering effect. The calculation of the DBI index involves the average distance of samples within the cluster and the average distance between different clusters, providing an intuitive quantitative measure for the performance of clustering algorithms.

[0132] Considering that a single metric may not comprehensively reflect the comprehensive performance of clustering algorithms, the disclosed embodiments of the present invention introduce WS for comprehensive evaluation. By integrating SC and DBI and assigning them appropriate weights ω1 and ω2, a more comprehensive evaluation of clustering quality is provided. The calculation method of the weighted score takes into account the importance and influence of different metrics, enabling a more accurate reflection of the performance of clustering algorithms in different aspects.

[0133] In addition, the method further includes: evaluating the effect of clustering analysis through the adjusted Rand coefficient and normalized mutual information.

[0134] It should be noted that the calculation of internal evaluation metrics (SC, DBI, WS) is mainly used to find the optimal number of clusters for track clustering analysis, while the calculation of external metrics (ARI, NMI) is used for the final evaluation of the clustering effect of track data (validation set data). Among them, ARI is the adjusted Rand coefficient (Rand Index, RI), which is a metric for evaluating the clustering effect. Its value ranges from -1 to 1, and a higher value indicates that the clustering result is more consistent with the true labels.

[0135] The calculation of ARI is based on a confusion matrix, which contains two clustering results: the true labels (T) and the clustering labels (C). The elements in the confusion matrix represent the number of samples with true label and clustering label . Through this matrix, the following four values can be calculated: is the number of pairs of samples in the same cluster and also in the same class in the true labels; is the number of pairs of samples in the same cluster but not in the same class in the true labels; is the number of pairs of samples not in the same cluster but in the same class in the true labels; is the number of pairs of samples not in the same cluster and not in the same class in the true labels.

[0136]

[0137] Among them, is the expected value of RI, and max(RI) is the maximum possible value of RI, usually 1.

[0138] Normalized Mutual Information (NMI) is also an index for evaluating the clustering effect, with a value range from 0 to 1. The higher the value, the more consistent the clustering result is with the true labels. The calculation of NMI is based on Mutual Information (MI) and Entropy. MI measures the mutual dependence between two random variables, while Entropy measures the uncertainty of a random variable. By normalizing the mutual information, NMI makes the result independent of the data scale, thus enabling the comparison of the similarities of different datasets or different clustering results.

[0139]

[0140] Among them, is the mutual information, representing the mutual information quantity between the random variables and ; and are the entropies of and respectively; is the joint probability that the random variable takes the value and takes the value ; and are the marginal probabilities of and respectively.

[0141] To illustrate the calculation method in this step more clearly, the following is an example:

[0142] To determine the weights of the two coefficients in the comprehensive score of the clustering result, the present invention applied three weight combinations of ω1 = 0.6, ω2 = 0.4; ω1 = 0.5, ω2 = 0.5; ω1 = 0.4, ω2 = 0.6 respectively during the case study, and conducted a visual analysis of the clustering results. Finally, the weight combination of ω1 = 0.5, ω2 = 0.5 was determined. This weight combination has the following advantages - Balanced evaluation: It can consider the cohesion within the clusters and the separation between the clusters in a balanced manner, ensuring that the clustering result is both tight and has good separation. Comprehensiveness: Combining the two indicators can more comprehensively reflect the complexity and multi-dimensional characteristics of the track data, thereby improving the accuracy and reliability of the clustering evaluation. Flexibility: In different situations, the weights can be adjusted to adapt to different clustering requirements and data characteristics, improving the flexibility of the clustering evaluation. Therefore, setting the weights of the silhouette coefficient and the Davies coefficient to 0.5 can ensure that the clustering evaluation considers both the tightness within the clusters and the separation between the clusters, making the evaluation of the clustering effect more scientific and accurate. For the optimal number of clustering clusters k, the present invention based on the maximum number of labels in the data sets the iteration space of the k value to [2, 60] during the model testing stage.

[0143] Based on the above analysis, by mining and analyzing the frequency-domain characteristic quantities of the track data (verification and data), the analysis results of the following three indicators can be obtained, and the variation of each indicator with the number of clustering clusters is as Figure 6 shown.

