CN110032709B - Positioning and estimation method for abnormal point in geographic coordinate conversion - Google Patents
Positioning and estimation method for abnormal point in geographic coordinate conversion Download PDFInfo
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Abstract
The invention discloses a method for positioning and estimating abnormal points in geographic coordinate conversion, which comprises the steps of influencing the posterior variance subjected to least square adjustment through abnormal values, obviously reducing the posterior variance after eliminating the abnormal values, and eliminating observed values (coordinate points) one by one to calculate the corresponding posterior variance; after a complete search, temporarily regarding the observation value corresponding to the minimum post-test variance as an abnormal observation value; the method can automatically find and position a plurality of abnormal points in the process of solving the conversion parameters, simultaneously estimate and correct the abnormal values, and directly obtain the conversion parameters after correction of the abnormal values.
Description
Technical Field
The invention relates to the fields of surveying and mapping science and technology, geographic information science and remote sensing image processing, in particular to a method for positioning and estimating abnormal points in geographic coordinate conversion.
Background
At present, the main technology for realizing coordinate transformation is to utilize coordinates of coincident points under different coordinate systems and solve transformation parameters under the support of least square method (LS for short), provided that the precision of the coordinates of the coincident points is reliable. In fact, the precision of the coordinate point is comprehensively influenced by accidental error accumulation, earth crust motion deformation and regional system errors, so that the coordinate point deviates from the real position of the coordinate point, if the deviation value of the control point is abnormal, the reliability of the solving parameter is seriously influenced, and the precision of the conversion result is finally reduced.
The robust estimation method is to construct a weight function formula to resist the influence of an abnormal point on parameters in the process of parameter solving, and the essential function of the weight function is to give a smaller weight to an abnormal observed value in the process of parameter solving, wherein a tuning coefficient is an important parameter for constructing the weight function formula, but the uncertainty of the tuning coefficient brings inconvenience to the detection of the abnormal value, which is the disadvantage of the method.
In practical application, which points are reliable cannot be judged, for identification of abnormal points, after LS parameter solving, a coincident point conversion residual is calculated, if the residual of a certain coordinate component is larger than 3 times of median error, the coordinate point is considered to be an abnormal point, the principle of the method is a 3-time standard deviation criterion, and the abnormal point is judged by applying the 3-time standard deviation criterion in technical specifications, which is effective for larger abnormal values.
For the processing of abnormal points, the main measure is to eliminate the abnormal points without participating in solving the conversion parameters, and the result is to change the number and the spatial distribution of the coincident points.
Disclosure of Invention
The present invention is directed to a method for locating and estimating outliers in a geographic coordinate transformation, which solves the above-mentioned deficiencies of the prior art.
The technology of the invention is realized by the following technical scheme: a method for locating and estimating outliers in geographic coordinate transformation comprises the following steps:
obtaining coordinate values of coincident points in different geographic coordinate systems as observed values of coordinate conversion;
setting the number of the observation values as n, rejecting one of all the observation values each time, performing n-1 dimensional complete search by using the rest observation values, calculating to obtain the post-test variances with the number equal to the observation values, and determining the observation value corresponding to the minimum value of the obtained post-test variances as the assumed abnormal observation value;
performing next complete search on the remaining n-1 observation values, calculating to obtain posterior square differences equal to the remaining observation values in number, and determining the observation value corresponding to the minimum value of the obtained posterior square differences as an assumed abnormal observation value;
judging whether the minimum posterior square difference of two adjacent complete searches has a significant difference, if so, stopping performing the next complete search, and taking the coincidence point of the minimum posterior square difference obtained by each complete search corresponding to the observed value as an abnormal observed value;
constructing a positioning matrix according to the search result of the abnormal observed value, and directly deriving an estimation equation and a conversion parameter correction equation of the abnormal value according to the relation between the abnormal value and the residual error; and substituting the positioning matrix and the original observed value into an estimation equation to directly obtain the size of the abnormal value, substituting the estimation result of the abnormal value into a correction equation, and directly outputting the coordinate conversion parameters corrected by the abnormal value.
