Correlation

(Lírios-Vicent Van Gogh,1889)

setosa versicolor virginica

Iris data

• Fisher’s iris data

S.Length S.Width P.Length P.Width Species

1 5.1 3.5 1.4 0.2 setosa 2 4.9 3.0 1.4 0.2 setosa ………………. 49 5.3 3.7 1.5 0.2 setosa 50 5.0 3.3 1.4 0.2 setosa 51 7.0 3.2 4.7 1.4

versicolor 52 6.4 3.2 4.5 1.5

versicolor …………………. 99 6.2 2.9 4.3 1.3

versicolor 100 5.7 2.8 4.1 1.3

versicolor101 6.3 3.3 6.0 2.5

virginica …………………150 5.9 3.0 5.1 1.8

virginica

Scatter-plot matrix

Sepal.Length

2.0 3.0 4.0 0.5 1.5 2.5

4.55.5

6.57.5

2.0

3.0

4.0

Sepal.Width

Petal.Length

1234567

0.5

1.5

2.5

Petal.Width

4.5 5.5 6.5 7.5 1 2 3 4 5 6 7 1.0 2.0 3.0

1.0

2.0

3.0

Species

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

2.0

2.5

3.0

3.5

4.0

Sepal.Length

Sepal.Width

setosa

versicolorvirginica

Scatter plot (by group) and Trendlines

4.5 5.0 5.5

2.5

3.0

3.5

4.0

Sepal.Length

Sepal.Width

Scatterplot for setosa of iris data

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90

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negative relationship positive relationship

no apparent relationship

How to quantify the relationship ?

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count pairs

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-10 -30 0 30 10

count positive pairs

X

Y

XX YY

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50 70 40 10 30

-30 -20 0 20 30

10 30 0 -30 -10

count negative pairs

X

Y

XX YY

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50

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90

300 500 700 9003

05

07

09

0

Need to consider scale matters

-10, -2, 3, 5, 7, 9

5, -7, 10, -3, 8, 5

Maximize the sum of products of each pair.

-10, -2, 3, 5, 7, 9

-7, -3, 5, 5, 8, 10

positively matched, negatively matched

-10, -2, 3, 5, 7, 9

10, 8, 5, 5, -3, -7

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30 10 40 70 50

-30 -20 0 20 30

-10 -30 0 30 10

X

Y

XX YY

n

iii yyxx

nYXCov

1

* ))((1

1),(

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90

-

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90

0

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90

+

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-10 -30 0 30 10

Need to consider scale

XX YY

XX YY

n

iii yyxx

nYXCov

1

* ))((1

1),(

-300 -200 0 200 300

-20 -60 0 60 20XX YY

n

iii yyxx

nYXCov

1

* ))((1

1),(

n

iii xxxx

nXVar

1

* ))((1

1)(

)()(

),(),(

**

**

YVarXVar

YXCovYXCorr

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-20 -60 0 60 20

22 XX

YY

),(10),10(),( 1*

1*

2* YXCovYXCovYXCov

)(10)10()( 12

1*

2* XVarXVarXVar

),(),( 1*

2* YXCorrYXCorr

-30 -20 0 20 30

-20 -60 0 60 2011 XX

YY

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50

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90 -1

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05

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09

0 +1

Cauchy-Schwartz inequality

1),(*1 YXCorr

(very strong) negative linear relationship(very strong) positive linear relationship

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Exercise

X

Y

?),(* YXCov

YXYX XYEYXEYXCov )())((),(

Sample version

n

iii yyxx

nYXCov

1

* ))((1

1),(

Population version

)(,)( YEXE YX

2 YX

Population

covariance

1 2 3 Total

1 0 0.25 0 0.25

2 0.25 0 0.25 0.5

3 0 0.25 0 0.25

Total 0.25 0.5 0.25 1

XY X Y XY f

1 2 2 0.25

2 1 2 0.25

2 3 6 0.25

3 2 6 0.25

3 3 9 0

… … … …

0224)())((),( YXYX XYEYXEYXCov

425.0625.02)( XYE

Population

covariance

1 2 3 Total

1 1/3 0 0 1/3

2 0 1/3 0 1/3

3 0 0 1/3 1/3

Total 1/3 1/3 1/3 1

XY

?)())((),( YXYX XYEYXEYXCov

Exercise

1 2 3 Total

1 0 0 1/3 1/3

2 0 1/3 0 1/3

3 1/3 0 0 1/3

Total 1/3 1/3 1/3 1

XY

))((),( YX YXEYXCov

dydxyxfYX YX ),())((

i j

jiYjXi yxfyx ),())((

iiiYiXi yxfyx ),())((

22YX

XYXY

ss

sr

n N

n

iiiXY yyxx

ns

1

))((1

1),())((

1ii

N

iYiXiXY yxfyx

22YX

XYXY

Covariance is a measure of linear association

between two variables. Covariance is not a measure

curved association.

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050

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150

x

y

Covariance is affected by scales of variables,

but correlation is not, except of sign of scale.

Covariance may be any real value,

but correlation is a value only in [-1,1].

cXZ /)(

cYXCovYZCov /),(),(

0,),(

0),,(),(

cYXCorr

cYXCorrYZCorr

YX

XY

YXYX

YXYX

YXCovYXCovZZCov

),(),(

),(

XXX XZ /)(

Correlation is the covariance for standardized variables

YYY YZ /)(

),( YXCorr

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300 500 700 900

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Covariance = 189

Correlation = 0.78

Covariance = ?

Correlation = ?

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50

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90

-90 -80 -70

60

80

10

01

20

Covariance = 189

Correlation = 0.78

Covariance = ?

Correlation = ?

-60

Grouped by Zip Code

Gathering groups with + corr's does not give + corr.

High correlation does not mean that one variable

is the cause of the other.

Correlation is a measure of linear association,

but not a causation.

Correlation and causality

The more STBK stores, the higher will APT price increase ?

The more Starbucks, the higher APT price !

APT prices in Seoul

STBK

APT price

강남구 45 1030

강동구 2 530

중구 24 520

중랑구 0 330

STBK: number of Starbucks stores

APT price: Average APT price by a 1 m2

The more Starbucks, the deeper financial crisis are !

Thank you !!