[0144] Analysis of the silhouette coefficient (SC):

[0145] As Figure 6 shown, as the number of clustering clusters increases, the silhouette coefficient initially shows an upward trend. This is because when the number of clustering clusters is small, the samples within the cluster may not be tight enough. However, as k increases, the samples are more carefully assigned to smaller and more similar clusters, thereby enhancing the cohesion within the cluster. When k is in the interval [26, 30], the value of SC changes slightly, and its range (maximum value minus minimum value) is 0.0044, which is almost negligible. This indicates that the clustering effect within this interval is excellent. When k = 30, SC reaches the maximum value of 0.8159, which marks that at this number of clustering clusters, the cohesion within the cluster and the separation between clusters reach the best balance. After that, as the number of clustering clusters continues to increase, there are sharp fluctuations when k is in the interval [33, 59]. This fluctuation indicates that within this interval, the clustering effect is constantly adjusted and optimized, but it never exceeds the SC value when k = 30. After that, when k continues to increase, since the number of samples in each cluster decreases, the cohesion within the cluster decreases, so the SC value shows a downward trend instead.

[0146] Analysis of the Davies-Bouldin index (DBI):

[0147] As Figure 6 shown, it is observed that the DBI value shows a trend of first decreasing and then increasing as k increases. This phenomenon reveals a clustering principle: increasing the number of clusters within a certain range can enhance the separation between clusters, but when the number of clusters increases beyond a certain threshold, too many clusters may lead to a decrease in the separation of data points within the cluster, thereby reducing the overall performance of clustering. At the beginning, as the number of clusters increases, the DBI value gradually decreases, and the clustering effect continuously improves, and reaches the minimum value when k is in the interval [23, 30]. When k = 30, the DBI value drops to the lowest point of 0.4008, indicating that at this number of clusters, the clustering effect reaches the optimal state. However, when the k value continues to increase, the DBI value shows a fluctuating upward trend instead. This may be due to the fact that too large a number of clusters leads to a decrease in the separation of data points within the cluster, resulting in too few data points within the cluster to effectively distinguish different clusters, thus causing the clustering effect to deteriorate.

[0148] From the relationship between the DBI and the number of clusters k alone, we can conclude that when k = 30, the clustering effect reaches the best. This number of clusters not only ensures the compactness of the data points within the clusters, but also maintains the distinctiveness between the clusters, achieving the optimal performance of the clustering algorithm.

[0149] Analysis of the weighted score (WS):

[0150] Determining the optimal number of clusters from a single indicator perspective is too one-sided. The present invention determines the optimal number of clustering clusters k based on the comprehensive weighted score of SC and DBI. By analyzing the relationship between the weighted score (WS) and the number of clustering clusters k, we find that WS first decreases and then increases with the increase of k, which indicates that the number of clustering clusters has a significant impact on the clustering performance. When k is small, increasing the number of clustering clusters helps to improve the cohesion within the clusters (SC increases) and the separation between the clusters (DBI decreases). As k gradually increases, WS also continuously increases, and when k = 30, WS reaches the maximum value of 0.2076, which indicates that at this number of clustering clusters, after comprehensively considering SC and DBI, the clustering effect is the best. As the value of k further increases, when the number of clusters exceeds the optimal range supported by the internal structure of the data, the number of samples in each cluster decreases, resulting in a decrease in the similarity within the clusters, and the distinctiveness between the clusters also decreases accordingly. This excessive subdivision phenomenon weakens the discrimination ability of the clustering algorithm, and thus damages the overall clustering effect, which is reflected in the weighted score (WS) as a decrease in performance.

[0151] By carefully analyzing the relationship between WS and the number of clustering clusters k, we can conclude that when k = 30, the weighted score reaches 0.2076. At this time, SC = 0.8159 and DBI = 0.4008, both of which are at their respective optimal levels. Therefore, after comprehensive evaluation, it can be concluded that when the number of clustering clusters k = 30, the clustering effect is the best. Using the obtained optimal number of clustering clusters K = 30, the frequency domain characteristic quantities of the track data (verification set data) are finally clustered and analyzed, and the external indicators are calculated. We can get ARI = 0.3705 and NMI = 0.6718. Therefore, to a certain extent, it can be considered that the results of track clustering have relatively high accuracy and credibility, and the performance of the two external indicators is sufficient for further data analysis and decision support.

