CN110764124A - Efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method - Google Patents

Efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method Download PDF

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CN110764124A
CN110764124A CN201911043261.0A CN201911043261A CN110764124A CN 110764124 A CN110764124 A CN 110764124A CN 201911043261 A CN201911043261 A CN 201911043261A CN 110764124 A CN110764124 A CN 110764124A
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CN110764124B (en
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章浙涛
蒋弥
何秀凤
沈月千
吴怿昊
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Hohai University HHU
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/40—Correcting position, velocity or attitude
    • G01S19/41—Differential correction, e.g. DGPS [differential GPS]
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/42—Determining position

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Abstract

The invention discloses a high-efficiency and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method, which comprises the steps of obtaining multi-frequency multi-mode GNSS observation values; constructing a first covariance matrix according to the multi-frequency multi-mode GNSS observation values; according to the relation between the observation value type and the multi-frequency multi-mode GNSS observation value during positioning, introducing mathematical correlation into the first covariance matrix through a covariance propagation law to obtain a second covariance matrix; extracting a physical correlation coefficient of the second covariance matrix; carrying out significance test on the extracted physical correlation coefficient; preserving the physical correlation coefficients passing the significance test, and forming a third covariance matrix based on the preserved physical correlation coefficients passing the significance test; transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation mode to obtain a final covariance matrix of the observed value; and substituting the final covariance matrix of the observed values into the GNSS resolving mathematical model. The invention has the advantages of high calculation efficiency, high reliability and the like, and provides precise satellite navigation positioning service for users.

Description

Efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method
Technical Field
The invention relates to the technical field of observed value covariance matrix construction in the satellite navigation positioning technology, in particular to a high-efficiency and reliable multi-frequency multi-mode GNSS observed value covariance matrix estimation method.
Background
In gnss (global Navigation Satellite system) Navigation positioning, determining a stochastic model, namely constructing an observation value covariance matrix, is a very important step. Only by constructing a correct covariance matrix, high-precision and high-reliability satellite navigation positioning can be realized. With the development of the multi-frequency multi-mode GNSS, the number of observed values is greatly increased, so that the dimension of a corresponding covariance matrix is multiplied, and the difficulty in constructing the multi-frequency multi-mode GNSS observed value covariance matrix is increased.
Currently, the elements in the observed covariance matrix are often estimated empirically and passively. As in Real-Time Kinematic (RTK) positioning, only mathematical correlations are considered into the covariance element, since double-difference solutions cause mathematical correlations. In order to obtain reliable Variance-covariance components, especially when there are multiple observation types, a Variance-covariance Component Estimation (VCE) method is often used, and the like. When a more realistic covariance matrix is needed, physical correlation, i.e., spatial correlation, cross correlation, and temporal correlation, also need to be considered.
However, for multi-frequency multi-mode GNSS observations, the corresponding covariance matrix is estimated efficiently and accurately, and there is still a problem to be solved. First, in the covariance matrix, since there are too many physical correlations to be estimated, it is difficult for the user to estimate all of these correlations, especially in the multi-frequency and multi-mode GNSS application scenario, since VCE may be used, these estimated variances or covariance elements will be unstable or even meaningless. Secondly, inverting the covariance matrix results in a huge amount of computation, and thus the time-dependent covariance matrix needs to be converted to a non-time-dependent block-diagonal matrix.
Disclosure of Invention
Aiming at the problems, the invention provides an efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method, which practically improves the usability and accuracy of GNSS application.
In order to achieve the technical purpose and achieve the technical effects, the invention is realized by the following technical scheme:
an efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method comprises the following steps:
acquiring a multi-frequency multi-mode GNSS observation value;
constructing a first covariance matrix according to the multi-frequency multi-mode GNSS observation value;
according to the relation between the observation value type during positioning and the multi-frequency multi-mode GNSS observation value, introducing mathematical correlation into the first covariance matrix through a covariance propagation law to obtain a second covariance matrix;
extracting physical correlation coefficients of the second covariance matrix;
carrying out significance test on the extracted physical correlation coefficient;
preserving the physical correlation coefficients passing the significance test, and forming a third covariance matrix based on the preserved physical correlation coefficients passing the significance test;
transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation mode to obtain a final covariance matrix of the observed value;
and substituting the final covariance matrix of the observed values into a GNSS resolving mathematical model to finish navigation positioning.
