A Method for Detecting Outliers of Time Difference of Arrival Based on Multidimensional Scaling Method

Abstract

The present invention relates to a method for detecting outliers of time difference of arrival based on multidimensional scaling method, including: an offline acquisition stage, before the signal source sends a signal, according to the known sensor position coordinates, the variance of the Gaussian noise of the time difference of arrival measurement value, and the signal propagation speed, calculating the empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix of all sensor groups, wherein the scalar product matrix is constructed based on the multidimensional scaling method; an online detection stage, when the signal source sends a signal, measuring the time difference of arrival measurement values between all pairs of sensors, and using the empirical probability distribution function, detecting the outliers in the time difference of arrival measurement values by means of hypothesis testing. Compared with the prior art, the present invention has the advantages of wide application range, high detection accuracy, etc.

Description

Technical Field

[0001] The present invention relates to the fields of wireless network and mobile computing in information technology, and particularly to a method for detecting outliers of time-difference-of-arrival (TDOA) based on multidimensional scaling (MDS). Background Art

[0002] In application fields such as radar, sonar, mobile communication, multimedia, and wireless sensor networks, it is often necessary to measure the time-difference-of-arrival information for further processing (such as positioning, calibration, synchronization, etc.). The so-called time-difference-of-arrival refers to the difference in the time when a signal emitted by a signal source arrives at one sensor and the time when it arrives at another sensor, which are received by sensors distributed in space with known positions and synchronized in time. Due to factors such as interfering signal sources and multipath propagation, the measured time-difference-of-arrival information sometimes has outliers. An outlier refers to a measured value whose measurement error amplitude is much larger than that of normal Gaussian white noise measurement error. Such a measured value is called an outlier. For outliers, appropriate methods must be taken for detection and elimination, otherwise it will have a huge adverse impact on subsequent processing.

[0003] After a search of the prior art, it was found that J. Velasco et al. proposed an outlier detection method based on the TDOA matrix in the paper "TDOA matrices: algebraic properties and their application to robust denoising with missing data" published in Volume 64, Issue 20, 2016 of the journal IEEE Transactions on Signal Processing. According to the algebraic properties of the TDOA matrix, a hard decision method was used for outlier detection. However, this technique is only applicable to the case where the number of outliers is small, and the upper bound of the number of outliers needs to be known in advance. M. Compagnoni et al. proposed an outlier detection method based on geometry and hypothesis testing in the paper "A geometrical - statistical approach to outlier removal for TDOA measurements" published in Volume 65, Issue 15, 2017 of the journal IEEE Transactions on Signal Processing. According to the geometric information of the sensors, a hypothesis testing method was used for outlier detection. The advantage of this technique is that it does not require the number of outliers to be small, and it also does not need to know the upper bound of the number of outliers in advance. However, this technique can only utilize the geometric information of no more than 3 sensors in a single detection, resulting in room for further improvement in detection accuracy.

[0004] Chinese patent document CN111982121B, with the authorization announcement date of February 26, 2021, discloses a high - precision positioning method in a hybrid line - of - sight and non - line - of - sight environment in the field of wireless networks and mobile computing. By substituting the high - precision solution into the basic positioning equation to calculate the estimated value of the observable, and then combining the characteristic that the non - line - of - sight error is positive and much larger than Gaussian noise, the difference between the estimated value and the observed value is used to obtain the residual vector, which can be divided into a line - of - sight residual part and a non - line - of - sight residual part. When all the residual values corresponding to the non - line - of - sight base stations are less than 0 and the amplitude is greater than the maximum absolute value of the line - of - sight residual, the set of the best base - station combination and the best positioning solution is obtained. Finally, the best positioning result is selected by comparing the distances of the two - step positioning solutions at the previous and current moments. Although this technique can detect some outliers caused by the non - line - of - sight environment during the positioning process, it can only detect outliers caused by the non - line - of - sight environment and cannot detect outliers caused by other reasons. At the same time, this technique needs to be combined with specific positioning applications to be used and cannot be continued to be used in other application fields unrelated to positioning. Summary of the Invention

[0005] The object of the present invention is to provide a time difference of arrival outlier detection method based on multidimensional scaling method with a wide application range and high detection accuracy, so as to overcome the defects existing in the above-mentioned prior art.

