Optimization method based on minimization m-Order central moments used in surveying engineering problems

is a PDF file of an unedited manuscript that has been publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A new optimization method presented in this work – the Least m-Order Central Moments method, is a generalization of the Least Squares method. It allows fitting a geometric object into a set of points in such a way that the maximum shift between the object and the points after fitting is smaller than in the Least Squares method. This property can be very useful in some engineering tasks, e.g. in the realignment of a railway track or gantry rails. The theoretical properties of the proposed optimization method are analyzed. The computational problems are discussed. The appropriate computational techniques are proposed to overcome these problems. The detailed computational algorithm and formulas of iterative processes have been derived. The numerical tests are presented, in order to illustrate the operation of proposed techniques. The results have been analyzed, and the conclusions were then formulated.


Introduction
Optimization techniques are sometimes applied in engineering tasks. In such applications, the problem is usually formulated as fitting a geometrical figure into a set of points in 2D or 3D space. In contrast to many estimation problems where the solution has to be free of outliers (CASPARY 1990, CHANG, GUO 2005, HAMPEL et al. 1986, HUBER 1981, KAMIŃSKI, WIŚNIEWSKI 1992, KOCH 1996, YANG 1999, YANG et al. 2002, XU 1989, ZHONG 1997, ZHU 1996, the opposite problem is presented here: an optimization method preferring outliers. This feature is beneficial in engineering applications, where constraints concerning maximum shifts appear. In some cases, the maximum shifts cannot exceed a critical value. Such constraints can appear e.g. during the realignment of a railway track (SKAŁA-SZYMAŃSKA et al. 2014). These constraints result from a limited structure gauge, i.e. the width of tunnels, bridges or distances to railway platforms, buildings or other objects. The constraints, mentioned above, have to be taken into account during the construction process when some elements of a structure are installed in a limited space. An example of such civil engineering task is to fit elevator guide rails inside the elevator shaft. One of the tools for solving the problem of limited shifts can be the min-max algorithm from game theory (VON NEUMANN 1947). Also, the Least Squares (LS) method with constraints can be applied (LIEW 1976, MEAD, RENAUT 2010, WERNER 1990). An alternative method is presented here. In the proposed method, the special form of the objective function is applied. This form is related to the m-order central moments (with m ≥ 2). Thus, the proposed method is a generalization of the LS method and the Least Fourth Powers (LFP) method (CELLMER 2014). Special attention must be paid to the optimization technique. So far, a lot of interesting optimization techniques were proposed e.g. (FLETCHER 1987, NOCEDAL, WRIGHT 1999, AVRIEL 2003. It was shown in (CELLMER 2014) that if an inappropriate technique of searching for the minimum of the objective function is applied, then the computational process is not convergent. In such a case, two alternate solutions are obtained in consecutive iterations like in the M-split estimation method (DUCHNOWSKI, WIŚNIEWSKI 2012, WIŚNIEWSKI 2009, 2010. In the M-split estimation method, this effect was obtained deliberatelyit resulted from the theoretical foundations of this method. However, in the method considered here, a single, unique solution is required. The technique of searching for a solution should have the property of skipping local minima and pursue the global one. This property was described e.g. in (MARTINS, TSUZUKI 2009).
In the next section, the objective function of the Least m-Order Central Moments (LmOCM) method is presented. The justification for applying this form of the objective function was carried out using certain concepts of estimation theory and was illustrated using the plot of weight function for different values of

Objective function of the LmOCM method and its properties
The optimization method proposed in this article is based on minimizing the objective function Ψ( ): where are elements of the vector in the simple, linear model: where: yvector of entries that are fitted into the A model, One of the properties of the optimization method can be described by the weight-function (KADAJ 1988).
The form of this function is: The plots of the weight function with the values of the m-parameter: 2, 4, 6, 8, 10 and 12 are depicted in Figure 1. The black line depicts the weight function of the LS method (m = 2). As follows from the definition of the weight function (5), in the case of the LS method, it is constant. All observations are treated equally. (1) In the case of large values of the m-parameter, there is a range around the zero value, where the trajectory of the plot of the function is almost horizontal. Outside this range, the function value rapidly increases. This means that if there are no outliers in the data set, this optimization method provides results similar to the results obtained with the LS method. However, if there are some outliers in the data set, they have more impact on the solution than other observations. The consequence of this is a smaller maximal value of the residuals in the optimization process.

