Track abnormality detection clustering method based on hamming distance similarity

Abstract

The application discloses a track abnormality detection clustering method based on Hamming distance similarity, which comprises the following steps: obtaining track data in a set region and a specified time period and performing data preprocessing; performing track time sequence information feature extraction on the preprocessed track data based on a long short-term memory (LSTM) self-encoder to obtain track features; converting the track features into binary hash codes based on an iterative quantization hash algorithm, calculating the Hamming distance between the hash codes of each track feature, and calculating the affinity matrix of the track features based on the Hamming distance similarity; performing iterative clustering of the track features through a domination set clustering algorithm; and outputting track features that cannot be clustered as abnormal tracks in the iterative clustering process. The application can obtain the affinity matrix for clustering quickly and efficiently without relying on artificial marking of similarity relations, and realizes track abnormality detection in a large-scale data environment.

Description

Technical Field

[0001] The present invention relates to the technical field of track detection, and in particular to a clustering method for track anomaly detection based on Hamming distance similarity. Background Art

[0002] Hashing algorithms can map data from a high-dimensional feature space to a low-dimensional Hamming space, taking up less space and reducing query complexity. Numerous studies have investigated hashing algorithms for image or single feature retrieval. Unsupervised hashing preserves the similarity between samples in the original feature space, similar to the Euclidean distance used in clustering algorithms, but with significantly less computational complexity. While there's no universal, strict definition of clustering, data points within a cluster are generally highly similar, while points outside the cluster are highly dissimilar. Graph clustering algorithms, such as dominating set clustering, use a weighted undirected graph to represent the similarity between data points. By calculating the weights between points within and outside the cluster, clusters with high internal correlation are generated. Dominating set clustering algorithms have the advantages of not requiring a predefined number of cluster centers, not classifying random points, and allowing for overlapping clusters.

[0003] Currently, the technical implementation approaches for detecting abnormal ship tracks at home and abroad fall into two main categories: 1. Building ship behavior distribution and models based on historical Automatic Identification System (AIS) data, treating samples that do not conform to the normal model as abnormal trajectory samples; 2. Manually calibrating abnormal behaviors based on manually set rules, and defining those that meet the conditions as abnormal tracks. However, due to the current difficulties in obtaining abnormal ship data in the maritime regulatory field and the confidentiality and closed nature of ship management systems, most research has relied on unsupervised models based on AIS data to analyze abnormal tracks. Summary of the Invention

[0004] In order to overcome the defects and shortcomings of the existing technology, the present invention provides a track anomaly detection clustering method based on Hamming distance similarity. The present invention performs hash coding on the basis of data cleaning and segmentation and long short-term memory artificial neural network (LSTM) preprocessing track data set, proposes an affinity matrix based on Hamming distance similarity, and uses a dominating set clustering algorithm to cluster the data according to the affinity matrix to realize ship track anomaly detection. The similarity measurement based on Hamming distance in the present invention has the characteristics of discreteness and fast binary bit calculation. It can quickly and efficiently obtain the affinity matrix for clustering without relying on manual labeling of similarity relationships, thereby realizing track anomaly detection in a large-scale data environment.

[0005] In order to achieve the above object, the present invention adopts the following technical solutions:

[0006] The present invention provides a clustering method for track anomaly detection based on Hamming distance similarity, comprising the following steps:

[0007] Acquire track data in the set area and specified time period and perform data preprocessing;

[0008] The track time series information features of the pre-processed track data are extracted based on the long short-term memory network LSTM autoencoder to obtain the track features;

[0009] The track features are converted into binary hash codes based on the iterative quantized hash algorithm, the Hamming distance between the hash codes of each track feature is calculated, and the affinity matrix of the track features based on the similarity of the Hamming distance is calculated;

[0010] The track features are iteratively clustered using the dominating set clustering algorithm. During the iterative clustering process, the track features that cannot be clustered are output as abnormal tracks.

[0011] As a preferred technical solution, the acquisition of track data in a set area and a specified time period and the data preprocessing specifically include:

[0012] Obtain the geographic location information of all ships in the set area and specified time period, set filtering conditions, delete and clean the data;

[0013] The AIS data are sorted by ship number and arranged in chronological order to obtain the track data of each ship within the specified time period.

