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Establishing a Measure for Robust Estimation Methods in its Application on Precise Geodetic Networks

Khaled Mohamed Zaky Hassan

Associate Professor of Surveying and Geodesy

Surveying Department, Shoubra Faculty of Engineering, Benha University, Cairo, Egypt

Mail address: 108 Shoubra Street, Cairo, Egypt

E-mail address: [email protected],

Phone Number: +2 01201222780

ABSTRACT

This research discusses the application of robust estimation and least squares methods in the adjustment procedure of precise geodetic networks. In the past, only the least squares method was applied for adjusting geodetic observations. Observations contain blunders or gross errors leads to incorrect or biased solution of precise geodetic networks i.e. results of solution are far from their true values. The research also discusses the establishment of a measure for the robust estimation methods in its application on geodetic networks. The results of this application indicate that, the variances of the least squares estimates and their corresponding robust estimates are approximately the same (close to each other) in the case of observations free from gross errors or contain small departures. In the case of observations include relatively large departures, the least squares estimates were far from those of the robust estimates and their true values while the robust estimates were very close from their corresponding true values. Finally, it is recommended to use robust estimation methods in the case of observations contain blunders/departures.  

Keywords: Precise Geodetic Networks, Robust Estimation, Blunders, Robust Measure.

1. INTRODUCTION

Robust estimation methods have been used in different works such as triangulation and leveling networks for the first time at 1968 especially in the adjustment procedures [2]. Several weight functions have been developed for the robust estimators. Although most of these weight functions have no theoretic base [1]. So that, most weight functions were empirical. The weight functions could be selected to robust estimates if only the values of them became less while residuals became larger. The explanation of the selected functions could be only understood through numerical examples. That is because, the measures of robustness for robust estimate is not unique. Then, there is no robust estimate has the same position as the least squares estimate in surveying adjustment. Based on the least squares estimate, study of the robustness, define a measure for it and deduce the corresponding robust estimate will be taken place in this research.

2. ROBUSTNESS MEASURE

Robustness can be defined as 'the properties of a procedure which renders the answers and gives insensitivity to departures (blunders) may be occur in practice [7]'. Another definition given by Huber [5]: 'the robustness signifies insensitivity to small deviations from the assumptions'. The explanation of robustness is given as:

' The robust estimate or procedure should have a reasonable optimal or nearly optimal efficiency at the assumed model.

' The robust estimate should be robust in the sense that small deviations from the model assumptions those impair the performance only slightly, that is, the later (described, say in terms of the asymptotic variance of an estimate, or of the level and power of a test) should be close to the normal value computed at the model.

' Some larger deviations from the model should not cause catastrophe.

This definition and explanation mean that the robustness is the value of robust estimate or procedure which should be as close as possible from the computed value in the assumed model. To judge on robustness of an estimate, the measure of robustness must be selected. In 1964, the superior of asymptotic variance has been selected by Huber [6] to be a measure for robustness. Under this measure and the normal distribution, Huber shows that the often robust estimate was uniquely determined and corresponding to ('') as follows [6]:

  (1)

Where V is the residual of observation and k is a selected value (critical value).

In 1986, another robust estimate method was also proposed by Hampel [4]. This method called as 'influence function'. The robustness may be described by some characteristics of this function. Based on the influence function, robust estimate can be computed as follows:

  (2)

Where: V is the residual,

 a, b, c are selected as critical values for residual.

Robustness can be described by the following three methods [3]:

1- Qualitative robustness: this method is equivalent to weak continuity to the empirical distribution function (T). In other words, qualitative robustness requires a slight change in the assumption or the model which should result in only a small change in the distribution of the statistics (T).

2- Quantitative robustness: a value of quantitative description to how a small change in the model or assumption will change the results of estimates. In this method, there are two types of measures:

' The maximum bias and

' The maximum variance.

3- Infinitesimal robustness: this method is related to the influence function. From this function, there are three types of robustness's measures as follows [4]:

' The gross error sensitivity measure,

' The local shift sensitivity measure,

' The rejection point sensitivity measure.

Robustness measures are unique and not desired to statisticians. So that, there is no unique or at least there is no distinguished optimal robust method. As a result, there is no any robust estimate has the same position as the least squares estimates in the scientific applications. Any of the above mentioned measures of robustness signifies an aspect of robustness but can not soundly reflect the implication of the definition of robustness. For instance, the maximum bias could expresses the sensitivity of estimates to the deviation from the ideal assumptions but it can not applicable whether 'it has a reasonable good efficiency at the assumed model', that is, the first one of the three properties for the sensitivity given by Huber [6]. The maximum variance can reflect the efficiency at the assumed model, but can not reflect the sensitivity of the procedure to the deviation. The three measures derived from the influence function also do the same. Always, if there is no departure in the practical model from the simplified assumption or no blunder in observations, the most estimates would be the least squares estimate (T). If there are departures from the assumption on observations, the corresponding estimate is (Te), then the difference between T and Te given by:

' =   Te ' T (3)

This difference reflects the effect of departure on the estimate, then:

D = 'T ' = (Te ' T)T . (Te ' T) (4)  

The last equation is a kind of measure, the mean distance will be as follows:

Dm = E ('T ') = E (Te ' T) T . (Te ' T)   (5)

If there are no departures from the assumptions, Dm could be expresses the difference between the robust estimate (Te) and the least squares estimate (T). That is meaning, distance Dm could be expresses that if the robust estimate is of good reasonable efficiency in the assumed model. If there are departures in assumptions, the distance Dm could be illustrate whether the robust estimate is close to the computed value in the model. If there are large departures from the assumption, then Dm could be illustrate how large the corresponded 'catastrophe' will be according to the explanation of robustness, the above distance Dm is a sound measure of robustness.

3. Robust Estimate Fundamentals [6]

The general form of observation equation is as follows:

L + V = A . X (6)

Where L is the vector of observations, V is the vector of residuals, A is the coefficient matrix and X is the unknown vector.

Also, observation equation can be written in the following form:

L ~ N (E (L) , Q) (7)

Where, Q is the variance of observation matrix.

The least squares estimate of unknown (X) in the equation (6) will be as follows:

X = (AT PO A)-1  AT PO L (8)

And PO = Q-1 (9)

Where, Q-1 is the inverse of variance matrix of observation.

Or T = X = q PO   L (10)

Where, q =(AT PO A)-1  AT (11)

Or q = (q1 , q2 , q3 ,  ''', qn)

Where, qi = (q1 , q2 , q3 ,'')= column matrix

If there are departures (blunders) in the model of equation (7), then the corresponding observation equation will be as follows:

Le = L + '' (12)

Where Le is the corresponding observation of departures and '' is vector of departures.

In this case, the least squares estimate method will be affected by ''. There is only one method for this case in robust estimate. This method is to change the weight of the corresponding observation Le. This method is called W-estimator [9].

Another robust estimate used in surveying adjustment and can be written as [8]:

X = (AT P A)-1  AT P L (13)

Where PO of the least squares estimates is replaced by P in the robust estimate. P is a function of the residuals and:

q =(AT P A)-1  AT (14)

Where q is assumed be approximately invariable.

Or Te = Xe = q P L (15)

And the difference between Te and T given by:

'' = T - Te = q Po L ' q P Le

Or '' = q (Po L - q P Le) (16)

And the mean distance between Te and T will be as follows:

Dm = E ('T ') = tr (qT . q ((Po  - P) . Q . (Po ' P) + P . E ('' '') .P)) (17)  

Where tr is the trace (sum of diagonal elements) of a matrix.

If the change in (P) of observation (i) or (Pi) from (Po) for the observation (i) or (Poi) could make (Dm) to be minimum i.e. Dm = minimum and the corresponding estimation must be reasonable estimate as follows:

  (18)

And (19)

And (20)

The true value of blunder or departure (''i) is not known, thus we have to decide or determine (Pi) on posterior information. In surveying adjustment, there are two types in the error model to describe the blunders or departures. One of them is called 'Stochastic model' in which, the blunder is considered 'Zero'. The observations contain blunders are considered have the same expected value like natural observation free from blunders but they have different variances as follows:

(21)

Where  is the variance of observation and the estimate of:

  (22)

In which ri is the local redundancy number.

In order to check or test the confidence of  , we have to form the following:

 ~ '' (23)

Then, the following will be accepted:

(24)

Where

  (25)

If the estimate  replaced by  in equation (25) and  by Fa and substituting equation (24) in equation (20), we get the following:

  (26)

The second type of error model is called 'mean shift model'. In this model,

E (''i . ''i) =  

In the adjustment stage, the estimate of blunder ''i can be derived by the following equation:

''i = Vi / ri

And we will use 'Baarda's test statistic' as follows:

  ~ N (0, 1) (27)

To check or prove that confidence of ''i at confidence level (1 ' ''), ''i may expresses and accepted in the limit of the following equation:

  (28)

4. Numerical Example

The observed horizontal angles between sides and lengths of these sides of a geodetic network are known. This network consists of 6 points. To check the reliability of estimates, a simulation study and statistical tests have been applied on this network. If the observation denoted by L and their corresponding departures are known, then the following estimates can be determined as:

   ,

   ,

  and

    

Where:

' T1 is the least squares estimates of observations free from gross errors.

' T2 is the robust estimates of observations free from gross errors.

' T3 is the least squares estimates of observations contain gross errors.

' T4 is the robust estimates of observations contain gross errors.

' Le = L + ''

' P is determined using equation (26).

