16 August 2017 A simple method for processing data with least square method
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Proceedings Volume 10452, 14th Conference on Education and Training in Optics and Photonics: ETOP 2017; 104523A (2017) https://doi.org/10.1117/12.2270972
Event: 14th Conference on Education and Training in Optics and Photonics, ETOP 2017, 2017, Hangzhou, China
Abstract
The least square method is widely used in data processing and error estimation. The mathematical method has become an essential technique for parameter estimation, data processing, regression analysis and experimental data fitting, and has become a criterion tool for statistical inference. In measurement data analysis, the distribution of complex rules is usually based on the least square principle, i.e., the use of matrix to solve the final estimate and to improve its accuracy. In this paper, a new method is presented for the solution of the method which is based on algebraic computation and is relatively straightforward and easy to understand. The practicability of this method is described by a concrete example.
Chunyan, Liqun, Yongxiang, and Guangning: A simple method for processing data with least square method

1.

INTRODUCTION

In order to obtain the most reliable results, the number of measurements is often more than the numbers of unknown parameters, which means that the number of residual equations is more than the number of unknown numbers. The method of general solving algebraic equations cannot be used to solve these problems. The least squares method convert the residual equation into an algebraic equation with a definite solution and the unknown parameters can be solved. This algebraic equation with definite solution is called the normal equations of least squares estimation. In this paper, we build the basis of linear algebra, and propose another method for solving the least squares estimator.

2.

THE INTRODUCTION OF THE LEAST SQUARES PRINCIPLE

The least square method is a method that estimates the regression equation using the sample data according to the least squares criterion.

Suppose Li is the i-th sample observed value and is the corresponding i-th sample best estimate. The residual between Li and is denoted by ei.

The criteria that apply the quadratic sum of the residual of all the observed values 00107_PSISDG10452_104523A_page_1_1.jpg is minimized[4], 00107_PSISDG10452_104523A_page_1_2.jpg to determine the estimates of the unknown parameter x1, x2, x3,…, xk. And the the least squares criterion is

00107_PSISDG10452_104523A_page_1_3.jpg

and

00107_PSISDG10452_104523A_page_1_4.jpg

The standard deviation[5] of the direct measurement result is

00107_PSISDG10452_104523A_page_2_1.jpg

The corresponding standard deviation estimate can be obtained according to the equation

00107_PSISDG10452_104523A_page_2_2.jpg

where djj(j = 1,2,3,4,…) is the value of diagonal of (AT A)−1 obtained above.

3.

Two methods of calculating xi and σxi

3.1

First method

Assume observation equation is:

00107_PSISDG10452_104523A_page_2_3.jpg

where aii (i = 1,2,3)—constant, lj (j = 1,2,3,4)—measurement result, which is also a constant. List the coefficient matrix A and the measurement result matrix L [3]

00107_PSISDG10452_104523A_page_2_4.jpg
00107_PSISDG10452_104523A_page_2_5.jpg

Eq.(1) shows that AT A and AT L should be known, then determine the inverse matrix C of AT A:

00107_PSISDG10452_104523A_page_2_6.jpg
00107_PSISDG10452_104523A_page_2_7.jpg

The least square estimator of are obtained by multiplying C by AL.

3.2

Second method

First, make a table according to the observation equation, where aii (i = 1,2,3,4) and lj (j = 1,2,3,4) are all four numbers of the corresponding columns of the constants Ai(i = 1,2,3) and Y, which are given by the four formulas above are assumed (Note: They are not a vector).

Next, make another form firstly. Ai × Aj in first row represents the four numbers, which multiply the corresponding the four numbers that are represented by Ai and Aj, respectively. For example, A1 × A1 means the number of a11 × a11a21 × a21a31 × a31a41 × a41. And ∑i(i = 1,2,3,…) in last row represent the sum of the first four digits of the corresponding column.

Table 1

Calculation table of normal equation coefficient

A1 × A1A1 × A2A1 × A3A2 × A1A2 × A2A2 × A3A3 × A1A3 × A2A3 × A3A1 × YA2 × YA3 × Y
a11 × a11a11 × a12          
a21 × a21           
a31 × a31           
a41 × a41           
∑ 1∑ 2∑ 3∑ 4∑ 5∑ 6∑ 7∑ 8∑ 9∑ 10∑ 11∑ 12

Then the last line of the table three per group divided into four groups and the three equations are written

00107_PSISDG10452_104523A_page_3_1.jpg

Third, the least squares estimator could be calculated by solving the equations obtained in the second step.