Wherein, in the step of performing a complete search on the observed value, the method comprises the steps of:
supposing n observed values, before positioning and searching abnormal values, carrying out least square method LS to solve parameters and estimating posterior square difference
The subscript 0 represents the posterior variance obtained by all observed values participating in the least square method LS calculation;
eliminating the ith observation value L i And performing LS parameter calculation by using the rest observation data and calculating the posterior square difference after each parameter calculation
Wherein, the subscript n-1 indicates that n-1 observed values participate in LS parameter calculation, and the subscript i indicates that the i-th posterior variance is eliminatedObserved value L i Calculating; an estimate of n posterior variances is then obtained>
Finding the minimum value of the n posterior variances->
After n-1 dimensional complete search, temporarily combining L j Regarded as an abnormal observation value, using a positioning vector E j To represent L j The position of (2):
wherein E is j Is an n × 1 vector, the jth element is set to 1, and represents the position of abnormal data in n observations;
continue to perform the next full search, through m (m)>After = 2) complete searches,
is the minimum posterior variance obtained for an n-m +1 dimensional full search, is->
Is the minimum posterior variance obtained through n-m dimension complete search, if all abnormal values are just searched, if->
And &>
The search is considered to be over, with a significant difference.
Wherein, in the judgment
And &>
Compared with the steps with obvious difference, the method comprises the following steps:
judging whether the next complete search is continued or not by using the minimum variance ratio of two adjacent complete searches, wherein the minimum variance ratio is defined as:
wherein ρ m Is the variance ratio, m is the number of complete searches;
using F test to test if there is significant difference between the variances of two independent sampling processes, if formula (1) satisfies the principle of F test, the probability of F test is
P{ρ m >F a/2 (f 1 ,f 2 )}＝a (2)
In the formula: f. of 1 ＝r-m+1，f 2 ＝r-m，f 1 -f 2 =1,r is the number of redundant observations, P is the probability of an event occurring, a =0.05 is a given significance level, f 1 And f 2 Is a degree of freedom;
if the formula (2) is satisfied, then
And &>
There were significant differences.
Wherein the error equation model is a gaussian markov model, and the formula is expressed as:
L+Δ＝BX,
wherein, L is an observation vector of n multiplied by 1, B is a full rank array design matrix of n multiplied by t, X is a parameter vector of t multiplied by 1, delta is an observation error vector, P is a prior weight matrix of an observed value, which is a symmetrical positive definite matrix of n multiplied by n,
is a unit weight variance factor;
to find the state parameter vector, equation (3) is rewritten as an error equation:
wherein, the first and the second end of the pipe are connected with each other,
is an estimate of the parameter X, V is the nx1 residual vector;
the relation between the residual error and the observation value is expressed as an expression (5) in a matrix form;
V＝-RL＝-RΔ (5)
wherein R = I-B (B) T PB) -1 B T P, R is an idempotent matrix of nxn, R is related to B and P;
the observation error set in formula (1) is divided into two groups, one group is composed of observation errors with abnormal values, namely an abnormal group, and the other group is composed of observation errors without abnormal values, namely a random error group, and the observation error vector is expressed as:
Δ＝Δ ε +GΔ g (6)
wherein the content of the first and second substances,
wherein, delta g Is a m × 1 anomaly group vector, Δ ε Is an n × 1 random error group vector, G is an n × m positioning matrix, which is composed of each positioning vector (E), and non-zero elements in G represent the positions of abnormal observed values;
substituting formula (6) for formula (5) to obtain:
V＝-RΔ＝-RΔ ε -RGΔ g ＝V ε -RGΔ g (7)
wherein, V ε ＝-RΔ ε And then have
V ε ＝V+RGΔ g (8)
At V ε T V ε Calculating an abnormal value Δ under the condition of = min g Is estimated by
Comprises the following steps:
in the formula (I), the compound is shown in the specification,
rank(R g )＝m。
wherein, after estimating the magnitude of the abnormal value, positioning the matrix G and
the method is used for correcting the abnormal observed value, and the correction equation is as follows:
wherein, I is a unit array,
is a corrected observed value, and the optimal parameter estimation is as follows:
in the formula (I), the compound is shown in the specification,
is a t × 1 parameter vector corrected by an abnormal value.
