数学建模中的回归分析法-毕业论文

本科毕业论文题目名称：数学建模中的回归分析法学院：数学与统计学院专业年级：数学与应用数学2009级（精算与风险管理）学生姓名：李雨函班级学号：200911030139指导教师：王艺霏二O一三年五月二十四日I摘我们要现实生活中,由于客观事物内部规律的复杂性及人们认识程度的限制,人们常搜集大量的数据,基于数据的统计分析建立合乎机理规律的数学模型,然后通过计算得到的模型结果来解释实际问题.回归分析法是数学模型中常用解决问题的有效方法.它是研究某个变量关于另一些变量的具体依赖关系的计算方法.主要内容是从一组样本数据出发,确定变量之间的数学关系式对这些关系式的可信程度进行各种统计检验,并从影响某一特定变量的诸多变量中找出哪些变量的影响显著,哪些不显著.利用所求的关系式,根据一个或几个变量的取值来预测或控制另一个特定变量的取值,并给出这种预测或控制的精确程度.本文介绍线性回归模型和非线性回归模型的概念、基本原理和应用步骤,并最后通过实例分析介绍从数据出发建立、检验回归模型的步骤和模型结果中具体每个符号的实际意义.结果表明,在实际生活各个领域,回归分析是很好的预测分析方法.关键字:回归分析;线性回归模型;非线性回归模型IIAbstractInreallife,thecomplexityoftheinternallawofthingsandawarenessofthelimits,peoplecollectedalargeamountofdataandbasedonthestatisticalanalysisofdatatosetupmechanismmodel,andthenthroughthecalculatedmodelresultstoexplainthepracticalproblems.Regressionanalysisiscommonlyusedinmathematicalmodelistheeffectivemethodtosolvetheproblem.Itisthestudyofonevariableonothervariablesdependonthespecificcalculationmethod.Themaincontentfromasetofsampledata,determinethemathematicalrelationshipbetweenthevariablesoftherelationbetweenthecredibledegreeofvariousstatisticaltests,andfromtheinfluenceofaparticularvariablevariablestofindouttheinfluenceofwhichvariablessignificantly,whichwasnotsignificant.Theuseofpetitions,accordingtothevalueofoneorseveralvariablestopredictorcontroltheotherofaparticularvariablevalues,andgivetheaccuratepredictorcontrol.Thispaperintroducestheconceptofthelinearregressionmodelandnonlinearregressionmodel,basicprincipleandapplicationsteps,andfinallythroughtheinstanceanalysisisintroducedfromdatasetup,testingprocedureandmodeloftheregressionmodelresultsinthepracticalsignificanceofthespecificeachsymbol.Theresultsshowthattheregressionanalysisisagoodwaytoforecastanalysis.Keyword:RegressionAnalysis;LinearRegressionModel;NonlinearRegressionModelIII目录中文摘要························································································Ⅰ英文摘要························································································Ⅱ目录························································································Ⅲ1.引言·······················································································12.