Optimization techniques
The objective function in minimization problem (1) can be presented in the following matrix notation: The solution, which is at the point of a minimum of the objective function (6) can be found by zeroing the gradient of this function. As was shown in (CELLMER 2014), this technique cannot be employed in its simple, classic form. The gradient of the objective function (6) is derived, based on (2) and (7): where diag( m-2 ) denotes a diagonal matrix, containing elements of the m-2 -vector on a diagonal. Thence the system of normal equations is obtained: where: The solution can be obtained in an iterative process: The W matrix is formed according to (11) in each iteration. However, as was shown in (CELLMER 2014), in the case of the objective function (6), process (12) does not work properly. This technique does not ensure proper convergence to the correct, unique solution. As a result of using formulas (12), two alternate solutions are obtained. They appear alternately in consecutive iterations. A similar effect is obtained when 6 using the M-split estimation proposed by (WIŚNIEWSKI 2009(WIŚNIEWSKI , 2010. However, the unique solution of the minimization process of the objective function (6) exists. CELLMER (2014) proposed applying one of two optimization techniques for obtaining a unique solution in the considered problem. The first one is based on the modification of the process (12). The second is based on applying the Newton technique of optimization. In the first technique, the iterative process (12) is modified to the form: where the m-2( ) vector in the last row is formed according to (8). In this technique, in contrast to process (12), the results obtained from consecutive iterations are accepted as the starting point (approximate value of the parameter) in the next iteration. This is performed by updating the dy and p vectors. Additionally, the reduction parameter k is introduced. This parameter reduces the length of every single step in the iterative process of searching for the solution. This was imposed to eliminate the effect of jumping over the global solution in consecutive iterations. The k parameter has to take the value from the range: (0;1).
Determination of the optimal value of this parameter is presented in the next section.
In the second technique, the Newton method of optimization is applied in its classic form: where G and H are, appropriately, the gradient and Hessian of the objective function . The gradient of the objective function (8) is expressed by formula (9). Let us denote the gradient G as a transpose of (9): The Hessian can then be formed as: Thus, in this case the optimization process can be presented as follows: ( ) = Ap ( )y for = 1, 2, … and W (0) = I The correction vector in the first iteration ( (0)) is calculated using formula (12). Table 1 contains the coordinates of the points. The axes are to be fitted into each set of these points using criterion (1). In the first example the formula of the fitted line is:

Numerical examples
whereas in the second example, it is: The A matrix and the y vector in each example are formed based on the following observation equations: Hence, in the first example, the A is the column vector of x-coordinates, whereas in the second example this is a two-column matrix: the first column contains x-coordinates and the second column is a vector of ones. At the start point of optimization, the y-vector contains the y-coordinates of the points listed in Table   1. The first test was performed in order to determine the optimal value of the k-factor in process (13).
Therefore, this process was carried out for different values of k, and afterwards, the speed of convergence was analyzed for each of them. The results are listed in Table 2. It was shown in (CELLMER 2014) that if k = 1, the solution splits into two results, repeating alternately in consecutive iterations. Therefore, the kfactor values have been assumed here as: In process (13), the k-factor has been assumed as: In the case analyzed here, both methods (13) and (18) Table 4.

Summary and Conclusions
In the paper, a generalization of the Least Squares optimization method is proposed. A new method is based on the criterion of minimization of a sum of misclosures raised to the power of 'm'. This criterion allows one to reduce the maximum misclosure in the optimization problem in comparison to the LS method.
A crucial problem in the use of this method is ensuring proper convergence of the computational process.
Two techniques of finding the minimum of the objective function were tested: the simple gradient zeroing method and the Newton method. Both methods employ an iterative process. The formula of the computational process of the first technique contains the k-factor. This factor warrants the proper convergence of the computational process. It is proposed here, that the value of this factor is calculated as k = (m−1) -1 if the sum of misclosures to the power of 'm' is the optimization criterion. In the case study considered here, the two optimization techniques provided the same results. The results of this case study    Table 3. Results of optimization (y = ax). Table contains the value of the a parameter and maximum shift m after fitting axis into a set of points Table 4. Results of optimization (y=ax+b). Table contains the value of the a and b parameters and maximum shift m after fitting axis into a set of points