Mousavi R topics in Farsi

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(1)CNwt. QwBU=@a. Qmi. C. O}yW. =. =Q u H. xm. jQ@ w ?. = x@. u s v. CavY. QO xS} w EL=@t. 100 Q=DWwv. CU=Q} w. |vWwOv|wUwt O}aUO}U. s_mousavi@pwut.ac.ir 1391. R}}=B. x=oWv=O.

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(3) ?r=]t CUQyi O 1 1 1 1 3 5 6 6 7 7 8 8 9 10 10. Q=DioV}B v tR |=y|QU pw= pYi. | =. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . |= Q . |rY Q}}e O | = Q s} Q . . . . = Q s} Q . . . | = Q Ov . |rY Q}}e O . . . . . . . . . . . . . . . | = = Q x RH ............... = O . . . . . . . . . . R = xi w x RH . . . . . . . . . . . . . . . . . . . . |oDU@t . . . . . . . T = u}o =} h Qa . . . . . |oDU@t T = w h Qa . |oDU@t w T = w w ` w h Qa . |oDU@t w x@ =L . . . . =o |oDU@t. v tR | U. i C=. D w. vwQ '. y| U. v tR | U. i C=. D w. v tR | U. U D. U D. 1111. J. 2111. vwQ. 3111. 111. v tR | y| U. yp t. QO. y. r -t. }. D. }. D. y w. yO N w. 221. v t. }. D. 131. v } Q=w m. }. D. 231. }. D. 331. yO N. Q. v. hr=. v } Q= m D=. U. @= D. t. 1331. y. 2331. 21. 121. y. v } Q=w w. 11. 31.

(4) 16 16 16 17 17 17 19 19 19 20 21 21 21 23 23 24 25 26 26 27 27 27 28. 30 30 30 32 34 34 35 35 36 37 37. } B |iO=YD |=ypOt swO pYi. x =. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. O. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. =ov|oDU@ty. ......... . . . . . x Ok . . . . h Qa R = x}@. 112. D. 212. W. 312. X= N. 412. }. QO |R U. w swO. x@DQt. O}iU xiwv. t. t. w. . . . . . . . . . . . . . . .| . . . . . . . . . . . . . x Ok . . . . . . . . . . . . h Qa . . . . . . . QU =kD Qort . | =Y = x@ Q w . . . . . . . . . p =i Qort .......... = x}@ . . . . . . . . . . . . . . w} Q . . . . . . . . . . . . h Qa . w} Q w | =DU = | =DU . . AR(1) O x@ Q w . . . AR(1) Ov Q =o |oDU@t . . . . | R |oDU@t w ` = .......... = x}@ . . . . . . xD = Q = O. =YD. iO. 122. D. 222. a. 322. D t X= N. 422. }. w. iO. B p. D uORs o swO. =. uORs o. t. t. v=. D. a. 522. |R U. W. 622. . w. =. O. U oQ D= | yp t D. 132. }=. 232. D t X= N. 332. v. y. 432. yO N. @ D. 532. |R U. W. 632. i } VR= @ | yp t. 732. }. U oQ D= QO. }= v w. }. p t swO. } i Q. } H. = x}@W. xDi=}. . . . .. Q. x W |R U. .... . . . .. =. | yxO=O. Q. VR= @. AR. Q@ xDi=} O. Q. }. =. O. 832. Q. 932. VR= @ | yp t. |v=yH. p t %. =. | tO | U. 12. 22. 32. }= |=ypOt swU pYi. =DU. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . QLD u}o =} . . . . w h Qa MA(q) Ov Q = x}@ =o |oDU@t R = =F . . . . . . . . . . . . . xD = Q MA . O = x}@ Q x xD = Q O . . . . . . . . . . ARMA Ov Q |@} Q . . . . . . . . . . . . . . . h Qa . . . . . . . . . O x@ Q w . . . . . . . . . . | QH p}rL ARMA . . . . . . . . . . Q = x}@ l. X= N w. |R U. W w Q. }. t. y %. 113. | yp t. 213. i } VR= @. x W |R U. W | U. @. i } VR= @ p t } i %. @. D. O. =. O. m D | yp t. D t X= N. 233. D %. W. 13. 23. 123. 133. VR= @ w |R U. ?. =. | yp t. D. }. p t swO. O. } i. D %. v. =. v t | yp t. =. O. | yp t 143. 33. 43.

(5) 37. 39 40 40 41 42 43 44 44 45 47 47 50 50 51 52 53 54. . . . . . . . . . . . . . . . . . . . . . . . . . . . p O@ }. Q. Q. D Mv | U. 243. }==v |=ypOt sQ=yJ pYi. =DU. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . |rY Q} ARIMA . . . . . . . Q O} w Q p =i . . . . . . . . . . . . |k}ir O . . . . . . . . . . = =F h Qa . . . . . . . . . Q = x}@ . . . . . . . . . . . . |rY ARIMA = . . . . . . . . . . . . . . h Qa ............ Q x . . . . . . . . . . . . . . . . ARCH = . . . . . . . . . SP500 = Q ARCH O h Qa O = = Q O . GARCH = O x w Q = \U xD = Q GARCH O = x}@ . . . . . . . . SP500 Q Q Q . . . . . . . . s}r Q O== . . = x}@ |v} V} GARCH i. j @. e. O. p t. D. 114. D p t. 214. r D | U w. . yp t w. }. VR= @ w |R U. D. 314. W. 414. i. }. VR= @. O. | yp t. D. 124. } wQ. 224. O. | yp t. p t. }. | y| U. 134. D %|Q= } B v uO m p t. 234. @ \ @ t | y. @. 334. p t w |R U. W. 434. @ VR= @. 534. k= | U QO |Q= } B v. 634. | yp t. i } VR= @. | U. |R U. W w. 56. @. B QO. 14. 24. 34. 734. H=Qt. `. G.

(6) Q=DioV}B OW x=Q= OvDUy. OD@=. x=Q |=. QO w | ". OD@t xv}tR u}=. CU=. =. xm |v=Um |}=vW Qw_vt x@. QO. =} p@=k u=o}=Q. |R UxO B. R. R. Q. Q |twta Q=DWwv. Q= i=s v |= @. w x xm CiQo Q=Qk Q} R. CQ Y @. x@. TQO. R C=. } U QO w. http://cran.r-project.org/doc/contrib/Mousavi-R-lang_in_Farsi.pdf j} Q]. R= =Q. Q}kL sy w |t. "O W. O. |=x a w. CU= xDiQo Q=Qk. RoU=BU =yv. |Q=. =H u}ty. R=. Ovtxkqa. u= R=. =iDU=. xO. w. O. xDio xDWwv xm OUQ|t Q_v x@. OQ t x W. xm OvO=O Q=Qk V} wN C@Lt. h]r. w. w l}vwQDmr= CUB. OQ t. EL=@t

(7) u=wva Q} R Q =L Q=DWwv u}=Q@=v@ "sO=Di= R OOt x@ X=N EL=@t R= |=xQ=B x}yD Qmi x@ x=Q xt=O= QO xOv@ Q |t KQ]t xOv@ lOv= Ca=@ OL. "OO o. x@ EL=@t u}= Q@. w |t s}OkD pYi OvJ. w O W. Q xm OwW|t u=Wv Q]=N =Hv}=. Z i. CU=. ". =kDQ=. u <. w K. QO . xO. wtv ?Um. qY=. Q. =Q. R. QO u K W. xm. w. | D. Q. QO "O W. R. = u}tR. O. w Q t. pY=L. '| L. OQ=O. w. Q}O=kt '|v=tR. |v=tR. =. Q. | y| U. Q Kkvt xDWwv =D. Q_v Q=y_= xvwo Qy. =. Q Ovv=t. | y| U. x@. =YDN=. X. Q |t V}=Q} w. OO o. Q = x}rw= |}=vW 'xS} w EL=@t ?}kaD. Q= i=s v @. =yvW}B. |= @ |O. :::. w. R}}=B. QyD. u=. O. EL@t u}rw=. O@=}|t xt=O=. Q. Qw t. Q |t=Qo xOvv=wN xm CU= u}=. OvU} wv 'CU}v prN. R= x. R=. |r=N Q =L xR}Hw ,=trUt. wtv Oy=wN. 1391. S. QO x } w. |= @. "O. |vWwOv|wUwt O}aUO}U. u. R. . xO. =iDU=. w p. =@kDU=.

