1

数据的矩阵描述

2

Array of Data

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样本

3

Descriptive Statistics

Summary numbers to assess the information contained in data Basic descriptive statistics

Sample mean Sample variance Sample standard deviation Sample covariance Sample correlation coefficient

4

Sample Mean and Sample Variance

( )1

22

1

1

1

1, 2, ,

n

k jkj

n

k kk jk kj

x xn

s s x xn

k p

=

=

=

= = −

=

∑

∑!

5

Sample Covariance and Sample Correlation Coefficient

( )( )

( )( )

( ) ( )

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n

jkjk

n

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n

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=

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,,,2,1;,,2,1

1

1

2

1

2

1

1

!!

6

Standardized Values (or Standardized Scores)

Centered at zero Unit standard deviation Sample correlation coefficient can be regarded as a sample covariance of two standardized variables

kk

kjk

s

xx −

7

Properties of Sample Correlation Coefficient

Value is between -1 and 1 Magnitude measure the strength of the linear association Sign indicates the direction of the association Value remains unchanged if all xji’s and xjk’s are changed to yji = a xji + b and yjk = c xjk + d, respectively, provided that the constants a and c have the same sign

8

Arrays of Basic Descriptive Statistics

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p

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9

Random Vectors and Random Matrices

Random vector Vector whose elements are random variables

!Random matrix

Matrix whose elements are random variables

10

Expected Value of a Random Matrix

11 12 1

21 22 2

1 2

( ) ( ) ( )( ) ( ) ( )

( )

( ) ( ) ( )

( )( )

( )

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11

Population Mean Vectors1 2

1 2

2 2

1 2

Random vector '

Joint probability density function ( ) ( , , , )

Marginal probability distribution ( )( )

( )

( )

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i i

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12

Covariance

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13

Statistically Independent

general)in not true is converse (thetindependen are , if 0),Cov(

)()()(),,,(

)()()(][][] and[

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14

Population Variance-Covariance Matrices

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15

Population Correlation Coefficients

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16

Standard Deviation Matrix

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17

Correlation Matrix from Covariance Matrix

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18

Partitioning Covariance Matrix

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19

Partitioning Covariance Matrix

'1221

1,1

,11,1,11,1

1,1

11,1111

2221

1211

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)')((

ΣΣ

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20

Linear Combinations of Random Variables

)Cov( and )( where')'Var( ariance v

')'(mean has

'n combinatioLinear 11

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21

Example of Linear Combinations of Random Variables

ΣccXcVar

µcXcXc

'][)'(

')'(],['],,['2

)]()([

)]()[()Var(

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1211

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22

Linear Combinations of Random Variables

')Cov()()(

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23

Sample Mean Vector and Covariance Matrix

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24

Partitioning Sample Mean Vector

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25

Partitioning Sample Covariance Matrix

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1,1

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1211

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|||

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SS

SSS

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26

Population and Sample

总 体（随机变量或向量）——分布

统计量（随机变量或向量）——分布

数 据

推 导

计 算

⽬目 标

桥 梁

出发点

27

Random Matrix

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'1

21

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p

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!

28

Random SampleRow vectors X1’, X2’, …, Xn’ represent independent observations from a common joint distribution with density function f(x)=f(x1, x2, …, xp) Mathematically, the joint density function of X1’, X2’, …, Xn’ is

),,,()()()()(

21

21

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n

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29

Random SampleMeasurements of a single trial, such as Xj’=[Xj1,Xj2,…,Xjp], will usually be correlated The measurements from different trials must be independent The independence of measurements from trial to trial may not hold when the variables are likely to drift over time

30

Result 1

) of estimatepoint unbiasedan as 1

(

)1

(

1)(,

1)Cov(

) of estimatepoint unbiasedan as (,)(

then,matrix covariance and r mean vecto hason that distributi

joint a from sample random a are ,,, 21

ΣSS

ΣS

ΣSΣX

µXµXΣ

µXXX

n

n

n

n

nn

nnE

nnE

n

E

−=

=−

−==

=

!

31

Proof of Result 1

( ) ( )

( )( )

( )( )'1)')(()Cov(

'1

'11

)')((

)(1

)(1

)(1

)111

()(

1 12

1 12

11

21

21

µXµXµXµXX

µXµX

µXµXµXµX

µXXX

XXXX

−−=−−=

−−=

"#

$%&

'−""

#

$%%&

'−=−−

=+++=

+++=

∑∑

∑∑

∑∑

= =

= =

==

!!

!!

!!

!

!

n

j

n

j

n

j

n

j

nn

jj

n

n

En

E

n

nn

En

En

En

nnnEE

32

Proof of Result 1

( )( )

( )( )

( ) ( )

( )( )

2 21

1

1

' '

1 1 1

'

( )( ) ' 0 for because of independence.

1 1 1Cov( ) ( )( ) '

1( ) ( ')

'

' '

( ) ( ') '

j

n

j jj

n

n j jj

n

j jj

n n n

j j j j jj j j

j j j j

E j

E nnn n

E En

n

E E

=

=

=

= = =

− − = ≠

= − − = =

= − −

− −

= − − − = −

= − + − + = +

∑

∑

∑

∑ ∑ ∑

Xµ X µ

X X

µ X µ Σ Σ

S X X X X

X X X X

X X X X X X X X XX

X X X

µ µ X µ µ Σ µµ! !

33

Proof of Result 1

( )( )

( )

'

1

1

1( ') ( ') '

1( ) ( ')

1 1 1' '

1 ( )( ) '1 1

n

n j jj

n

n j jj

E En

E E nn

nn nn n nnn n

=

=

= − + − + = +

= −

" # −" #= + − − =$ %$ %& '& '

= = − −− −

∑

∑

XX Xµ µ X µ µ Σ µµ

S X X XX

Σ

µµ Σ µµ Σ

S S X X X X

34

Some Other Estimators

( )( )1

The expectation of the ( , ) entry of 1

( ) ( )1

( ) , ( )

Biases ( ) and ( ) can usually

be ignored if size is moderately large

n

ik ji i jk k ikj

ii ii ik ik

ii ii ik ik

i k th

E s E X X X Xn

E s E r

E s E rn

σ

σ ρ

σ ρ

=

= − − =−

≠ ≠

− −

∑

S