CN1244654A - Quasi-accrate detection approach for measurement coarse error - Google Patents

Abstract

The invention discloses a quasi-quasi-verification method for measurement gross error, and relates to the processing technology of measurement and other experimental observation data. The present invention is a new method for identifying, locating and estimating gross errors in observed data. Taking the true error of the observed value as the research object, by adding the condition that "the norm of the true error of the quasi-observation is extremely small", the rank deficient equations about the true error are directly solved; according to the distribution characteristics of the true error estimation (grouping phenomenon ), accurately locate the gross error, then estimate the size of the gross error and correct it. An implementation plan for effectively selecting quasi-observations in two stages (primary selection and secondary selection) is designed. The outstanding advantages of the present invention are: ①high accuracy in identifying gross errors; ②can locate multiple gross errors at the same time. The more gross errors, the more obvious the effect of fast calculation.

Description

Measure the accurate checking method of plan of rough error

The invention discloses the accurate checking method of plan of measuring rough error, relate to the treatment technology of measurement and other experimental observation data.

In measurement or other experiments, there is rough error unavoidably in observation data.Do not get rid of rough error and disturb, the result can produce bias, even is distorted, and causes the waste of human and material resources and financial resources.Tackled rough error, except following tight measurement or experiment standard and taking the adequate measures, adopt two class methods mostly: a class is " statistical test " in the past, and the Data Snooping that proposes as the Baarda that measures boundary Holland obtains more extensively adopting; Another kind of is to study more " anti-difference is estimated " over nearly one, 20 year (Robustestimation), to manage statistical circles Huber in full, and Hampel and Rousseeuw have established the theoretical foundation that anti-poor M estimates.People such as Zhou Jiangwen, the Huang Youcai of the measurement Kubik of circle, Caspary and China, Yang Yuanxi have done further investigation.

Though it is wider that statistical test is used, when having a plurality of rough error in measurement data, effect is bad.Mainly be that LS is the ability of anti-rough error not, thereby makes this method very unreliable because the statistic of statistical test is based upon on the basis of classical least square (LS).

Anti-difference estimates that (mainly being that M estimates) through years development, becoming more effective a kind of method aspect the deduction and exemption rough error.Its key is the effective weight function of structure, yet does not also find the general weight function that is suitable for different situations at present, and particularly general relevant anti-difference weight function structure is very difficult.For the multidimensional situation, also there is not the tight accuracy assessment way that is suitable at present.

A common ground of method in the past all is that residual error with observed reading is a research object.And the residual error that classical least square method is tried to achieve both had been subjected to the influence of rough error in the observed reading, was subjected to the restriction of system architecture again, so-called lever observation problem occurred.This is a thorny problem that does not always better solve.

Along with science and technology development, the progress of society, more and more higher to the accuracy requirement of measurement and other experimental datas.People also more and more pay attention to the influence of rough error in the data, and new more effective Gross Error Detection method is studied and invented in an urgent demand, eliminate or reduce or remit the influence of rough error, improve the quality of data.

The objective of the invention is to adapt to this situation, on the basis of deep study and analysis existing methods advantage and deficiency, the thinking and the method for a kind of brand-new detection rough error of proposition.

The object of the present invention is achieved like this, and promptly the true error with observed reading is an object.The analytic relationship formula of determining according to true error and observed reading, proposed directly to find the solution the thinking of true error valuation, implementing method by the unique selection " quasi accurate observation " of a cover, under the condition of additional " norm of the true error of quasi accurate observation is minimum ", solved the problem of finding the solution about the rank defect equation of true error; Distribution characteristics according to the true error valuation is differentiated rough error, and location rough error and then estimation rough error size and precision are revised the observed reading that contains rough error then, improve the accuracy and the precision of parameter estimation.

Intend the ultimate principle of accurate checking method:

If linearizing measurement observation equation is expressed as

AX=L+ Δ (1) wherein A is that n * m maintains matrix number, and X is a m dimension parameter vector to be estimated, and L is a n dimension observation vector, and Δ is a n dimension true error vector.(1) valuation form is

, V is a residual error.