[0152] As Figure 7 shown, some management implications can also be obtained by analyzing the number distribution of tracks in different clusters. Please refer to Table 2. Cluster 1 contains 2,755 tracks, which account for about 23.29% of the total number of flights. Since the number of such tracks is large, they play an important role in the overall flight flow and have an important impact on air traffic management and route planning. Therefore, it is necessary to pay special attention to the aircraft operation activities here.

[0153] Table 2 Distribution of the number of tracks in each cluster

[0154]

[0155] As Figures 8 - 9 shown, the present invention selects some representative clusters and uses plane and three-dimensional visualization techniques to intuitively present the clustering results of the arrival and departure flight tracks of aircraft in the terminal area. Through this intuitive display method, it can be clearly observed that after analyzing the arrival and departure flight tracks of aircraft in the terminal area using this clustering method, the flight tracks of each aircraft in its respective approach or departure direction are clearly reflected and distinguished. The clustered flight tracks show a high degree of concentration, indicating an obvious aggregation trend, which shows that the clustering effect is significant. The flight tracks within each cluster are closely connected, while the flight tracks between clusters are relatively independent. This high degree of separation further confirms the effectiveness of the clustering. In addition, through this method, we can also efficiently identify abnormal flight tracks in each direction. These abnormal flight tracks often appear as outliers in the clustering graph, thus providing convenience for further analysis and processing.

[0156] In step 106, the clustering analysis result of the target flight track data is visually displayed.

[0157] Exemplarily, as Figure 5 shown, a multi-level visual display of the flight track clustering result is performed, including but not limited to the two-dimensional plane and three-dimensional distribution of the flight tracks included in each clustering cluster, the number of flight tracks included in each clustering cluster, the variation of multi-dimensional evaluation indexes with the number of clustering clusters, etc.

[0158] Specifically, Figure 8 is the two-dimensional plane visual display of the flight track. As Figure 8 shown, through two-dimensional plane analysis, the rationality of the airway setting can be intuitively analyzed. For example, Figure 8 -e)8-f) show that the flight tracks in cluster 21 and cluster 27 are very messy and the phenomenon of aircraft flying around frequently occurs, indicating that there are problems with the arrival and departure route settings here and it may be necessary to re-evaluate and plan; while Figure 8 -a) and 8-d) show that the flight tracks included in cluster 0 and cluster 18 are relatively regular. Although there are still some flight tracks flying around in cluster 0, generally speaking, it can still show that the route setting here is reasonable and the aircraft can obtain reasonable route allocation and flight activities here. In addition, through the two-dimensional visualization of the flight track clustering result, abnormal flight tracks can be identified more quickly. For example, Figure 8 -b) and 8-c) can clearly and intuitively display the outliers and abnormal flight tracks in cluster 5 and cluster 16, which can provide support for further analysis of the abnormal reasons.

[0159] Figure 9 is the three-dimensional visual display of the flight track. AsFigure 9 As shown, only studying the planar characteristics of the flight trajectory will ignore the multi-dimensional characteristics of the trajectory points and also lead to limitations in further exploring abnormal flight tracks of aircraft. Therefore, while combining planar analysis, it is more necessary to conduct research around the three-dimensional spatial distribution of the aircraft flight trajectory. Taking Cluster 0 as an example, since it has a considerable number of flight tracks, more detailed flight track management and traffic control strategies are required to avoid potential conflicts and improve airspace utilization. However, as Figure 8 shown in -a), observing only from a two-dimensional perspective is too messy and not conducive to further analysis. At this time, it is necessary to supplement with Figure 9 the three-dimensional spatial visualization shown in -a) for multi-level analysis, so as to comprehensively grasp the situation of the aircraft flight trajectory. In addition, from the two-dimensional display diagram of Cluster 18 ( Figure 8 -d)), its flight tracks are relatively concentrated and regular; but from a three-dimensional perspective ( Figure 9 -d)), there is still a relatively large-scale phenomenon of flying around in Cluster 18, which can only be better presented in three-dimensional space. For Cluster 21, even though its two-dimensional flight tracks are relatively messy, they are still relatively concentrated and regular in three-dimensional space. For Clusters 15 and 16, supplementing with three-dimensional analysis of the clustering results can also more quickly and efficiently identify abnormal flight tracks. Therefore, only analyzing the two-dimensional clustering results of flight tracks has certain limitations, which is easy to mislead the analysis and lead to wrong conclusions.