As a further improvement of the invention, the multifrequency and multimode GNSS observation values comprise various types of raw observation values of different satellite systems and different frequencies.
As a further improvement of the invention, variance elements are arranged on the main diagonal lines of the first covariance matrix, and covariance elements are arranged on the non-main diagonal lines.
As a further improvement of the present invention, the expression of the second covariance matrix is:
D1＝KD0KT
where K is the coefficient matrix, D1As a second covariance matrix, D0Is a first covariance matrix.
As a further improvement of the present invention, the physical correlation coefficient includes spatial correlation, cross correlation, and temporal correlation coefficient.
As a further improvement of the present invention, the spatial correlation and the cross-correlation between the observed values are estimated by using the cross-correlation coefficient, as shown in the following formula:
wherein ,ρijIs ai and ljCross correlation coefficient of (1), pjiIs aj and liCross correlation coefficient of (a)i and σjAre respectively observed values li and ljCorresponding residual vi and vjStandard deviation of (d); 'Cov' stands for covariance operator;
estimating the time correlation between the observed values by using the autocorrelation coefficient as shown in the following formula:
wherein τ is a time interval and satisfies
Figure BDA0002253431150000023
c0C when τ is 0τN is the number of observation residuals, v (k) and v (k + τ) are the k and k + τ th observation residuals,
Figure BDA0002253431150000024
is the mean of the residuals of the n observations.
As a further improvement of the present invention, in extracting the spatial correlation:
for the spatial correlation, as in the double-difference positioning mode, a single-difference residual error SD of any nth satellite is obtained by adding an independent limiting conditionn；
Figure BDA0002253431150000031
wherein ,ωnRepresenting weighting of the nth satellite by an altitude weighting function, and theta representing the corresponding altitude and satisfying omegan＝sin2(theta) and Σ ωnSDn＝0；DDmnRepresenting the double difference residuals of satellites m and n. Based on this single-difference residual, a spatial correlation is estimated that is not affected by the mathematical correlation.
As a further improvement of the present invention, the significance test of the extracted physical correlation coefficient specifically includes the following sub-steps:
coefficient of physical correlation { ρ1,…,ρKAre random variables satisfying independent equal distribution, and sample mean
Figure BDA0002253431150000037
Considered as a normal distribution;
using zero mean test and setting original hypothesis H0And alternative hypothesis H1Are respectively H0:ρ＝0，H1:ρ≠0；
Will be provided withBy performing the normalization, one can obtain:
Figure BDA0002253431150000032
where μ is 0, and the standard deviation σ of the physical correlation coefficient is determinedρAnd a significance level α, and estimating a corresponding confidence interval according to a central limit theorem to complete significance test.
As a further improvement of the present invention, the transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation method to obtain a final covariance matrix of the observed values specifically includes:
the function model for adjacent GNSS observations is set as follows:
L*＝B*X*+E*
wherein ,
Figure BDA0002253431150000033
B*＝blkdiag([Ai-1,Ai])；
Figure BDA0002253431150000034
li-1 and liIs an observed value of the i-1 th and i epochs, Ai-1 and AiDesign matrix for i-1 and i epochs, xi-1 and xiUnknown parameters containing position coordinates for the i-1 th and i epochs, ei-1 and ei(ii) observed noise for the i-1 and i epochs;
the covariance matrix of adjacent GNSS observations satisfies Q (i-1) ═ Q (i) ═ Q', and thus:
wherein ,
Figure BDA0002253431150000036
represents a kronecker inner multiplicative operator; to obtain independent observations, the following transformations are performed:
first, let matrix R satisfy the following equation:
URUT＝D
wherein
Figure BDA0002253431150000041
Then, the two sides of the above formula are multiplied by
Figure BDA0002253431150000042
Where m is the number of observations observed at one time, thus yielding a transformed function model:
wherein ,
Figure BDA0002253431150000044
the new covariance matrix at this time is:
Figure BDA0002253431150000046
obviously, the new covariance matrix is a block diagonal matrix.
