CN107341514B

CN107341514B - Abnormal point and edge point detection method based on joint density and angle

Info

Publication number: CN107341514B
Application number: CN201710548763.3A
Authority: CN
Inventors: 李孝杰; 吴锡; 吕建成; 周激流; 李莉丽; 王强
Original assignee: Chengdu University of Information Technology
Current assignee: Chengdu University of Information Technology
Priority date: 2017-07-07
Filing date: 2017-07-07
Publication date: 2020-07-21
Anticipated expiration: 2037-07-07
Also published as: CN107341514A

Abstract

The invention relates to an abnormal point and edge point detection method based on joint density and angle, which is based on the idea that edge points and abnormal points have lower local density and smaller angle variance change, combines the joint information of angle and density of a data set, utilizes joint measure to judge the degree of sample points belonging to the abnormal points and the edge points, and automatically determines special points by setting a threshold value. The method for detecting the abnormal points and the edge points is stable, improves the performance of detecting the edge points and the abnormal points, can better reflect the characteristics of a data set, can detect noise data and better removes noise. The defect that the effect of detecting special points in a complex data set is poor and unstable in the prior art is overcome.

Description

Abnormal point and edge point detection method based on joint density and angle

Technical Field

The invention relates to the field of data detection, in particular to an abnormal point and edge point detection method based on joint density and angle.

Background

Traditional clustering, classification, pattern recognition techniques are directed to finding general patterns, while detection of specific points, including edge points and outlier points, is often used to identify valid, interesting and potentially valuable patterns in the data. Detection of a special spot is often a more meaningful task than detection of a normal spot. In addition, most algorithms are affected by outliers. For example, the famous nonlinear manifold learning dimension reduction algorithm Isomap itself does not describe the problem related to abnormal point detection, but the code provided by the author includes the abnormal point detection process. Therefore, how to correctly detect the abnormal point in the complex space is a real problem to be solved urgently, and is an important task in data preprocessing.

Researchers at home and abroad propose various detection algorithms from different technical perspectives. From the method of determining outliers, a global model and a local model can be divided. And the global model performs binary judgment on all the observation points so as to judge whether the current observation point is an abnormal point. While local models typically assign a certain metric (e.g., an angular change factor) to an observation point for estimating the degree to which the point belongs to an outlier. Data tag information can be classified into supervised, semi-supervised and unsupervised algorithms depending on whether they are needed. Currently, most algorithms generally employ 5-class shallow technique approaches based on statistics (Statistical-based), Distance-based, neighborhood or Density (Density-based), Clustering (Clustering-based), or bias (development-based).

Distance-Based methods are the most common methods currently used because of geometric clarity, the method uses Distance as a measure, points with no "enough" neighbors are judged as anomalous data, and the idea of statistical-Based inconsistency checking is extended.

Where m is the dimension of the data, dist_maxAnd dist_minThe distance of the current point to its farthest neighbor and nearest neighbor, respectively. As m increases, the above equation will go to zero. The high dimensional spatial data ubiquitous to distance metric processing is therefore more or less non-compliant. To address this problem, professor Hans-Peter Kriegel, university of munich, germany, proposes an Angle-Based Outlier Detection (ABOD) algorithm. The method solves the problems to a certain extent, and introduces a new neighbor relation problem.

The density method usually adopts local anomaly factors (L cal Outlier Factor, L OF) to judge the anomaly point, if the density OF the area where the data point is located is lower, the probability that the current point becomes the anomaly point is higher, if the value OF L OF is larger, the more classical methods such as L OF method, INF L O method, inverse KNN method and the like.

Kriegel proposed an angle-based anomaly detection Algorithm (ABOD) in 2008, which was independent of the parameter selection problem and alleviated the dimension disaster problem to some extent. However, the ABOD algorithm only considers the current point's relationship to neighbors and not more of its neighbors, resulting in the algorithm easily identifying the wrong outliers. The edge point is determined by obtaining label information, namely prior information, but some application environments cannot obtain the prior information, so that the application range of the algorithm is limited.

