Ming-Feng Yeh 3-1

2. Grey Relational Analysis2. Grey Relational Analysis

x

k

x1

x2

x3

Ming-Feng Yeh 3-2

x0={x0(1), x0(2),…, x0(n)}: reference sequence

xi={xi(1), xi(2),…, xi(n)}: comparative sequence

i = 1,2,…,m.

Grey relational coefficient: (x0(k), xi(k))

(x0(k), xi(k)) = [min+max] [0i(k) +max]

0i(k)=x0(k) xi(k), : distinguish coefficient

max=maxi maxkx0(k) xi(k), 0 1,

min=mini minkx0(k) xi(k).

2.1:2.1: Grey Relational Analysis Grey Relational Analysis

Ming-Feng Yeh 3-3

Grey Relational AnalysisGrey Relational Analysis

Grey relational grade: (x0, xi)

0 (x0(k), xi(k)) 1.

0 (x0, xi) 1.

Describes the posture relationships between one main factor (reference series) and all other factors (comparison series) in a given system.

1 ,))(),((),( 11 00 nk k

nk iki wkxkxwxx

Ming-Feng Yeh 3-4

Axioms of GRAAxioms of GRA

Norm Interval

(x0(k), xi(k))(0,1], k.

(x0(k), xi(k)) = 1, iff x0(k) = xi(k), k.

(x0(k), xi(k)) = 1, iff x0, xi .

Duality Symmetric

(x0(k), xi(k)) = (xi(k), x0(k)) , iff X = {x0, xi}.

Ming-Feng Yeh 3-5

Axioms of GRAAxioms of GRA

Wholeness

(x0(k), xi(k)) (xi(k), x0(k)) almost always,

iff X = {xj j = 0,1,…,m, m 2}.

Approachability (x0(k), xi(k)) decreases along with (k) increasing,

where (k) = [(x0(k) xi(k))2]1/2 =x0(k) xi(k).

Ming-Feng Yeh 3-6

2.2:2.2: Grey Generating Space Grey Generating Space

Based on the concept and generating schemes of grey system theory, the disorderly raw data canbe turned to a regular series for grey modeling.be transferred to a dimensionless series for grey analyzing.be changed into a unidirectional series for decision making.

Ming-Feng Yeh 3-7

灰關聯因子集灰關聯因子集假設 X 為序列 xi = {xi(1), xi(2),…, xi(n)}, 其中 i = 1,2,…, m, 所構成之集合。若 P(X) 為一灰關聯因子集，則 xi P(X) 。為使序列具有可以比較之特性，以利灰關聯分析的進行，則序列 xi必須滿足下列三個條件：無因次性 (Normalization) ：不論因子 xi(k) 之測度單位為何，必須經過處理使其成為無因次性（去除單位）。同等級性 (Scaling) ：各序列 xi中之 xi(k) 值均屬同等級或等級相差不大（等級相差不超過 2 ）。同級性 (Polarization) ：序列中的因子描述應為同方向。

Ming-Feng Yeh 3-8

Grey Generating OperationsGrey Generating Operations

An original sequence x = {x(1), x(2),…, x(n)}

The generating sequence y = {y(1), y(2),…, y(n)}Initializing operation: y(k) = x(k) x(1)

Averaging operation: y(k) = x(k) xave,

Maximizing operation: y(k) = x(k) xmax

Minimizing operation: y(k) = x(k) xmin

Intervalizing operation:

y(k) = [x(k) xmin] [xmax xmin]

n

kave kx

nx

1

)(1

Ming-Feng Yeh 3-9

Example 2.1Example 2.1

x = {4, 2, 6, 8}; xave= 5, xmax= 8, xmin= 2.

y(1) y(2) y(3) y(4)

Initializing 1.00 0.50 1.50 2.00

Averaging 0.80 0.40 1.20 1.60

Maximizing 0.50 0.25 0.75 1.00

Minimizing 2.00 1.00 3.00 4.00

Intervalizing 0.33 0.00 0.67 1.00

Ming-Feng Yeh 3-10

An original sequence x(0)={x(0)(1), x(0)(2),…, x(0)(n)}, x(0)(k) 0.≧The 1st order AGO (1-AGO): AGO•x(0) = x(1)

The jth order AGO (j-AGO):

Accumulated Generating Accumulated Generating Operation (AGO)Operation (AGO)

)()1()()( )0()1(

1

)0()1( kxkxmxkxk

m

k

m

jj mxkx1

)1()( )()(

Ming-Feng Yeh 3-11

Inverse AGO (IAGO)Inverse AGO (IAGO)

(0)(x(r)(k)) = x(r)(k).

