CN110222299A - Contain the method and apparatus of the straight line fitting problem of error for bivariate - Google Patents
Contain the method and apparatus of the straight line fitting problem of error for bivariate Download PDFInfo
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Abstract
The present invention provides a kind of method and apparatus of straight line fitting problem for containing error for bivariate, the method comprise the steps that establishing the function model and stochastic model of indirect adjustment；Initialize X0And solve parameter reduction；Undated parameter estimated value simultaneously iteratively solves until convergence；Execute accuracy assessment.Using the solution of the present invention, solve the problems, such as that bivariate contains the best estimate of straight slope and intercept under error condition, and complete accuracy assessment.This method avoid the complex models and Iteration of weighting Least Square (Weighted Total Least Squares, WTLS) method, and clear model, mathematical form is simple, is easily programmed calculating.
Description
Technical field
This invention relates generally to measurement adjustment fields.More particularly, the present invention relate to solve bivariate containing wrong
The method and apparatus of the straight line fitting problem of difference.
Background technique
Straight line fitting frequently encounters in Various types of data is analyzed or handled, it can be described as: for one group of given sight
Measured data (xi, yi), (an optimal fitting a straight line y=kx+b (or straight slope k and intercept b is found in i=1,2 ..., n)
Best estimate).It, can if the observation of independent variable x is free of error (only the observation of y includes error) to the above problem
Directly using y=kx+b as observation model, solved by classical least square adjustment theory.However, if the observation of x and y
Value contains error, and straight line fitting problem just becomes more complicated, and above-mentioned observation model is no longer applicable in.
In fact, there is the straight line fitting of error in bivariate, it is that " observation vector, coefficient matrix are all in measurement adjustment
There are errors " such issues that special case in straight line fitting.For this kind of measurement adjustment problem, Golub and Van Loan is proposed
Singular value decomposition (Singular Value Decomposition, SVD) algorithm, and use Least Square (Total
Least Squares, TLS) name, it is different from classical least-squares estimation, TLS is suitable for observation data equally accurate, independence
The case where；For data usually unequal accuracy, related situation is observed in practical problem, Schaffrin and Wieser are proposed
Least Square (Weighted Total Least Squares, the WTLS) method of weighting, many researchers also propose in succession
Some iterative algorithms.
Although bivariate has the straight line fitting problem in the case of error that can correctly solve by the theory and model of WTLS,
But the characteristics of its mathematical model and Iteration are complicated, also do not make full use of this special case problem and theoretical excellent of classical adjustment
Gesture.
Summary of the invention
It is at least to solve above-mentioned technical problem, in an aspect, the present invention provides one kind to contain for solving bivariate
There is the method and apparatus of the straight line fitting problem of error, the method comprise the steps that
Establish the function model and stochastic model of indirect adjustment；
Initiation parameter X0And solve parameter reduction
Undated parameter estimated valueAnd it iteratively solves until restraining, wherein the estimates of parameters isAnd
Execute accuracy assessment.
In one embodiment, the process of the function model for establishing indirect adjustment and stochastic model includes: cut-off line
Slope k, intercept d and n observation point x coordinate be parameter, i.e.,Take n observation point
X coordinate, y-coordinate are observed quantity, i.e.,It, will according to indirect adjustment model
Observed quantityTo parameter?Place's linearisation, to obtain the mistake of indirect adjustment
Eikonal equationThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantity
Power battle array are as follows:
In another embodiment, the initiation parameter X0And solve parameter reductionProcess include: first with
(x1, y1)、(xn, yn) solve K0、B0Such as following formula (3):
And it takesComplete X0Initialization.Then A and l is calculated according to formula (1), is counted by following formula (4)
Calculate parameter reduction
In another embodiment, the undated parameter estimated valueAnd it iteratively solves until convergent process includes: to sentence
Disconnected parameter reduction(vector) the 1st, 2 elements, i.e. straight slope reductionIntercept reductionAbsolute value and convergence door
Limit Tk、TbSize:
If 1,AndExit iteration；
2, it otherwise, takesAs new initialization value X0, A and l is calculated according to formula (1), is calculated and is corresponded to according to formula (4)
The parameter reduction of new initialization valueAt the end of iteration, the estimated value of parameter is obtained:
The parameter (vector) the 1st, 2 elements, i.e., straight slope, intercept estimated value.