(8) pw= pYi. |v=tR |=y|QU |v=tR. x@ Ov=wD|t =@. "OO o. x@. C= y. O =Wt C@F pta "Ovvm|t Q}}eD u=tR. C= y. Q |t O}m =D. =Ut |v=tR pY=wi =@ xDUUo C@F. |w. O =Wt xawtHt. w. 1,2, ,n. w. CQ Y. w |t. "O W. xm CU= |}=yOv} Qi C@F. QO. |wQ. =Hv}=. QP. =Wv. x1 x2. |rYi. OvDQ=@a |v=tR. =Hv= xDUw}B =}. QO "O } B s. x@ |v=tR xawtHt '?U=vt |v=tR. xO=O u. R=. xn =. C=Q}}eD w. T. =}kt. w. w. |v=tR. g. CU= xOW C@F. ". 1949-1960. =. xQwO QO Q t. wva Q} R R OwN QO |}=yxO=O p}=i R=. u}= "OyO|t u=Wv. :::. pmW. 1 1 1. 1 1 1 1. y|QU s}UQD. u=. =NDv=. ?. n. =. u} Qi=Ut O=OaD xm OwW|t xO=iDU= AirPassengers. w. O@t. s}UQD. Q. CQ Y. <=. xt : t = 1 2. '|v=tR |QU. =. | y| U. xDUUo. } w f. OvwQ. 1 1. |QU. w. s}UQD |=Q@. Qiv Q=Ry ?UL Q@ xv=y=t |rrtr=u}@. =Q . %p=Ft. > data(AirPassengers) 1.

(9) 1391 '. |vWwOv|wUwt. 2. > AP <{ AirPassengers > plot(AP, ylab = "Passengers (1000's)") O w| x w window() = x R = t. m OQ=O O Hw. s v. @. u @ R QO |. Qo}O `@=D "CU= q=@. = O. Q. | y ) m |= H=. pY=L. 11. pmW. 400 300 100. 200. Passengers (1000’s). 500. 600. v= D. 1950. 1952. 1954. 1956. 1958. 1960. Time. u} Qi=Ut xv=y=t |v=tR. Q. | U. V}=tv. %11. O}vm xHwD Q} R p=Ft x@ "OyO CUOx@. ". pmW =Q. |v=tR. Q. x wtHt Q} R l}. | U R= |= a. > x <{ AirPassengers > y <{ window(x, 1950, 1951) CU= Q} R. ". w. CQ Y. x@. u. xH}Dv xm. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1950 115 126 141 135 125 149 170 170 158 133 114 140 1951 145 ". O}vm xHwD Q} R p=Ft x@ x=t ,qFt 'O}W=@ xDW=O pQDvm p=U. R=. |rYi. |wQ. O}y=wN@ Qo =. > x <{ AirPassengers > y <{ window(x, 1950, c(1951,0)) CU= Q} R. ". w. CQ Y. x@. u. xH}Dv xm. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1950 115 126 141 135 125 149 170 170 158 133 114 140 O}vm xHwD Q} R p=Ft x@ =}. ". > x <{ AirPassengers > y <{ window(x, c(1950,2), c(1951,3)).

(10) |v=tR. 3. =. Q. | y| U. 1. pYi. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1950 126 141 135 125 149 170 170 158 133 114 140 1951 145 150 178 O M} Q=D xm O}vm xHwD Q} R. x = = |v=tR. "OQ= v. V1 V2 V3 V4 V5 3.59 2.20 4.53 7.01 10.45 9.60 19.12 21.93 24.20 25.22 3.13 3.97 17.52 11.77 9.65 2.76 4.17 6.71 10.57 9.87 1.22 2.81 6.25 9.07 28.12 ... ... ... .... 1 2 3 4 5 x@. wtv sUQ. "O. =Q. =yu TBU. V6 43.86 42.28 18.26 7.58 20.00 .... Q sy QU CWB. w xO=O Q= k ". CU= Q} R KQW x@. V7 59.97 37.32 72.26 15.45 36.48 .... =. =. Q. | U. x@ uwvm =. V8 V9 V10 V11 V12 92.29 47.00 19.16 12.45 8.84 14.11 6.75 4.40 2.62 2.82 32.45 12.09 5.12 3.03 2.46 29.85 4.25 1.74 1.12 1.04 45.32 12.19 2.98 2.85 2.19 ... ... ... ... .... =. O}=@ =OD@= xv=y=t. =Q |Q t | yp U. w. | v y t. = O. pw= CQ Y | y ) m "O=O s. =Hv=. =. | yxO=O. = u}=. s}UQD. w |t. =Q Q m. Q. |= @. w. u= D. CQ Y wO. > x <{ matrix(scan("F:/R_les/data/ghar.txt"), ncol=12, byrow=T) > y <{ t(x) > y <{ matrix(y,ncol=1, byrow=T) > ts.plot(y) CU= Q} R KQW x@. ". w. = O. swO CQ Y | y ) m. > x <{ read.table("F:/R_les/data/ghar.txt") > y <{ as.matrix(y) > y <{ t(x) > y <{ matrix(y, ncol=1, byrow=T) > ts.plot(y) 2 1 1 1. v tR |QU OvJ. | =. O}rwD p}=i. 1990. u}=. w. =. p U. =D. 1958. u)D ?UL Q@. "OQ=O O Hw . =. =}r=QDU=. qmW. w . p U R= C. qFt. QO ,. =Wv. "O=O u. w. w x. s-= D CQ Y @ =Q. |v=tR. QD}r uw}r}t ?UL Q@ Q}aWr=<=t 'Ca=U w |t xOv=wN. "O W. RO. ) m. \UwD. Q. | U. OvJ. w |t. u= D. wr}m ?UL Q@. C=w. online. w x. Q. CQ Y @ | U. Q. j @. xU. > "http://www.massey.ac.nz/~pscowper/ts/cbe.dat" > CBE <{ read.table(www, header = T) CBE1:4, ]. CU= Q} R. ". w x. =. = O |HwQN. CQ Y @ q @ | y ) m.

(11) 1391 '. |vWwOv|wUwt. 4. choc beer elec 1 1451 96.3 1497 2 2037 84.4 1463 3 2477 91.2 1648 4 2785 81.9 1595. |=Q=O w. Q. OvQ=Ov M} Q=D Qo}O. wtv p}O@D |v=tR. |= @ "O. =@ x@ "OvDUy x=t. CQ a. Q. | U. x@. ts() ` =. =. w p U. =iDU= =@. @ D R= xO. Ok=i q=@ =}. =Q < W=. =. w |t x_Lqt xm Qw]u=ty. | yxO=O O W. u}= O}=@ u}=Q@=v@ "OvDU}v |v=tR O}vm xHwD Q} R. ". Q. =D =. | U Q N U. = O x@ Q=m u}=. | y ) m. > "http://www.massey.ac.nz/~pscowper/ts/cbe.dat" > CBE <{ read.table(www, header = T) > Elec.ts <- ts(CBE, 3], start = 1958, freq = 12) > Beer.ts <- ts(CBE, 2], start = 1958, freq = 12) > Choc.ts <- ts(CBE, 1], start = 1958, freq = 12) > plot(cbind(Elec.ts, Beer.ts, Choc.ts)) ". CU= q=@. = O. 10000 14000 6000 150 6000 2000. Choc.ts. 100. Beer.ts. 200 2000. Elec.ts. cbind(Elec.ts, Beer.ts, Choc.ts). 1960. 1965. 1970. 1975. 1980. 1985. Time. |v=tR. 1. trend. 2. seasonal variation. Q. | U. OvJ V}=tv. %21. pmW. Q. | y ) m |= H=. 1990. pY=L. wtv. 21 Q=O.