Most of the above classical abnormal point detection algorithms have good effects under specific conditions or specific fields, and when the dimensionality of data is high, the effects of the algorithms are not ideal and the generalization capability is weak. Most of the existing abnormal point detection algorithms only consider single characteristics of data, such as density, angle and the like, to detect abnormality, and aim at the problem that a complex data set has unstable performance of the detection algorithm.

Therefore, how to improve the performance and stability of the abnormal point and edge point detection algorithm becomes an urgent problem to be solved in the field of data detection.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides an abnormal point and edge point detection method based on joint density and angle, which comprises the following steps:

step 1 input data set X ∈ R^m×nWhere X is a data set of m × n, each column of X representing oneData samples, i.e. X, comprising n samples, each sample having m dimensions, X_i∈R^mI ∈ {1,2, … n }, m representing a sample dimension, n representing the number of samples of the dataset;

step 2: setting the number k of neighbors required for calculating the joint angle information as floor (5log10 (n));

and step 3: calculating local density information ρ ═ ρ for data points₁,ρ₂,…ρ_n]For each sample x_iCalculating the corresponding local density value as rho according to the formula (1)_i：

Wherein d is_ijRepresents a sample x_iAnd sample x_jOf the Euclidean distance between d_cFor the cutoff distance, formula (1) is according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density value ρ of_i；

And 4, step 4: calculating the joint angle information ζ ═ ζ₁,ζ₂,…ζ_n]For each sample point x_iCalculating the corresponding joint angle information value zeta_iThe joint angle information value ζ_iIncluding a first local angle measure_iAnd a second local angle measure τ_iCalculating the joint angle information value ζ_iThe method comprises the following steps:

step 41: data processing, for each data point x_iI ∈ {1,2, … n }, selecting its k neighbors for normalization and decoupling operations, denoted as X_iThe specific operation is shown as formula (2):

order to

Then

Step 42: calculating a first local angle measure_iSample x_iIs/are as follows_iThe values are calculated as follows:

step 43: calculating a second local angle measure τ_iFor each sample x_iAnd X corresponding thereto_iCalculating X using the mean method_iIs approximated by a normal vector v_iI.e. v_i＝mean(X_i) (ii) a Calculating tau according to equation (3) based on the relation between normal and angle_i：

Step 44, for each data sample x_iI ∈ {1,2, … n }, represented by the formula ζ_i＝_i/τ_iCalculating a joint angle value;

and 5: judging the special points, comprising the following steps:

step 51: the threshold value y is determined and,

step 52: for sample point x_iIf ρ is_i,/γ < min (mean (. rho.)/γ, max (. zeta.)), then x_iIs judged as an abnormal point; for each sample point, step 52 is repeated to determine whether it is an outlier or a normal point.

According to a preferred embodiment, the method of calculating local density values in step 3 comprises:

step 31, calculating the Euclidean distance between any two samples in the data set X, and obtaining a distance matrix D ∈ R after calculation aiming at the data set X^n×nWherein the element D in the distance matrix D_ijRepresents a sample x_iAnd sample x_jThe Euclidean distance between;

step 32: setting cutoff distance d_cArranging D by sorting n × n elements in the distance matrix D from small to large according to the value of the n elements into a vector sd._cSelecting the p-th element value in the sd vector, wherein p is round ((n-1) n percent/100);

step 33: according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density value ρ of_iFrom the formula (1), when d_ijIs less than d_cThen x is considered to be_jIs the current point x_iOtherwise it is not its neighbor.

According to a preferred embodiment, the method further comprises verifying the special points, including:

step 6, visual display, namely, for the data set X ∈ R^m×nWhen m is more than or equal to 1 and less than or equal to 3, marking and displaying the judged special point in the visual space to judge whether the special point is at the edge position of the data set, and carrying out X ∈ R on the data set^m×nAnd when m is larger than 3, reducing the data set X to a visual space by adopting a classical dimension reduction method, and then performing label display.

The invention has the beneficial effects that:

1. the invention provides a stable method for detecting the abnormal points and the edge points based on the idea that the edge points and the abnormal points have lower local density and smaller angle variance change and the joint information of the angle and the density of the data set, so that the performance of detecting the edge points and the abnormal points is improved, the characteristics of the data set can be better reflected, the noise data can be detected, the noise can be better removed, and the defects of poor and unstable effect of detecting special points in a complex data set in the prior art are overcome.

