OUTLIER DETECTION IN PARTIAL ERRORS-IN-VARIABLES MODEL

Detecção de outliers usando um modelo de observações de erro

Gauss-Markov (G-M) model and least-squares (LS) method are widely used in geodetic science. Most of time, the elements of the coefficient matrix may be consisting of the observations possessing the statistical properties in many applications such as the coordinate transformation (Akyilmaz, 2007; Li et al., 2012; Li et al., 2013; Fang, 2014), and the estimates of the unknown parameters derived by the LS method would not be optimal because the statistical properties of the elements in the coefficient matrix are ignored. The errors-in-variables (EIV) model and so called total least-squares (TLS) method named by Gloub et al. (1980) are more rigorous than the LS method. There are many algorithms to compute the TLS estimate (Gloub et al.,1980; Schaffrin, 2006) or weighted TLS (WTLS) estimate (Schaffrin and Wieser, 2008; Shen et al., 2011; Xu et al., 2012; Amiri-Simkooei and Jazaeri, 2012; Mahboub, 2012; Fang, 2013; Jazaeri et al., 2014).

Unfortunately, like the LS estimate, the WTLS estimate is also extremely vulnerable to the outliers in the EIV model. Although many methods for detecting the outliesr in the G-M model are investigated extensively (Baarda, 1968; Pope, 1976; Kok, 1984; Huber 1981; Hekimoglu, 2005; Gui et al. 1999, 2005a, 2005b, 2007, 2011; Guo et al., 2007; Hekimoglu and Erenoglu, 2009; Lehmann, 2013; Hekimoglu et al., 2014), they cannot be directly employed to deal with the outliers in the EIV model. Schaffrin and Uzun (2011) have generalized the mean-shift method to detect a single outlier located either in the observations or in the coefficient matrix in the EIV model. The reliability was also analyzed (Schaffrin and Uzun, 2012). Amiri-Simkooei and Jazaeri (2013) applied the data-snooping procedure to identify the outliers based on the WTLS method formulated with the standard LS theory (Amiri-Simkooei and Jazaeri, 2012). However, the test procedure is required to be implemented more than once while there are some repeated random elements in the different locations of the coefficient matrix like the two-dimensional affine transformation.

The partial EIV model is a generalized EIV model and can avoid considering the correlations between the repeated random elements in the coefficient matrix (Xu et al., 2012). Therefore, it is a more proper model to be used to deal with the case where the coefficient matrix follows a structured characteristic. Unfortunately, the test statistics for detecting the outliers cannot be clearly derived through the existing WTLS method. For this reason, a new two-step iterated approach of computing the WTLS estimates under the framework of LS theory is developed in this paper so that some test statistics of identifying the outliers for the partial EIV model can be constructed.

The remaining of the paper is organized as follows. In Section 2, a two-step iterated method for the partial EIV model taking advantage of LS theory is proposed. In Section 3, the corresponding w-test statistics are constructed to detect the outliers while the observations, coefficient matrix or both are contaminated with the outliers and an algorithm for detecting outliers in the partial EIV model is designed. If the variance factor is not known, we will employ the least median squares (LMS) method to estimate it. In a latter section, a simulated data and a real data about two-dimensional affine transformation are used to verify the validity of the proposed method. In the end, some concluding remarks are presented.

As a matter of fact, not all elements of the coefficient matrix are random and there are some repeated random elements in the different locations of the coefficient matrix such as the coordinate transformation. As a result, their correlations between the repeated random elements must be taken into account. The five rules (Mahboub, 2012) can be used to determine the variance-covariance matrix of the coefficient matrix. However, if the partial EIV model proposed by Xu et al. (2012) is considered, the correlations can be avoided so that the additional burden is reduced. Therefore, the partial EIV model is more superior to be adopted. The function model is shown as following:

(1)

Where X= t×1 vector of unknown parameters; L= n×1 vector of observations; I_n = n×n identity matrix; h= nt×1 vector that is consisting of zero and fixed elements of the coefficient matrix A;B =nt×s known structured matrix; s=the number of different random elements of A = invec(h+Ba); = s ×1 true values vector of a; e = s ×1 random errors vector of a; Δ= n ×1 vector of random errors of observations; invec is a mathematic function for transforming an nt×1 vector to an n × t matrix; =Kronecker product operator. The stochastic model is expressed as follows:

(2)

Where QL= n×n cofactor matrix of L; Qa= s×s cofactor matrix of a; σ2=unknown variance factor.