(12) |v=tR. 5. =. Q. | y| U. 3 1 1 1. i C=Q}}eD w OvwQ. |rY. 3 CQB Q}O=kt '2 |rYi. = =}. ,v L= w. |rm Cr=L. '. u}@D =. Q t. Q. QO w. Q}}eD '1 OvwQ xmr@ OyO|t CUOx@. Q. C=. CU= |rYi. Q}}eD p=U Qy. C=. O u} QDxO=U "Ov} wo OvwQ 'OvwW|tv Qy=_ |@ w=vD w |t 'OvwQ K =w. u= D. = |t Qy=_ R}v. QO 'OR U. w x xm |v=tR. |= @ p t. `}tHD =@ |rYi C=Q}}eD Q}F-=D. wor= =yvD xv |v=tR. =Q | U |. wor= Q=QmD q=@ p=Ft. QO. CQ Y @. uOQw. CUOx@. Q. =Q. Q. sUQ. | U. =]N Q}O=kt. l}D=tDU}U. | U QO. pYi. 1. Q}}eD. C=. Q CU= |]N Vy=m w V}=Ri= 'OvwQ. |= @ ". Qy Q}O=kt xYqN "OwW|t s=Hv= aggregate() `@=D R= xO=iDU= =@ R QO Q=m u}= "OQ@ u}@ R= xvq=U Q=t x@ xv=y=t = |t u}at =yxO=O R= l} Qy |=Q@. "OR U. pYi cycle() `@=D w OwW|t XNWt boxplot() `@=D \UwD pYi. =Q. > data(AirPassengers) > AP <{ AirPassengers > layout(1:2) > plot(aggregate(AP)) > boxplot(AP ~ cycle(AP)) ". Q |t x_Lqt xm Qw]u=ty. w |t xOv=wN. "O W. RO. ) m. \UwD. online. = O. Q. | y ) m |= H=. w x xv=y=t. CQ Y @. pY=L. =m}@. |Q. wtv. 31 Q=O. =. | yxO=O. %p=Ft. 2000. 5000. aggregate(AP). OO o. CU= q=@. 1950. 1952. 1954. 1956. 1958. 1960. 100. 400. Time. 1. =yv. ts() ` =. @ D R=. |v=tR. Q. | U. boxplot. x@ =yv p}O@D. `}tHD `@=D \UwD xvq=U \UwDt. 2. 3. O. w x W. Q. |= @ w. 4. 6. 7. `}tHD xvq=U. u QO. 8. 9. Q. V}=tv. | U. OvDU}v |v=tR. Q |t x_Lqt. "OO o. 5. Q. 10. | U. freq. 11. 12. %31. pmW. =yxO=O u}= "CU=. =v. Q. ?w D xQwO w `w W. unemploy Q}eD. QDt=Q=B xm. w |t. 'O W. =iDU=. xO. w |t u}at s}UkD Qorta. "O W. > www <{ "http://www.massey.ac.nz/~pscowper/ts/Maine.dat" > Maine.month <{ read.table(www, header = TRUE). 3. outliers. =. t s v. w.

(13) 1391 '. |vWwOv|wUwt. 6. > Maine.month.ts <{ ts(unemploy, start = c(1996, 1), freq = 12) > Maine.annual.ts <{ aggregate(Maine.month.ts)/12 > layout(1:2) > plot(Maine.month.ts, ylab = "unemployed (%)") > plot(Maine.annual.ts, ylab = "unemployed (%)") CU= q=@. = O. Q. | y ) m |= H=. pY=L. wtv. 41 Q=O. 3 4 5 6. unemployed (%). ". 1996. 1998. 2000. 2002. 2004. 2006. 4.5 3.5. unemployed (%). Time. 1996. 1998. 2000. 2002. 2004. Time. =yv \UwDt. =m}@. w |Q. Q. | U. V}=tv. %41. pmW. |v=tR. |=y|QU. 2 1. x} RHD. 1 2 1. =ypOt. =ypOt u}=Q@=v@. w |t xOy=Wt |rYi. "O W. O = |t Q} R. " W @. Q}}eD =}. C=. w. OvwQ =y|QU. w x xm 'CU= 4 |atH pOt '|v=tR. CQ Y @. Q. R=. |r}N =. | U xO U. O. QO ' W. x_Lqt ,q@k xm Qw]u=ty. x} RHD l} "OvDUy |}=yxirw-t. |=Q=O. xt = mt + st + zt s. xirw-t "CU= |rYi C=Q}}eD xOv}=tv t. w. OvwQ xv=Wv. mt xi w. r -t ". CU= xOW xOy=Wt. Q. | U. O. =. w u tR. |t u=Wv. " yO u. xt. =Q. t. u QO. xm. z. =]N R}v t. =Wv Q} R CQwYx@ xm 'CU= 5 |@ Q pOt l} ?U=vt pOt 'OwW Qy=_ OvwQ l} CQwYx@ |rYi C=Q}}eD Qo = w |t. "O W. xt = mt :st:zt 0. 4. additive. 5. multiplicative. 0. 0. xO=O.

(14) |v=tR. 7. Q |t p}O@D |atH pOt x@ |@ Q pOt 'Q}N= x]@=Q. "OO o. R=. =. Q. | y| U. pYi. 1. sD} Q=or uDiQo =@ OwW|t x_Lqt xm Qw]u=ty. yt = ln(xt ) = ln(mt ) + ln(st ) + ln(zt ) = mt + st + zt 0. 0. 0. R QO Qo = "Ovm|t x@. Q. OQw @ l. w Q. =F. QLDt u}ov=}t Q. =iDU= =@. VwQ R= xO. Q |t O=H}=. \ @ t xO=O p t |= @ "OO o. =yDv=. |. w |t s}UQD. QO w O W. 51. Q=O. =Q. |rYi. Q}}eD. C=. wtv xU =@ pmW l} x=ov. pmW |@ Q. w. |atH pOt. OvwQ. w. w x Q} R. CQ Y @. decompose() ` = R plot() ` = p w `= @ D '. Q}o Q=Qk. 'O. 2 2 1. =yxirw -t x} RHD. @ D. = O. N=O j i. =. =F. QO @ D. Q O}rwD. | y ) m QO 'q @ p t j @. pmW OOQo|t x_Lqt =Hm} |rYi C=Q}}eD w OvwQ Q=Owtv l} |wQ xQNq=@ w OwW|t p=@vO |@ Q pOt =yO)m "61. trend seasonal. 14000 8000 12000 2000 1.10 2000 6000 1.00 0.94. random. 1.06 0.90. 1.00. observed. Decomposition of multiplicative time series. 1960. 1965. 1970. 1975. 1980. 1985. 1990. Time. Q. = xirw-t V}=tv. | U | y. %51. pmW. 3 1. |oDU@ty. |oDU@ty u}vJ O}=@ =yv p}rLD =. | yxO=O. j} Q]. w. x} RHD. Q. w Ovy=wN xDU@ty |r=wDt. |= @ "O @. w |t h} QaD |oDU@ty `@=D \UwD |v=tR. R= w O W. =. Q. | y| U QO. = Q}eDt. | y. =. 'Cq L R=. |oDU@ty Q=DN=U Q |t. "OO o. |r}N. QO. Q u}at. "OO o. Q O. O =Wt. OQw @ x W x y.