[0160] Figure 10 is a structural block diagram of a flight track analysis system that fuses frequency domain features and multi-dimensional evaluation indicators shown according to an exemplary embodiment. As Figure 10 shown, the flight track analysis system 1000 includes:

[0161] An abnormal flight number filtering module 1010 filters the flight track data collected by the flight track data acquisition device to obtain the filtered first flight track data, and the first flight track data is arranged in time sequence;

[0162] A frequency domain conversion module 1020 is connected to the abnormal flight number filtering module 1010, traverses the first flight track data through fast Fourier transform, performs frequency domain conversion on the first flight track data to obtain second flight track data, and extracts the high-frequency energy feature and peak frequency feature in the second flight track data;

[0163] A flight track data screening module 1030 is connected to the frequency domain conversion module 1020, and screens out the target flight track data required for analysis from the high-frequency energy feature and peak frequency feature through a unified standardization processing method;

[0164] The optimal number of clustering clusters determination module 1040, connected to the track data screening module 1030, adaptively determines the interval where the optimal number of clustering clusters is located according to the number of label data in the target track data;

[0165] The clustering analysis module 1050, connected to the optimal number of clustering clusters determination module 1040, traverses each number of clustering clusters in this interval through a fusion traversal algorithm, determines the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, and performs clustering analysis on the target track data according to the optimal number of clustering clusters K through the K-Means++ algorithm;

[0166] The visualization display module 1060, connected to the clustering analysis module 1050, visually displays the clustering analysis results of the target track data.

[0167] Figure 11 is based on Figure 10 shows a structural block diagram of an abnormal flight number filtering module, as Figure 11 shown, the abnormal flight number filtering module 1010 includes:

[0168] The label data extraction unit 1011 extracts the flight number and the corresponding feature label data for each flight number corresponding to the track nodes in the track data;

[0169] The empty dictionary filling unit 1012, connected to the label data extraction unit 1011, fills the flight number and the corresponding feature label into the created empty dictionary;

[0170] The dictionary search unit 1013, connected to the empty dictionary filling unit 1012, after the empty dictionary is filled, detects the number of label data corresponding to the flight number in the filled dictionary through a dictionary search algorithm;

[0171] The abnormal flight number determination unit 1014, connected to the dictionary search unit 1013, if the number of label data is greater than 1, determines the flight number as an abnormal flight number;

[0172] The abnormal flight number filtering unit 1015, connected to the abnormal flight number determination unit 1014, filters the abnormal flight numbers in the track data to obtain the first track data.

[0173] Figure 12 is based on Figure 10 shows a structural block diagram of a track data screening module, as Figure 12 shown, the track data screening module 1030 includes:

[0174] The mean and standard deviation calculation unit 1031 calculates the mean and standard deviation of the features within each combination based on the combination of the high-frequency energy features and the combination of the peak frequency features;

[0175] The track data screening unit 1032, which is connected to the mean and standard deviation calculation unit 1031, for each feature, subtracts the mean corresponding to the feature from the feature to obtain a difference, and then divides the difference by the standard deviation corresponding to the feature to obtain a quotient value, so as to obtain the target track data screened from the high-frequency energy features and the peak frequency features according to the quotient value corresponding to each feature.

[0176] In summary, the present invention discloses a track analysis method and system that fuses frequency domain features and multi-dimensional evaluation indicators, including: filtering abnormal flight numbers of the track data collected by the track data acquisition device; obtaining the second track data through fast Fourier transform, and extracting high-frequency energy features and peak frequency features; screening out the target track data to be used for analysis through a unified standardization processing method; adaptively determining the interval where the optimal number of clustering clusters is located; traversing each number of clustering clusters within this interval, and determining the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, so as to perform clustering analysis according to the optimal number of clustering clusters K through the K-Means++ algorithm; visually displaying the clustering analysis results of the target track data. It can provide a theoretical reference for the clustering analysis of the arrival and departure tracks of aircraft in the terminal area through an analysis method that fuses frequency domain features and multi-dimensional evaluation indicators, making the optimization results conform to the actual situation and having more practical application potential.