[0014] As a preferred technical solution, the long short-term memory network LSTM autoencoder extracts track time series information features from the preprocessed track data, specifically including:

[0015] The preprocessed track data is input into the long short-term memory network LSTM network and mapped to the hidden layer to generate track hidden features. Then it is restored from the hidden layer through the long short-term memory network LSTM and mapped back to track data of the same dimension to extract the track hidden features. The specific expression is:

[0016] D=encoder(P)

[0017] P′=decoder(D)

[0018] Where P represents the preprocessed track data, D is the track hidden feature, P' is the restored track data, encoder(.) and decoder(.) are the encoder and decoder LSTM networks of the autoencoder respectively.

[0019] As a preferred technical solution, the long short-term memory network LSTM autoencoder uses the mean square error loss function to calculate the error between the input and output of each track data. The mean square error loss function is expressed as:

[0020]

[0021] Among them, d is the characteristic dimension of the track at each time point, n is the number of time points set in a track, and p q and p q ' are the original input data at the qth time point of the track and the data restored by the autoencoder.

[0022] As a preferred technical solution, the track features are converted into binary hash codes based on an iterative quantized hash algorithm, specifically including:

[0023] Perform principal component analysis on the track characteristics to obtain its first k eigenvectors, randomly generate an orthogonal rotation matrix R to balance the information in the eigenvectors, and minimize the quantization error Q to perform iterative quantization optimization, which can be specifically expressed as:

[0024]

[0025] Where D represents the track characteristics, W represents the number of feature vectors, and B is the hash code of the track data.

[0026] As a preferred technical solution, the affinity matrix of track features based on Hamming distance similarity is calculated, specifically including:

[0027] The number of different bits of the hash code of each track feature is calculated as the Hamming distance. The affinity in the affinity matrix based on the Hamming distance similarity is expressed as:

[0028]

[0029] Among them, a ij is the affinity between track features i and j, Ham ij is the Hamming distance between two track features i and j, and σ is the preset parameter for affinity calculation.

[0030] As a preferred technical solution, iterative clustering of track features is performed using a dominating set clustering algorithm, specifically including:

[0031] Use the affinity matrix A based on Hamming distance similarity composed of affinity elements sim =(a ij ), perform iterative clustering of track features, expressed as:

[0032]

[0033] Among them, y i and(·) i represents the weight and corresponding vector value of the i-th track, t represents the number of iterative calculations in a clustering, and y is the vector composed of the weights of all track feature data;

[0034] The weights of all N tracks in the first clustering are initialized as For each track feature:

[0035]

[0036] Among them, N T Represents the total number of unclustered samples in the Tth clustering. When T=1 is the first clustering, N1=N.

[0037] The present invention also provides a track anomaly detection clustering system based on Hamming distance similarity, which is used to implement the above-mentioned track anomaly detection clustering method based on Hamming distance similarity. The system includes: a track data acquisition module, a data preprocessing module, a feature extraction module, a hash coding module, a Hamming distance calculation module, an affinity matrix calculation module, an iterative clustering module, and an abnormal track output module;

[0038] The track data acquisition module is used to acquire track data in a set area and a specified time period;

[0039] The data preprocessing module is used to perform data preprocessing on the track data;

[0040] The feature extraction module is used to extract track time series information features from the preprocessed track data based on the long short-term memory network LSTM autoencoder to obtain track features;

[0041] The hash coding module is used to convert the track characteristics into binary hash codes based on an iterative quantized hash algorithm;

[0042] The Hamming distance calculation module is used to calculate the Hamming distance between the hash codes of each track feature;

[0043] The affinity matrix calculation module is used to calculate the affinity matrix of the track features based on the Hamming distance similarity;

[0044] The iterative clustering module is used to perform iterative clustering of track features through a dominating set clustering algorithm;

[0045] The abnormal track output module is used to output track features that cannot be clustered as abnormal tracks during the iterative process of clustering.

[0046] The present invention also provides a computer-readable storage medium storing a program, which, when executed by a processor, implements the above-mentioned track anomaly detection clustering method based on Hamming distance similarity.