In another embodiment, the process for executing accuracy assessment includes:
According to indirect adjustment is theoretical and formula, variance of unit weight after (6) digital simulation according to the following formula:
The middle error of (7) calculating parameter according to the following formula:
∑=σ0(ATPA)-1 (7)
The diagonal of a matrix the 1st, 2 elements, i.e., straight slope, intercept estimated value middle error.
In another aspect, the straight line fitting that the present invention also provides a kind of to contain error for solving the problems, such as bivariate
Equipment, the equipment includes:
Module is established, is disposed for establishing the function model of indirect adjustment and stochastic model；
Initialization module is disposed for initializing variable X0And solve parameter reduction
It updates and solves module, be disposed for undated parameter estimated value and iteratively solve until restraining, wherein the ginseng
Counting estimated value isAnd
Accuracy assessment module is disposed for executing accuracy assessment.
In one embodiment, the module of establishing is configured to for executing:
The x coordinate of the slope k of cut-off line, intercept d and n observation point is parameter；
Taking the x coordinate of n observation point, y-coordinate is observed quantity
According to indirect adjustment model, by observed quantityTo parameter?Place
Linearisation, to obtain the error equation of indirect adjustmentThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantity
Power battle array are as follows:
In another embodiment, the initialization module is configured to for executing:
First with (x1, y1)、(xn, yn) solve K0、B0Such as following formula (3):
And it takesComplete X0Initialization.
Then, A and l is calculated according to formula (1), by following formula (4) calculating parameter reduction
In another embodiment, the update solves module and is configured to for executing:
Judge parameter reduction(vector) the 1st, 2 elements, i.e. straight slope reductionIntercept reductionIt is absolute
Value and convergence threshold Tk、TbSize:
If 1,AndExit iteration；
2, it otherwise, takesAs new initialization value X0, A and l is calculated according to formula (1), is calculated and is corresponded to according to formula (4)
The parameter reduction of new initialization valueAt the end of iteration, the estimated value of parameter is obtained:
The parameter (vector) the 1st, 2 elements, i.e., straight slope, intercept estimated value.
In another embodiment, the execution module is configured to for executing:
According to indirect adjustment is theoretical and formula, variance of unit weight after (6) digital simulation according to the following formula:
The middle error of (7) calculating parameter according to the following formula:
∑=σ0(ATPA)-1 (7)
The diagonal of a matrix the 1st, 2 elements, i.e., straight slope, intercept estimated value middle error.
It is theoretical that the present invention is based on classical least square adjustments, in the case where considering that bivariate contains error, proposes new
Straight line fitting observation model solve bivariate by rebuilding the function model of straight line fitting and contain under error condition
The best estimate problem of straight slope and intercept, and complete accuracy assessment.
This method uses Indirect Adjustment Method, using the advantage of particularity and the classical adjustment theory of straight line fitting problem,
The best estimate of straight slope and intercept is obtained, and completes accuracy assessment, avoids the complex model and iteration lattice of WTLS method
Formula, clear model, mathematical form is simple, is easily programmed calculating.This method can be used for solving many necks such as mapping, physics, space flight
The bivariate in domain contains the straight line fitting problem of error, and the linear relationship between quantitative analysis bivariate solves corresponding field
Specific actual techniques problem.
Detailed description of the invention
By read be provided by way of example only and with reference to attached drawing carry out being described below, be better understood with the present invention and
Its advantage, in which:
Fig. 1 is the process for showing the method for the straight line fitting according to the present invention for being used to solve the problems, such as that bivariate contains error
Figure；
Fig. 2 is the equipment composition frame for showing the straight line fitting according to the present invention for being used to solve the problems, such as that bivariate contains error
Figure；And
Fig. 3 is to show the result that using method of the invention one group of observation data are carried out with straight line fitting.
Specific embodiment
The present invention contains the straight line fitting problem of error for bivariate, proposes a kind of new line fitting method, the party
Method uses indirect adjustment principle, obtains the best estimate of straight slope and intercept, and complete accuracy assessment.
The following describes the present invention in detail with reference to the accompanying drawings and specific embodiments.
Fig. 1 is the flow chart element of the method for the straight line fitting according to the present invention for being used to solve the problems, such as that bivariate contains error
Figure.As shown in Figure 1, the method for the invention 100 first begins to step 101, in this step, the letter of indirect adjustment is established
Exponential model and stochastic model；Then method 100 enters 102 steps, in this step, initiation parameter X0And it solves parameter and changes
Positive quantityThen method 100 proceeds to 103 steps, in this step, undated parameter estimated valueAnd iteratively solve up to convergence,
Wherein the estimates of parameters isFinally, method 100 executes 104 step of accuracy assessment.