2. The invention does not need to acquire prior information in advance from an unknown environment, overcomes the defect that the prior art needs to acquire the prior information, and ensures that the application range of the algorithm is wider.

Drawings

FIG. 1 is a flow chart of a method of determining a singularity according to the present invention;

FIG. 2 is a diagram illustrating the effect of step 41 after data processing;

FIG. 3 is a diagram illustrating the effect of the method of the present invention on the two-dimensional Flame data for the singular point determination;

FIG. 4 is a diagram of the effect of the method of the present invention on a two-dimensional XOR dataset on a special point decision;

FIG. 5 is a diagram showing the effect of the method of the present invention on the two-dimensional Banana data set on the decision of a singular point;

fig. 6 is a diagram showing the effect of the method of the present invention on the decision of a singular point for a two-dimensional PanelB dataset with noise.

Detailed Description

The following detailed description is made with reference to the accompanying drawings.

The data set in the present invention is a matrix of m × n dimensions, in which each column represents a sample.

The abnormal point in the present invention means: observation points that are suspected of being far from other points are considered to be from different data generation mechanisms.

The edge points in the present invention mean: points located on the edges of the data set of the higher density distribution.

The special points in the present invention include edge points and outliers.

The neighbors in the present invention are: a point that is similar to the current point by some measure.

The cutoff distance in the present invention means: the critical value.

The meaning of the local density value ρ in the present invention is: and counting the number of neighbors with the distance to the current point smaller than the critical value.

Joint angle value ζ in the present invention_iThe meaning of (A) is:_i/τ_ithe ratio of the two angular measures.

The meaning of the first local angle value in the present invention is:_iis the variance transformation of the inner product of the neighbor vectors. The smaller the variance transformation, the less likely it is to be an outlier.

In the present inventionThe meaning of the second local angle value is: tau is_iThe number of inner product positive values of the neighbor vector and the normal is counted. If the inner product value is that the regular included angle is an acute angle, the included angle is a neighbor in the included angle measurement. The larger the second local angle value is, the smaller the possibility that the current point is an outlier is.

The meaning of the joint measure in the present invention is: the variance transformation is smaller and more neighbors of included angles are jointly determined._iAnd 1/tau_iWhen both are small, the more likely the current point is an outlier.

The invention provides a stable abnormal point and edge point detection method based on the thought that edge points and abnormal points have lower local density and smaller angle variance change, combines the joint characteristics of angles and densities of a data set, utilizes joint measure to judge the degree of sample points belonging to the abnormal points and the edge points, and automatically determines special points by setting a threshold value gamma, so that the performance of edge point and abnormal point detection is improved, and the characteristics of the data set are better reflected.

FIG. 1 is a flow chart of a method for determining a special point according to the present invention. As shown in fig. 1, the method for detecting outliers and edge points based on joint density and angle of the present invention includes the following steps:

step 1 input data set X ∈ R^m×nEach column in the matrix X represents a sample, i.e. X comprises n samples, each sample having m dimensions, i.e. X_i∈R^mI ∈ {1,2, … n }. Each sample corresponds to a point in m-dimensional space, such as a 3-dimensional column vector, corresponding to a point in 3-dimensional space, i.e., the point can be represented by (X, y, z) coordinates.

Where X represents m × n dataset matrix X ═ (X)₁,x₂,x₃…x_n) Each column of which represents one data sample x_i∈R^mI ∈ {1,2, … n }, i.e., X, comprises n samples, each sample having an m dimension, R represents the data space, m represents the sample dimension, and n represents the number of samples of the data set.

For example, 10 pictures of size 20 × 20 can be processed into a column vector of size 400 × 1 for each image, and then 10 images form a data set X ∈ R^400×10X is a matrix of size 400 × 10.

Step 2: the number k of neighbors required for calculating the joint angle information is set to floor (5log10 (n)). The floor (. cndot.) function is a rounded down function, e.g., floor (5.4) results in 5. Compared with the method for setting the number of the neighbors through manual experience, the method for automatically setting the number of the neighbors better combines the distribution characteristics of data and has better adaptability.