Compared with the prior art, the invention has the beneficial effects that:
1. according to the efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method, the constructed GNSS observation value covariance matrix fully considers time correlation and physical correlation coefficients, has the advantage of high reliability, and can provide precise satellite navigation positioning service for users.
2. And (4) carrying out significance test on the physical correlation coefficient, and keeping the physical correlation coefficient passing the significance test. The scheme not only considers the physical correlation coefficients of cross correlation, spatial correlation and time correlation, but also maximally reduces the insignificant physical correlation coefficients, thereby simplifying the structure of a covariance matrix, improving the subsequent computational feasibility and efficiency and having the advantage of high efficiency.
3. And transforming the covariance matrix into a block diagonal matrix by using a matrix transformation mode to obtain the final covariance matrix of the observed value. The scheme not only considers the time correlation, but also changes the time-correlated covariance matrix into a time-independent covariance matrix, thereby simplifying the calculation and being applicable to the field of GNSS real-time dynamic navigation positioning.
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In order that the present disclosure may be more readily and clearly understood, reference is now made to the following detailed description of the present disclosure taken in conjunction with the accompanying drawings, in which:
fig. 1 is a schematic flow chart of an efficient and reliable covariance matrix estimation method for observed values of a multi-frequency multi-mode GNSS according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of the invention.
The following detailed description of the principles of the invention is provided in connection with the accompanying drawings.
The invention provides an efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method, which specifically comprises the following steps as shown in FIG. 1:
(1) acquiring a multi-frequency multi-mode GNSS observation value;
the multi-frequency multi-mode GNSS observation values comprise various original observation values of different satellite systems and different frequencies; in a specific embodiment of the present invention, an RTK Positioning mode is taken as an example, and the invention relates to observations of different frequencies and types such as gps (global Positioning system), bds (beidou Navigation Satellite system), glonass (global Navigation Satellite system), and Galileo;
(2) constructing a first covariance matrix according to the multi-frequency multi-mode GNSS observation value;
in a specific embodiment of the present invention, variance elements are on the main diagonal lines and covariance elements are on the non-main diagonal lines of the first covariance matrix; the specific calculation methods of the variance elements and the covariance elements are the prior art, so that redundant description is not needed in the invention; in this embodiment, the variance element of the observation value can be estimated by using an index such as an altitude angle, i.e., by using the altitude angle as a function of the independent variable.
(3) According to the relation between the observation value type during positioning and the multi-frequency multi-mode GNSS observation value, introducing mathematical correlation into the first covariance matrix through a covariance propagation law to obtain a second covariance matrix;
in an embodiment of the present invention, the expression of the second covariance matrix is:
D1＝KD0KT
where K is the coefficient matrix, D1As a second covariance matrix, D0Is a first covariance matrix.
(4) Extracting physical correlation coefficients of the second covariance matrix, wherein the physical correlation coefficients comprise spatial correlation, cross correlation and time correlation coefficients;
the spatial correlation and the cross correlation are essentially cross correlation coefficients, but the positions of the cross correlation coefficients are different, and represent cross correlation coefficients with different properties, some are spatial correlation coefficients, and some are cross correlation coefficients, for this reason, in one embodiment of the present invention, the spatial correlation and the cross correlation between the observed values are estimated by using the cross correlation coefficients, as shown in the following formula:
Figure BDA0002253431150000051
wherein ,ρijIs ai and ljCross correlation coefficient of (1), pjiIs aj and liCross correlation coefficient of (a)i and σjAre respectively observed values li and ljCorresponding residual vi and vjStandard deviation of (d); 'Cov' stands for covariance operator;
estimating the time correlation between the observed values by using the autocorrelation coefficient as shown in the following formula:
Figure BDA0002253431150000052
wherein τ is a time interval and satisfies
Figure BDA0002253431150000053
c0C when τ is 0τN is the number of observed value residuals, v (k) and v (k + tau) are the k and k + tau observed value residualsThe difference is that the number of the first and second,is the mean of the residuals of the n observations.