Can obtain the relational expression of determining of Δ and L through deriving

RA=-RL (2) is R=I-A (A here ^TA) ^-1A ^T

Because the order of R is n-m, is rank deficient matrix, generally speaking, can not try to achieve the true error valuation by (2) formula Determine separate.

Intending accurate checking method different from the past is the Gross Error Detection method of research object with residual error V.It is research object with the true error.The research thinking is that on the basis of equation (2), additional one group of condition that meets objective reality can be in the hope of the valuation of true error

Determine separate, tell rough error thus.

In fact, the observed reading great majority in measurement and other scientific experiments are normal, have only minority observation to contain rough error.We claim basic normal but wait to confirm be observed quasi accurate observation, claim to contain the observation that peels off of being observed of rough error.If select r (the individual quasi accurate observation of r＞m), additional " the true error Δ of quasi accurate observation _rNorm minimum " condition, promptly

The ‖ Δ _r‖ ²=min (3) can be tried to achieve the valuation of true error by equation (2)

Get m * n dimension matrix G _Q=(0, A _r ^T), wherein, m * r ties up matrix A _r ^TFor in the transposed matrix of factor arrays A corresponding to the partitioned matrix of this r quasi accurate observation.Solve

Association's factor battle array $Q_{\widehat{\mathrm{\Δ}}}=\mathrm{FRQRF}={(R+G_Q^T{G}_Q)}^{-1}{R(R+G_Q^T{G}_Q)}^{-1}...\left(5\right)$

Usually in advance and do not know which observation is normal observation (being quasi accurate observation), which contains rough error (observation promptly peels off), therefore the key how correctly selected quasi accurate observation is the accurate checking method of plan.

The present invention has designed the embodiment of dividing two stages to select quasi accurate observation:

1) the primary election quasi accurate observation according to grasp in advance about the information of this batch observed reading quality or calculate some specific index, tentatively choose quasi accurate observation, and calculate the valuation of true error by formula (4) and (5);

2) the final election quasi accurate observation calculate on the basis of the initial valuation of true error, calculate some specific indexs, through electing quasi accurate observation additional member or changing the iterative computation of quasi accurate observation, determine quasi accurate observation, and then calculate more accurately

With

According to what determine With

, calculate specific indexes, through and given dividing value relatively, index is judged to be greater than those observations of dividing value and contains rough error.So just find and located rough error.

On the basis of the correct location of rough error, suppose to find b rough error, obtain b n dimension vector of unit length e _j=(0 ..., 0,1,0 ..., 0) ^T, corresponding j the observation that rough error is arranged, its j component is 1, all the other are 0.(j＝1，…，b)。With rough error additional parameter _bNew observation equation is constructed in expression $A\hat{X}+C_b{\widehat{\▿}}_b=L+V_b.....\left(6\right)$ Wherein n * b ties up Matrix C _b=(e ₁..., e _b)

Obtain after the adjustment

The rough error valuation ${\widehat{\▿}}_b={\left(C_b^T{\mathrm{RC}}_b\right)}^{-1}{C}_b^T\mathrm{RL}........\left(7\right)$

Its association's factor battle array $Q_{{\widehat{\▿}}_b}={\left(C_b^T{\mathrm{RC}}_b\right)}^{-1}........\left(8\right)$

Surplus poor $V_{\▿}=V+{\mathrm{RC}}_b{\widehat{\▿}}_b......\left(9\right)$ At this moment variance of unit weight valuation is ${\widehat{\mathrm{\σ}}}_0^2=V_{\▿}^T{V}_{\▿}/(n-m-b).....\left(10\right)$

Can try to achieve the parameter estimation under the situation of eliminating the rough error influence simultaneously

, at this moment Quality (valuation and precision) all improve.Further specify below in conjunction with drawings and Examples: Fig. 1 intends accurate checking method program flow diagram for the present invention, and Fig. 2 is the angle measurement network synoptic diagram, and Fig. 3 is the leveling network synoptic diagram.Wherein 0-begins the valuation of 1-primary election quasi accurate observation 2-calculating true error among Fig. 1

And association's factor battle array 3-parameter W _i ^I(and C ₁) and W _i ^II(and C ₂) 4-check W _i ^IAnd W _i ^IIWhether greater than dividing value 5-replacing or gravity treatment quasi accurate observation 6-comparison