And step 3: calculating local density information ρ ═ ρ for data points₁,ρ₂,…ρ_n]For each sample point x_iCalculating the corresponding local density value as rho according to the formula (1)_i；

Wherein d is_ijRepresents a sample x_iAnd sample x_jOf the Euclidean distance between d_cFor the cutoff distance, formula (1) is according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density value ρ of_i. Wherein z is d_ij-d_c。

The existing density detection algorithm needs to determine more parameters such as the number of nearest neighbors and the radius of a local area to a great extent. When the density difference of each cluster in the data set is large, and particularly when the high-dimensional data distribution is sparse, the performance of most density-type methods becomes worse. The technical scheme of the invention only adopts the unique parameter d when calculating the local density information_cAnd the external influence is reduced as much as possible, and the detection performance is further optimized.

Step 31: between any two samples in the calculation datasetThe Euclidean distance of (2) is calculated to obtain a distance matrix D ∈ R for the data set X^n×nWherein the element D in the distance matrix D_ijRepresents a sample x_iAnd sample x_jThe euclidean distance between them. The value range of i is 1 to n, and the value range of j is 1 to n.

Step 32: setting cutoff distance d_cArranging D by sorting n × n elements in the distance matrix D from small to large according to the value of the n elements into a vector sd._cAnd (p) selecting the p-th element value in the sd vector. Where p is round ((n-1) n percent/100), percent is typically set to 0.2. round (·) is a rounding function.

Step 33: according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density value ρ of_iCalculating the local density value ρ according to equation (1)_i. According to the formula (1), when d_ijIs less than d_cThen x is considered to be_jIs the current point x_iOtherwise it is not its neighbor.

And 4, step 4: calculating the joint angle information ζ ═ ζ₁,ζ₂,…ζ_n]For each sample point x_iCalculating the corresponding joint angle value zeta_iZeta, value of the joint angle_iIncluding a first local angle value_iAnd a second local angle value tau_iCalculating the joint angle value ζ_iThe method comprises the following steps:

step 41: data processing, for each data point x_i∈R^mI ∈ {1,2, … n }, selecting its k neighbors for normalization and decoupling operations, denoted as X_iThe specific operation is shown as the following formula:

order to

Then

Most detection methods in the future rely heavily on distance measures, and especially in high-dimensional space, the separability of distance-based measures is poor. In order to eliminate the influence of the distance, a normalization and decoupling preprocessing method is adopted to eliminate the influence of the distance on the angle measurement as much as possible.

After step 41, x is advantageously treated_iIs pulled to a unit circle, i.e. arbitrarily

To x_iIs 1, the influence of the distance is removed.

Fig. 2 is an effect diagram after the data processing of step 41. As can be seen from fig. 2, x_iAll neighbors to x_iAre all 1.

Step 42: calculating a first local angle value_iSample x_iIs/are as follows_iThe values are calculated as follows:

due to the fact that

And

therefore, it is not only easy to use

Thus, the measure of the first local angle value may be converted into an inner product measure. Wherein theta is_ijRepresents the ith sample

And j sample

The included angle therebetween.

Refers to X in the formula (2)_iThe var () function is a function that takes the variance.

In the general case of the above-mentioned,_ithe smaller the value, x_iThe greater the probability of becoming an edge point. However, some special edge points, such as the part points between the two classes, have higher values_i. Because of the point in the middle of the two classes, when k is larger, its neighbors can be selected from the two classes instead of the single one, which results in a larger change in its angle.

In order to solve the problem that partial points between two data classes are wrongly judged as normal points to a certain extent, the invention provides a method for joint angle measurement detection.

Step 43: calculating a second local angle value τ_iFor each sample x_iAnd X corresponding thereto_iCalculating X using the mean method_iIs approximated by a normal vector v_iI.e. v_i＝mean(X_i). Calculating tau according to equation (3) based on the relation between normal and angle_i：

Step 44: for each data sample point x_i∈R^mI ∈ {1,2, … n }, represented by the formula ζ_i＝_i/τ_iA joint angle value is calculated.

The invention adopts two angle measurement values to ensure that the robustness of the algorithm is better,_iis based on the variance information, tau, of the inner product of the current point and the neighbor vector_iIs defined according to the angle value between the vector and the normal.