Assuming that one group of observation data is (xi, yi) (i=1,2 ..., n), variances of these observation data are denoted as
In one embodiment, the process of the function model and stochastic model of establishing indirect adjustment includes: the oblique of cut-off line
The x coordinate of rate k, intercept d and n observation point are parameter, i.e.,Take the x coordinate, y of n observation point
Coordinate is observed quantity, i.e.,
According to indirect adjustment model, by observed quantityTo parameter?Place
Linearisation, thus the error equation of indirect adjustmentThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantity
Power battle array are as follows:
In another embodiment, X is initialized0And solve parameter reduction102 process includes: initialization X0, and ask
Solve parameter reductionUtilize (x1, y1)、(xn, yn) solve K0、B0It is as follows:
And it takesComplete X0Initialization.
A and l is calculated according to formula (1), by following formula (4) calculating parameter reduction
In another embodiment, undated parameter estimated value and iterative solution are until the process of convergence 103 includes: judgement ginseng
Number reduction(vector) the 1st, 2 elements, i.e. straight slope reductionIntercept reductionAbsolute value and convergence threshold
Tk、TbSize:
If 1,AndExit iteration；
2, it otherwise, takesAs new initialization value X0, A and l is calculated according to formula (1), is calculated and is corresponded to according to formula (4)
The parameter reduction of new initialization value
At the end of iteration, the estimated value of parameter is obtained:
The parameter (vector) the 1st, 2 elements, i.e., straight slope, intercept estimated value.
In another embodiment, the process of execution accuracy assessment 104 includes:
According to indirect adjustment is theoretical and formula, variance of unit weight after (6) digital simulation according to the following formula:
The middle error of (7) calculating parameter according to the following formula:
∑=σ0(ATPA)-1 (7)
The diagonal of a matrix the 1st, 2 elements, i.e., straight slope, intercept estimated value middle error.
Fig. 2 is the equipment composition frame for showing the straight line fitting according to the present invention for being used to solve the problems, such as that bivariate contains error
Figure.As shown in Fig. 2, equipment 200 of the present invention includes: to establish module 201, it is disposed for establishing the function of indirect adjustment
Model and stochastic model；Initialization module 202 is disposed for initializing variable X0And solve parameter reductionUpdate is asked
Module 203 is solved, undated parameter estimated value is disposed for and is iteratively solved until restraining, wherein the estimates of parameters isAnd accuracy assessment module 204, it is disposed for executing accuracy assessment.
In one embodiment, the module 201 of establishing is configured to for executing:
The x coordinate of the slope k of cut-off line, intercept d and n observation point is parameter
Taking the x coordinate of n observation point, y-coordinate is observed quantity
According to indirect adjustment model, by observed quantityTo parameter?Place
Linearisation, to obtain the error equation of indirect adjustmentThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantity
Power battle array are as follows:
In another embodiment, the initialization module 202 is configured to for executing:
First with (x1, y1)、(xn, yn) solve K0、B0Such as following formula (3):
And it takesComplete X0Initialization.
Then, A and l is calculated according to formula (1), by following formula (4) calculating parameter reduction
In another embodiment, the update solves module 203 and is configured to for executing:
Judge parameter reduction(vector) the 1st, 2 elements, i.e. straight slope reductionIntercept reductionIt is absolute
Value and convergence threshold Tk、TbSize:
If 1,AndExit iteration；
2, it otherwise, takesAs new initialization value X0, A and l is calculated according to formula (1), is calculated and is corresponded to according to formula (4)
The parameter reduction of new initialization valueAt the end of iteration, the estimated value of parameter is obtained:
The parameter (vector) the 1st, 2 elements, i.e., straight slope, intercept estimated value.
In another embodiment, the execution module 204 is configured to for executing:
According to indirect adjustment is theoretical and formula, variance of unit weight after (6) digital simulation according to the following formula:
The middle error of (7) calculating parameter according to the following formula:
∑=σ0(ATPA)-1 (7)
The diagonal of a matrix the 1st, 2 elements, i.e., straight slope, intercept estimated value middle error.
The realization principle of the method for the invention is illustrated below with reference to the result schematic diagram of Fig. 3.For convenience of and other
Method compares, using with document Schaffrin B and Wieser A.On weighted total least-squares
Adjustment for linear regression [J] .Journal of Geodesy, 2008,82 (7): in 415-421.
Identical data are as follows:
Based on above data, straight line fitting is carried out according to following process.