[0047] The present invention also provides a computing device, comprising a processor and a memory for storing a program executable by the processor, wherein when the processor executes the program stored in the memory, the above-mentioned track anomaly detection clustering method based on Hamming distance similarity is implemented.

[0048] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0049] (1) The present invention cleans and divides the ship trajectory segments, improves the quality of track features through LSTM autoencoder, and thus improves the subsequent anomaly detection effect and clustering efficiency.

[0050] (2) Current clustering algorithms have problems such as the need to preset the number of clusters, long clustering time for large-scale data, and the need to give pairwise similarity. The present invention calculates hash codes for time-series continuous track features. Based on the discrete and efficient characteristics of the hash algorithm, it can quickly obtain an affinity matrix based on Hamming distance similarity in the case of unlabeled large-scale data for subsequent anomaly detection clustering.

[0051] (3) The present invention automatically clusters unsupervised ship tracks, leaving behind non-clusterable ship tracks to achieve anomaly identification, and improves the clustering algorithm based on Hamming distance, thereby enhancing the algorithm's data mining capability and analysis speed, enabling it to more accurately identify complex actual maritime track situations. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 Schematic diagram of the implementation architecture of the clustering method for track anomaly detection based on Hamming distance similarity of the present invention;

[0053] Figure 2 This is a schematic diagram of clustering of track data instances used in the present invention;

[0054] Figure 3 The figure is a schematic diagram of an abnormal track detected by the present invention and an example of one of the tracks. DETAILED DESCRIPTION

[0055] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0056] Example 1

[0057] like Figure 1As shown, this embodiment provides a track anomaly detection clustering method based on Hamming distance similarity, comprising the following steps:

[0058] S1: First, the track data of each ship in a specific area (such as a river or sea basin) and a specified time period is cleaned and segmented. Then, an autoencoder with a long short-term memory (LSTM) network structure is used to extract time series information features from the track data. The data processing includes the following steps:

[0059] S11: Extraction and cleaning of ship data in target sea area;

[0060] Use the data from the Automatic Identification System (AIS) to obtain the geographic location information of all ships in a specified sea area (sea area near a river or sea) within a certain period of time, including the longitude, latitude, speed, and direction of travel of each ship at each time point. Delete and clean the data that does not contain any information or is obviously erroneous, including: stationary ship data points with a speed of 0, track data outside the longitude and longitude of the specified sea area, track data with incorrect format, data with an absolute value of longitude exceeding 180°, an absolute value of latitude exceeding 90°, and a heading less than 0° or greater than or equal to 360°. Figure 2 As shown, the geographical location information of a river basin near a certain river or sea is determined, that is, the tracks that are not within the scope of this geographical location information are removed.

[0061] S12: Ship track data segmentation and construction;

[0062] The AIS data is sorted by ship number and arranged in chronological order to obtain the track data of each ship within the specified time period. In order to detect track anomalies and prevent the distance and time span of a single track from being too large, for ships with too long time intervals, they are divided into multiple separate tracks according to a certain number of time points. A track is set to consist of n time point data, which can be set to 100 time points. In each segmented track, each time point of the data contains the ship's longitude, latitude, speed, direction of travel, and the time interval between the current time point and the previous time point. One track is recorded as P, and the data at the qth time point is recorded as p q .

[0063] S13: Extract track temporal information features based on long short-term memory network (LSTM) autoencoder;

[0064] An autoencoder based on a long short-term memory (LSTM) network is used to capture the temporal information of track data. The overall structure of the autoencoder is as follows: the track data is used as input, mapped to a low-dimensional hidden layer through the LSTM network to generate track hidden features. This hidden layer is then restored and mapped back to track data of the same dimension through the long short-term memory (LSTM) network to extract hidden features that represent the input track data information:

[0065] D=encoder(P)

[0066] P′=decoder(D)

[0067] Where D is the hidden feature of the track, P' is the restored track data, encoder(.) and decoder(.) are the encoder and decoder LSTM networks of the autoencoder respectively.

