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1、第32卷第1期作物学报Vol32,No12006年1月1~6页ACTAAGRONOMICASINICApp1-6Jan,2006回归分析中的病态矩阵及其改进莫惠栋(扬州大学数量遗传研究室,江苏扬州225009)-1摘要:在回归分析中,信息矩阵XX的行列式值det(XX)如果近于0,就会造成其逆阵(XX)的极度膨胀,进而大大增加回归系数的误差均方,影响回归配合的稳健性和精确度。因而det(XX近于0的XX被称为病态矩阵!。本文提出以X变数的相关矩阵R的行列式值为综合指标,当det(R)在区间[-001,001]和[-00001,00
2、001]但非0时,可分别认为其对应的XX是病态的!和严重病态的!。XX的病态源于X矩阵的高度列依赖,可用简单相关系数、多重决定系数和状态指数度量其列依赖程度。为了改进或消除XX的病态,建议选用(1)简化原回归模型,(2)增加新的资料,-(3)对回归系数添加限制条件,(4)采用诸如脊回归、广义逆M回归等非常规回归程序。简要讨论了病态诊断的重要性和病态改进的评价。关键词:回归分析;病态矩阵;病态的诊断和改进+中图分类号:Q332;S114IllconditionedMatrixandItsImprovementinRegressionMOHuiDong(Labor
3、atoryofQuantitativeGenetics,YangzhouUniversity,Yangzhou225009,Jiangsu,China)-1Abstract:Inregressionanalysis,theinformationmatrixXXisanimportantfactorbecauseofb=(XX)XY.If-1thedeterminantvalueofXX,det(XX),isclosetozero,theinverseoftheXX,(XX),willextremelyinflate,theerrormeansquarefo
4、rregressioncoefficientwilllargelyincrease,andinconsequencetheregressionfittingwillbepoorrobustnessandlowprecision.ThusthematrixXXofdet(XX)∀0iscalledillconditionedmatrix!.InthispaperthedeterminantvalueofcorrelativematrixRofXvariables,det(R),isusedasasyntheticindexforillconditioning,i.
5、e.ifthedet(R)liesintheintervals[001,001]and[00001,00001]butnonzero,thecorrespondingmatrixXXcanberegardedasillconditionedandseriouslyillconditioned,respectively.TheillconditionedXXresultsfromthelineardependencyamongcolumnsinXmatrix.Threediagnosticcriteria,includinglinearcorrelati
6、oncoefficient,multipledeterminationcoefficientandconditionindex,canmeasurethedegreeofthecolumndependency.InordertoimproveoreliminatetheillconditioningofXX,fourmethods,i.e.(1)toreducetheoriginalregressionmodel,(2)tocollectthenewdata,(3)toaddtherestrictiveconditionforregressioncoefficien
7、tsand(4)toadoptthenoncustomaryregressionproceduresuchasthe-ridgeregressionandthegeneralizedinverseMregression,aresuggested.Theimportanceofdiagnosingtheillconditioningandtheevaluationforimprovedillconditioningarealsodiscussedbriefly.Keywords:Regressionanalysis;Illcondi