Step 1: establishing the function model and stochastic model 101 of indirect adjustment.According to above data, n=10, i.e.,By observed quantity
To parameter?Place's linearisation, so that A and l are as follows:
Above-mentioned observed quantityPower battle array are as follows:
In above formula, diag () indicates to construct diagonal matrix by the element on diagonal line of the element in bracket.
Step 2: initialization X0And solve parameter reduction102.Utilize (x1, y1)、(x10, y10) solve K0、B0It is as follows:
And it takesComplete X0Initialization.
A and l, and calculating parameter reduction are calculated according to formula (9)
Step 3: undated parameter estimated value simultaneously iteratively solves until convergence 103.If convergence threshold Tk、TbIt is 1 × 10-10,
1 × 10 is all larger than due to 0.130852989,0.503485017-10, need to takeAs new initialization value X0, i.e.,
X0=[- 0.463741606 5.396514983 ...]T (19)
A and l, while calculating parameter reduction are calculated according to formula (9)And judge convergent.Such iteration is reciprocal, until
It obtains the 13rd time and calculates resulting parameter reduction
Meet the condition of convergence at this time, exits iteration, in turn
Thus straight slope is obtained, the estimated value of intercept is respectively -0.480533407 and 5.479910224.
Step 4: accuracy assessment is executed, including variance of unit weight after digital simulation:
The middle error of calculating parameter according to the following formula:
Thus straight slope, intercept estimated value middle error be respectively 0.07,0.36.
It calculates acquired results according to the present invention and the comparison of bibliography, true value is as follows:
Fig. 3 is that the result for carrying out straight line fitting to one group of observation data as above using method of the invention compares with true value
Compared with schematic diagram.It is completely the same with bibliography acquired results that the present invention can be seen that when Fig. 3 from above, and approach true
Value, illustrates that model and method of the invention are correct.
The method of the straight line fitting problem for containing error for bivariate of the invention can be by computer-readable
Recording medium is embodied with computer-readable code.Computer readable recording medium includes that storage can be interpreted by computer system
Data all kinds recording medium.The recording medium for example can include but is not limited to read-only memory (ROM, " Read
Only Memory "), random access memory (RAM, " Random Access Memory "), disk, disk, CD, flash memory
Deng.Further, these computer-readable recording mediums can (including computer communication network, honeycomb be logical by communication network
Communication network or local field communication network) it propagates or spreads between each communication entity, so that arbitrary mode can also be passed through
To run the computer-readable instruction being stored on computer readable storage medium or computer-executable code.
Although the mode that the present invention is implemented is as above, the content is implementation that is of the invention for ease of understanding and using
Example, the range and application scenarios being not intended to limit the invention.Technical staff in any technical field of the present invention, not
Be detached from disclosed herein spirit and scope under the premise of, can make in the formal and details of implementation any modification with
Variation, but scope of patent protection of the invention, still should be subject to the scope of the claims as defined in the appended claims.
Claims (10)
1. a kind of method for the straight line fitting problem for containing error for bivariate, comprising:
Establish the function model and stochastic model of indirect adjustment；
Initiation parameter X0And solve parameter reduction
Undated parameter estimated valueAnd it iteratively solves until restraining, wherein the estimates of parameters isAnd
Execute accuracy assessment.
2. according to the method described in claim 1, wherein the function model for establishing indirect adjustment and stochastic model include:
The x coordinate of the slope k of cut-off line, intercept d and n observation point is parameter
Taking the x coordinate of n observation point, y-coordinate is observed quantity
According to indirect adjustment model, by observed quantityTo parameter?Place's linearisation,
To obtain the error equation of indirect adjustmentThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantityPower
Battle array are as follows:
3. according to the method described in claim 1, the wherein initiation parameter X0And solve parameter reductionInclude:
Utilize (x1, y1)、(xn, yn) solve K0、B0, and takeComplete X0Initialization；And
According to indirect adjustment theory and formula, A and l, and then calculating parameter reduction are calculated。
4. according to the method described in claim 1, the wherein undated parameter estimated valueAnd it iteratively solves until convergence includes:
Judge parameter reductionThe 1st element and the 2nd element absolute value and convergence threshold Tk、TbSize, wherein the described 1st
Element and the 2nd element are respectively straight slope reductionWith intercept reduction
IfAndThen exit iteration；Otherwise, it takesAs new initialization value X0, A and l are recalculated, and
The parameter reduction of corresponding new initialization value
At the end of iteration, the estimated value of parameter is obtained
Wherein the 1st element and the 2nd element of the parameter are the estimated value of straight slope and intercept respectively.