Introducing abnormal data relative proximity RAD to measure the estimation accuracy of an abnormal value, wherein the estimation value is closer to a true value when the RAD is larger; wherein RAD is defined as:
different from the prior art, the method for positioning and estimating the abnormal points in the geographic coordinate conversion has the advantages that the posterior variance after LS (least squares) adjustment is influenced by the abnormal values, the posterior variance after the abnormal values are removed is obviously reduced, and the observed values (coordinate points) are removed one by one to calculate the corresponding posterior variance; after a complete search, temporarily taking the observation value corresponding to the minimum post-test variance as an abnormal observation value; the method can automatically find and position a plurality of abnormal points in the process of solving the conversion parameters, simultaneously estimate and correct the abnormal values, and directly obtain the conversion parameters after the abnormal values are corrected.
Drawings
Fig. 1 is a schematic flowchart of a method for locating and estimating outliers in a geographic coordinate transformation according to the present invention.
Fig. 2 is a comparative graph of transformation residuals obtained by different experimental methods in a method for locating and estimating outliers in a geographic coordinate transformation according to the present invention.
Detailed Description
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. This invention may be embodied in many different forms than those herein set forth and should be readily appreciated by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.
Next, the present invention is described in detail by using schematic diagrams, and when the embodiments of the present invention are described in detail, the schematic diagrams are only examples for convenience of description, and should not limit the protection scope of the present invention.
Referring to fig. 1, fig. 1 is a schematic flowchart of a method for locating and estimating outliers in a geographic coordinate transformation according to the present invention. The method comprises the following steps:
s110: and obtaining coordinate values of coincident points in different geographic coordinate systems as an observed value of coordinate conversion.
S120: setting the number of the observation values as n, rejecting one of all the observation values each time, performing n-1 dimensional complete search by using the rest observation values, calculating to obtain the posterior variance with the number equal to the observation values, and determining the observation value corresponding to the minimum value of the posterior variance to be used as the abnormal observation value.
S130: and performing next complete search on the remaining n-1 observation values, calculating to obtain posterior square differences with the number equal to that of the remaining observation values, and determining the observation value corresponding to the minimum value of the obtained posterior square differences to serve as an abnormal observation value.
S140: and judging whether the minimum posterior square difference of two adjacent complete searches has a significant difference, if so, stopping performing the next complete search, and taking the coincidence point of the observed values corresponding to the minimum posterior square difference obtained by each complete search as the abnormal observed value.
S150: constructing a positioning matrix according to the search result of the abnormal observed value, and directly deriving an estimation equation and a conversion parameter correction equation of the abnormal value according to the relation between the abnormal value and the residual error; and substituting the positioning matrix and the original observed value into an estimation equation to directly estimate the size of the abnormal value, substituting the estimation result of the abnormal value into a correction equation, and directly outputting the coordinate conversion parameters corrected by the abnormal value.
Wherein, in the step of performing a complete search on the observed value, the method comprises the steps of:
supposing that n observed values are provided, before positioning and searching abnormal values, least square method LS is carried out to solve parameters, and post-test square difference is estimated
The subscript 0 represents the posterior variance obtained by all observed values participating in the least square method LS calculation;
eliminating the ith observation value L i LS parameter calculation is carried out again by using the rest observation data and the posterior variance after each parameter calculation is calculated
Wherein, the subscript n-1 indicates that n-1 observed values participate in LS parameter calculation, and the subscript i indicates that the i-th observed value L is removed from the posterior variance i Calculating; an estimate of n posterior variances is then obtained>
Finding the minimum value of the n posterior variances->
After n-1 dimensional complete search, temporarily combining L j As an anomalous observation, using a location vector E j To represent L j The position of (2):
wherein E is j Is an n x 1 vector, the jth element is set to 1, representing the location of anomalous data in the n observations;
continue to perform the next full search, through m (m)>After = 2) complete searches,
is the minimum post-test variance obtained for an n-m +1 dimensional completion search, and->
Is the minimum posterior variance obtained through n-m dimension complete search, if all abnormal values are just searched, if->
And &>
The search is considered to be over, with a significant difference.
