1、1數(shù)據(jù)的矩陣描述2Array of Datanpnknnjpjkjjpkpkxxxxxxxxxxxxxxxx2121222221111211x樣本3Descriptive StatisticsSummary numbers to assess the information contained in dataBasic descriptive statisticsSample meanSample varianceSample standard deviationSample covarianceSample correlation coefficient4Sample Mean and Sa

2、mple Variance1221111, 2,nkjkjnkkkjkkjxxnssxxnkp5Sample Covariance and Sample Correlation Coefficientkiikkiiknjkjknjijinjkjkijikkiiikiknjkjkijiikrrsspkpixxxxxxxxsssrxxxxns, 2, 1;, 2, 111212116Standardized Values (or Standardized Scores)Centered at zeroUnit standard deviationSample correlation coeffic

3、ient can be regarded as a sample covariance of two standardized variableskkkjksxx7Properties of Sample Correlation CoefficientValue is between -1 and 1Magnitude measure the strength of the linear associationSign indicates the direction of the associationValue remains unchanged if all xjis and xjks a

4、re changed to yji = a xji + b and yjk = c xjk + d, respectively, provided that the constants a and c have the same sign 8Arrays of Basic Descriptive Statistics111,2122111221222211121121ppppppppppnprrrrrrsssssssssxxxRSx9Random Vectors and Random MatricesRandom vectorVector whose elements are random v

5、ariablesRandom matrixMatrix whose elements are random variables10Expected Value of a Random Matrix111212122212()()()()()()( )()()()()()()()( )ijppnnnpijijijijijijijijall xE XE XE XE XE XE XEE XE XE Xx fx dxE Xx pxEEXAXBAX B11Population Mean Vectors12122212Random vector Joint probability density func

6、tion ( )(,)Marginal probability distribution ()()()( )ppiiiiiipXXXff x xxf xE XE XEXxX12CovarianceikxallxallkiikkkiikikiikkkiikkiikiikxxpxxdxdxxxfxxXXEXX ),()(),()()(),Cov(13Statistically Independentgeneral)in not true is converse (thetindependen are , if 0),Cov()()()(),()()()( and22112112,kikippppk

7、kiikiikkkiikkiiXXXXxfxfxfxxxfxfxfxxfxXPxXPxXxXP14Population Variance-Covariance MatricesppppppppppXXXXXXEE21222211121122112211)Cov()(XXX15Population Correlation Coefficients1212222111211iikkiiikikpppppp16Standard Deviation Matrix2/12/12/12/122112/1000000VVVVVpp17Correlation Matrix from Covariance Ma

8、trix15/15/15/116/15/16/115/10003/10002/150003000200000025323912142/12/12/13322112/1333231232221131211VVVV18Partitioning Covariance Matrix)()()()()(,)2()2()2()2()1()1()2()2()2()2()1()1()1()1()1()1()2()1()2()1(11XXXXXXXXXXXXXpqqXXXX19Partitioning Covariance Matrix12211,1, 11, 1, 11 , 11,111, 111122211

9、211|)(XXppqppqppqqqqqqqpqqqqqpqqE20Linear Combinations of Random Variables)Cov( and )( where)Var( ariance v)(mean hasn combinatioLinear 11XXccXccXcXcEEXcXcpp21Example of Linear Combinations of Random VariablesccXcVarcXcXc)()(,2)()()()()Var()()()(221212112112222112222112212121212121babaEXXbaabbaXbXaE

10、babXaXEbXaXbaXbEXaEbXaXE22Linear Combinations of Random Variables)Cov()()(212222111211CCCXCCXZCXXZXZXZEEcccccccccpppppp23Sample Mean Vector and Covariance Matrixnjpjpnjpjpjnjpjpjnjjppppnpxxnxxxxnxxxxnxxnssssxxx1211111112111111211111,Sx24Partitioning Sample Mean Vector)2()1(11xxxpqqxxxx25Partitioning

11、 Sample Covariance Matrix12211,1, 11, 1, 11 , 11,111, 111122211211|SSSSSSSppqppqppqqqqqqqpqqqqqpqqnssssssssssssssss26Population and Sample總總 體體（隨機(jī)變量或向量）（隨機(jī)變量或向量）分布分布統(tǒng)計量統(tǒng)計量（隨機(jī)變量或向量）（隨機(jī)變量或向量）分布分布數(shù)數(shù) 據(jù)據(jù)推推 導(dǎo)導(dǎo)計計 算算目目 標(biāo)標(biāo)橋橋 梁梁出發(fā)點出發(fā)點21212222111211nnpnnppXXXXXXXXXXXXX),()()()()(2121jpjjjnxxxfffffxxxx) of esti

12、matepoint unbiasedan as 1( )1(1)(,1)Cov() of estimatepoint unbiasedan as (,)(then,matrix covariance and r mean vecto hason that distributijoint a from sample random a are ,21SSSSXXXXXXnnnnnnnnEnnEnE1)()Cov(111)()(1)(1)(1)111()(112112112121XXXXXXXXXXXXXXXXXXnjnjnjnjnnjjnnEnEnnnEnEnEnnnnEE22111111()()0for because of independence.111Cov( )()()1()()()()jnjjjnnjjjnjjjnnnjjjjjjjjjjjjEjEnnnnEEnnEEX XXX XSXXXXXXXXXX XXX XX XXXX XXX111()()1()()1111()()11nnjjjnnjjjEEnEEn