[0006] The object of the present invention can be achieved by the following technical solutions:

[0007] A time difference of arrival outlier detection method based on multidimensional scaling method, comprising:

[0008] In the off-line acquisition stage, before the signal source sends a signal, according to the known sensor position coordinates, the variance of the Gaussian noise of the time difference of arrival measurement value and the signal propagation speed, the empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix of all sensor groups is calculated, and the scalar product matrix is constructed based on the multidimensional scaling method;

[0009] In the on-line detection stage, when the signal source sends a signal, measure the time difference of arrival measurement values between all pairs of sensors, and use the empirical probability distribution function to detect outliers in the time difference of arrival measurement values through a hypothesis testing method.

[0010] Further, the off-line acquisition stage specifically includes:

[0011] Construct sensor groups, with one sensor as the reference sensor in each sensor group, and assume that the position of the signal source is the geometric center of the sensor group;

[0012] Randomly generate measurement errors of the time difference of arrival measurement values that conform to the known Gaussian distribution, calculate the corresponding time difference of arrival measurement values, and construct a scalar product matrix according to the multidimensional scaling method;

[0013] Calculate the sum of the squares of the specific eigenvalues of the scalar product matrix to obtain the empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix.

[0014] Further, the on-line detection stage specifically includes the following steps:

[0015] Obtain the time difference of arrival measurement values of the signal source arriving at pairs of sensors measured online, and denote the sensor set composed of all sensors as Denote the sensor pair composed of sensor m as the first sensor and sensor n as the second sensor as All sensor pairs The set of sensor pairs formed is denoted as Construct all possible sensor groups, and denote the set of sensor groups formed by all sensor groups as

[0016] Select a certain sensor pair from the sensor pair set in

[0017] Select a subset of sensor groups from the set of sensor groups The subset of sensor groups Each sensor group in this subset of sensor groups contains sensor m and sensor s, and uses sensor s as the reference sensor, denoted as Among them, represents a subset of sensors composed of several sensors, means the subset of sensors contains sensor m;

[0018] For each sensor group in the subset of sensor groups Based on the time difference of arrival measurement values, construct a scalar product matrix according to the multi-dimensional scaling method, use the sum of the squares of the specific eigenvalues of this scalar product matrix as a statistic, and calculate the probability value that the time difference of arrival measurement values of the sensor group do not contain outliers by comparing with the empirical probability distribution function; The probability value that the time difference of arrival measurement values of the sensor group

[0019] After obtaining the probability values of all sensor groups in the subset of sensor groups correct all the probability values, and calculate the minimum value of the corrected probability values and the combined value of the uncorrected probability values. These two values are respectively called the probability minimum value and the probability combined value corresponding to the sensor pair ;

[0020] After calculating the probability minimum value and the probability combined value corresponding to each sensor pair in the set of sensor pairs use all the probability minimum values corresponding to all sensor pairs to determine whether there are outliers in the remaining time difference of arrival measurement values. If so, use all the probability combined values corresponding to all sensor pairs to determine which sensor pair the time difference of arrival measurement value of is an outlier, then remove this outlier, and at the same time remove the sensor pair from the set of sensor pairs ; Repeat the above steps for the remaining time difference of arrival measurement values until there are no outliers in the remaining time difference of arrival measurement values.

[0021] Furthermore, the number of sensors in the sensor group is at least 6.

[0022] Furthermore, the scalar product matrix is denoted as

[0023] where S represents the number of sensors in the sensor group, and the scalar product matrix ​ The element in the \(i\)-th row and \(j\)-th column is:

[0024]

[0025] where \((x m , y m , z m ) represents the position coordinates of sensor \(m\), respectively represent the measured values of the difference in arrival distances of sensor \(m i and \(m j relative to the reference sensor \(s\). Among them, based on the known signal propagation speed, the measured value of the difference in arrival distances is obtained by multiplying the measured value of the difference in arrival times by the signal propagation speed.

[0026] Furthermore, the specific eigenvalue is two specific eigenvalues selected from the eigenvalues of the scalar product matrix.

[0027] Furthermore, denote all the eigenvalues of the scalar product matrix arranged in ascending order as where \(S\) is the number of sensors in the sensor group, represents the set of real numbers, then the specific eigenvalues are and

[0028] Furthermore, the specific correction of all probability values is as follows: Sort all the probability values in ascending order to obtain which is the \(i\)-th smallest probability value among all ;

[0029] The minimum value among the corrected probability values is denoted as:

[0030]

[0031] The combined value of the uncorrected probability values is denoted as:

[0032]

[0033] Furthermore, the specific determination of whether there are outliers in the remaining measured values of the difference in arrival times is as follows:

[0034] Determine whether there exists If so, there are outliers.