(15) |vWwOv|wUwt. 8. 2000. 4000. 6000. 8000 10000. 14000. 1391 '. 1960. 1965. 1970. 1975. 1980. 1985. 1990. Time. |rYi. Q}}eD. C=. w. OvwQ V}=tv. pmW. %61. Tv=}. Q}eDt u}ov=}t x@ xm CU=. . E (x) u Q =v. CU= xat=H u}ov=}t `k=w. w |t. }= @ @ ". w. QO 'O W. E (x )2 ]. = QLv= `@ Qt u}ov=}t. ;. p L C i=. w |t. "O W. xO=O. =@. Q=w w. xO=O u. =Wv. w |t. xO=O. CQ a "O W. V}=tv R}v. 2 =. @. CU= h} QaD p@=k Q} R. ". E. Q. =@ xm | =} Q O}t=. h L. V}=tv R}v. xm OwW|t xDN=vW. |oDU@ty. w. 1 3 1. u}ov=}t h} QaD. Tv=}. Q=wwm. w x Tv=} Q=wwm s=v x@ |twyit 'O}W=@ xDW=O. CQ Y @. = x x Q}eD @. CU=. m. t. x. Tv=} Q=w 2 3 1. h} QaD. (x y) Q}eD. t wO. Qo =. (x y) = E (x x)(y y )] ;. R=. n. O x@. xR= v=. x wtv l} Qo =. |= v. CU= Q} R. ". "O. Q}o|t. w x. CQ Y @ u. O. xR= v= =Q. w. -1 u}. @. O. w. ?} Q u}= "Ovm|t u}at Oa)@ uwO@. "OQ= v O Hw. (x y). (x y) Q}eD. x]@=Q xm Owtv. Cov(x y) =. ;. t wO. Q. OQw @ =Q. u}@ |]N. =@. R}t `k=w. \ DQ= u=. xvwtv Tv=} Q=wwm. QO. Tv=} Q=wwm. w |t O}W=@ xDW=O. u= D. (xi yi). X. (xi ; x)(yi ; y)=n. w x. CQ Y @ =Q. (x y) Q}eD. t wO. u}@. =@. R}t |oDU@ty ?} Q. \ DQ= u=. = Q}eDt u}@ |]N \=@DQ= xm CU= |vat u}O@ OW=@ QiY ?} Q u}= Qo =. | y. w |t h} QaD Q} R. "O W. w x. CQ Y @. (x y). = Q}eDt u}@. | y. (x y) = E (x x)(y y )] = (xy) x y x y ;. ;. |va. }. Q. "OQ=O Q= k. +1. xat=H |oDU@ty.

(16) |v=tR. 9. Q |t x@U=Lt Q} R. "OO o. =. Q. | y| U. pYi. 1. w x xvwtv |oDU@ty ?} Q. CQ Y @. Cov(x y) sd(x) sd(y). Cor(x y) =. |oDU@tyOwN. w. Tv=}. Q=wmwD=. 3 3 1. `@=wD h} QaD. w x |v=tR |QU l} u}ov=}t `@=D "OwW|t xDN=OQB u}ov=}t x@ =OD@= |v=tR |=y|QU |oDU@ty h} QaD |=Q@. CQ Y @. O = |t. " W @. (t) = E (xt ) Q. OQw @ w. CU= 6 =DU}= u}ov=}t. QO. |v=tR. Q. =ov 'OW=@v. | U x. x = O = |t Q} R. " W @. =. u tR R=. t. |a@=D. CU= Q} R. w. |a@=D |va} 'OW=@ C@=F u}ov=}t `@=D Qo = CU= Q} R. w x. ". n X. R=. x wtv. CQ Y @ u |= v. xt =n. i=1. w x CU= =DU}= u}ov=}t. CQ Y @ '. QO. xm |v=tR. Q. | U. l} Tv=} Q=w `@=D. 2 (t) = E (xt )2] ;. w x. x wtv OQw Q@ "OOQo|t. CQ Y @ u |= v. 2 C =. x@ p}O@D. @ F. Var(x) = =. =. x. =ov 'OW=@ =DU}= R}v Tv=} Q=w QO |v=tR |QU Qo =. P. CU= Q} R. ". (xt ; x) n;1. xDU@ty CU= umtt =yQ}eDt =t= "OvOw@ =DU}= Tv=} Q=w |oDU@ =yv. 2(t). w. 2. u}ov=}t. QO. =iD x@ \ki =yQ}eDt u}@ |oDU@ty xm CU=. | ys o Cw. ". Ov} wo 7 Q}N-=D. xm O}Owtv x_Lqt swO =Q. x@DQt. Q Qo = "Ovt=v 9 |B=}B |oDU@ty =} w 8 |oDU@tyOwN =Q hrDNt |=yu=tR w x. CQ Y @ =Q. CU=. k. |v=tR. =DU}= |v=tR. |. =yQ}eDt u}@. | U. =Q. =. =. | ys o O=. w. Q. wvm =D. | U u. Q. OvW=@ R}v. | U ". OaD "OW=@ xDW=O. = Q}eDt l} |oDU@ty. QO VO N @. . |a@=D xm k |va} (acvf) Tv=} Q=wwmwD= `@=D u=wD|t x=ov 'OW=@ swO x@DQt |=DU}= |v=tR. R=. wtv u=}@ Q} R. "O. k = E (xt )(xt+k )] ;. `@=D "CU=. E (xt ) = E (xt+k ) = |va. ;. CU}v u=tR x@ xDU@=w u}ov=}t =Q} R. } '. w |t h} QaD Q} R. "O W. w x. CQ Y @. k Q}. . O. =. = =. (acf) |oDU@t. OQ= v u tR N-D @. x@ |oDU@ k `@=D w. yO N. k = k2 O = |t. " W @. 6. stationary. 7. lag. 0 = 2. Q. = }R '. 8. Autocorrelation. CU=. 0 = 1 x. m. CiQo xH}Dv. 9. Serial correlation. w |t q=@ h} QaD. u= D. =iDU= =@. R= xO.