In particular, in extracting the spatial correlation, the method further comprises:
for the spatial correlation, as in the double-difference positioning mode, a single-difference residual error SD of any nth satellite is obtained by adding an independent limiting conditionn；
Figure BDA0002253431150000062
wherein ,ωnRepresenting weighting of the nth satellite by an altitude weighting function, and theta representing the corresponding altitude and satisfying omegan＝sin2(theta) and Σ ωnSDn＝0；DDmnRepresenting the double difference residuals of satellites m and n. Based on this single-difference residual, a spatial correlation is estimated that is not affected by the mathematical correlation.
(5) Carrying out significance test on the extracted physical correlation coefficient;
in an embodiment of the present invention, the significance checking of the extracted physical correlation coefficient specifically includes the following sub-steps:
coefficient of physical correlation { ρ1,…,ρKAre random variables satisfying independent equal distribution, and sample mean
Figure BDA0002253431150000064
Considered as a normal distribution;
using zero mean test and setting original hypothesis H0And alternative hypothesis H1Are respectively H0:ρ＝0，H1:ρ≠0；
Will be provided withBy performing the normalization, one can obtain:
Figure BDA0002253431150000063
where μ is 0, and the standard deviation σ of the physical correlation coefficient is determinedρAnd a significance level α, and estimating a corresponding confidence interval according to a central limit theorem to complete significance test.
(6) Preserving the physical correlation coefficients which pass the significance test (namely, preserving the significant spatial correlation, cross correlation and time correlation coefficients and deleting the insignificant spatial correlation, cross correlation and time correlation coefficients), and forming a third covariance matrix based on the preserved physical correlation coefficients which pass the significance test;
(7) transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation mode to obtain a final covariance matrix of the observed value;
in a specific embodiment of the present invention, the transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation method to obtain a final covariance matrix of the observed values specifically includes:
the function model for adjacent GNSS observations is set as follows:
L*＝B*X*+E*
wherein ,
Figure BDA0002253431150000071
B*＝blkdiag([Ai-1,Ai])；
Figure BDA0002253431150000072
li-1 and liIs an observed value of the i-1 th and i epochs, Ai-1 and AiDesign matrix for i-1 and i epochs, xi-1 and xiUnknown parameters containing position coordinates for the i-1 th and i epochs, ei-1 and ei(ii) observed noise for the i-1 and i epochs;
the covariance matrix of adjacent GNSS observations satisfies Q (i-1) ═ Q (i) ═ Q', and thus:
Figure BDA0002253431150000073
wherein ,
Figure BDA0002253431150000074
represents a kronecker inner multiplicative operator; to obtain independent observations, the following transformations are performed:
first, let matrix R satisfy the following equation:
URUT＝D
wherein
Figure BDA0002253431150000075
Then, the two sides of the above formula are multiplied by
Figure BDA0002253431150000076
Where m is the number of observations observed at one time, thus yielding a transformed function model:
Figure BDA0002253431150000077
wherein , the new covariance matrix at this time is:
Figure BDA00022534311500000710
obviously, the new covariance matrix is a block diagonal matrix.
(8) Substituting the final covariance matrix of the observation values into a GNSS resolving mathematical model to complete navigation and positioning; the GNSS calculating mathematical model can be any one of the prior art as long as navigation and positioning can be realized.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (9)

1. An efficient and reliable multi-frequency multi-mode GNSS observation value covariance matrix estimation method is characterized by comprising the following steps:
acquiring a multi-frequency multi-mode GNSS observation value;
constructing a first covariance matrix according to the multi-frequency multi-mode GNSS observation value;
according to the relation between the observation value type during positioning and the multi-frequency multi-mode GNSS observation value, introducing mathematical correlation into the first covariance matrix through a covariance propagation law to obtain a second covariance matrix;
extracting physical correlation coefficients of the second covariance matrix;
carrying out significance test on the extracted physical correlation coefficient;
preserving the physical correlation coefficients passing the significance test, and forming a third covariance matrix based on the preserved physical correlation coefficients passing the significance test;
transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation mode to obtain a final covariance matrix of the observed value;
and substituting the final covariance matrix of the observed values into a GNSS resolving mathematical model to finish navigation positioning.
2. The method according to claim 1, wherein the method comprises the following steps: the multi-frequency multi-mode GNSS observation values comprise various original observation values of different satellite systems and different frequencies.