[0068] It is expected that the hidden features can represent the corresponding track data so that the restored track data is the same as the corresponding track data. The autoencoder uses the mean square error loss function L m Calculate the error between the input and output of each track:

[0069]

[0070] Where d is the characteristic dimension of the track at each time point, i.e. 5, n is the number of time points set in a track, i.e. 100, p q and p q ' are the original input data and the data restored by the autoencoder at the qth time point of the track, respectively. This embodiment uses the gradient descent method to train the autoencoder. After the autoencoder network is trained, the track data is input into the encoder encoder(.) to obtain the track hidden features, which are used for subsequent encoding.

[0071] S2: Use the iterative quantized hash algorithm to convert the extracted ship track features into binary hash codes, calculate the Hamming distance between the hash codes of each track feature, and then calculate the affinity matrix of the track data based on the Hamming distance similarity, including the following steps:

[0072] S21: Iterative quantized hash coding calculation of track characteristics;

[0073] Perform principal component analysis on the track feature D to obtain its first k eigenvectors. For example, k can be set to 64, denoted as W. Randomly generate an orthogonal rotation matrix R to balance the information in the eigenvector, and then minimize the following quantization error Q for iterative quantization optimization:

[0074]

[0075] Where B is the hash code of the track data.

[0076] In iterative quantization optimization, R is first fixed, so D, W, and R are known. The optimal hash code B = sgn(DWR) is calculated, where sgn(.) is a sign function that represents values ​​greater than 0 as 1 and all other values ​​as -1. B is then fixed, and the optimal R is calculated using singular value decomposition. After multiple iterations of optimization, the optimal hash code B is obtained. The optimal solution is achieved when the hash code and the rotation matrix remain unchanged during the iterations. Generally, a near-optimal solution is achieved after 50 iterations.

[0077] The huge original data space is mapped into the Hamming space through the hash function, so that the probability that similar data in the original space (expressed as points with similar distances in the track feature space in this invention) are also similar in the new Hamming space is very high, and vice versa is very small, so that the hash codes of similar track features are also similar, and the hash codes of dissimilar track features are very different.

[0078] S22: Calculate the affinity matrix of track features based on Hamming distance similarity;

[0079] The Hamming distance is the number of different bits in the hash code of each track feature, that is, the degree of difference in the hash code. The larger the distance, the lower the affinity. An element in the affinity matrix based on Hamming distance similarity is expressed as the following formula, which is used to calculate the affinity between track features:

[0080]

[0081] Where a ij is the affinity between two points i and j, Ham ij is the Hamming distance between two track features i and j, σ is the parameter for affinity calculation, which is an empirical value and is generally set to the maximum value of the Hamming distance between the data. The matrix diagonal is set to 0, that is, when i = j, the corresponding element of the affinity matrix is ​​0.

[0082] S3: Finally, the track features are used to generate an affinity matrix based on the Hamming distance similarity, and the dominating set clustering algorithm is used to automatically iteratively cluster the tracks. During the iterative clustering process, regular track data with high similarity that can be clustered is removed, and track data that cannot be clustered is detected as abnormal tracks. This includes the following:

[0083] After calculating the affinity matrix based on Hamming distance similarity, a dominating set clustering algorithm is used to identify anomalous tracks by leaving out anomalous data samples without clustering. The track features and affinity matrix are considered a graph, with the track feature values ​​representing the points in the graph and the affinity matrix representing the edges. Track feature sets that meet the dominating set criteria are considered to have a high degree of similarity. In a large amount of actual track data, normal tracks far outnumber abnormal tracks, and in reality, the normal routes of ships tend to be relatively fixed. Through iterative clustering, the algorithm identifies anomalous tracks that differ from normal routes and therefore cannot form a highly aggregated set. The track data is processed through cyclical iterative calculations. Each time a dominating set with a high degree of aggregation is found, this portion of track data is removed from the set, and the next round of search is performed. After several rounds of cyclic clustering, any remaining data that cannot be clustered is considered anomalous data.

[0084] Use the affinity matrix A based on Hamming distance similarity composed of affinity elements sim =(a ij ), the ship tracks are automatically clustered by iterative calculation using the following formula:

[0085]

[0086] Where y i and(·) i Represents the weight and corresponding vector value of the i-th track, t represents the number of iterative calculations in a clustering, and the maximum value of t is limited to 1000. Generally, the weight does not change after the update within the maximum value, that is, the iteration is completed. y is the vector composed of the weights of all track feature data. The weights of all N tracks in the first clustering are initialized to That is, for each track feature:

[0087]

[0088] Where, N T Represents the total number of unclustered samples during the T-th clustering, that is, N1=N during the first clustering T=1.