(17) 1391 '. |vWwOv|wUwt. 10. 1 3 3 1. yOwN x@U=Lt. |oDU@t. O =Wt CiH. x y. n 1 ;. w |t "OW=@ xDUUo |v=tR. Q. u= D. x1 x2 . l}. | U. xn. n O}v. O =Wt. x y. Q. wvm =. m Z i u. |va}. (x1 x2 ) (x2 x3 ) Q. Z i swO. Q}eDt. wva x@. u=. O =Wt u}twO. =Q x y. Q}eDt. w pw=. (xn 1 xn) ;. u=. wva x@. Qy. GwR. O =Wt u}rw= Qo = "CN=U. CU= Q} R. QO =Q x y. w. ". nP ;1 ;. r1 = s. t=1 nP ;1 ;. xt x(1). t=1. Ovt=v. ". ;. O =Wt u}@ |oDU@tyOwN ?} Q. C= y. ;. w |t. nP ;1 t=1. k Q}. = =. N-D @ =Q. rk =. t=1. t=1 R= =Q. |v=tR. Q. = q]=. | U C a. R=. |r}N. ;. =. =. w. n P. w. ;. t. R=. @. y. W @ nQ @. u QO. m. k=w QO. m. i m xR= v=. @. o=. (xt ; x)2 w |t j} Q] u}ty x@ "CU=. u= D. Q. n P. x = n1. t=1. xt. u QO. xm. k = 0 1 2. 2. u}=. nP ;1. xt x1 = n 1 1 xt x t=2 t=1 x Q}eD u} |oDU@t ` x r1 O = R | = O x n Q. (xt ; x) (xt+1 ; x). (xt ; x). CQ Y. 2. | yu tR QO. (xt ; x) (xt+k ; x) n P. n P. =iDU= Q} R pwtQi. |oDU@tyOwN ?} Q nP ;k. ;. xO. t=1. wtv x@U=Lt. xt+1 x(2). x2 = n 1 1. CU= hrDNt. "O W. "O. t=1. CU=. =Q u w. ". r1 =. . ;. ;1 ; 2 nP. x@ |oDU@ty ?} Q 'O}vm. CQ Y. xt x(1) xt+1 x(2) ;. =Q. Q x@U=Lt. QO = } R "O m. =. rk O =@ k > n4 Q. =Q y. =kt. }O. } v. Q. wtat. |= @ ,q. O}yO|t CUO. ". w |t xDiQo Q_v. "O W. 2. QO p. n. Q=. Okt 5%. w. r. |vat K]U =@ k. uO @ Q=O. |=. xrY=i. Q. 2 3 3 1. 10 Q=ov|oDU@ty. w |t. "O W Q. =iDU= =ypw] QwLt. xO. k ?U. Q =. Q. r. wLt k. L @ yZ a Q. Q=O. wtv. R= '. |v=tR. =. Q. | y| U. Q}UiD. Q. OQw @ |= @. w. Q}@aD. Q. |= @. wLt "CU= Oa)@ uwO@ OQ=O Q=Qk |wQ |oDU@tyOwN Q}O=kt xm =yZQa QwLt "Ovt=v Q=ov|oDU@ty =Q Q=Owtv u}=. 10. correlogram.

(18) |v=tR. 11. Q}oxvwtv |v=tR xrY=i x@ |oDU@ OyO|t u=Wv. "OQ=O |. =F. p t u=. wvax@. w |t. "O W. xO. R. =iDU=. lag.plot() ` =. QO. @ D R=. sy x@ C@Uv hrDNt. =. =Q. W OO o. N-D Q. U D. t. J. D. | yxO=O '. pYi. 1. =yQ}N-=D xm =ypw]. = Q}N-=D s}UQD. Q. | y. > lag.plot(LakeHuron, lag=4, labels=F, do.lines=F, diag.col = "red") C lag=4 Q pm Q | s} Q Q} = =y = LakeHuron = |= @ "71. Q. | y| U. U=. w. =. |= @. =F. u J 'q @ p t QO. LakeHuron. 576 577 578 579 580 581 582 LakeHuron. 576 577 578 579 580 581 582. LakeHuron lag 3. lag 4 576 577 578 579 580 581 582. Q}N-=D. CU= Q} R. ". w x. CQ Y @ u. 576 577 578 579 580 581 582. lag 2. LakeHuron. lag 1. |rm pmW. w |t. "O W. =. | yQ=O. =iDU=. xO. wtv. %71. pmW. acf() ` = R @ D R=. QO Q. =ov|oDU@ty s}UQD. w. rk x@. =Lt. U. acf(x, lag.max=NULL, type=c("correlation","covariance","partial"), plot=TRUE) u QO. x w Q} = Oa lag.max Q | x@ =L 10log(N) = QD = xD type CU= =yxO=O. ". x]@=Q. |. R=. xm CU=. Q V}B. Z i. O = |t. " W @. "correlation". Q. R. |=Q=O. 10. P. =v@t. |. = w. 'OO o h L u t oQ. Q. = w. P. 'OO o h L u t oQ QO u. sD} Q=or. u}= Qo = "CU= Q_v. = x wtv. w y v. u}= Qo = "Ovm|t u}at. =Q. O. xR= v=. acf ` =. N-D O=. OQ t. Nx. m 'OO o. w. CU=. @ D ` v w. xm. t. y ". U. Q. Q=O @ %. D %. t. m =Q m R= |=. WQ %. CU= `@=D u}=. Q V}B. Z i. Q |t sUQ Q=ov|oDU@ty Q=Owtv CQwY u}= QO w CU= TRUE ZQiV}B |=Q=O |k]vt u=twoQ u}= %plot. "OO o. w |t. "O W R=. =. A J. =yQ}N-=D. =yv x@ |UQDUO. k=0 Q. o= ". Q. |=R=. x@. acf Q. }O. =kt. Q |t xQ}NP. |= @ w OO o. w |tv sUQ xOW xDio. wtv 'OW=@. w O W. acf(x)$acf. Q=O. w x. Q. CQ Y @ |Q=O @ QO. FALSE. = w. u}= Qo =. u t oQ. Q}O=kt u}= xm O}W=@ xDW=O xHwD. CU= acf$acfk+1] CQwYx@ q=@ Q=OQ@ u}=Q@=v@ "OwW|t xO=iDU= Q=OQ@ T}Ov= u=wva x@ Q}N-=D u=tR ". O}}=tv x_Lqt. ". =Q. acf$acf1] Ok acf ` = =F wv x. CU= l} Q@=Q@. Q} R. = O. | y ) m. Q=. @ D p t u=. O =. t ' W @. wvm =. a @ u.

(19) 1391 '. |vWwOv|wUwt. 12. > www <{ "http://www.massey.ac.nz/~pscowper/ts/wave.dat" > wave.dat <{ read.table (www, header=T) > par(mfrow=c(2,1), mar=c(2,4,2,2)) > plot(ts(waveht), ylab = 'wave height (mm)') > acf(waveht, main="") CU= q=@. = O. Q. | y ) m |= H=. pY=L. wtv. 81 Q=O. −500. 0. wave height (mm). 500. ". 100. 200. 300. 400. −0.5. 0.0. ACF. 0.5. 1.0. 0. 0. 5. =ov|oDU@ty. u Q. 10. w. 15. |v=tR. Q. | U. w |t x@U=Lt Q} R. "O W. > acf(waveht)$acf2] 1] 0.4702564 w |t x@U=Lt Q} R. "O W. w x O =. CQ Y @ ' W @. 20. V}=tv. 25. %81. pmW. CQ Y @. k=1. k=1. Tv=} Q=wwmwD= x@U=Lt Qw_vt Qo = ,=OOHt. w x. QO. acf. QO. Q=. Okt p@k p=Ft xt=O=. QO. > acf(waveht, type = c("covariance"))$acf2] 1] 33328.39 r O N. xrY=i xv=Wv xm OvQ=O OwHw =y k u}}=B O = |t =yxvwtv. " W @. r. xR= v=. u}@ R= k Q}O=kt Qo =. "O. x@ O}=@ =Hv}=. xD@r=. "O m OQ =Q. uwQO. Q}O=kt xm. \N. wO. QO. xrY=i. Q}o|t Q=Qk Q. xO. =. u QO. xm 'CU=. =iDU= OQwt. %5 K]U. u}J]N. w q @ QO. QO. Q. | U. Q@=Q@ %5. 2. p. w x \N. CQ Y @. N. w. }. QiY. CU= |vFDUt xOa=k u}=. u swO ". O}vm|t x_Lqt xm Qw]u=ty. |vat K]U =@. uO @ Q=O. |oDU@ty u}}aD |=Q@. k = 0 |va. wO. Q. u R=. xm CU=. R=. =D =. |Q N U. w |t x=ov 'Ot. Z i u= D. OvDUy u=v}t]=. w. Q}@ Q}N=. uw. r0 = 1 x. u}=. =D. wO ' yxO=O O=. m. pw= ". rk Q. =kt. }O. w]N. \. CW=O xHwD xDmv xU. Q}N-=D 40 |=Q@ ,qFt %5 K]U QO xm u swU "OvQ=Ov |Q=O|vat Cw=iD QiY =@ =t= OvW=@v QiY ,=k}kO CU= umtt " Q. O}vmv. =. |Qw=O u T U=. =ov|oDU@ty. O. QO " yO. Q@. |t. O}v=Ov =]N pwtWt. w. =Q . =Wv xDUy Vy=m l}. u. r0. <. RH@. w x. Q}@ =D. uO i= uw. CQ Y @ Q. =ov|oDU@ty. =. OaD =@ ?U=vDt. w. QO =Q O N. =yxO=O OvwQ ,qwtat.

