最小二乘曲線擬合及MATLAB實現(xiàn)

摘  要

介紹曲線擬合的基本理論，對最小二乘原理進行了全方位的理論闡述，同時也闡述了曲線擬合的基本原理及多項式曲線擬合模型的建立。詳細的解答了曲線擬合中的最小二乘法，并介紹了部分的正交最小二乘法理論。重點講解多項式擬合的具體步驟，同時也介紹了非線性方程的最小二乘擬合，在建立理論的基礎(chǔ)上對最小二乘曲線擬合法的MATLAB實現(xiàn)方法進行研究，利用MATLAB2012b的平臺對測量數(shù)據(jù)進行最小二乘曲線擬合，介紹MATLAB的具體構(gòu)造和曲線擬合工具。利用MATLAB中的ployfit函數(shù)對實測數(shù)據(jù)進行多項式曲線擬合，并給出曲線擬合MATLAB實現(xiàn)的源程序，給出擬合曲線，并評定擬合的精度證明該方法是行之有效的。

關(guān)鍵詞：最小二乘法，曲線擬合，MATLAB，測量數(shù)據(jù) 

Curve Fitting in Least-Square Method

and Its Realization with Matlab

Abstract

   To introduce the basic theory of curve fitting and discuss the least squares principle in this paper, what’s more, we also discuss the basic principle of curve fitting and the establishment of polynomial curve fitting model. Meanwhile, we also introduce the least-square method of curve fitting in detail and part of the theory of orthogonal least square method. We mainly discuss the specific steps of polynomial fitting, and also introduces the nonlinear equation of the least squares fitting at the same time, which established on the theory of least squares curve fitting in MATLAB in order to realize the method to do research. Using MATLAB2012b platform to achieve the goal of measuring data and introducing the special structure of MATLAB and curve fitting tool. We can use ployfit function in MATLAB to polynomial curve fitting of experimental data, and get the MATLAB source program about curve fitting and the fitting curve. Finally, we need to prove the method of assessing the precision of the fitting is effective.

Key words: least square method;  curve fitting;  MATLAB, metrical data

最小二乘曲線擬合及MATLAB實現(xiàn) I

摘  要 I

CURVE FITTING IN LEAST-SQUARE METHOD AND ITS REALIZATION WITH MATLAB II

ABSTRACT II

第一章 引 言 1

1.1研究背景 1

1.1.1 歷史理論原理 1

1.1.2 現(xiàn)代研究 1

1.2 問題定義 2

1.2.1 曲線擬合的思想 2

1.2.2 多項式擬合 3

1.2.3 利用Matlab的polyfit函數(shù)進行多項式擬合 3

1.3 論文結(jié)構(gòu) 3

第二章 數(shù)據(jù)曲線擬合 4

2.1測量數(shù)據(jù) 4

2.2擬合模型 4

2.3最小二乘原理 5

2.3.1最小二乘法 5

2.3.2最小二乘估計與極大似然估計 7

2.4數(shù)據(jù)擬合 9

2.4.1曲線擬合理論 9

2.4.2最小二乘法線性擬合原理 10

2.4.3最小二乘非線性擬合 12

2.4.4正交多項式 13

2.4.5正交最小二乘曲線擬合 15

2.5曲線擬合精度評定 17

第三章MATLAB 19

3.1MATLAB概述 19

3.1.1MATLAB簡介 19

3.1.2MATLAB的主要組成部分 21

3.2MATLAB2012B的運行簡介 23

3.2.1啟動和退出MATLAB2012b 23

3.2.2MATLAB2012b桌面系統(tǒng) 24

3.2.3MATLAB函數(shù)調(diào)用系統(tǒng) 26

3.2.4MATLAB2012b的幫助系統(tǒng) 27

3.2.5附件管理系統(tǒng) 28

3.2.6數(shù)據(jù)交換系統(tǒng) 28

3.2.7MATLAB 中的其他系統(tǒng) 29

3.3最小二乘曲線擬合法的MATLAB實現(xiàn) 30

第四章 最小二乘法曲線擬合的MATLAB實現(xiàn) 32

4.1 使用POLYFIT函數(shù)實現(xiàn)多項式擬合 32

4.2 二次多項式的曲線擬合 33

4.3三次多項式的曲線擬合 34

4.4 四次多項式曲線擬合 35

4.5數(shù)據(jù)處理和精度評定 36

第五章 總結(jié) 40

參考文獻 41

附錄1： 43

MATLAB語言編程源代碼 43

附錄2： 45

各次擬合的擬合曲線方程 45

致謝 46

外文翻譯 47

外文部分 47

翻譯部分 54