Prove: The matrix X is grouped

00107_PSISDG10452_104523A_page_3_2.jpg

So 00107_PSISDG10452_104523A_page_3_3.jpg (a is a column vector of 4 × 1, b is a row vector of 1 × 4.), and so

00107_PSISDG10452_104523A_page_3_4.jpg
00107_PSISDG10452_104523A_page_3_5.jpg

According to the basic characteristics of transposed and matrix

00107_PSISDG10452_104523A_page_3_6.jpg

If ∑1 = a1b1, the sum of the above table corresponds to the elements in the XT X and XT Y matrices. So the finally corresponding equation set in the third step is:

00107_PSISDG10452_104523A_page_3_7.jpg

Forth, the standard deviation S of the measured value is also obtained first, and the value of djj is calculated by the equation set obtained by the above form

00107_PSISDG10452_104523A_page_3_8.jpg

And the value of C1 obtained by the above formula is the value of d11. Similarly, replace the value of the right side of the equation by 0 1 0 when calculating the value of d22, and replace the value of the right side of the equation by 0 0 1

when calculating the value of 00107_PSISDG10452_104523A_page_4_1.jpg

Finally, the corresponding standard deviation is obtained by 00107_PSISDG10452_104523A_page_4_2.jpg

Prove: In the calculation of the matrix of the inverse matrix, the defining equation is CC−1 = E, from the above we already know

00107_PSISDG10452_104523A_page_4_3.jpg

(This is a symmetric matrix, and its inverse matrix is symmetric matrix as well.) Assume its inverse matrix is 00107_PSISDG10452_104523A_page_4_4.jpg, then E is a unit matrix of 3 × 3. By matrix division, whenC is multiplied by the first column of C−1, the resulting value is the first column of E,

00107_PSISDG10452_104523A_page_4_5.jpg

The values of t1、 t2、 t3 are the diagonal of the inverse matrix, that is, the value of djj which we need.

4.

Conclusion

As can be seen from the above two proofs, the last value obtained by the second method is exactly the same as the matrix algorithm. The first method of the least squares calculation given above is entirely based on the solution of the matrix, but the second method is more use of our more familiar algebra method. In the calculation of large amount of data, usually based on the guidance of the first method using MATLAB[1]- [2] and other computer software to solve the final value, when you encounter small data, or in the answer needs, the second method if you master, it will greatly reduce the difficulty of the calculation, and not easy to miscalculate, very practical. Finally, I would like to offer this document to teachers and students puzzled by the least squares of the matrix algorithm.

REFERENCES

[1] 

CHEN Lanfeng, YANG Jingyu. [MATLAB simulation of curve fitting based onleast-squares] Journal of Shenyang Normal University (Natural Science Edition), (2014)Google Scholar

[2] 

XU Yitang, [Curve Fitting Based on Least Squares Method and Its Application in MATLAB] Jiangsu University of Science and Technology, School of Electronic and Information (2012)Google Scholar

[3] 

WANG Peng, “Strategy of Stimulating Studying Interest for Professional Course Teacher in Food Science and Engineering,” Journal of Anqing Teachers College (Natural Science Edition), 21(1) (2015)Google Scholar

[4] 

YUE Kui, “Programming for Evaluation of Roundness Error by Least Square Mean Circle Method,” Journal of Hefei University of Technology,4, (2006)Google Scholar

[5] 

LIU Zhi-ping, SHI Lin-ying, “The Principle of Least Square Algorithm and its Achievement by MATLAB,” 33–34,7(19),(2008)Google Scholar

© (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Chunyan Wang, Chunyan Wang, Liqun Qi, Liqun Qi, Yongxiang Chen, Yongxiang Chen, Guangning Pang , Guangning Pang , } "A simple method for processing data with least square method", Proc. SPIE 10452, 14th Conference on Education and Training in Optics and Photonics: ETOP 2017, 104523A (16 August 2017); doi: 10.1117/12.2270972; https://doi.org/10.1117/12.2270972
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