[0035] Furthermore, the specific determination of which measured value of the difference in arrival times is an outlier is as follows:

[0036] Take the measured value corresponding to the maximum value of the combined value of the uncorrected probability values as the outlier.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] 1. The present invention is applicable to the situation where the number of outliers is either large or small, and there is no need to know in advance the upper bound of the number of outliers;

[0039] 2. The present invention can detect outliers caused by various reasons and is not limited by the application field;

[0040] 3. In a single detection, the present invention can utilize the geometric information of more than 5 sensors, making the detection accuracy higher than that of the prior art.

[0041] 4. Since the positions of the sensors are fixed, and in the absence of outliers, the variance of the Gaussian noise of the time difference of arrival measurement values is also fixed, the empirical probability distribution function can be used for a long time once it is obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is a schematic flowchart of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0043] The present invention will be described in detail below with reference to the drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and the detailed implementation manners and specific operation processes are given, but the protection scope of the present invention is not limited to the following embodiments.

[0044] Refer to Figure 1 As shown, this embodiment provides a method for detecting outliers of time difference of arrival based on multidimensional scaling method, including an offline acquisition stage and an online detection stage.

[0045] 1. Offline acquisition stage

[0046] The offline acquisition stage refers to, before the signal source sends a signal, calculating the empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix of all sensor groups based on the known sensor position coordinates, the variance of the Gaussian noise of the time difference of arrival measurement values, and the signal propagation speed, where the scalar product matrix is constructed based on the multidimensional scaling method.

[0047] Specifically, one sensor is taken as the reference sensor, and at least five other sensors are combined with it to form a sensor group. Assume that the position of the signal source is the geometric center of the sensor group. Randomly generate the measurement errors of the time difference of arrival (TDOA) measurements that conform to the known Gaussian distribution. According to the known sensor position coordinates, calculate the corresponding TDOA measurement values, construct a scalar product matrix according to the multidimensional scaling method, calculate the sum of the squares of the specific eigenvalues of this matrix, and finally obtain the empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix, and save it for use in the online detection stage. Since the positions of the sensors are fixed, and in the absence of outliers, the variance of the Gaussian noise of the TDOA measurements is also fixed, so this empirical probability distribution function can be used for a long time once it is obtained.

[0048] The work in the offline acquisition stage is as follows:

[0049] Given:

[0050] (1) There are M≥6 sensor devices in space, and their position coordinates are known, denoted as

[0051]

[0052] (2) It is known that the measurement errors of the TDOA measurements conform to a Gaussian distribution with a mean of 0 and a variance of .

[0053] (3) The signal propagation speed is known to be c.

[0054] Find:

[0055] The empirical probability distribution function of the sum of the squares of the specific eigenvalues of the scalar product matrix for all sensor groups.

[0056] The specific process in the offline acquisition stage includes:

[0057] Step 1: Take S∈{6,7…,M} (the default value is S = 6).

[0058] Step 2: Consider taking one sensor as the reference sensor, and combining the other S - 1 sensors with the reference sensor s to form a sensor group, denoted as

[0059] Step 3: Assume that the position of the signal source is the geometric center of the sensor group, and its position coordinates are Calculate the true value of the distance m , from the signal source u to each sensor u Calculate the true value d of the arrival distance difference of each sensor relative to the reference sensor s m,s = d m-d s , The time difference of arrival measurement value can be obtained based on the signal propagation speed.

[0060] Step 4: Randomly generate S - 1 Gaussian random variables q with a mean of 0 and a variance of , where m,s , m ≠ s. Let q s,s = 0. Thus, a set of arrival distance difference measurement values Construct a scalar product matrix according to the multidimensional scaling method where the element in the i-th row and j-th column of the matrix is:

[0061]

[0062] Denote the eigenvalues of the matrix as where The specific eigenvalues specifically selected are and Calculate and save it.

[0063] Step 5: Repeat Step 4 10,000 times to obtain 10,000 Thus, the sum of the squares of the specific eigenvalues of the scalar product matrix of the sensor group is obtained, and the empirical probability distribution function of is denoted as

[0064] Step 6: Repeat Steps 1 to 5 until the empirical probability distribution functions of the sums of the squares of the specific eigenvalues of the scalar product matrices of all sensor groups are obtained. Save the empirical probability distribution functions of all sensor groups for use in the online detection stage.

[0065] 2. Online Detection Stage

[0066] The online detection stage refers to when the signal source sends a transmission signal, measuring the time difference of arrival measurement values between all pairs of sensors, and using the empirical probability distribution function to detect outliers in the time difference of arrival measurement values through a hypothesis testing method.