3. The method according to claim 1, wherein the method comprises the following steps: the main diagonal of the first covariance matrix is provided with variance elements, and the non-main diagonal is provided with covariance elements.
4. The method according to claim 1, wherein the method comprises the following steps: the expression of the second covariance matrix is:
D1＝KD0KT
where K is the coefficient matrix, D1As a second covariance matrix, D0Is a first covariance matrix.
5. The method according to claim 1, wherein the method comprises the following steps: the physical correlation coefficients include spatial correlation, cross correlation, and temporal correlation coefficients.
6. The method according to claim 5, wherein the method comprises the following steps:
estimating the spatial correlation and the cross correlation between the observed values by using the cross correlation coefficient as shown in the following formula:
Figure FDA0002253431140000011
wherein ,ρijIs ai and ljCross correlation coefficient of (1), pjiIs aj and liCross correlation coefficient of (a)i and σjAre respectively observed values li and ljCorresponding residual vi and vjStandard deviation of (d); 'Cov' stands for covariance operator;
estimating the time correlation between the observed values by using the autocorrelation coefficient as shown in the following formula:
Figure FDA0002253431140000021
wherein τ is a time interval and satisfies
Figure FDA0002253431140000022
c0C when τ is 0τN is the number of observation residuals, v (k) and v (k + τ) are the k and k + τ th observation residuals,
Figure FDA0002253431140000023
is the mean of the residuals of the n observations.
7. The method according to claim 1, wherein the method comprises the following steps: when extracting spatial correlation:
for the spatial correlation, as in the double-difference positioning mode, a single-difference residual error SD of any nth satellite is obtained by adding an independent limiting conditionn；
Figure FDA0002253431140000024
wherein ,ωnRepresenting weighting of the nth satellite by an altitude weighting function, and theta representing the corresponding altitude and satisfying omegan＝sin2(theta) and Σ ωnSDn＝0；DDmnRepresenting pairs of m and n of satellitesAnd a difference residual, based on the single difference residual, estimating the spatial correlation which is not influenced by the mathematical correlation.
8. The method according to claim 1, wherein the method comprises the following steps: the significance test is carried out on the extracted physical correlation coefficient, and the significance test specifically comprises the following substeps: coefficient of physical correlation { ρ1,…,ρKAre random variables satisfying independent equal distribution, and sample mean
Figure FDA0002253431140000025
Considered as a normal distribution;
using zero mean test and setting original hypothesis H0And alternative hypothesis H1Are respectively H0:ρ＝0，H1:ρ≠0；
Will be provided with
Figure FDA0002253431140000026
By performing the normalization, one can obtain:
Figure FDA0002253431140000027
where μ is 0, and the standard deviation σ of the physical correlation coefficient is determinedρAnd a significance level α, and estimating a corresponding confidence interval according to a central limit theorem to complete significance test.
9. The method according to claim 1, wherein the method comprises the following steps: the transforming the third covariance matrix into a block diagonal matrix by using a matrix transformation manner to obtain a final covariance matrix of the observed value specifically includes:
the function model for adjacent GNSS observations is set as follows:
L*＝B*X*+E*
wherein ,
Figure FDA0002253431140000031
B*＝blkdiag([Ai-1,Ai])；li-1 and liIs an observed value of the i-1 th and i epochs, Ai-1 and AiDesign matrix for i-1 and i epochs, xi-1 and xiUnknown parameters containing position coordinates for the i-1 th and i epochs, ei-1 and ei(ii) observed noise for the i-1 and i epochs; the covariance matrix of adjacent GNSS observations satisfies Q (i-1) ═ Q (i) ═ Q', and thus:
wherein ,
Figure FDA0002253431140000034
represents a kronecker inner multiplicative operator; to obtain independent observations, the following transformations are performed:
first, let matrix R satisfy the following equation:
URUT＝D
wherein
Figure FDA0002253431140000035
Then, the two sides of the above formula are multiplied by
Figure FDA0002253431140000036
Where m is the number of observations observed at one time, thus yielding a transformed function model:
Figure FDA0002253431140000037
wherein ,
Figure FDA0002253431140000038
Figure FDA0002253431140000039
the new covariance matrix at this time is:
Figure FDA00022534311400000310
obviously, the new covariance matrix is a block diagonal matrix.
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