An adaptive moving total least squares method for curve fitting
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The document presents the adaptive moving total least squares (amtls) method for curve fitting, highlighting its advantages over traditional moving least squares (mls) and moving total least squares (mtls) methods. It outlines a procedure for determining the parameter λ in the amtls method, which enhances fitting accuracy, and includes various results and tables demonstrating performance comparison of the methods. Ultimately, the amtls method is shown to be more adaptable and provides better solutions for curve fitting tasks.
![Adaptive moving total least squares (AMTLS) METHOD :
The following procedure is carried out in this paper to determine the parameter k in
AMTLS method:
• Step 1: Add the known random error (di, ei) to the measured data (xim, yim)
for getting tested data (xit, yit).
• Step 2: k is set to 0.
• Step 3: Fit the tested data (xit, yit) using AMTLS for getting fitting value (xif,
yif).
• Step 4: According to Ref. [18], calculate the fitting errors of the measured
value yim and fitting value yif by
S = summation of (i=1 to n) | yim- yif | and record the value of s.
• Step 5: Vary k with an increment of certain step h.
• Step 6: Repeat Step 3–Step 5 until s reaches the minimum value.
• Step 7: Record k corresponding to the minimum value of s.
• Step 8: Repeat Step 1–Step7 for r times.
• Step 9: Average the recorded values of k and take it as the final value for λ in
moving total least squares (AMTLS).](https://image.slidesharecdn.com/cmme170050750006-180320035921/85/An-adaptive-moving-total-least-squares-method-for-curve-fitting-4-320.jpg)
An adaptive moving total least squares method for curve fitting
- 1. A Project Presentation on “An adaptive moving total least squares method for curve fitting” MECHANICAL DEPARTMENT Prepared By:- Kaumil Shah -- 170050750006 (PG – AMS) Guided By:- Prof. RAVI K BRAR
- 2. •The moving least square (MLS) approximation is an important method to form the shape function in the meshless methods •It is evolved from the ordinary least squares method, and the corresponding meshless method, in which the shape function is obtained with the moving least square approximation, can achieve a very precise solution •The MLS approximation has been largely documented in the literature and used by many scholars for optimization problems •As a fitting technology, MLS is one way for building the regression model •MLS method for the numerical solution has been applied especially Introduction
- 3. in the engineering literatures for the prediction of machined surface quality and directed projecting onto point clouds. •This method starts with a weighted least squares formulation for an arbitrary fixed point, and then move this point over the entire parameter domain, where a weighted least squares fit is calculated and evaluated for each measured point individually •In this paper, a curve fitting approach called adaptive moving total least squares (AMTLS) method is proposed for EIV model considering the direction of local approximants
- 4. Adaptive moving total least squares (AMTLS) METHOD : The following procedure is carried out in this paper to determine the parameter k in AMTLS method: • Step 1: Add the known random error (di, ei) to the measured data (xim, yim) for getting tested data (xit, yit). • Step 2: k is set to 0. • Step 3: Fit the tested data (xit, yit) using AMTLS for getting fitting value (xif, yif). • Step 4: According to Ref. [18], calculate the fitting errors of the measured value yim and fitting value yif by S = summation of (i=1 to n) | yim- yif | and record the value of s. • Step 5: Vary k with an increment of certain step h. • Step 6: Repeat Step 3–Step 5 until s reaches the minimum value. • Step 7: Record k corresponding to the minimum value of s. • Step 8: Repeat Step 1–Step7 for r times. • Step 9: Average the recorded values of k and take it as the final value for λ in moving total least squares (AMTLS).