[0177] The preferred embodiments of the present disclosure have been described in detail above with reference to the accompanying drawings. However, the present disclosure is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present disclosure, various simple modifications can be made to the technical solutions of the present disclosure, and these simple modifications all belong to the protection scope of the present disclosure.

[0178] In addition, it should be noted that, in the case of no contradiction, the various specific technical features described in the above specific embodiments can be combined in any suitable way. To avoid unnecessary repetition, the present disclosure does not separately describe various possible combination methods.

[0179] Furthermore, any combination can be made between different embodiments of the present disclosure as long as it does not violate the idea of the present disclosure, and it should also be regarded as the content disclosed by the present disclosure.

Claims

1. A track analysis method that fuses frequency domain features and multi-dimensional evaluation metrics, characterized in that The method includes: Filtering abnormal flight numbers from the track data collected by the track data acquisition device to obtain the first track data after filtering, and arranging the first track data in time sequence; Traversing the first track data through fast Fourier transform, performing frequency domain conversion on the first track data to obtain second track data, and extracting the high-frequency energy feature and peak frequency feature in the second track data; Selecting the target track data to be analyzed from the high-frequency energy feature and peak frequency feature through a unified standardization processing method; Adaptive determining the interval where the optimal number of clustering clusters is located according to the number of label data in the target track data; Traversing each number of clustering clusters in the interval through a fusion traversal algorithm, determining the optimal number of clustering clusters K according to the silhouette coefficient and Davies-Bouldin index, and performing clustering analysis on the target track data according to the optimal number of clustering clusters K through the K-Means++ algorithm; Visualizing the clustering analysis result of the target track data.

2. The track analysis method that fuses frequency domain features and multi-dimensional evaluation indicators according to claim 1, wherein The filtering abnormal flight numbers from the track data collected by the track data acquisition device to obtain the first track data after filtering includes: For each flight number corresponding to a track node in the track data, extracting the flight number and the corresponding feature label data; Filling the flight number and the corresponding feature label into an empty dictionary created; After the empty dictionary is filled, detecting the number of label data corresponding to the flight number in the filled dictionary through a dictionary search algorithm; If the number of label data is greater than 1, determining the flight number as an abnormal flight number; Filtering the abnormal flight numbers in the track data to obtain the first track data.

3. The track analysis method that fuses frequency domain features and multi-dimensional evaluation indicators according to claim 1, characterized in that The traversing the first track data through fast Fourier transform, performing frequency domain conversion on the first track data to obtain second track data, and extracting the high-frequency energy feature and peak frequency feature in the second track data includes: Traverse the first track data through fast Fourier transform , is the sequence length of the first track data; Perform a frequency-domain conversion on the first track data to obtain second track data It can be expressed by the following formula: ， ， ， Among them, is the th frequency component of the sequence, is the imaginary unit, is the frequency index from 0 to ( ), is the FFT result of the even-index vector, is the FFT result of the odd-index vector, is the twiddle factor in the FFT; Extract the high-frequency energy features in the second track data, where the high-frequency energy features can be represented by the following formula: ， Among them, and are the lower and upper limit indexes of the high-frequency cut-off frequency respectively; Extract the peak frequency feature in the second track data, and the peak frequency feature can be expressed by the following formula: ， Among them, is when it is the maximum value index, is the sampling frequency.

4. The track analysis method that fuses frequency domain features and multi-dimensional evaluation indexes according to claim 1, wherein The selecting the target track data to be analyzed from the high-frequency energy feature and peak frequency feature through a unified standardization processing method includes: Based on the combination of the high-frequency energy feature and the combination of the peak frequency feature, calculating the mean and standard deviation of the features in each combination; For each feature, obtaining the difference by subtracting the mean corresponding to the feature from the feature, and then dividing the difference by the standard deviation corresponding to the feature to obtain the quotient value, and obtaining the target track data selected from the high-frequency energy feature and peak frequency feature according to the quotient value corresponding to each feature.