[0089] After multiple iterative calculations, when the weight of each track does not change, the weight of some track feature data is higher than 0, while the weight of the remaining track data is 0, which means that these track data have a high degree of aggregation compared with all other data and can be removed as a group of regular tracks. After the algorithm has removed multiple groups of regular tracks through multiple cycles of clustering, abnormal tracks will remain and cannot be clustered. Track data that cannot be clustered or has too few clusters will be marked as abnormal tracks to monitor ship track anomalies. Figure 3As shown in the figure, the tracks that cannot be clustered are abnormal tracks. The specific situation of one track is as follows: Figure 3 As shown in , there is a big difference from the smooth curve of ships sailing on normal routes.

[0090] Example 2

[0091] This embodiment provides a track anomaly detection clustering system based on Hamming distance similarity, which is used to implement the track anomaly detection clustering method based on Hamming distance similarity in the above embodiment 1. The system includes: a track data acquisition module, a data preprocessing module, a feature extraction module, a hash coding module, a Hamming distance calculation module, an affinity matrix calculation module, an iterative clustering module, and an abnormal track output module;

[0092] In this embodiment, the track data acquisition module is used to acquire track data in a set area and a specified time period;

[0093] In this embodiment, the data preprocessing module is used to perform data preprocessing on the track data;

[0094] In this embodiment, the feature extraction module is used to extract track time series information features from the preprocessed track data based on the long short-term memory network LSTM autoencoder to obtain track features;

[0095] In this embodiment, the hash coding module is used to convert the track features into binary hash codes based on an iterative quantized hash algorithm;

[0096] In this embodiment, the Hamming distance calculation module is used to calculate the Hamming distance between the hash codes of each track feature;

[0097] In this embodiment, the affinity matrix calculation module is used to calculate the affinity matrix of the track features based on the Hamming distance similarity;

[0098] In this embodiment, the iterative clustering module is used to perform iterative clustering of track features using a dominating set clustering algorithm;

[0099] In this embodiment, the abnormal track output module is used to output track features that cannot be clustered as abnormal tracks during the iterative process of clustering.

[0100] Example 3

[0101] This embodiment provides a storage medium, which may be a ROM, RAM, disk, CD, or other storage medium. The storage medium stores one or more programs. When the program is executed by a processor, the track anomaly detection clustering method based on Hamming distance similarity of embodiment 1 is implemented.

[0102] Example 4

[0103] This embodiment provides a computing device, which can be a desktop computer, a laptop computer, a smart phone, a PDA handheld terminal, a tablet computer, or other terminal device with a display function. The computing device includes a processor and a memory, and the memory stores one or more programs. When the processor executes the program stored in the memory, the track anomaly detection clustering method based on Hamming distance similarity of Example 1 is implemented.

[0104] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A clustering method for track anomaly detection based on Hamming distance similarity, characterized in that: The steps include: Acquire track data in the set area and specified time period and perform data preprocessing; The track time series information features of the pre-processed track data are extracted based on the long short-term memory network LSTM autoencoder to obtain the track features; The track features are converted into binary hash codes based on the iterative quantized hash algorithm, the Hamming distance between the hash codes of each track feature is calculated, and the affinity matrix of the track features based on the similarity of the Hamming distance is calculated; The track features are iteratively clustered using the dominating set clustering algorithm. During the iterative clustering process, the track features that cannot be clustered are output as abnormal tracks.

2. The track anomaly detection clustering method based on Hamming distance similarity according to claim 1 is characterized in that: The acquisition of track data in a set area and a specified time period and the data preprocessing specifically include: Obtain the geographic location information of all ships in the set area and specified time period, set filtering conditions, delete and clean the data; The AIS data are sorted by ship number and arranged in chronological order to obtain the track data of each ship within the specified time period.