- 5. 0.00001 0.001 0.4994485 0.4994527 0.4994432 0.0001 0.001 0.4997744 0.4997986 0.4997637 0.0005 0.001 0.5005451 0.5005579 0.5005358 0.001 0.001 0.5001428 0.5002001 0.5001057 0.005 0.001 0.5488532 0.5489189 0.5485946 0.001 0.0001 0.4872520 0.4872455 0.4871978 0.005 0.0001 0.5360100 0.5358991 0.5355786 0.001 0.00001 0.4903655 0.4903647 0.4903223 0.005 0.00001 0.5517421 0.5517160 0.5514604 0.005 0.000005 0.5258014 0.5257660 0.5254073 Variance Δi εi MLS MTLS AMTLS So • The fitting accuracy of MTLS changes from worse than that of MLS to better than that of MLS when the value di/ei increases, whereas the fitting accuracy of MLS is always better than that of MTLS in table 1. • The fitting results of MLS and MTLS show complicated phenomenon. So it is difficult to choose the proper fitting method, MLS or MTLS, for the better result. Table - 1
- 6. The fitting curve of Adaptive moving total least squares
- 7. Variance Δi εi MLS MTLS AMTLS So 0.00001 0.001 0.157699073 0.157699394 0.157698970 0.0001 0.001 0.158111890 0.158112201 0.158111843 0.0005 0.001 0.157346612 0.157346908 0.157346532 0.001 0.001 0.155128267 0.155128590 0.155427299 0.005 0.001 0.158256446 0.158256741 0.158256204 0.001 0.0001 0.155630269 0.155630530 0.155630164 0.005 0.0001 0.155198790 0.155199051 0.155198721 0.001 0.00001 0.155377381 0.155377646 0.155377326 0.005 0.00001 0.155332683 0.155332940 0.155332654 0.005 0.000005 0.155359226 0.155359488 0.155359226 Table - 2
- 8. The fitting curve of Adaptive moving total least squares
- 9. Variance Δi εi MLS MTLS AMTLS So 0.015 0.0005 1.41793895 1.41840855 1.41734746 (λ = 1.62) •The results of Tables 1 and 2 show that the direction of local approximants is also influenced by the weight function with compact support,which defines the influence domain of x and attributes a weight to each discrete point. In AMTLS, the direction of the local approximants can be determined by the parameter λ based on the above-mentioned procedure. •So the fitting accuracy of AMTLS is always better than those of MLS and MTLS as shown in Tables 1 and 2. •However, the extra time is required for determining the parameter k in AMTLS. •In Example 1, when n = 160, d= 0.3/10, h= 0.01 and r = 100, the time required for determining λ is about 2 h. Table - 3
- 10. • The repetitive positioning error of X axis is about 0.015 mm and the repetitive error of the sensor is about 0.0005 mm. • MLS,MTLS and AMTLS are all applied to fit the measured data. Figure shows the fitting curve of AMTLS for measured data. •The results for three methods are listed in Table 3. •Table 3 indicates that the fitting accuracy of AMTLS is better than those of MLS and MTLS when the direction of the local approximants is determined by λ,where λ = 1.62.
- 11. The fitting curve of Adaptive moving total least squares
- 12. Conclusions •The advantage of the moving least squares and moving total least squares approximation is to obtain the shape function with higher order continuity and consistency by employing the basis functions with lower order and choosing a suitable weight function with compact support. •This paper presents a curve fitting method called Adaptive Moving Total Least Squares (AMTLS) method, Different from moving least squares and moving total least squares method, a parameter λ is introduced to determine the direction of local approximation in Adaptive Moving Total Least Squares (AMTLS) method & Considering the practical need for measurement, a procedure is given to obtain the parameter λ. moving least squares and moving total least squares can be regarded as the special cases of Adaptive Moving Total Least Squares (AMTLS). Adaptive Moving Total Least Squares AMTLS method can be converted to MLS method and MTLS method by setting different values of parameter λ. •The curve fitting results by AMTLS method for the discrete points generated by numerical simulation or obtained by measurement are compared with those
- 13. by moving least squares (MLS) and moving total least squares (MTLS) method on the same condition. •Adaptive Moving Total Least Squares (AMTLS) method has better adaptability than MLS method and moving total least squares (MTLS) method for curve fitting,which confirms the validity of the proposed AMTLS method for curve fitting, which confirms the validity of the proposed Adaptive Moving Total Least Squares (AMTLS) method. •This paper only considers the direction of local approximants is influenced by the error of variables for certain weight function.