A two-step iterated method of computing the WTLS estimate for the partial EIV model is proposed in order to develop an outlier detection method suitable for the partial EIV model. For any given X(0), the model (1) can be transformed as follows:

(3)

Furthermore, the model (3) can be rewritten as

(4)

where. If we denote

(5)

the estimate ofcan be derived by the LS principle (Koch, 1999). As a result, we have

(6)

The residual vector of a is

(7)

Insertinginto the first equation of the model (1) yields

(8)

If the inverse transformation of the mathematic operator vec (invec) is used, we can obtain

(9)

Then the model (8) is easily rewritten as follows:

(10)

Similarly, based on the LS principle (Koch, 1999), the estimate of X is

(11)

and the residual vector of L is

(12)

The posterior estimate of the variance factor, which can be obtained from Equation 7 and Equation 12, is

(13)

The data-snooping method suggested by Baarda (1968) is employed extensively in geodetic data processing for detecting the outliers (Kok, 1984; Koch, 1999). If the observations or coefficient matrix in the partial EIV model are contaminated with the outliers, the following w-test statistics can be constructed based on Equation 6 or Equation 11 to detect the outliers:

(14)

(15)

where,,;

andare an unit vector with the ith and jth element equal to 1, respectively; N(0,1) represents the standard normal distribution.

In general, when the variance factor is unknown, its posterior estimatecan be adopted (Pope, 1976). Then we have

(16)

and

(17)

Where τn = τ distribution with n degree of freedom. The computation about τ distribution can be found in Baselga (2007) and Guo and Zhao (2012).

The robust method is an efficient one to estimate the variance factor. By employing the least median squares (LMS) method (Rousseeuw and Leroy, 1987), the variance factor may be estimated by

(18)

or

(19)

So the test statistics (14) and (15) with (18) and (19) become

(20)

and

(21)

The superiority of the above two test statistics is that they are very robust to the outliers so that it is more reliable for them to be used for detecting the outliers. It is to be noted here that they do not strictly follow a normal distribution. Therefore, it is very hard to give the exact probability distributions of them. In order to simplify the computation of the threshold value which is used to identify the outliers, the upper percentage point of the standard normal distribution is still used when the principle of identifying the outliers is established.

The implemented procedure for detecting the outliers in the partial EIV model is summarized as follows:

Step1. Give a,L,h,B,QL,Qe and define.

Step2. Set the initial value.

Step3. For any k, compute

, ,

Step4. Compute.

Step5. Computeand

Step6. If, the iteration will be stopped, whereis a given value. Otherwise, return to Step 3.

Step7. Compute,and .

Step8. According to the data-snooping procedure, for single outlier, if and are satisfied simultaneously, one can judge that the outlier locates in the observation equation containing the observation L_j and coefficient matrix element a_i . For multiple outliers, if and we will deem that the corresponding observation equation containing the observations L_j and coefficient matrix elements a_i is contaminated with outlier. But one still can’t confirm that the outliers locate in the observations or coefficient matrix, or both. Here u_α is the upper α-percentage point of the standard normal distribution.

Step9. If multiple outliers exist in the observations or coefficient matrix, the above procedure of Step 1 to Step 8 should be repeated until all the w-test statistics are smaller than the threshold value.

The WTLS estimate of the partial EIV model may strongly be influenced by the outliers. The aim of this paper is to develop an approach to detect the outliers in the partial EIV model. Firstly, we propose a two-step iterated method of computing the WTLS estimates for the partial EIV model based on the standard LS theory. Then the corresponding w-test statistics are constructed to detect the outliers while the observations, coefficient matrix or both are contaminated with the outliers. If the variance factor is unknown, it may be estimated by the LMS method. Making using of the proposed two-step iterated method, the implement algorithm for detecting the outliers in the partial EIV model is proposed. Through the numerical results with the two-dimensional affine transformation, the identification of outliers is implemented only once through the proposed procedure compared with previously approach while single outlier is considered. For multiple outliers, the repeated test with step by step is suggested. However, we still can’t discriminate that the outliers locate in the observation or coefficient matrix or both, which is a very open problem to be discussed in the future

The authors are grateful to the editor and two anonymous reviewers for their constructive comments and suggestions so that the paper is substantial improved. This research has been supported jointly by the National Natural Science Foundation of China (Grant No. 41174005, 414714009) and State Key Laboratory of Geodesy and Earth’s Dynamics (SKLGED 2017-3-2-E)