[0067] ​Specifically, consider the measured value of the difference in the arrival time from the signal source to sensor m and the arrival time from the signal source to sensor s; a sensor group consisting of no less than 6 sensors including sensor m and sensor s, with sensor s as the reference sensor, is formed. A scalar product matrix is constructed according to the multidimensional scaling method, and the sum of the squares of the specific eigenvalues of this matrix is calculated as a statistic. By comparing with the empirical probability distribution function of this statistic obtained in the offline acquisition stage, the probability value that the arrival time difference measurement value of the sensor group does not contain outliers can be calculated; after calculating the probability values that the arrival time difference measurement values of all sensor groups including sensor m and sensor s and with sensor s as the reference sensor do not contain outliers, all these probability values are corrected, and the minimum value of the corrected probability values and the combined value of the uncorrected probability values are calculated; using the minimum value of the corrected probability values, it is judged whether there are outliers in the remaining arrival time difference measurement values. If so, then using the combined value of the uncorrected probability values, it is judged which arrival time difference measurement value is an outlier and it is removed; repeating the above steps can find all outliers.

[0068] The work in the online detection stage is as follows:

[0069] It is known that:

[0070] (1) There are M≥6 sensor devices in space, and their position coordinates are known, denoted as

[0071]

[0072] (2) The signal propagation speed is known to be c.

[0073] (3) Denote the signal source as u0, and its position coordinates are unknown. It is known that the measured arrival time from the signal source u0 to the mth sensor u m and the arrival time from the signal source u0 to the nth sensor u n The difference is

[0074] Seek:

[0075] Detect outliers from all the measured arrival time differences .

[0076] The specific process in the online detection stage includes:

[0077] Step 1: Calculate m≠n. Let Let

[0078] Step 2: Take \(S\in\{6,7,\ldots,M\}\) (default value \(S = 6\)). Consider the sensor group composed of the sensor \(s\) as the reference sensor and another \(S - 1\) sensors, and denote the whole of such sensor groups as

[0079] Step 3: For each group Execute Steps 4 to 6. If Steps 4 to 6 have been executed for all then execute Step 7.

[0080] Step 4: Consider the measured value of the difference between the arrival time from the signal source to the sensor \(m\) and the arrival time from the signal source to the sensor \(s\), and denote the whole of the sensor groups containing this measured value as

[0081] Step 5: For each sensor group Construct a scalar product matrix according to the multidimensional scaling method where the element in the \(i\)-th row and \(j\)-th column of the matrix is:

[0082]

[0083] Denote the eigenvalues of the matrix as where Calculate Use the empirical probability distribution function calculated in the "offline acquisition stage" Calculate Calculate the probability value that the arrival time difference measurement value of the sensor group does not contain outliers Thus obtain all

[0084] Step 6: Correct all the probability values and calculate the minimum value among the corrected probability values, as well as the combined value of the uncorrected probability values. That is, sort all in ascending order to obtain where is the \(i\)-th smallest number among all Calculate the minimum value of the corrected probability values Calculate the combined value of the uncorrected probability values Save and Go back to Step 3.

[0085] Step 7: Use the minimum value among the corrected probability values to judge whether there are outliers in the remaining arrival time difference measurement values. That is, if Then perform step 8; otherwise, perform step 9.

[0086] Step 8: Use the combined value of the uncorrected probability values to determine which time difference of arrival measurement is an outlier. Specifically, use the measurement corresponding to the maximum value of the combined value of the uncorrected probability values as the outlier and remove it. That is, let Let Let where the symbol \ means to exclude from the set. Return to step 3.

[0087] Step 9: All outliers are

[0088] If the above method is implemented in the form of software functional units and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or part of this technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to enable a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in various embodiments of the present invention. The aforementioned storage medium includes: various media such as USB flash drives, mobile hard disks, read-only memories (ROM, Read-Only Memory), random access memories (RAM, Random Access Memory), magnetic disks, or optical discs that can store program codes.

[0089] The preferred specific embodiments of the present invention have been described in detail above. It should be understood that those of ordinary skill in the art can make many modifications and variations based on the concept of the present invention without creative efforts. Therefore, all technical solutions that can be obtained by those skilled in the art in the technical field of the present invention based on the concept of the present invention through logical analysis, reasoning, or limited experiments on the basis of the prior art should be within the protection scope determined by the claims.