(1)(x(r)(k)) = (0)(x(r)(k)) (0)(x(r)(k1)).

( j)(x(r)(k)) = ( j1)(x(r)(k)) ( j1)(x(r)(k1)).

IAGO•x(1) = x(0) = (1)(x(1))

x(0)(1) ＝ x(1)(1),x(0)(k) ＝ x(1)(k) x(1)(k1) , k ＝ 2,3,…,n

Mean generating operation: z(1)(k) ＝ 0.5[ x(1)(k) ＋ x(1)(k1)], k ＝ 2,3,…,n

- ｒ ＡＧＯ

-ｒ ＩＡＧＯ

)0(x )(rx

Ming-Feng Yeh 3-12

Example 2.2Example 2.2

x(0)={x(0)(1), x(0)(2), x(0)(3), x(0)(4)}={1,2,1.5,3}

x(1)={x(1)(1), x(1)(2), x(1)(3), x(1)(4)}={1,3,4.5,7.5}

1 2 3 4 k

x(0)(k)

1

2

3

1 2 3 4 k

x(1)(k)

1

3

4.5

7.5

Ming-Feng Yeh 3-13

AGO EffectAGO Effect

The non-negative, smooth, discrete function can be transferred into a series, extended according to an approximate exponential law (grey exponential law).

The disorderly raw data can be turned to a regular series for grey modeling.

Ming-Feng Yeh 3-14

Example 2.3Example 2.3

項 目 1 月 2 月 3 月 4 月 5 月 6 月 7 月 8 月 9 月 10月

11月

12月

降雨量( 公釐 )

131.9 155.1 192.2 151.8 207.8 229.3 149.7 212.1 279.4 187.9 142.0 108.7

降雨天數( 天 )

16 15 16 13 15 13 8 10 12 12 14 14

平均氣溫( 攝氏度 )

14.9 15.2 17.2 21.2 24.6 26.9 28.9 28.7 26.9 23.8 20.4 16.8

相對濕度( 百分比 )

82 84 84 83 83 83 79 79 79 78 79 81

Ming-Feng Yeh 3-15

Weather AnalysisWeather Analysis

0

100

200

300

1月 2月 3月 4月 5月 6月 7月 8月 9月 10月

11月

12月

降雨量 降雨天數 平均氣溫 相對濕度

Ming-Feng Yeh 3-16

Weather AnalysisWeather Analysis

Step 1: Data Processing – Initializing

降雨量 : x0={1,1.176,1.457,1.151,1.575,1.738,1.135,1.608,2.118,1.425,1.077,0.824}

降雨天數 : x1={1,0.938,1,0.813,0.938,0.813,0.500,0.625,0.750,0.750,0.875,0.875}

平均氣溫 : x2={1,1.020,1.154,1.423,1.651,1.805,1.940,1.926,1.805,1.597,1.369,1.128}

相對濕度 : x3={1,1.024,1.024,1.012,1.012,1.012,0.963,0.963,0.963,0.851,0.963,0.988}

Ming-Feng Yeh 3-17

Weather AnalysisWeather Analysis

0.00

0.50

1.00

1.50

2.00

2.50

1月 2月 3月 4月 5月 6月 7月 8月 9月 10月

11月

12月

降雨量 降雨天數 平均氣溫 相對濕度

Ming-Feng Yeh 3-18

Weather AnalysisWeather Analysis

Step 2: Compute 0i(k)=x0(k) xi(k) and then Find max and min

01={0,0.238,0.457,0.338,0.638,0.926,0.635,0.983,1.368,0.675,0.202,0.051}

02={0,0.156,0.303,0.272,0.076,0.067,0.805,0.318,0.313,0.173,0.293,0.303}

03={0,0.152,0.433,0.139,0.563,0.726,0.172,0.645,1.155,0.473,0.113,0.164}

max=1.368, min=0

Ming-Feng Yeh 3-19

Weather AnalysisWeather Analysis

Step 3: Find the Grey Relational Coefficients Let =0.5 r(x0(k),x1(k)) r(x0(k),x2(k)) r(x0(k),x3(k))