Wherein, in the judgment
And &>
The method comprises the following steps of:
judging whether the next complete search is continued or not by using the minimum variance ratio of two adjacent complete searches, wherein the minimum variance ratio is defined as:
wherein ρ m Is the variance ratio, m is the number of complete searches;
using F test to test whether the variance of two independent sampling processes has significant difference, if formula (1) satisfies the principle of F test, the probability of F test is
P{ρ m >F a/2 (f 1 ,f 2 )}＝a (2)
In the formula: f. of 1 ＝r-m+1，f 2 ＝r-m，f 1 -f 2 =1,r is the number of redundant observations, P is the probability of an event occurring, a =0.05 is a given significance level, f 1 And f 2 Is a degree of freedom;
if the formula (2) is established, then
And &>
There were significant differences.
Wherein the error equation model is a gaussian markov model, and the formula is expressed as:
L+Δ＝BX,
wherein, L is an observation vector of nx1, B is a full-rank array design matrix of nxt, X is a parameter vector of tx1, delta is an observation error vector, P is a prior weight matrix of an observed value, and is a symmetrical positive definite matrix of nxn,
is a unit weight variance factor;
to find the state parameter vector, equation (3) is rewritten as an error equation:
wherein the content of the first and second substances,
is an estimate of the parameter X, V is the nx1 residual vector;
the relation between the residual error and the observation value is expressed as an expression (5) in a matrix form;
V＝-RL＝-RΔ (5)
wherein R = I-B (B) T PB) -1 B T P, R is an idempotent matrix of nxn, R is related to B and P;
the observation error set in formula (1) is divided into two groups, one group is composed of observation errors with abnormal values, namely an abnormal group, the other group is composed of observation errors without abnormal values, namely a random error group, and the observation error vector is expressed as follows:
Δ＝Δ ε +GΔ g (6)
wherein the content of the first and second substances,
wherein, delta g Is a m × 1 anomaly group vector, Δ ε Is an n × 1 random error group vector, G is an n × m positioning matrix, which is composed of each positioning vector (E), and non-zero elements in G represent the positions of abnormal observed values;
substituting formula (6) for formula (5) to obtain:
V＝-RΔ＝-RΔ ε -RGΔ g ＝V ε -RGΔ g (7)
wherein, V ε ＝-RΔ ε And then have
V ε ＝V+RGΔ g (8)
At V ε T V ε Calculating an abnormal value Δ under the condition of = min g Is estimated value of
Comprises the following steps:
in the formula (I), the compound is shown in the specification,
rank(R g )＝m。
wherein, after estimating the magnitude of the abnormal value, positioning the matrix G and
the method is used for correcting the abnormal observed value, and the correction equation is as follows:
wherein, I is a unit array,
is a corrected observed value, and the optimal parameter estimation is as follows:
in the formula (I), the compound is shown in the specification,
is the t × 1 parameter vector corrected by the abnormal value.
Introducing relative proximity RAD of abnormal data to measure estimation accuracy of an abnormal value, wherein the estimation value is closer to a true value when the RAD is larger; wherein RAD is defined as:
the above is a method for locating, estimating and correcting abnormal values of control points (Location and Estimation outputs, referred to as LEO in the present invention).
The LEO can realize the positioning and estimation of abnormal points in coordinate transformation, the effectiveness and the advancement of the LEO can be proved by the following three experiments, the positions and the sizes of abnormal data in the experiments are the same and are random simulation values, and the contents of the three experiments are as follows:
LS experiment: in the experiment, only classical LS is used for solving parameters, coincident point coordinates (observed values) only have random errors, namely coordinate conversion parameters and residual errors after parameter solving by the LS are not polluted by abnormal values.
LS + Outlier experiment: the experiment still uses the LS adjustment, but 4 abnormal values are simulated in the observed values, and the final coordinate transformation parameters and the residual errors are polluted by the abnormal values.
LEO + Outlier experiment: the experiment adopts the abnormal values with the same numerical value and the same position as the first two schemes, and the LEO method is applied to position and estimate the abnormal values, and the result is reliable conversion parameters corrected by the LEO abnormal values, so that the pollution of the abnormal values on the conversion parameters is avoided.
