Ming-Feng Yeh Grey Relational Analysis x k x1x1 x2x2 x3x3

Ming-Feng Yeh3-6 x 0 ={x 0 (1), x 0 (2),…, x 0 (n)}: reference sequence x i ={x i (1), x i (2),…, x i (n)}: comparative sequence i = 1,2,…,m. Grey relational coefficient:  (x 0 (k), x i (k))  (x 0 (k), x i (k)) = [  min +  max ]  [  0i (k) +  max ]  0i (k)=  x 0 (k)  x i (k) ,  : distinguish coefficient  max =max i max k  x 0 (k)  x i (k) , 0    1,  min =min i min k  x 0 (k)  x i (k) . 2.1: Grey Relational Analysis

Ming-Feng Yeh3-7 Grey Relational Analysis Grey relational grade:  (x 0, x i ) 0   (x 0 (k), x i (k))  1. 0   (x 0, x i )  1. Describes the posture relationships between one main factor (reference series) and all other factors (comparison series) in a given system.

Ming-Feng Yeh3-8 Axioms of GRA Norm Interval  (x 0 (k), x i (k))  (0,1],  k.  (x 0 (k), x i (k)) = 1, iff x 0 (k) = x i (k),  k.  (x 0 (k), x i (k)) = 1, iff x 0, x i . Duality Symmetric  (x 0 (k), x i (k)) =  (x i (k), x 0 (k)), iff X = {x 0, x i }.

Ming-Feng Yeh3-9 Axioms of GRA Wholeness  (x 0 (k), x i (k))   (x i (k), x 0 (k)) almost always, iff X = {x j  j = 0,1,…,m, m  2}. Approachability  (x 0 (k), x i (k)) decreases along with  (k) increasing, where  (k) = [(x 0 (k)  x i (k)) 2 ] 1/2 =  x 0 (k)  x i (k) .

Ming-Feng Yeh : Grey Generating Space Based on the concept and generating schemes of grey system theory, the disorderly raw data can be 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 Yeh3-11 灰關聯因子集 假設 X 為序列 x i = {x i (1), x i (2),…, x i (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 x i  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 x i 必須滿足下列三個條件：  無因次性 (Normalization) ：不論因子 x i (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 x i 中之 x i (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 Yeh3-12 Grey 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)  x ave, Maximizing operation: y(k) = x(k)  x max Minimizing operation: y(k) = x(k)  x min Intervalizing operation: y(k) = [x(k)  x min ]  [x max  x min ]

Ming-Feng Yeh3-13 Example 2.1 x = {4, 2, 6, 8}; x ave = 5, x max = 8, x min = 2. y (1) y (2) y (3) y (4) Initializing Averaging Maximizing Minimizing Intervalizing

Ming-Feng Yeh3-14 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 Operation (AGO)

Ming-Feng Yeh3-15 Inverse AGO (IAGO)  (0) (x (r) (k)) = x (r) (k).  (1) (x (r) (k)) =  (0) (x (r) (k))   (0) (x (r) (k  1)).  ( j) (x (r) (k)) =  ( j  1) (x (r) (k))   ( j  1) (x (r) (k  1)). IAGO x (1) = x (0) =  (1) (x (1) ) x (0) (1) ＝ x (1) (1), x (0) (k) ＝ x (1) (k)  x (1) (k  1), k ＝ 2,3, …,n Mean generating operation : z (1) (k) ＝ 0.5[ x (1) (k) ＋ x (1) (k  1)], k ＝ 2,3, …,n

Ming-Feng Yeh3-16 Example 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} k x (0) (k) k x (1) (k)

Ming-Feng Yeh3-17 AGO 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 Yeh3-18 Example 2.3 項 目 1月1月 2月2月 3月3月 4月4月 5月5月 6月6月 7月7月 8月8月 9月9月 10 月 11 月 12 月 降雨量 ( 公釐 ) 降雨天數 ( 天 ) 平均氣溫 ( 攝氏度 ) 相對濕度 ( 百分比 )

Ming-Feng Yeh3-19 Weather Analysis

Ming-Feng Yeh3-20 Weather Analysis Step 1: Data Processing – Initializing 降雨量 : x 0 ={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 : x 1 ={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 : x 2 ={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 : x 3 ={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 Yeh3-21 Weather Analysis

Ming-Feng Yeh3-22 Weather Analysis Step 2: Compute  0i (k)=  x 0 (k)  x i (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 Yeh3-23 Weather Analysis Step 3: Find the Grey Relational Coefficients Let  =0.5 r(x 0 (k),x 1 (k))r(x 0 (k),x 2 (k))r(x 0 (k),x 3 (k))

Ming-Feng Yeh3-24 Weather Analysis Step 4: Calculate the Grey Relational Grades Average the grey relational coefficients then r(x 0,x 1 )=0.619, r(x 0,x 2 )=0.755, r(x 0,x 3 )=0.687 Step 5: Sort the Grey Relational Grades r(x 0,x 2 )  r(x 0,x 3 )  r(x 0,x 1 ) Note: x 0 = 降雨量, x 1 = 降雨天數, x 2 = 平均氣溫, x 3 = 相對濕度

Ming-Feng Yeh3-25 Example 2.4 Data Pre-processing: x 1 = {1.0000, , , } x 2 = {1.1759, , , } x 3 = {1.4572, , , } x 4 = {1.1509, , , } 項 目 1月1月 2月2月 3月3月 4月4月 降雨量 ( 公釐 ) 降雨天數 ( 天 ) 平均氣溫 ( 攝氏度 ) 相對濕度 ( 百分比 )

Ming-Feng Yeh3-26 Weather Analysis 2

Ming-Feng Yeh3-27 Weather Analysis 2 Grey Relational Coefficients:  = 0.8 r 12 = , r 21 =  Among four months, January and February are very alike.  In general, r ij  r ji

Ming-Feng Yeh3-28 Multi-Reference Sequences Reference sequences: y i ={y i (1), y i (2),…, y i (n)} Comparison sequence: x j ={x j (1), x j (2),…, x i (n)} i=1,2,…,p; j=1,2,…,q. Grey relational coefficient:  (y i (k), x j (k))  (y i (k), x j (k)) = [  min +  max ]  [  ij (k) +  max ]  ij (k) =  y i (k)  x j (k) ,  : distinguish coefficient  max =max i max j max k  ij (k), 0    1,  min =min i min j min k  ij (k). Grey Relational Grade:  (y i, x j )

Ming-Feng Yeh3-29 Example 2.5 Data Pre-processing: x 1 = {1, 0.889, 0.865, 0.849}  y 1 x 2 = {1, 1.010, 1.017, 1.027}  y 2 x 3 = {1, 0.990, 1.086, 1.042} x 4 = {1, 1.529, 1.467, 1.510} k1234 農業 x 工業 x 運輸業 x 商業 x

Ming-Feng Yeh3-30 Numerical 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  (y 2, x 3 ) = 最大，故運輸業 x 3 對工業 y 2 之影響最大。 ，最強參考列 y 2 。 ，最強比較列 x 3 。