回归模型的建立·············································································22.1回归分析模型一般形式···································································32.2多元线形回归的模型······································································32.2.1多元线形回归的模型······························································32.2.2多元线形回归的假设······························································32.2.3多元线形回归的求解······························································32.2.4多元线形回归的检验······························································42.3曲线回归模型··········································································52.3.1可化成线形回归的曲线回归·····················································52.3.2不可转化的非线性回归模型·····················································63.回归分析模型的实际应用·································································6致谢························································································11参考文献························································································1211.引言当需要从定量的角度分析和研究一个实际问题时,人们就要在深入调查研究、了解对象信息、做出简化假设、分析内在规律等工作的基础上,用数学的符号和语言,把它表述为数学式子,也就是数学模型,然后通过计算得到的模型结果来解释实际问题,并接受实际的检验[1].1983年，数学建模作为一门独立的课程进入我国高等学校,20多年来,数学建模工作发展的非常快,许多高校相继开设了数学建模课程,我国1992年国家教委高教司提出在全国普通高等学校开展数学建模竞赛,旨在”培养学生解决实际问题的能力和创新精神,全面提高学生的综合素质”.近年来,数学模型和数学建模这两个术语使用的频率越来越高,而数学模型和数学建模也被广泛地应用于其他学科和社会的各个领域.在数学建模中常用的方法有很多种,本文主要介绍最常用的有效方法——回归分析法.回归分析方法是统计分析的重要组成部分,回归分析的主要内容,一是从一组数据出发,确定这些变量间的回归模型;二是对模型的可信度进行统计检验;三是从有关的许多变量中,判断变量的显著性(即哪些是显著的,哪些不是,显著的保留,不显著的忽略);四是应用结果是对实际问题作出的判断.根据回归模型中回归的特征,常见的回归模型有:一元线性回归模型、多元线性回归模型、非线性回归模型.近年来国内外学者应用回归分析法解决了实际中一系列问题.周新宇,孙凡雷在《因素回归分析法在不良债权价值分析中的应用》中对小金额债权采用相关因素回归分析法进行价值分析可以较好地解决金额小且户数众多的资产价值分析.马瑞民,姚立飞在《回归分析在数学建模中的应用--基于上海世博会参观人数的预测分析模型》中对参加世博会参观人数进行预测,与实际相差很小.澳大利亚学者SalinaHishama,CheRozidMamat等人在《马来西亚华人脚部身高测量人体形态学的回归分析》中给出利用人脚的尺寸预测身高的回归模型[2].为了更好的指导回归分析在实际中的应用,本文主要讨论回归分析法的分类和各种建模及其应用.22.回归模型的建立2.1回归分析问题的一般形式设有p个自变量pxxx,,,21和1个因变量y,它们之间有下列关系),,,;,,,(2121pPaaaxxxFy.其中F是函数形式已知的p元函数,paaa,,,21是常数,是函数F中的未知参数,是表示误差的随机变量,一般可认为～),0(2N,0.对Pxxx,,,21,y进行n次观测,得到观测值),,,,(21iPiiiyxxx,ni,,2,1.对每一次观测来说,同样有下列关系ipimiiiaaaxxxFy),,,;,,,(2121,其中),,2,1(nii是第i次观测时的随机误差.回归分析目标是从观测数据出发,求出paaa,,,21的估计paaaˆ,,ˆ,ˆ21,使得下列平方和Q达到最小.nipmiiiiaaaxxxFyQ122121]),,,;,,,([.由于估计的目标是使一个平方和达到最小,而平方又称为“二乘”,所以,这种估计称为最小二乘估计(LSE),求这种估计的方法称为最小二乘法[3].把paaaˆ,,ˆ,ˆ21代入Q表达式,就得到Q的最小值nipmiiiiaaaxxxFyQ122121min])ˆ,,ˆ,ˆ;,,,([.Q的最小值称为残差平方和,残差平方和越小,说明回归方程表达变量之间统计相关关系的精确程度越高,也就是回归分析的效果越好.3【】2.2线形回归模型的建立2.2.1多元线形回归的一般形式设随机变量y与一般变量pxxx21,的线性回归模型为ppxxxy22110,其中，p,,,,210是1p个未知参数，0称为回归常数，p,,,,210称为回归系数.参数称为随机误差;y称为被解释变量(因变量),1x,pxx2是p个可以精确测量并控制的一般变量，称为解释变量(自变量).1p时,该回归模型为一元线性回归模型;当2p时,就称该式为多元线性回归模型.2.2.2多元线性回归模型的基本假设(1)解释变量1x,pxx,,2是确定性变量,不是随机变量,且要求npXrank1)(,并要求样本量的个数应大于解释变量的个数.(2)随机误差项具有零均值和等方差,即niEi,,2,1,0)((3)对于自变量pxxx,,,2