And 5: judging the abnormal points and the edge points, comprising the following steps:

step 51: the threshold value y is determined and,

The method for verifying the special points comprises the following steps:

step 6, visual display, namely, for the data set X ∈ R^m×nWhen m is more than or equal to 1 and less than or equal to 3, marking and displaying the judged special point in the visual space to judge whether the special point is at the edge position of the data set, and carrying out X ∈ R on the data set^m×nAnd when m is larger than 3, the data set X is high-dimensional data, the data set X is reduced to a visual space by adopting a classical dimension reduction method, and then the data set X is marked and displayed.

The current dimension reduction algorithm mainly includes Principal Component Analysis (PCA), local linear embedding (L annular linear embedding, LL E), and placian feature mapping (L annular eigenmaps, L E).

The method for verifying the special points can also verify the feasibility of the special point judgment method by judging whether the searched special points can improve the performance of a clustering or classifying algorithm.

After finding out the special points and the edge points, data analysis can be performed to analyze the potentially valuable patterns in the data set, that is, the characteristics of the data, such as people who are between normal people and patients with liver diseases and who are about to get ill but not ill, the analysis of the special population will help to research the characteristics of the liver diseases, and the people who are at the edge of the liver diseases should attract attention of people. Therefore, the analysis of the special points has very important research significance.

In order to further illustrate the effect of the BPDAD algorithm provided by the invention on detecting special points in the data sets, different data sets are adopted for testing experiments.

The dots in the data set are shown as small circles, with the special dots detected being marked as filled dots by a number.

FIG. 3 is a graph of the results of detecting distinctive points using the BPDAD algorithm on a two-dimensional Flame dataset, as shown in FIG. 3, with the points marked with solid gray being the distinctive points detected using the BPDAD algorithm proposed by the present invention.

Fig. 4 is a result diagram of detecting special points by using the BPDAD algorithm for the two-dimensional XOR dataset, and as shown in fig. 4, the points marked with solid gray are the special points detected by using the BPDAD algorithm, and it can be seen from fig. 4 that the detected special points are all located at the edge of the dataset, and the detection effect is good.

Fig. 5 is a result diagram of detecting special points by using the BPDAD algorithm for the two-dimensional Banana data set, and as shown in fig. 5, points marked with solid gray are special points detected by using the BPDAD algorithm, and it can be seen from fig. 5 that the detected special points are all located at the edge of the data set, so that the detection effect is good.

Fig. 6 is a graph of the results of detecting a distinctive point using the BPDAD algorithm for a two-dimensional PanelB dataset with noise. As shown in fig. 6, the dots filled with gray are the special dots and noise data detected by the BPDAD algorithm, and it can be seen from fig. 6 that the BPDAD algorithm can not only detect the special dots located at the edge of the data set, but also effectively remove the noise data in the data set.

In order to objectively explain that the BPDAD (boundary-edge Detection with Angle analysis) algorithm provided by the invention can more accurately identify abnormal Points and special Points, special point Detection is carried out on the same data by respectively using a BEPS (boundary-edge Pattern Selection algorithm), an IDD algorithm and an IDD whole algorithm, and the effectiveness of the BPDAD algorithm in detecting the special Points is further verified by using K-Means and SMCE (sparse Man Cluster and embedding) Clustering algorithms. The experimental data are shown in table 1.

TABLE 1

Method	#REMOVED BOUNDARY POINTS	CLUSTER	P(％)
						KMEANS	70.22
BEPS	43	KMEANS	68.89
				IDD	0	KMEANS	70.22
IDDWHOLE	0		53.37
				BPDAD	95	KMEANS	73.49
		SMCE	70.22
				IDD	0	SMCE	69.10
BEPS	43	SMCE	62.96
				BPDAD	95	SMCE	63.86

The test data in table 1 is high-dimensional Wine data, "REMOVED BOUNDARY POINTS" in table 1 represents the number of special POINTS detected by using a corresponding method, "C L user" in table 1 represents the clustering algorithm used, such as K-Means or SMCE clustering algorithm, "P (%)" in table 1 represents the clustering precision, and the higher the value of P (%), the better the effectiveness of detecting special POINTS.