(20) |v=tR. 13. w. CU=. Q. x@=Wt Q}O=kt. | U QO. |W=v xm C@Ft. R=. R Q}O=kt. O}vm|t x_Lqt. Q. OvDUy syx@ l}ORv. ". 1. acf(AirPassengers). pmW. 91. w nQ @. =. | y| U. =. | yxO=O. =. =Q ". pYi. =. | yu tR QO. −0.2. 0.0. 0.2. 0.4. ACF. 0.6. 0.8. 1.0. Series AirPassengers. 0.0. 0.5. 1.0. 1.5. Lag. AirPassengers =. Q. xvq=U. QO ' yxO=O | U. =v. =. | yxO=O Q. =ov|oDU@ty. w |t Qy=_ Q=ov|oDU@ty. ?w D "O W. %91. pmW. O = xDW=O OwHw R}v |rYi. |wQ ' W @. Q}}eD Qo =. C=. x]@=Q xm CU= p=Um} Q}N-=D QO |oDU@ty Q=Okt QFm =OL "OwW|t x_Lqt ?w=vD u}ty =@ R}v u Q=ov|oDU@ty =D. w. Tmar=@ "OyO|t u=Wv. Q iQ u J. =@ |ivt x]@=Q xm Ov=xOW =OH sy ". xm |rYi P. h L. Ov=xOW =OH sy. =Q. =. R= p U. 0.5. =. x t. =. R= x t. 6. 12 \. Q}}eD. w. OvwQ. P. h L R=. TB |v=tR. Q. Q |t s}UQD. 111. pmW. | U QO. =ov|oDU@ty. QO Q. pkDUt. July. D. D w. w x. U=. =. W. U. m y. QO. R=. |rY=. =. w 101. }= t. =iDU= =@ Q} R O)m. xO. pmW. m. QO. Q. TB Q=}at. =}at. QLv= xm O}vm|t x_Lqt. h=. QLv=. h=. w. CU=. 41 O. Q O. |= @ " vQ=O. 111. P. vwQ h L R=. w. =Hv=. |. TB 'Q=}at. h=. O. h y. CU= "Ov=xOW OQw Q@ Q. | U. pmW. TBU. QLv= "CU=. Q |t. "OO o. =ov|oDU@ty. QO Q. w@v QFw-t ,qt=m. xO. QDW}@ |UQQ@ Qo = "Ovm|t. 'O W s. sy. QO ". |iO=YD xirw-t. OyO|t u=Wv. Q}}eD. C=. |oDU@tyOwN X}NWD Q=ov|oDU@ty |rY=. = xirw-t x@ x} RHD =Q} R OUQ|tv CUQO Q_v x@ Q}@aD u}= "CU=. | yxO=O | U Q R=. u}@ |]N C@Ft. xm q=@ Q}O=kt p=Ft. CQ Y @ =Q x t xOR=wO ' y. Q}}eD K}LYD. C=. =DU@=D. MQ u. > data(AirPassengers) > AP <{ AirPassengers > AP.decom <{ decompose(AP, "multiplicative") > plot(ts(AP.decom$random7:138])) > acf(AP.decom$random7:138]) |rY Q}}e K}LY C pm | wv}U = rk | Q} x i C=. GwR. w x Q}O=kt 'CU= |Uwv}U ,=@} QkD |rYi. OvwQ xirw-t w K}LYD |rYi C=Q}}eD decompose() `@=D R= "OO o. (xt xt+12 ). CQ Y @. CU= u=DUtR lJwm Q}O=kt p=@vO x@ OyO|t. C=. w xm. U D. Q. OQw @. 109 Q Q June = @= @. D.

(21) |vWwOv|wUwt. 14. 0.90. 0.95. 1.00. ts(AP.decom$random[7:138]). 1.05. 1.10. 1391 '. 0. 20. 40. 60. 80. 100. 120. Time. AirPassengers |. =YD xirw-t. iO. O}vm xHwD Q} R. ". = O x@ =yQ=}at. | y ) m. > AP <{ AirPassengers > AP.decom <{ decompose(AP, "multiplicative") > sd(AP7:138]) 1] 109.4187 > sd(AP7:138] - AP.decom$trend7:138]) 1] 41.11491 > sd(AP.decom$random7:138]) 1] 0.0333884. %101. pmW. QLv= x@U=Lt. h=. Q CU=. |= @ ". 0.03 Q Q. @= @. |rYi.

(22) |v=tR. 15. 0.4 −0.2. 0.0. 0.2. ACF. 0.6. 0.8. 1.0. Series AP.decom$random[7:138]. 0. 5. 10. 15. 20. Lag. AirPassengers |. =YD xirw-t Q=ov|oDU@ty. iO. %111. pmW. =. Q. | y| U. 1. pYi.

(23) swO pYi. x}=B |iO=YD |=ypOt |}=yQorta ,=vt "OwW|t EL@ 'OQ=O |Oa@ pwYi QO |O=} R OQ@ Q=m xm |iO=YD |=ypOt R= |=xQ=B pYi u}= QO O QD?U=vt X}NWD. p t. Q |t QDK =w. "OO o. w. Q. OO. R. |= @ | } H Q= @= w. O@=}|t. Ovvm|t. =iDU= |r}N =yv. xO. QDW}@ \U@ R}v xDWPo. |. = EL@. | y. R=. |v=tR. =. Q. | y| U. = u}vJty. R= |=xQ B. xm. Q |t |iQat. OO o. Q}o|t Q=Qk |UQQ@. 1 O}iU xiwv. Ovm|t u=}@. xt |va. t. =. = O =t}k=@. } yx v. O. O =Wt. =. u tR QO =Q p t w x y. Q. w. | U CQ Y. u}=. Q. | yxO=O | U. CU=. QO ". Zw. u}@. Cw. =iD. R=. y. |W=v. Qit pOt \UwD ^t |va}. |. =]N O. O l}. 'p t. 1 1 2. O =t}k=@. QO x v. Q Q}O=kt. u x W OQw @. w. CU= Q} R. ". xt = yt y^t ;. 1. White noise 16. OQ t. 1 2. xtOkt. ". w. "O. yt. Q. | U. w x. CQ Y @.