Claims

1. A method for detecting outliers in time difference of arrival based on multidimensional scaling method, characterized in that, Including: In the off-line acquisition stage, before the signal source sends a signal, according to the known sensor position coordinates, the Gaussian noise variance of the time difference of arrival (TDOA) measurement values, and the signal propagation speed, calculate the empirical probability distribution function of the sum of squares of specific eigenvalues of the scalar product matrix for all sensor groups, where the scalar product matrix is constructed based on the multidimensional scaling method; In the on-line detection stage, when the signal source sends a signal, measure the TDOA measurement values between all pairs of sensors, and use the empirical probability distribution function to detect the outliers in the TDOA measurement values through a hypothesis testing method. The on-line detection stage specifically includes the following steps: Obtain the time difference of arrival measurement values from the signal source to each pair of sensors for online measurement, and denote the set of all sensors as Denote the sensor pair formed by taking sensor m as the first sensor and sensor n as the second sensor as All sensor pairs Denote the set of sensor pairs formed as Construct all possible sensor groups, and denote the set of all sensor groups formed as Select a sensor pair from the set of sensor pairs from the set From the set of sensor groups select a subset of sensor groups In each sensor group of this subset of sensor groups both sensor m and sensor s are included, and sensor s is used as a reference sensor, denoted as wherein represents a subset of sensors composed of a number of sensors means that the subset of sensors includes sensor m For each subset of sensor groups in the sensor group Based on the time difference of arrival measurements, construct a scalar product matrix according to the multi-dimensional scaling method, use the sum of squares of specific eigenvalues of the scalar product matrix as a statistic, and calculate the probability value that the time difference of arrival measurements of the sensor group do not contain outliers with reference to the empirical probability distribution function; After obtaining the probability values of all the sensor groups in the sub - set of sensor groups all the probability values are corrected, and the minimum value among the corrected probability values and the combined value of the uncorrected probability values are calculated. These two values are respectively called the probability minimum value and the probability combined value corresponding to the sensor pair ; For each sensor pair in the set of sensor pairs after calculating its corresponding minimum probability value and probability combination value, all the minimum probability values corresponding to all sensor pairs are used to determine whether there are outliers in the remaining time difference of arrival measurements. If so, all the probability combination values corresponding to all sensor pairs are used to determine which sensor pair has an outlier in its time difference of arrival measurement, and then the outlier is removed. At the same time, the sensor pair is removed from the set of sensor pairs ; ​ Repeat the above steps for the remaining TDOA measurement values until there are no more outliers in the remaining TDOA measurement values.

2. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to claim 1, characterized in that, The off-line acquisition stage specifically includes: Construct sensor groups, with one sensor in each sensor group as a reference sensor, and assume that the position of the signal source is the geometric center of the sensor group; Randomly generate measurement errors of TDOA measurement values that conform to the known Gaussian distribution, calculate the corresponding TDOA measurement values, and construct a scalar product matrix according to the multidimensional scaling method; Calculate the sum of squares of specific eigenvalues of the scalar product matrix to obtain the empirical probability distribution function of the sum of squares of specific eigenvalues of the scalar product matrix.

3. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to any one of claims 1-2, characterized in that, The number of sensors in the sensor group is at least 6.

4. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to any one of claims 1-2, characterized in that, The scalar product matrix is expressed as where S represents the number of sensors in the sensor group, and the scalar product matrix The element in the i-th row and j-th column is:[ Among them, (x m , y m , z m ) represents the position coordinates of sensor m, respectively represent the measured values of the difference in arrival distances of sensor m i , m j relative to the reference sensor s. Among them, based on the known signal propagation speed, the measured value of the difference in arrival distances is obtained by multiplying the measured value of the difference in arrival times by the signal propagation speed.

5. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to any one of claims 1-2, characterized in that, The specific eigenvalues are two specific eigenvalues selected from the eigenvalues of the scalar product matrix.

6. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to claim 5, characterized in that, Denote the scalar product matrix and arrange all its eigenvalues in ascending order as where S is the number of sensors in the sensor group, denotes the set of real numbers, then the specific eigenvalue is and 7. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to claim 1, characterized in that, The correction of all probability values specifically is: sort all probability values in ascending order to obtain is the i-th smallest probability value among all ; The minimum value in the corrected probability values is expressed as: The combined value of the uncorrected probability values is expressed as:

8. The method for detecting outliers in time difference of arrival based on multidimensional scaling method according to claim 1, characterized in that,Specifically, judging whether there are outliers in the remaining TDOA measurement values is: Determine whether there is If so, there are outliers.

9. The time difference of arrival outlier detection method based on multidimensional scaling method according to claim 1, characterized in that, Specifically, judging which TDOA measurement value is an outlier is: Take the measurement value corresponding to the maximum value of the combined value of the uncorrected probability values as the outlier.

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