With

7-increase quasi accurate observation number, 8-parameter W _i ^III(and C ₃) 9-differentiate rough error 10-relatively the contain observation number b of rough error ^(m)And b ^(m-1)11-estimate rough error size, calculating parameter valuation and evaluate precision, output

And

12-finish

The step of " intending accurate checking method " is as follows: the beginning of the 0th step

The observation data of measurement or other experiments is input to hard disc of computer, makes least square adjustment, obtain error in residual error V=-RL and the weight unit

The 1st step primary election quasi accurate observation

If 1., select quasi accurate observation according to these information in advance relevant for the quality information of this batch observed reading.For example some (individual) observed reading may contain rough error, and these (individual) observation is not elected to be quasi accurate observation so.

If 2. there is not prior imformation

1) with ν _iBe expressed as $v_i=-R_{\mathrm{ii}}{l}_i-\underset{j\≠i}{\overset{n}{\mathrm{\Σ}}}{R}_{\mathrm{ij}}{l}_j={-a}_i-b_i$ , l wherein _i, l _jBe the component of observation vector L, R _IjBe the element of matrix R, i, j=1 ..., n

Calculate | a _i| median,

Calculate ${\widehat{\mathrm{\σ}}}_0=\underset{i}{\mathrm{med}}\left|v_i\right|,$

Calculate $u_i=\frac{v_i}{{\widehat{\mathrm{\σ}}}_0\sqrt{R_{\mathrm{ii}}}}$

With u _iPress the order of magnitude ordering.

Calculate the Internal Reliability index

2) according to above calculated amount, observed reading is divided into 4 classes: " 0 " class: if $\left|a_i\right|-\left|b_i\right|>3\underset{j}{\mathrm{med}}\left|a_j\right|,$ Or | a _i| and | b _i| all

(j=1 ... n), it is bigger that this type of observation contains the rough error possibility, can not be elected to be quasi accurate observation." 1 " class: if

, here ${\mathrm{\λ}}_{\mathrm{mean}}=\underset{j\≠i}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\λ}}_{l_j}/(n-1)$

This type of observation structure is poor, and may be influences by force a little, is not selected into quasi accurate observation." 3 " class: if $\left|a_i\right|-\left|b_i\right|<-3\underset{j}{\mathrm{med}}\left|a_j\right|$

This type of can think that the possibility that contains rough error is little, is chosen as quasi accurate observation." 2 " class: remove above-mentioned three class special circumstances, all the other observations all are included into this type of, look particular problem.This

In the class observation, u _iLess be selected into quasi accurate observation.

If u _iValue apparent in view " hiving off " appears, promptly near a certain numerical value, a part of u _iObviously greater than another part, u so _iThe less relatively that part of quasi accurate observation that is chosen as of value.

If u _iValue is not obviously hived off, and generally speaking, selects u _iThe individual quasi accurate observation that is observed of r=m+1 (m is the dimension of parameter X) that value is less.

After just selecting r 〉=m+1 quasi accurate observation, entered for (2) step.(4) and (5) the calculating true error valuation by formula of the 2nd step and association's factor battle array thereof are promptly calculated

With , pressing order of magnitude then will Ordering is big

Preceding, little after.The 3rd step needed to calculate following index for the more accurate quasi accurate observation of choosing: , wherein $C_i=1.483\underset{i}{\mathrm{med}}\left|{\widehat{\mathrm{\&Delta;}}}_i\right|$ (i＝1，…，n)， $W_i^{\mathrm{II}}={\widehat{\mathrm{\&Delta;}}}_i/C_2,$ $C_2={(\mathrm{trace}{Q}_{{\widehat{\mathrm{\&Delta;}}}_r}/r)}^{1/2},$ It is quasi accurate observation

The true error valuation

Association's factor battle array, be included in

In.Trace () expression is asked

Mark.If the quasi accurate observation of the 4th step primary election is correct, calculate

Generally present obviously and hive off one

Part

Numerical value obviously greater than another part.At this moment check W _i ^IAnd W _i ^IIWhether

Greater than predefined dividing value (generally getting 3.0 or 2.5).If

Hive off obviously,

And part W _i ^IOr W _i ^II＞dividing value entered for (6) step, otherwise forwards (5) to