(24) x}=B |iO=YD. 17. Tmavt Q=ov|oDU@ty. QO. |rYi. OvW=@ xDU@ty O}=@v =yxOv=t}k=@. Q}}eD =}. C=. Q. w. OvwQ Ovv=t |v=tR. Q. Q |t Q} R h} QaD ?Hwt xO}= u}= "OyO u=Wv. "OO o. 2. =. Q. = w xm OW xO}O u}vJty. wn. xH}Dv. p. w |t. u= D. w |t. wor= `wv I}y O}=@v =yv Q=ov|oDU@ty =Pr. |. w |t x=ov 'Ovm. u= D. w : t = 1 2. n. = Q}eDt Qo = 'Ov} wo 2 (DWN) xDUUo O}iU xiwv =Q f t. | y. qkDU= C}Y=N |w. 2 T =. OvW=@. R= ". v } Q=w w. Q}B p=tQv p=tDL= |r=oJ `@=D. QiY u}ov=}t =@ u=Um} `} RwD. g |v=tR |QU. pkDUt sy. |=Q=O w. CU=. ". m. m. W=O. Uw o. U. |a@. O \UwD xOW. =Q p t. = x}@W. Q. |R U. w |t. | U "O W. =iDU= =yxO=O 3 |R=Ux}@W. xO. Q R}=tDt xOW xOy=Wt. "OO o. Q. w |t. |= @ u R= u= D. rnorm(100). w. O}=tv O=H}= xOv}. =F. Q O = |t. p t |= @ " @ }. Ovm O=H}=. ". =Q. Q. |= @ =Q. = =. |}=yw} Q=vU Ov=wD|t. R. |@ wN@ Q=m u}=. u t U. OvDUy. =. | yxO=O R=. i v | U. 3 1 2. Q xDi=}. VR= @ p t. O =D. = Q. =F. wtv. QO "O. Q CU=. =iDU=. O. xO. =Um} `} RwD. OQ= v U= p t v u. |= @. |=Q=O ". O}it Ov} Qi = QDt=Q=B. p t | y. = Qy. QO. xm |vat u}O@ 'OW=@ u=Um}. = |}=D100 hrDNt. | y. = Q. | y= H=. = x}@W. |R U. =@ OwW. P. h L ". Q. = Qy. |= H= Q @. `@=D u}= Qo = =t=. CU=. 12. Q. `w W. pmW q=@. 2 T =. v } Q=w w. =v}t]=. =. xR @. = O x@ uwvm =. = O. xH}Dv. Q. w |t. v= D. | y ) m. set.seed() ` = 100 x = Q Q. u |= @. =Um} OOa. Q=ov|oDU@ty w swO. Tv=} Q=wwmwD= `@=D |=Q=O 'OW=@. u. O}vm xHwD Q} R. u. Q. |R U. 100 O. Q |t O=H}=. | y ) m |= H=. =. | yxO=O. xm pkDUt Q}eDt. x]kv xm CU=. "OO o. O. = x}@W. |. > set.seed(1) > w <{ rnorm(100) > plot.ts(w, type = "l") Q @. Q. =D Ov} wo R}v 4 |oDN=U. 'p t |= @ ". i o. x}@W. =. =. R=. Cor(wi wj ) = 0 i 6= j x C Q wt N (0 2) x C | = O}i x w Q. R QO |R U C kw=. w. 2 1 2. = Q}eDt Qo = "CU=. R= y. QO. "O W. h} QaD. w1 w2. pYi. = xirw-t xm OW x_Lqt p@k pYi. | y| U ,a v. |YNWt. =Q. O. | U | y. =t= 'OvDUy xDU@ty |v=tR. | U. =. | yp t. @ D. t v @ |= H=. Q |t pY=L. "OO o. x@DQt. =iDt. Cw. 4 1 2. X=wN. w. QiY u}ov=}t |=Q=O w xDU@ty=v w f t g CQwY x@ |iO=YD Q}eDt Qo =. 8 > < 2. (k) = > :0. CU= Q} R. ". k=0 k=0 6. Q. = }R. 2. Discrete white noise. Cov(wt wt+k ) = E (wt ; )(wt+k ; )]. 3. Simulation. 4. Synthetic.

(25) |vWwOv|wUwt. 18. −2. −1. 0. w. 1. 2. 1391 '. 0. 20. 40. 60. 80. 100. Time. O}iU xiwv |v=tR. Q. V}=tv. | U. %12. pmW. = E (wt) = E (wt+k ) = 0. w |t xH}Dv x=ov 'CU= 8 > < ( 2). "O W. E wt = a2 E (wt wt+k ) = E (wt)E (wt+k ) = > :0 0. k=0 k=0. . %R=. 8 > <1. x@ pta. QO. =t= OvW=@ QiY Q@=Q@ O}=@. |vat u}O@ u}=. w. 6. OvwW|t `k=w u=v}t]=. Q}@ 'Q}N-=D 20 w %5. O. k=0. u t uw. w. uO @ Q=O. :0. |=R=. x@ xOW. 6. CUDQ=@a |oDU@tyOwN `@=D. 6. = x}@W O}iU xiwv. |R U. =. | yxO=O. r. = pN=O =t= 'CU}v QiY ,=k}kO =y k Q}O=kt. xR @. |oDU@tyOwN `@=D. Q}oxvwtv. |. r. O}vm xHwD Q} R p=Ft x@ uwvm = "CU= p=mW= q@ R}v =y k. OQ=O O Hw ". U=. @ D. QO. r= "OQ=O. @ D. |oORuwQ}@ l}. Ovm|t x@U=Lt. }= ". |. 7 Q} Q_v. O}vm xHwD Q} R p=Ft x@. ". Q}}eD Cra. C=. |vat K]U x@ xHwD =@ |DL "OvQ=Ov QiY =@ |Q=O|vat Cw=iD =yv xm CU= ". > set.seed(2) > acf(rnorm(100)) w |a = R xD@ C ARMAacf() ` = u. w. k=0 k=0. k = cck = > 0. w. u J. =. N-D QO. xm 'CU=. w x. CQ Y @ =Q "O. > wn <{ ARMAacf(ar=0, ma=0, lag.max=20). wtv s}UQD. 22. pmW q=@. = O. w l}. R= OQ t. Q. | y ) m |= H=. xH}Dv. r. O}iU xiwv l} k |oDU@tyOwN `@=D xm. plot() ` =. =. w |t. @ D @ u= D. =Q. `@=D u}= pY=L.

(26) x}=B |iO=YD. 19. =. O. 2. pYi. | y ) m |= H=. xH}Dv. | yp t. 0.4 −0.2. 0.0. 0.2. ACF. 0.6. 0.8. 1.0. Series rnorm(100). 0. 5. 10. 15. 20. Lag. O}iU xiwv |oDU@tyOwN `@=D V}=tv. > plot(wn, type='h') QiY Q@=Q@ Q}O=kt =yQ}N-=D Q}=U. Q. |= @ w. CU= l} Q@=Q@. k=0. QO. %22. pmW. xm 'CU=. pmW q=@. 32. = O. Q. CU=. ". 5 |iO=YD uORs=o. 1 2 2. xtOkt. Q}o =Qi pOt. |. Q |DL ,qwtat "CU= |iO=YD. R= u VR= @. = O. Q |@U=vt. | y vwQ |= @. Q =@ = |iO=YD. VR= @ , r e. h} QaD. Q. CU= |iO=YD. xt. =. uORs o f. g x=ov "CU=. |v=tR. Q. | U. l}. xt. f. =. uORs o. ARIMA. CU= QD?U=vt. ". % o=. 2 2. g. w. u J. 2 2 2. xm O}vm. Q. Z i. xt = xt 1 + wt ;. =@ u}vJty. = xrO=at. w q @. QO. xt 1 = x t 2 + w t ;. x. ;. 1. ;. |v} Ro}=H =@ "CU= O}iU xiwv. w |t xH}Dv 6 wQUB |v} Ro}=H. % m O W. 5. Random walk. VwQ. xt = wt + wt 1 + wt 2 +. 6. Back substitution. ;. ;. u}= xt=O=. w. Q. wt x xt 3 |v Ro =. | U f. |r@k. QO. g u QO. ;. }. m. } H.