It should be noted that the above-mentioned embodiments are exemplary, and that those skilled in the art, having benefit of the present disclosure, may devise various arrangements that are within the scope of the present disclosure and that fall within the scope of the invention. It should be understood by those skilled in the art that the present specification and figures are illustrative only and are not limiting upon the claims. The scope of the invention is defined by the claims and their equivalents.

Claims

1. An outlier and edge point detection method based on joint density and angle, characterized in that the method comprises the following steps:

step 1 input data set X ∈ R^m×nWhere X is a data set of m × n, each column of X representing a data sample, i.e. X comprises n samples, each sample having m dimensions, X_i∈R^mI ∈ {1,2, … n }, m representing a sample dimension, n representing the number of samples of the dataset, dataset X being a text or numerical image;

step 2: setting the number k of neighbors required in calculation of the joint angle information as floor (5log10(n)), wherein a floor (.) function is a function of rounding downwards, and the method for automatically setting the number of neighbors combines the distribution characteristics of data;

and step 3: calculating local density information rho ═ rho of sample points₁，ρ₂，…ρ_n]For each sample point x_iCalculating the corresponding local density value as rho according to the formula (1)_i：

Wherein d is_ijRepresents a sample point x_iAnd sample point x_jOf the Euclidean distance between d_cFor the cutoff distance, formula (1) is according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density value ρ of_i；

And 4, step 4: calculating the joint angle information ζ ═ ζ₁，ζ₂，…ζ_n]For each sample point x_iCalculating the corresponding joint angle information value zeta_iThe joint angle information value ζ_iIncluding a first local angle measure_iAnd a second local angle measure τ_iCalculating the joint angle information value ζ_iThe method comprises the following steps:

step 41: data processing, for each sample point x_iI ∈ {1,2, … n }, selecting its k neighbors for normalization and decoupling operations, denoted as X_iThe specific operation is shown as formula (2):

order to

Then

B is x_iIs pulled to a unit circle, i.e. arbitrarily

To x_iThe distance of (2) is 1, so that the influence of the distance is eliminated;

step 42: calculating a first local angle measure_iSample point x_iIs/are as follows_iThe values are calculated as follows:

refers to X in the formula (2)_iThe Var () function is a function taking the variance;

step 43: calculating a second local angle measure τ_iFor each sample point x_iAnd X corresponding thereto_iCalculating X using the mean method_iIs approximated by a normal vector v_iI.e. v_i＝mean(X_i) (ii) a Calculating tau according to equation (3) based on the relation between normal and angle_i：

Step 44, for each sample point x_iI ∈ {1,2, … n }, represented by the formula

Calculating a joint angle information value ζ_i；

And 5: judging the special points, comprising the following steps:

step 51: the threshold value y is determined and,

step 52: for sample point x_iIf, if

X is then_iIs judged as an abnormal point; for each sample point x_iRepeatedly executing step 52 to determine that the detected value is an abnormal point or a normal point;

step 6: the method for verifying the special points is as follows:

visual display of the data set X ∈ R^m×nWhen m is more than or equal to 1 and less than or equal to 3, marking and displaying the judged special points in the visual space to judge whether the special points are positioned on the edge position of the data set, and carrying out X ∈ R treatment on the data set^m×nAnd when m is larger than 3, reducing the data set X to a visual space by adopting a classical dimension reduction method, and then performing label display.

2. The inspection method of claim 1, wherein the method of calculating the local density values in step 3 comprises:

step 31, calculating the Euclidean distance between any two samples in the data set X, and obtaining a distance matrix D ∈ R after calculation aiming at the data set X^n×nWherein the element D in the distance matrix D_ijRepresents a sample point x_iAnd sample point x_jThe Euclidean distance between;

step 32: setting cutoff distance d_cSorting n × n elements in the distance matrix D into a vector sd according to the size of the values from small to large, and setting D_cSelecting the p-th element value in the sd vector, wherein p is round ((n-1) n percent/100), round () is rounding function, and percent is 0.2;

step 33: according to d_ijAnd d_cThe distance relation of (2) and the statistics of the current sample point x_iIs taken as x_iLocal density ofValue rho_iFrom the formula (1), when d_ijIs less than d_cThen x is considered to be_jIs the current sample point x_iOtherwise it is not its neighbor.