Introduction to Grey system theory
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...Dogan and Aydin (2011) as well as Ferreira and Borenstein (2012) introduced BN for handling existing uncertainty....
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...(14) is xð0ÞðkÞ þ azð1ÞðkÞ ¼ bðzð1ÞðkÞÞ2; ð15Þ xð0ÞðkÞ ¼ azð1ÞðkÞ þ bðzð1ÞðkÞÞ2: ð16Þ Similar to the GM(1,1) model ½a; b T ¼ ðBT BÞ 1BT Y; ð17Þ where Y ¼ ½xð0Þð2Þ; xð0Þð3Þ; . . . ; xð0ÞðnÞ T ; ð18Þ B ¼ zð1Þð2Þ ðzð1Þð2ÞÞ2 zð1Þð3Þ ðzð1Þð3ÞÞ2 zð1ÞðnÞ ðzð1ÞðnÞÞ2 2 6666666664 3 7777777775 : ð19Þ The solution of xð1ÞðtÞ at time k: xð1Þp ðkþ 1Þ ¼ axð0Þð1Þ bxð0Þð1Þ þ ða bxð0Þð1ÞÞeak : ð20Þ Applying the IAGO, the solution of xð0ÞðtÞ at time k: xð0Þp ðkÞ ¼ axð0Þð1Þða bxð0Þð1ÞÞ ðbxð0Þð1Þ þ ða bxð0Þð1ÞÞeaðk 1ÞÞ ð1 e aÞeaðk 2Þ ðbxð0Þð1Þ þ ða bxð0Þð1ÞÞeaðk 2ÞÞ : ð21Þ As can be seen, in Eq....
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...(7), the solution of xð1ÞðtÞ at time k: xð1Þp ðkþ 1Þ ¼ xð0Þð1Þ b a e ak þ b a : ð11Þ To obtain the predicted value of the primitive data at time (k + 1), the IAGO is used to establish the following grey model. xð0Þp ðkþ 1Þ ¼ xð0Þð1Þ b a e akð1 eaÞ ð12Þ and the predicted value of the primitive data at time (k + H): xð0Þp ðkþ HÞ ¼ xð0Þð1Þ b a e aðkþH 1Þð1 eaÞ: ð13Þ...
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...Finally, using the predicted value, the Inverse Accumulating Generation Operator (IAGO) is applied to find the predicted values of original data....
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...In order to smooth the randomness, the primitive data obtained from the system to form the GM(1,1) is subjected to an operator, named Accumulating Generation Operator (AGO) (Deng, 1989)....
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...…; zð1ÞðnÞÞ; ð4Þ where zð1ÞðkÞ is the mean value of adjacent data, i.e. zð1ÞðkÞ ¼ 0:5xð1ÞðkÞ þ 0:5xð1Þðk 1Þ; k ¼ 2;3; . . . ;n: ð5Þ The least square estimate sequence of the grey difference equation of GM(1,1) is defined as follows (Deng, 1989): xð0ÞðkÞ þ azð1ÞðkÞ ¼ b: ð6Þ The whitening equation is…...
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...Grey System theory was initiated in 1982 [7]....
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