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Ming-Feng Yeh3-5 2. Grey Relational Analysis x k x1x1 x2x2 x3x3
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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
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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.
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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 }.
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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) .
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Ming-Feng Yeh3-10 2.2: 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.
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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。
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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 ]
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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 1.00 0.501.502.00 Averaging0.800.401.201.60 Maximizing0.500.250.75 1.00 Minimizing2.00 1.00 3.004.00 Intervalizing0.33 0.00 0.67 1.00
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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)
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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
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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} 1 2 34k x (0) (k) 1 2 3 1234 k x (1) (k) 1 3 4.5 7.5
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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.
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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 ) 131.9155.1192.2151.8207.8229.3149.7212.1279.4187.9142.0108.7 降雨天數 ( 天 ) 16151613151381012 14 平均氣溫 ( 攝氏度 ) 14.915.217.221.224.626.928.928.726.923.820.416.8 相對濕度 ( 百分比 ) 8284 83 79 787981
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Ming-Feng Yeh3-19 Weather Analysis
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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}
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Ming-Feng Yeh3-21 Weather Analysis
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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
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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)) 11.000 20.7420.8150.819 30.5990.6930.613 40.6990.7160.832 50.5180.9010.549 60.4250.9110.485 70.5190.4600.800 80.4100.6830.515 90.3330.6860.372 100.5040.7980.591 110.7720.7010.858 120.9310.6930.807
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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
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Ming-Feng Yeh3-25 Example 2.4 Data Pre-processing: x 1 = {1.0000, 1.0000, 1.0000, 1.0000} x 2 = {1.1759, 0.9375, 1.0201, 1.0244} x 3 = {1.4572, 1.0000, 1.1544, 1.0244} x 4 = {1.1509, 0.8125, 1.4228, 1.0122} 項 目 1月1月 2月2月 3月3月 4月4月 降雨量 ( 公釐 ) 131.9155.1192.2151.8 降雨天數 ( 天 ) 16151613 平均氣溫 ( 攝氏度 ) 14.915.217.221.2 相對濕度 ( 百分比 ) 8284 83
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Ming-Feng Yeh3-26 Weather Analysis 2
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Ming-Feng Yeh3-27 Weather Analysis 2 Grey Relational Coefficients: = 0.8 r 12 = 0.8537, r 21 =0.8388 Among four months, January and February are very alike. In general, r ij r ji
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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 )
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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 1 39.8535.4434.4933.84 工業 x 2 44.5344.9845.3045.73 運輸業 x 3 3.593.563.903.74 商業 x 4 6.6710.209.7910.07
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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 ) = 0.932 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 。
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