The distance Dm can be determined using the following equations:

For this purpose, a computer program is designed to determine these estimates and distances using C++ programming language.

Let L1 has a departure ''1, ''1 is also normal error but has a variance '' = k . ''1. For each k,

results for k = 5, 10 are listed in the following tables:

Table (1): Results of robust measures, for k =5

Points T1 T2 T3 T4 Dm1 Dm2 Dm3

1

2

3

4

5

6 0.542

0.431

0.653

0.456

0.513

0.456 0.541

0.433

0.642

0.455

0.515

0.461 2.456

1.941

1.841

1.816

1.913

1.752 0.655

0.599

0.728

0.515

0.601

0.558

0.385

4.755

4.511

Table (2): Results of robust measures, for k =10

Points T1 T2 T3 T4 Dm1 Dm2 Dm3

1

2

3

4

5

6 0.569

0.555

0.724

0.578

0.673

0.723 0.568

0.553

0.753

0.577

0.671

0.524 4.613

2.315

1.450

1.504

0.958

0.735 1.759

0.859

0.785

0.815

0.851

0.654

0.0011

12.513

0.671

Let all departures ''i have a probability 0.05 to occur.

The corresponding results for k = 5, 10 listed in the tables 3 and 4.

Table (3): Results of robust measures, for k =5

Points T1 T2 T3 T4 Dm1 Dm2 Dm3

1

2

3

4

5

6 0.553

0.429

0.687

0.565

0.669

0.795 0.552

0.427

0.686

0.563

0.667

0.532 1.171

0.959

1.383

1.935

0.853

0.652 0.789

0.756

0.701

0.805

0.777

0.532

2.390

0.576

0.475

Table (4): Results of robust measures, for k =10

Points T1 T2 T3 T4 Dm1 Dm2 Dm3

1

2

3

4

5

6 0.601

0.458

0.653

0.593

0.701

0.821 0.601

0.460

0.648

0.594

0.705

0.621 1.185

1.001

1.395

1.853

0.724

0.523 0.654

0.757

0.695

0.953

0.782

0.542

2.551

0.672

0.594

4.1. Analysis of results

The results in tables 1 ' 4 showing that:

' The distance (Dm1) between the robust estimate T2 and the least squares estimate T1 is very small when the observation was free from gross errors.

' In case of observation contains relatively small gross errors, the variances of the least squares estimates and those of robust estimates are approximately the same.

' The distance (Dm3) between robust estimate T4 and least squares estimate T1 of observations free from blunders is also small.

' The variance of T4 is larger than that of T1 in the case of presence large departures while Dm3 still small.

' In the case of observations contain departures, the least squares estimate T3 is far from T1.  

' The final results of this analysis proved that the robustness and robust estimates are recommended to be used instead of least squares estimates especially in the case of observations contain gross errors.

5. Conclusions

The main aim of this paper is the establishment of a measure for the robust estimation methods in its application on precise geodetic networks with the least squares method. For this purpose, the robust estimation and least squares methods were applied on a precise network with and without observations include blunders. The results proved that the variances of the least squares estimates and their corresponding robust estimates are approximately the same in the case of observations free from contain small blunders. In the case of observations include relatively large blunders, the least squares estimates were far from those of the robust estimates and their true values while the robust estimates were very close from their corresponding true values. So that, it is recommend to use the robust estimation methods in the case of observations contain blunders/departures.

 

References

1. Andrews, D. F. & Bickel, P. J. & Hample, F. R. & Huber, P. J.  (1981). "Robust Estimates of Location." Survey and Advances, Princeton University Press, Princeton New Jersy.

2. Barrodale, I. (1968). "L1 Approximation and the Analysis of Data." Applied Styatistics, 17, 51-57.

3. Caspary, W. & Borutta, H. (1985). "Robust Estimation as Applied to Deformation Analysis." Proceeding of the 4th International Symposium on Geodetic Measurements of Deformations, Kattowice, 283-294.

4. Hample, F. R. (1986). "Robust Statistics: The Approach Based on Influence Functions." Wiley, New York.

5. Huber, P. J. (1964). "Robust Estimation of a Location Parameter." Ann. Math. Stat., 35, 79-101.

6. Huber, P. J. (1981). "Robust Statistics." Wily, New York.

7. Sergio Baselga (2007). "Global Optimization Solution of Robust Estimation." Journal of Surveying Engineering, 133(3), 123-128.

8. Springer Berlin, Heidelberg (2007). "On robust estimation with correlated observations." Journal of Geodesy, 63 (3).

9. Yuan-xi Yang, Tian-he Xu, and Li-jie Song (2005). "Robust Estimation of Variance Components with Application in Global Positioning System Network Adjustment." Journal of Surveying Engineering, 131 (4), 107-112.

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