Step.The 5th step was reselected quasi accurate observation: perhaps removable parts or all gravity treatment (if observed reading

And then entered for (2) step and calculate many words).The 6th the step will calculate specifically With calculated last time

Relatively, if there is bigger change

Change, move W on the boundary line of hiving off _i ^IOr W _i ^IIObservation greater than dividing value increases, this

In time, considered to increase the quasi accurate observation number, entered for (7) step.If

Change not quite,

Entered for (8) step.The 7th step is with those

Numerical value is little and corresponding W obviously _i ^IOr W _i ^IILess than the observation of dividing value,

Elect quasi accurate observation as.At this moment the quasi accurate observation of selecting is counted r to be increased, and changes (2) then

Step calculates. Before stable, the very poor observation of structure is not generally elected as and is intended accurate the sight

Survey.The 8th step determined quasi accurate observation preferably in final election after, foundation

Calculate

The 9th step check W _i ^III(i=1 ..., n) whether (generally get greater than dividing value given in advance

3.0), if certain W _i ^IIIGreater than dividing value, judge that this observation contains rough error.If

It is not obvious to hive off, and does not find W _i ^IIIGreater than the observation of dividing value, change once more over to

(5) step is changed quasi accurate observation.Otherwise entered for (10) step.The observation number b that contains rough error that the 10th step will be chosen specifically ^(m)With the b that chose last time ^(m-1)Ratio

, if change, changed for (7) step over to, with W _i ^IIILess than those of dividing value

Be observed quasi accurate observation, recomputate

With If b ^(m)Do not change,

Entered for (11) step.The 11st step was estimated the rough error size, and the calculating parameter valuation is also evaluated precision.

Calculate the rough error size by (7) and (8) formula

And association's factor battle array , calculate $\hat{X}={\left(A^{T}A\right)}^{-1}{A}^T(L-C_b{\widehat{\&dtri;}}_b)$ ${Q_{\hat{X}}=\left(A^{T}A\right)}^{-1}+{\left(A^{T}A\right)}^{-1}{A}^T{C}_b{Q}_{{\widehat{\&dtri;}}_b}{C}_b^T{A\left(A^{T}A\right)}^{-1}$

By (9), (10) formula is calculated V ,

Output

And Obtain behind the 12nd step correction rough error

When containing rough error

Obviously diminish.Tie at last

Fruit meets this feature, intends accurate checking method and finishes.

The implementing method design of this invention has one's own knack, and by retrieval, has not yet to see the report that similar invention is arranged both at home and abroad.

Outstanding advantage of the present invention is 1. by estimating that the true error valuation comes identification and judges rough error.The true error valuation presents the characteristics of obviously hiving off, thus identification and judgement rough error accurately, directly perceived; 2. can detect (n-m-1) (n is the observed reading number, and m is number of parameters to be estimated) individual rough error (if present) simultaneously, and the size and the precision of estimation rough error; 3. can overcome the influence of lever observation; 4. contrast has the result of way or the result that existing document is introduced, and it is quick to prove that the accurate checking method of plan detects a plurality of rough errors, and the location rough error is accurate, and the estimation rough error is accurate.And rough error is many more, and it is outstanding more to detect effect.Can also fast detecting go out the difficult rough error of finding of previous methods.Through correcting rough error, be eliminated or reduced or remitted the parameter estimation of rough error influence, reach the raising estimation accuracy, increase the purpose of credible result degree.With two embodiment the implementation process of intending accurate checking method and the effect that obtains are described below: many rough errors of embodiment 1. angle measurement networks are intended accurate calibrating

In order accurately to measure P ₁And P ₂2 coordinate, as shown in Figure 2 the angle measurement triangulation network of design, A, B, C, D are known points, by standard with precision observation 18 angles, the weight of observation battle array is got unit matrix.Relevant observed reading is listed in the 2nd hurdle of table 1, simulates 7 rough errors, sees the 4th hurdle, and the observation vector L that obtains after the observation equation linearization is listed in table 1 the 3rd hurdle.