5. The track analysis method that fuses frequency domain features and multi-dimensional evaluation indicators according to claim 1, wherein The traversing each number of clustering clusters in the interval through a fusion traversal algorithm, determining the optimal number of clustering clusters K according to the silhouette coefficient and Davies-Bouldin index includes: Traversing each number of clustering clusters in the interval, calculating the silhouette coefficient and Davies-Bouldin index of each number of clustering clusters, where the silhouette coefficient SC is represented by the following formula: ， Among them, , represents the silhouette coefficient of data point n, , represents the average distance from data point n to other data points in the same cluster, represents the data point to the minimum value of the average distance to all data points in other clusters, is the number of clustering clusters, The Davies-Bouldin index DBI is represented by the following formula: ， Among them, represents the average distance from the data points in each cluster to the cluster center, represents different clusters and the distance between them, represents the scatter of each cluster cluster, represents the cluster and all other clusters the maximum value of the scatter, represents the number of clusters, represents all clusters the average distance from the data points in the cluster to the cluster center, represents all clusters the average distance from the data points in the cluster to the cluster center; Calculate a weighted score based on the silhouette coefficient and the Davies-Bouldin index to determine the optimal number of clustering clusters K according to the weighted score, where the weighted score WS is represented by the following formula: , wherein, .

6. The track analysis method integrating frequency domain features and multi-dimensional evaluation indexes according to claim 5, characterized in that The method further includes: Evaluate the effect of clustering analysis by the adjusted Rand index and the normalized mutual information 7. The track analysis method that fuses frequency domain features and multi-dimensional evaluation indicators according to claim 1, wherein The visualization display of the clustering analysis result of the target track data includes: Visualize the clustering analysis result of the target track data from the visualization change situation, the number of tracks in each cluster, and the planar and spatial distribution of the tracks in each cluster.

8. A track analysis system that integrates frequency domain features and multi-dimensional evaluation indicators, characterized in that, The system includes: An abnormal flight number filtering module that filters the abnormal flight numbers in the track data collected by the track data acquisition device to obtain the first track data after filtering, and the first track data is arranged in time sequence; A frequency domain conversion module, connected to the abnormal flight number filtering module, traverses the first track data through fast Fourier transform, performs frequency domain conversion on the first track data, obtains the second track data, and extracts the high-frequency energy feature and peak frequency feature in the second track data; A track data screening module, connected to the frequency domain conversion module, screens out the target track data to be used for analysis from the high-frequency energy feature and peak frequency feature through a unified standardization processing method; An optimal number of clustering clusters determination module, connected to the track data screening module, adaptively determines the interval where the optimal number of clustering clusters is located according to the number of label data in the target track data; A clustering analysis module, connected to the optimal number of clustering clusters determination module, traverses each number of clustering clusters in the interval through a fusion traversal algorithm, determines the optimal number of clustering clusters K according to the silhouette coefficient and the Davies-Bouldin index, and performs clustering analysis on the target track data according to the optimal number of clustering clusters K through the K-Means++ algorithm; A visualization display module, connected to the clustering analysis module, visualizes the clustering analysis result of the target track data.

9. The track analysis system that fuses frequency domain features and multi-dimensional evaluation indexes according to claim 8, wherein The abnormal flight number filtering module includes: A label data extraction unit that extracts the flight number and the corresponding feature label data for each flight track node in the track data; An empty dictionary filling unit, connected to the label data extraction unit, fills the flight number and the corresponding feature label into the created empty dictionary; A dictionary search unit, connected to the empty dictionary filling unit, after the empty dictionary is filled, detects the number of label data corresponding to the flight number in the filled dictionary through a dictionary search algorithm; An abnormal flight number determination unit, connected to the dictionary search unit, determines the flight number as an abnormal flight number if the number of label data is greater than 1; An abnormal flight number filtering unit, connected to the abnormal flight number determination unit, filters the abnormal flight numbers in the track data to obtain the first track data.

10. The track analysis system integrating frequency domain features and multi-dimensional evaluation indexes according to claim 8, characterized in that, The track data screening module includes: A mean and standard deviation calculation unit that calculates the mean and standard deviation of the features in each combination based on the combination of the high-frequency energy features and the combination of the peak frequency features; A track data screening unit, connected to the mean standard deviation calculation unit, for each feature, subtracts the mean value corresponding to the feature from the feature to obtain a difference value, and then divides the difference value by the standard deviation corresponding to the feature to obtain a quotient value, so as to obtain target track data screened from the high-frequency energy feature and the peak frequency feature according to the quotient value corresponding to each feature.

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