3. The track anomaly detection clustering method based on Hamming distance similarity according to claim 1 is characterized in that: The long short-term memory network (LSTM) autoencoder extracts track time series information features from the preprocessed track data, specifically including: The preprocessed track data is input into the long short-term memory network LSTM network and mapped to the hidden layer to generate track hidden features. Then it is restored from the hidden layer through the long short-term memory network LSTM and mapped back to track data of the same dimension to extract the track hidden features. The specific expression is: D=encoder(P) P'=decoder(D) Where P represents the preprocessed track data, D is the track hidden feature, P' is the restored track data, encoder(.) and decoder(.) are the encoder and decoder LSTM networks of the autoencoder respectively.

4. The track anomaly detection clustering method based on Hamming distance similarity according to claim 1, characterized in that: The long short-term memory network LSTM autoencoder uses the mean square error loss function to calculate the error between the input and output of each track data. The mean square error loss function is expressed as: Among them, d is the characteristic dimension of the track at each time point, n is the number of time points set in a track, and p q and p q ' are the original input data at the qth time point of the track and the data restored by the autoencoder.

5. The track anomaly detection clustering method based on Hamming distance similarity according to claim 1 is characterized in that: The track features are converted into binary hash codes based on the iterative quantized hash algorithm, which includes: Perform principal component analysis on the track characteristics to obtain its first k eigenvectors, randomly generate an orthogonal rotation matrix R to balance the information in the eigenvectors, and minimize the quantization error Q to perform iterative quantization optimization, which can be specifically expressed as: Where D represents the track characteristics, W represents the number of feature vectors, and B is the hash code of the track data.

6. The track anomaly detection clustering method based on Hamming distance similarity according to claim 1, characterized in that: Calculate the affinity matrix of track features based on Hamming distance similarity, including: The number of different bits of the hash code of each track feature is calculated as the Hamming distance. The affinity in the affinity matrix based on the Hamming distance similarity is expressed as: Among them, a ij is the affinity between track features i and j, Ham ij is the Hamming distance between two track features i and j, and σ is the preset parameter for affinity calculation.

7. The track anomaly detection clustering method based on Hamming distance similarity according to claim 6 is characterized in that: Iterative clustering of track features is performed using the dominating set clustering algorithm, specifically including: Use the affinity matrix A based on Hamming distance similarity composed of affinity elements sim =(a ij ), perform iterative clustering of track features, expressed as: Among them, y i and(·) i represents the weight and corresponding vector value of the i-th track, t represents the number of iterative calculations in a clustering, and y is the vector composed of the weights of all track feature data; The weights of all N tracks in the first clustering are initialized as For each track feature: Among them, N T Represents the total number of unclustered samples in the Tth clustering. When T=1 is the first clustering, N1=N.

8. A clustering system for track anomaly detection based on Hamming distance similarity, characterized in that: A method for clustering flight track anomaly detection based on Hamming distance similarity according to any one of claims 1 to 7 is provided, the system comprising: a flight track data acquisition module, a data preprocessing module, a feature extraction module, a hash coding module, a Hamming distance calculation module, an affinity matrix calculation module, an iterative clustering module, and an abnormal flight track output module; The track data acquisition module is used to acquire track data in a set area and a specified time period; The data preprocessing module is used to perform data preprocessing on the track data; The feature extraction module is used to extract track time series information features from the preprocessed track data based on the long short-term memory network LSTM autoencoder to obtain track features; The hash coding module is used to convert the track characteristics into binary hash codes based on an iterative quantized hash algorithm; The Hamming distance calculation module is used to calculate the Hamming distance between the hash codes of each track feature; The affinity matrix calculation module is used to calculate the affinity matrix of the track features based on the Hamming distance similarity; The iterative clustering module is used to perform iterative clustering of track features through a dominating set clustering algorithm; The abnormal track output module is used to output track features that cannot be clustered as abnormal tracks during the iterative process of clustering.

9. A computer-readable storage medium storing a program, characterized in that: When the program is executed by a processor, the track anomaly detection clustering method based on Hamming distance similarity as described in any one of claims 1 to 7 is implemented.

10. A computing device comprising a processor and a memory for storing a program executable by the processor, characterized in that When the processor executes the program stored in the memory, the track anomaly detection clustering method based on Hamming distance similarity as described in any one of claims 1 to 7 is implemented.

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