Detect rough error with intending accurate checking method: 1) parameter at first | u _i|, will | u _i| ordering from big to small, see the 6th hurdle, primary election | u _i| little 5 (r=m+1) calculate as quasi accurate observation

With index W ^{I (1)}, at this moment

And W ^{I (1)}Obviously hive off, see the 8th, 9 hurdles.2) choosing | W _i ^{I (1)}| 11 of＜2 are observed quasi accurate observation, calculate

And W ^{I (2)}And W ^III, see the 11st, 12 and 13 hurdles.3) from table, can see

Numerical value obviously hives off, and 7 observations that contain rough error of simulation find that all the rough error valuation of being tried to achieve (the 14th hurdle) is approaching with the simulation rough error.

Table 2 has been listed when not containing rough error and containing rough error the coordinate correction after the least square adjustment and has been counted error estimator in valuation and the weight unit.Clearly, if the rough error influence is not got rid of, coordinate adjustment result and precision all are subjected to serious distortion.

Adopt the present invention, accurately find whole rough errors, the adjustment result after the correction, the unanimity as a result when not containing rough error together basically confirms that the solution of the present invention has stronger Quality Control Function.

Table 1.

????1	?????2	????3	????4	????5	????6	????7
							The angle numbering	Observed reading	Contain rough error L	Rough error	????No	Index | u|	????No
????1	???126°14′24.1	???-6.8	????-7.0	????1	???2.53	????8
							????2	???23??39??46.9	???.6		????8	???2.40	????1
????3	???30??05??46.7	???-10.1	????-7.0	????12	???2.09	????11
							????4	???117??22??46.2	???.9		????18	???2.02	????12
????5	???31??26??50.0	???7.5	?????7.0	????3	???2.02	????5
							????6	???31??10??22.6	???-2.6		????11	???1.44	????18
????7	???22??02??43.0	???3.1		????17	???1.41	????3
							????8	???130??03??14.2	???-14.1	????-5.6	????10	???1.29	????17
????9	???27??53??59.3	???1.9		????2	???1.22	????2
							????10	???65??55??00.8	???1.2		????4	???1.16	????10
????11	???67??02??49.4	???-12.9	???-10.0	????5	????.99	????15
							????12	???47??02??11.4	???10.3	?????7.0	????15	????.87	????9
????13	???46??38??56.4	???4.0		????7	????.75	????6
							????14	???66??34??54.7	???8.5		????14	????.68	????7
????15	???66??46??08.2	???-13.2		????13	????.41	????4
							????16	???29??58??35.5	???9.6		????16	????.23	????14
????17	???120??08??31.1	???-10.7		????9	????.23	????13
							????18	???29??52??55.4	???10.1	?????7.0	????6	????.00	????16
					????＝5

σ ₀＝1.4″

Continuous table 1

Table 2.

^*This routine observed reading with reference to " measurement adjustment basis " (what ancestor companion etc. is write, Mapping Press, nineteen eighty-three version, p.282) the accurate calibrating of the plan of embodiment 2. multidimensional rough errors

As shown in Figure 3, according to engine request, the leveling network that contains known spot elevation A and 10 spot elevations undetermined has been laid in design, discrepancy in elevation observation number n=19.Represent observation vector (unit is mm) with L, represent the distance (km of unit) of each line of level, be listed in the 2nd, 4 hurdles of table 3 respectively with S.6 rough errors of simulation are seen the 3rd hurdle in observed reading.

Utilize the way of the present invention with the observed reading classification, 6 observations that are easy to contain rough error in this leveling network are classified as " 0 " class, do not have the observation of " 1 " class. and remaining " 2 ", quasi accurate observation is all elected in the observation of " 3 " class as, calculates And W ^I, W ^IIAnd W ^III, all present fairly obvious branch group character.Only calculate once , 6 rough errors are just simultaneously by identification, location.Numerical value sees Table 3 the 9th, 10 and 11 hurdles, (W ^IIUnlisted).

This example shows that it is effective really that the accurate checking method of plan that the present invention adopts detects a plurality of rough errors, and calculated amount is little, and speed is fast, and the identification rough error is accurate, can play the effect of quality control fully to measurement data.

Table 3.