Experimental data are real coordinates in practice (because secret-related coordinates are processed), 5 coincident points are distributed on the experimental data by converting a GNSS coordinate system into a region independent coordinate system, table 1 is coincident point coordinates of two coordinate systems, bold numbers in the table represent 4 abnormal values randomly simulated at point 1 and point 5, and a seven-parameter model is adopted for parameter calculation.
TABLE 1 coincidence point coordinates
Table 2 shows the calculation results obtained by LEO through 4 complete searches, the positioning results of the abnormal values are correct, RAD is greater than 80%, and the estimation accuracy of the abnormal values is high.
TABLE 2LEO outlier location and evaluation results (cm)
Table 3 shows the statistical results of the conversion parameters of the three experiments, where 7 parameters obtained by LEO + Outlier and LS are close to each other, which indicates that LEO can automatically correct an abnormal value to ensure the reliability of the conversion parameters.
TABLE 3 comparison of transformation parameters for three experiments
Table 4 is the residual statistics of three experiments, where V x ，V y And V z The coordinate components are respectively, sigma is a standard deviation, the result shows that the residual error after LEO solution is obviously improved, and if the standard deviation is 3 times, the abnormal point can not be positioned.
TABLE 4 residual calculation (cm) for three experiments
It can be seen from the transformation residual comparison graph (fig. 2) that the invention can realize the positioning of the abnormal control point in the coordinate transformation process, and automatically correct in the process of solving the parameters, thereby ensuring the reliability of the transformation parameter result and improving the transformation precision.
Compared with the prior art, the positioning and estimating method for the abnormal points in the geographic coordinate conversion has the advantages that the posterior variance after LS (least squares) adjustment is influenced by the abnormal values, the posterior variance after the abnormal values are removed is obviously reduced, and the observed values (coordinate points) are removed one by one to calculate the corresponding posterior variance; after a complete search, temporarily regarding the observation value corresponding to the minimum post-test variance as an abnormal observation value; the method can automatically find and position a plurality of abnormal points in the process of solving the conversion parameters, simultaneously estimate and correct the abnormal values, and directly obtain the conversion parameters after the abnormal values are corrected.
Although the present invention has been described with reference to the preferred embodiments, it is not intended to limit the present invention, and those skilled in the art can make variations and modifications of the present invention without departing from the spirit and scope of the present invention by using the methods and technical contents disclosed above.
Claims (2)
1. A method for locating and estimating outliers in geographic coordinate transformation, comprising:
obtaining coordinate values of coincident points in different geographic coordinate systems as observed values of coordinate conversion;
setting the number of the observation values to be n, rejecting one of all the observation values each time, performing n-1 dimensional complete search by using the rest observation values, calculating to obtain n posterior variances equal to the observation values in number, and determining the observation value corresponding to the minimum value of the posterior variances to serve as the assumed abnormal observation value;
performing next complete search on the remaining n-1 observation values, calculating to obtain n-1 posterior square differences with the number equal to that of the remaining observation values, and determining the observation value corresponding to the minimum value of the obtained posterior square differences as an assumed abnormal observation value;
judging whether the minimum posterior square difference of two adjacent complete searches has a significant difference, if so, stopping performing the next complete search, and taking the coincidence point of the minimum posterior square difference obtained by each complete search corresponding to the observed value as an abnormal observed value;
constructing a positioning matrix according to the search result of the abnormal observed value, and directly deriving an estimation equation and a conversion parameter correction equation of the abnormal value according to the relation between the abnormal value and the residual error; substituting the positioning matrix and the original observed value into an estimation equation to directly obtain the size of an abnormal value, simultaneously substituting an abnormal value estimation result into a correction equation, and directly outputting a coordinate conversion parameter corrected by the abnormal value;