5. according to the method described in claim 1, wherein the execution accuracy assessment includes:
According to indirect adjustment is theoretical and formula, variance of unit weight after digital simulationAnd middle error ∑=σ of parameter0(ATPA)-1
(7),
Wherein the 1st element and the 2nd element of the diagonal of a matrix, the respectively middle error of straight slope and intercept estimated value.
6. a kind of equipment for solving the problems, such as straight line fitting that bivariate contains error, comprising:
Module is established, is disposed for establishing the function model of indirect adjustment and stochastic model；
Initialization module is disposed for initializing variable X0And solve parameter reduction
It updates and solves module, be disposed for undated parameter estimated value and iteratively solve until restraining, wherein the parameter is estimated
Evaluation isAnd
Accuracy assessment module is disposed for executing accuracy assessment.
7. equipment according to claim 6, wherein the module of establishing is configured to for executing:
The x coordinate of the slope k of cut-off line, intercept d and n observation point is parameter
Taking the x coordinate of n observation point, y-coordinate is observed quantity
According to indirect adjustment model, by observed quantityTo parameter?Place is linear
Change, to obtain the error equation of indirect adjustmentThe expression formula of middle A and l is as follows:
In error equationFor parameter reduction to be solved, taking variance of unit weight is 1, then above-mentioned observed quantityPower
Battle array are as follows:
8. equipment according to claim 6, wherein the initialization module is configured to for executing:
Utilize (x1, y1)、(xn, yn) solve K0、B0, and takeComplete X0Initialization；
According to indirect adjustment theory and formula, A and l, and then calculating parameter reduction are calculated。
9. equipment according to claim 6 is configured to wherein the update solves module for executing:
Judge parameter reductionThe 1st element and the 2nd element and convergence threshold Tk、TbSize, wherein the 1st element and
2 elements are respectively straight slope reductionWith intercept reduction
IfAndThen exit iteration；Otherwise, it takesAs new initialization value X0, A and l are recalculated, and
The parameter reduction of corresponding new initialization value
At the end of iteration, the estimated value of parameter is obtainedWherein the 1st element and the 2nd element of the parameter point
Not Wei straight slope and intercept estimated value.
10. equipment according to claim 6, wherein the accuracy assessment module is configured to for executing:
According to indirect adjustment is theoretical and formula, variance of unit weight after digital simulationAnd middle error ∑=σ of parameter0(ATPA)-1
(7), wherein the 1st element of the diagonal of a matrix and the 2nd element be respectively straight slope and intercept estimated value middle error.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112417683A (en) * | 2020-11-20 | 2021-02-26 | 中国人民解放军63921部队 | Data processing method and device for antenna on-orbit pointing calibration, electronic equipment and storage medium |
| CN112417682A (en) * | 2020-11-20 | 2021-02-26 | 中国人民解放军63921部队 | Parameter fitting method and device for far-field radiation power data of antenna |
| CN116108328A (en) * | 2023-04-13 | 2023-05-12 | 中国人民解放军63921部队 | Method for acquiring relative positions of different antenna reference points of juxtaposition station and storage medium |
-
2019
- 2019-05-05 CN CN201910367473.8A patent/CN110222299A/en active Pending
Cited By (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112417683A (en) * | 2020-11-20 | 2021-02-26 | 中国人民解放军63921部队 | Data processing method and device for antenna on-orbit pointing calibration, electronic equipment and storage medium |
| CN112417682A (en) * | 2020-11-20 | 2021-02-26 | 中国人民解放军63921部队 | Parameter fitting method and device for far-field radiation power data of antenna |
| CN112417683B (en) * | 2020-11-20 | 2022-09-13 | 中国人民解放军63921部队 | Data processing method and device for on-orbit pointing calibration of antenna, electronic equipment and storage medium |
| CN112417682B (en) * | 2020-11-20 | 2022-10-28 | 中国人民解放军63921部队 | Parameter fitting method and device for far-field radiation power data of antenna |
| CN116108328A (en) * | 2023-04-13 | 2023-05-12 | 中国人民解放军63921部队 | Method for acquiring relative positions of different antenna reference points of juxtaposition station and storage medium |
| CN116108328B (en) * | 2023-04-13 | 2023-09-05 | 中国人民解放军63921部队 | Method for acquiring relative positions of different antenna reference points of juxtaposition station and storage medium |
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