^*This routine observed reading and graphic reference what ancestor companion, Li Mingfeng: " location and definite value in the time of the multidimensional rough error ", Surveying ﹠ Cartography Scientific ﹠ Technological Univ., Wuhan's journal, 1996,21 (4): 323～329

Claims (4)

1. measure the accurate checking method of plan of rough error, be the rough error of a kind of discovery, location survey data and estimate rough error size and precision, the data processing method that is corrected then is characterized in that:

1. the true error with observed reading is a research object;

2. basic normal observation is called quasi accurate observation, the following embodiment of selecting quasi accurate observation of design divides two stages to select quasi accurate observation, it at first is the primary election quasi accurate observation, then on the basis of trying to achieve the true error valuation, according to the distribution characteristics of true error valuation, the final election quasi accurate observation:

3. by the condition of additional " norm of the true error of quasi accurate observation is minimum ", find the solution rank defect system of equations about true error;

The true error valuation is calculated as follows $\widehat{\mathrm{\&Delta;}}=-{(R+G_Q^T{G}_Q)}^{-1}\mathrm{RL}$ Its association's factor battle array is pressed $Q_{\widehat{\mathrm{\&Delta;}}}={(R+G_Q^T{G}_Q)}^{-1}{R(R+G_Q^T{G}_Q)}^{-1}$ Calculate;

4. establish and determined b rough error, obtain n * b dimension factor arrays C _b, rough error is tieed up parameter vector with b _bThe expression, utilize then additional parameter the indirect adjustment method can in the hope of

With its association's factor battle array

: ${\widehat{\&dtri;}}_b={\left(C_b^{T}R{C}_h\right)}^{-1}{C}_b^T\mathrm{RL}$ $Q_{{\widehat{\&dtri;}}_b}={\left(C_b^T{\mathrm{RC}}_b\right)}^{-1}$ 。

2. the accurate checking method of the plan of measurement rough error according to claim 1 is characterized in that the primary election quasi accurate observation follows these steps to carry out:

1. calculate , ν wherein _iBe least-square residuals,

(i=1 ... n);

2. with ν _iDecide into $v_i=-R_{\mathrm{ij}}{l}_i-\underset{j\&NotEqual;i}{\overset{n}{\mathrm{\&Sigma;}}}{R}_{\mathrm{ij}}{l}_j=-a_i-b_i$ , according to | a _i| and | b _i| with Relation and reliability index λ ₁=1/R _Ij, observed reading is divided into 4 classes: it is big that " 0 " class contains the rough error possibility, and " 1 " class formation is relatively poor, and a little less than the reliability, it is little that " 3 " class contains the rough error possibility, and all the other are " 2 " class;

3. according to observation classification situation and u _i, whether value exists obvious boundary, u in initial option " 3 " class and " 2 " class _iBeing worth little is quasi accurate observation; Select r 〉=m+1 quasi accurate observation;

4. after calculating the true error valuation, if

It is not obvious to hive off, and need reselect suitable quasi accurate observation.

3. the accurate checking method of the plan of measurement rough error according to claim 1 is characterized in that the final election quasi accurate observation follows these steps to carry out:

1. calculate following index

Wherein $C_1=1.483\mathrm{med}\left|{\widehat{\mathrm{\&Delta;}}}_i\right|,$ (i＝1，2，…n) $W_i^{\mathrm{II}}={\widehat{\mathrm{\&Delta;}}}_i/C_2,$ $C_2=\sqrt{\mathrm{trace}\left(Q_{{\widehat{\mathrm{\&Delta;}}}_r}\right)/r}$

2. foundation

Divide group character, with those

Numerical value is obviously little, and corresponding W _i ^IOr W _i ^IIObservation final election less than dividing value (generally getting 3.0 or 2.5) is a quasi accurate observation; Through electing or change the iterative computation of quasi accurate observation additional member, determine quasi accurate observation.

4. the accurate checking method of the plan of measurement rough error according to claim 1 is characterized in that following these steps to differentiate rough error:

1. parameter $W_i^{\mathrm{III}}={\widehat{\mathrm{\&Delta;}}}_i/C_3,C_3={[{\widehat{\mathrm{\&Delta;}}}_r^T{\widehat{\mathrm{\&Delta;}}}_r/(r-1)]}^{\frac{1}{2}}$

2. check W _i ^IIIWhether,, judge that this observation contains rough error if greater than dividing value greater than dividing value given in advance (generally getting 3.0).

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