1 1.000 1.000 1.000

2 0.742 0.815 0.819

3 0.599 0.693 0.613

4 0.699 0.716 0.832

5 0.518 0.901 0.549

6 0.425 0.911 0.485

7 0.519 0.460 0.800

8 0.410 0.683 0.515

9 0.333 0.686 0.372

10 0.504 0.798 0.591

11 0.772 0.701 0.858

12 0.931 0.693 0.807

Ming-Feng Yeh 3-20

Weather AnalysisWeather Analysis

Step 4: Calculate the Grey Relational Grades

Average the grey relational coefficients then

r(x0,x1)=0.619, r(x0,x2)=0.755, r(x0,x3)=0.687

Step 5: Sort the Grey Relational Grades

r(x0,x2) r(x0,x3) r(x0,x1)

Note: x0=降雨量 ,

x1=降雨天數 , x2=平均氣溫 , x3=相對濕度

Ming-Feng Yeh 3-21

Example 2.4Example 2.4

Data Pre-processing: x1 = {1.0000, 1.0000, 1.0000, 1.0000} x2 = {1.1759, 0.9375, 1.0201, 1.0244} x3 = {1.4572, 1.0000, 1.1544, 1.0244} x4 = {1.1509, 0.8125, 1.4228, 1.0122}

項 目 1 月 2 月 3 月 4 月降雨量 ( 公釐 ) 131.9 155.1 192.2 151.8

降雨天數 ( 天 ) 16 15 16 13

平均氣溫 ( 攝氏度 ) 14.9 15.2 17.2 21.2

相對濕度 ( 百分比 ) 82 84 84 83

Ming-Feng Yeh 3-22

Weather Analysis 2Weather Analysis 2

0

0.5

1

1.5

2

降雨量 降雨天數 平均氣溫 相對濕度

1月2月3月4月

Ming-Feng Yeh 3-23

Weather Analysis 2Weather Analysis 2

Grey Relational Coefficients: = 0.8

r12 = 0.8537, r21 =0.8388

Among four months, January and February are very alike.

In general, rij rji

0000.16727.07708.06861.0

6874.00000.17877.07713.0

7641.07693.00000.1

7001.07713.00000.1

0.8388

0.85371x

1x2x 3x

4x

2x

3x

4x

Ming-Feng Yeh 3-24

Multi-Reference SequencesMulti-Reference SequencesReference sequences: yi={yi(1), yi(2),…, yi(n)}

Comparison sequence: xj={xj(1), xj(2),…, xi(n)}

i=1,2,…,p; j=1,2,…,q.Grey relational coefficient: (yi(k), xj(k))

(yi(k), xj(k)) = [min+max] [ij(k) +max]

ij(k) =yi(k) xj(k), : distinguish coefficient

max=maxi maxj maxk ij(k), 0 1,

min=mini minj mink ij(k).

Grey Relational Grade: (yi, xj) 1 ,))(),((),( 11 nk k

nk jikji wkxkywxy

Ming-Feng Yeh 3-25

Example 2.5Example 2.5

Data Pre-processing: x1 = {1, 0.889, 0.865, 0.849} y1

x2 = {1, 1.010, 1.017, 1.027} y2

x3 = {1, 0.990, 1.086, 1.042} x4 = {1, 1.529, 1.467, 1.510}

k 1 2 3 4

農業 x1 39.85 35.44 34.49 33.84

工業 x2 44.53 44.98 45.30 45.73

運輸業 x3 3.59 3.56 3.90 3.74

商業 x4 6.67 10.20 9.79 10.07

Ming-Feng Yeh 3-26

Numerical ExampleNumerical Example

Compute 0i(k):

13={0, 0.101, 0.221, 0.193}; 14={0, 0.640, 0.602, 0.661}

23={0, 0.020, 0.067, 0.015}; 24={0, 0.519, 0.450, 0.483}

max = 0.661, min = 0.

If = 0.5, then

(y2, x3) = 0.932 最大，故運輸業 x3對工業 y2之影響最大。 ，最強參考列 y2 。 ，最強比較列 x3 。

555.0932.0

500.0749.0

),(),(

),(),(

4232

4131

xyxy

xyxy

)555.0,932.0()()(max 2 yRowyRow ii

)932.0,749.0()()(max jji xColxCol