【正文】 icients ?1, ?2, …., ?p using the sample auto correlation function rh by replacing rh with rh. Example Considering the data in example 1 (Sunspot Data) the time series was identified as an AR(2) time series . The autocorrelation at lag 1 is r1 = and the autocorrelation at lag 2 is r2 = . The equations for the estimators of the parameters of this series are 4 2 90?0 0 01?8 0 708 0 70?8 0 70?0 0 012121. . .. . .????????which has solution 6370?21. ?????Since d = m( 1 ?1 ?2) then it can be estimated as follows: Thus the identified model in this case is xt = xt1 xt2 + ut + ? ? ? ? 21 ??????? x ??dMaximum Likelihood Estimation of the parameters of an ARMA(p,q) Series The method of Maximum Likelihood Estimation selects as estimators of a set of parameters q1,q2, ... , qk , the values that maximize L(q1,q2, ... , qk) = f(x1,x2, ... , xN。 C o s i n e w a v e sD a m p e d E x p o n e n t i a l s a n d / o r C o s i n e w a v e sa f t e r q p .a f t e r p q .P ro c e s s M A ( q) A R (p ) A R M A (p ,q )P r op e r ti e s o f th e A C F a n d P A C F of M A , A R an d A R M A S e r i e sMore specically some typical patterns of the autocorrelation function and the partial autocorrelation function for some important ARMA series are as follows: Patterns of the ACF and PACF of AR(2) Time Series In the shaded region the roots of the AR operator are plex Patterns of the ACF and PACF of MA(2) Time Series In the shaded region the roots of the MA operator are plex Patterns of the ACF and PACF of ARMA() Time Series Note: The patterns exhibited by the ACF and the PACF give important and useful information relating to the values of the parameters of the time series. Summary: To determine p and q. Use the following table. MA(q) AR(p) ARMA(p,q) ACF Cuts after q Tails off Tails off PACF Tails off Cuts after p Tails off Note: Usually p + q ≤ 4. There is no harm in over identifying the time series. (allowing more parameters in the model than necessary. We can always test to determine if the extra parameters are zero.) Examples 2 0 01 0 00161718Ex a m pl e A : U nc o ntr o l l ed C o n c ent r a t i o n , Tw o H o url y R ea d i n g s : C he m i ca l P r o c es sThe data 1 17. 0 41 17. 6 81 16. 8 121 16. 9 161 17. 1 2 16. 6 42 17. 5 82 16. 7 122 17. 1 162 17. 1 3 16. 3 43 16. 5 83 16. 4 123 16. 8 163 17. 1 4 16. 1 44 17. 8 84 16. 5 124 17. 0 164 17. 4 5 17. 1 45 17. 3 85 16. 4 125 17. 2 165 17. 2 6 16. 9 46 17. 3 86 16. 6 126 17. 3 166 16. 9 7 16. 8 47 17. 1 87 16. 5 127 17. 2 167 16. 9 8 17. 4 48 17. 4 88 16. 7 128 17. 3 168 17. 0 9 17. 1 49 16. 9 89 16. 4 129 17. 2 169 16. 7 10 17. 0 50 17. 3 90 16. 4 130 17. 2 170 16. 9 11 16. 7 51 17. 6 91 16. 2 131 17. 5 171 17. 3 12 17. 4 52 16. 9 92 16. 4 132 16. 9 172 17. 8 13 17. 2 53 16. 7 93 16. 3 133 16. 9 173 17. 8 14 17. 4 54 16. 8 94 16. 4 134 16. 9 174 17. 6 15 17. 4 55 16. 8 95 17. 0 135 17. 0 175 17. 5 16 17. 0 56 17. 2 9 6 16. 9 136 16. 5 176 17. 0 17 17. 3 57 16. 8 97 17. 1 137 16. 7 177 16. 9 18 17. 2 58 17. 6 98 17. 1 138 16. 8 178 17. 1 19 17. 4 59 17. 2 99 16. 7 139 16. 7 179 17. 2 20 16. 8 60 16. 6 100 16. 9 140 16. 7 180 17. 4 21 17. 1 61 17. 1 101 16. 5 141 16. 6 181 17. 5 22 17. 4 62 16 .9 102 17. 2 142 16. 5 182 17. 9 23 17. 4 63 16. 6 103 16. 4 143 17. 0 183 17. 0 24 17. 5 64 18. 0 104 17. 0 144 16. 7 184 17. 0 25 17. 4 65 17. 2 105 17. 0 145 16. 7 185 17. 0 26 17. 6 66 17. 3 106 16. 7 146 16. 9 186 17. 2 27 17. 4 67 17. 0 107 16. 2 147 17. 4 187 17. 3 28 17 .3 68 16. 9 108 16. 6 148 17. 1 188 17. 4 29 17. 0 69 17. 3 109 16. 9 149 17. 0 189 17. 4 30 17. 8 70 16. 8 110 16. 5 150 16. 8 190 17. 0 31 17. 5 71 17. 3 111 16. 6 151 17. 2 191 18. 0 32 18. 1 72 17. 4 112 16. 6 152 17. 2 192 18. 2 33 17. 5 73 17. 7 113 17. 0 153 17. 4 193 17. 6 34 17. 4 74 16. 8 114 17. 1 154 17. 2 194 17. 8 35 17. 4 75 16. 9 115 17. 1 155 16. 9 195 17. 7 36 17. 1 76 17. 0 116 16. 7 156 16. 8 196 17. 2 37 17. 6 77 16. 9 117 16. 8 157 17. 0 197 17. 4 38 17. 7 78 17. 0 118 16. 3 158 17. 4 39 17. 4 79 16. 6 119 16. 6 159 17. 2 40 17. 8 80 16. 7 120 16. 8 160 17. 2 1 8 9 01 8 6 01 8 3 01 8 0 01 7 7 001 0 02 0 0