in the step of performing a complete search for the observed value, the method comprises the steps of:
supposing that n observed values are provided, before positioning and searching abnormal values, least square method LS is carried out to solve parameters, and post-test square difference is estimated
Wherein, subscript 0 represents the posterior variance calculated by all the observed values participating in the least square method LS;
eliminating the ith observed value L i And performing LS parameter calculation by using the rest observation data and calculating the posterior square difference after each parameter calculation
Wherein, the subscript n-1 indicates that n-1 observed values participate in LS parameter calculation, and the subscript i indicates that the i-th observed value L is removed from the posterior variance i Calculating; an estimate of n posterior variances is then obtained>
Finding the minimum value of the n posterior variances->
After n-1 dimensional complete search, temporarily combining L j Regarded as an abnormal observation value, using a positioning vector E j To represent L j The position of (c):
wherein E is j Is an n x 1 vector, the jth element is set to 1, representing the location of anomalous data in the n observations;
continuing to execute the next complete search, and after m complete searches, m>＝2，
Is n-m +1 dimensional complete search acquisitionIs based on the minimum post-test variance of->
Is the minimum posterior variance obtained by n-m dimension complete search, and if all abnormal values are just searched, the value is determined to be greater than or equal to>
And &>
The search is considered to be finished if the difference is significant;
in the judgment
And &>
The method comprises the following steps of:
judging whether the next complete search is continued or not by using the minimum variance ratio of two adjacent complete searches, wherein the minimum variance ratio is defined as:
wherein ρ m Is the variance ratio, m is the number of complete searches;
using F test to test whether the variance of two independent sampling processes has significant difference, if formula (1) satisfies the principle of F test, the probability of F test is
P{ρ m ＞F a/2 (f 1 ,f 2 )}＝a (2)
In the formula: f. of 1 ＝r-m+1，f 2 ＝r-m，f 1 -f 2 =1,r is the number of redundant observations, P is the probability of an event occurring, a =0.05 is a given significance level, f 1 And f 2 Is a degree of freedom;
if the formula (2) is satisfied, then
And &>
There were significant differences;
the error equation model is a Gaussian Markov model, and the formula is expressed as follows:
wherein, L is an observation vector of nx1, B is a column full rank design array of nxt, X is a parameter vector of tx1, delta is an observation error vector, P is a prior weight array of an observation value, is a symmetric positive definite array of nxn, and sigma is a symmetric positive definite array of nxn 0 2 Is a unit weight variance factor;
to find the state parameter vector, equation (3) is rewritten as an error equation:
wherein, the first and the second end of the pipe are connected with each other,
is an estimate of the parameter X, V is the nx1 residual vector;
the relation between the residual error and the observation value is expressed as an expression (5) in a matrix form;
V＝-RL＝-RΔ (5)
wherein R = I-B (B) T PB) -1 B T P, R is an idempotent matrix of nxn, R is related to B and P;
the observation error set in formula (1) is divided into two groups, one group is composed of observation errors with abnormal values, namely an abnormal group, and the other group is composed of observation errors without abnormal values, namely a random error group, and the observation error vector is expressed as:
Δ＝Δ ε +GΔ g (6)
wherein the content of the first and second substances,
wherein, delta g Is an mx 1 anomaly group vector, Δ ε Is an n × 1 random error group vector, G is an n × m positioning matrix, which is composed of each positioning vector (E), and non-zero elements in G represent the positions of abnormal observed values;
substituting formula (6) for formula (5) to obtain:
V＝-RΔ＝-RΔ ε -RGΔ g ＝V ε -RGΔ g (7)
wherein, V ε ＝-RΔ ε And then have
V ε ＝V+RGΔ g (8)
At V ε T V ε Calculating an abnormal value Δ under the condition of = min g The estimated values of (c) are:
in the formula (I), the compound is shown in the specification,
rank(R g )＝m；
after estimating the magnitude of the outliers, the matrices G and G are located
The method is used for correcting the abnormal observed value, and the correction equation is as follows:
wherein, I is a unit array,
is a corrected observation, the best parameter estimate is:
in the formula (I), the compound is shown in the specification,
is the t × 1 parameter vector corrected by the abnormal value.
2. The method for location and estimation of outliers in geographic coordinate transformation of claim 1,
introducing abnormal data relative proximity RAD to measure the estimation accuracy of the abnormal value, wherein the estimation value is closer to the true value when the RAD is larger; wherein RAD is defined as:
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