(27) |vWwOv|wUwt. 20. 0.0. 0.2. 0.4. wn. 0.6. 0.8. 1.0. 1391 '. 5. 10. 15. 20. Index. O}iU xiwv. |. Q_v |oDU@tyOwN `@=D V}=tv. P. w |t. %= r "O W. Q. `w W. t=1. %32. =. u tR R= w. xt = w1 + w2 +. pmW. CU}v C}=yv|@. w. Q. j i | U. pta. QO. + wt. w x Qorta l} u=wD|t u}=Q@=v@ "OOQo|t p=ta= R}v xO}J}B |v=tR |=y|QU |=ypOt |=Q@ wQUB |v} Ro}=H. CQ Y @. wtv h} QaD Q} R. "O. 7 wQUB p=kDv=. CU= Q} R. ". B (xt ) = xt. 3 2 2. Qorta. w x xm. CQ Y @. B Qort. a. 1. ;. =@ Q. O xDio Qorta Qo = "Ov} wo R}v Q}N-=D Qorta u x@ OQ=wt R= |=xQ=B QO xm 'OwW|t xO}t=v wQUB p=kDv= Qorta. , D t x W. =ov. %x. B n(xt ) = xt. Q. QmD. =. wvm =. 'OO o Q=. n. ;. w |t xDWwv Qorta u}= =@ |iO=YD. "O W. xt = B (xt ) + wt (1 B )xt = wt xt = (1 B ) 1 wt = (1 + B + B 2 + )wt xt = wt + wt 1 + wt 2 + ). ;. ). ). 7. Backward shift operator. ;. ;. ;. ;. uORs o u.

(28) x}=B |iO=YD. 21. |iO=YD. ". CU= Q} R. w x. CQ Y @. x = 0 x. @. k (t) = Cov(xt xt+k ) = Cov CU= Q} R. ". l}ORv R=. t X i=1. uORs=o swO. xHwD =@ |iO=YD. wi. t+k X j =1. !. wj =. x@DQt. =. X=wN. 4 2 2. = Ov} Qi. i=j. swO. w x |oDU@tyOwN ?} Q uwvm = "CU= =DU}==v Ov} Qi u}=Q@=v@ 'CU= u=tR. R=. Ov)m |r}N. }= @ @ ". w x. U=. J m |=. L. t. w Oy=wN C@Ft Q}O=kt. CQ Y @ w O @. @ k CQ Y @. |=Q=O. @. |iO=YD. v. Q=. =. Q. w. X= N. |a@=D Tv=} Q=wwm. 2 1 p k (t) = p Cov(xt xt+k ) = p 2 t = Var(xt ) Var(xt+k ) t (t + k)2 1 + k=t k Ok u Q =v C QDm w x_ q p = w x t x C@U k Ok = Qy t. x@DQt. Cov(wi wj ). CQ Y @. Q=. pYi. 2. uORs o. X. O. | yp t. t p L. R. =. t. @ nQ @ | y. =ov|oDU@ty. uORs o |= @ Q. w. Q. |= @. CU= l} x@. O = |t Vy=m l} Q=Okt. " @ }. 8 p=iD. x. 5 2 2. Qorta. xm OW=@ |iO=YD uORs=o f t g |QU Qo = p=Ft |=Q@ "Ovm p}O@D =DU}= |QU x@ =Q =DU}==v |QU Ov=wD|t Qorta u}=. xt xt 1 = wt ;. ;. w x. CQ Y @ u. x. x]@=Q xm Ovm|t O=H}= =Q O}iU xiwv |QU f t g pw= x@DQt p =iD 'CU= =DU}==v CU=. ". w |t h} QaD Q} R. "O W. xt = xt xt. r. Q. ;. =} QUB p=kDv= Qorta ?UL Q@ Ov=wD|t. "OO o u @ w. r. 1. u}=Q@=v@ "CU=. r. xt = (1 B )xt x. CU= Q} R. n. |va} p =iD Qorta. ;. ". r. w x. CQ Y @ r. ;. m. O}W=@ xDW=O xHwD. w x p =iD Qorta QDq=@. CQ Y @. = x@DQt. | y. = (1 ; B )n |R=Ux}@W. 6 2 2. w |t Qy=_ =yQ=Owtv QO pOt |rY= C=}YwYN w CU= O}it |v=tR |QU l} xar=]t |=Q@ |R=Ux}@W ,=@r=e. "O W. wkr=@ u=mt= l}. x. wvax@ Ov=wD|t pOt. u=. =Wv OwN. 'O=O u. R= =Q. x@=Wt. =} wYN |N} Q=D. C Y. xO=O. xm |Dkw u}=Q@=v@. Q |iQat. "OO o ". > set.seed(2) > x <{ w <{ rnorm(1000) 8. Di

(29) erence operator. Ovm. = x}@W. |R U. =Q. |iO=YD. = l} Ov=wD|t Q} R. uORs o. Q. u |= @. = O. | y ) m.

(30) 1391 '. |vWwOv|wUwt. 22. > for (t in 2:1000) xt] <{ xt - 1] + wt] > plot(x, type = "l") = for xkr Q| | Ok x x =vt. uORs o. xkrL. L "OO o. t. yOQ=. xm Owtv O=H}= |}=yO)m =@. R= u QO. @ ,. t. =Q. w. |iO=YD. OyO|t C@Uv =. = x}@W. uORs o |R U. wx. O}iU xiwv. @ =Q. Q. wDUO u}rw=. | U Q. w |t xD@r= "Ovm|t O=H}=. u= D. =Q. |iO=YD. O = OWv. " W @ x. =iDU=. xO. > set.seed(2) # so you can reproduce the results > v <{ rnorm(1000) # v contains 1000 iid N(0,1) variates > x <{ cumsum(v) # x is a random walk > plot(x, type = "l") Q. =ov|oDU@ty. CUOx@. uOQw. Q CU= xOW. |= @ ". =Wv. xO=O u. 42. pmW. O. =H}=. QO x W O. Q. | U |wQ. Qy@. 0. 20. x. 40. 60. | U Q. 0. 200. 400. 600. 800. 1000. Index. |iO=YD. = V}=tv. uORs o. %42. pmW. w |t xOy=Wt 52 pmW QO xH}Dv TBU "OOQo|t =QH= Q} R QwDUO 'p@k |=yxt=vQ@ xt=O= QO 'xOW |R=Ux}@W. "O W. > acf(x) Q |t O}iU xiwv. "OO o. p =iD Qorta p=ta=. '. 62. pmW. QO. =. Q. | yxO=O | U. x@ p}O@D p =iD l}. Q CU= jO=Y R}v xOW. |= @ ". xH}Dv TBU. = x}@W. |R U. w |t =QH= Q} R QwDUO 'Q}N=. "O W. QO. |iO=YD Q. = O. uORs o ' W. w. x_Lqt ,q@k xm Qw]u=ty. O xDio ?r]t xm O}O O}=@ uwvm =. | U OQ t QO x W. = x = Q xt=O=. | y t v @. CU= OwHwt. QO ". R di() ` = QO. @ D. w |t xOy=Wt. "O W. > acf(di(x)) u. =v}t]=. =. xR @ R=. |oORuwQ}@. w. w |tv x_Lqt Q=ov|oDU@ty. OQ t wO "O W. Q |YNWt. |= @. wor= 'Q}N= pmW. |. QO. = x}@W |QU CiQ|t Q=_Dv= xm Qw]uty u}=Q@=v@ "CU}v =vDa= p@=k %5 uOw@ Q=O|vat K]U QO xm OQ=O OwHw. |R U.

(31) x}=B |iO=YD. 23. =. O. | yp t. pYi. 2. 0.0. 0.2. 0.4. ACF. 0.6. 0.8. 1.0. Series x. 0. 5. 10. 15. 20. 25. 30. Lag. |iO=YD. =. =ov|oDU@tyOwN V}=tv. %52. uORs o Q. Ovm|t. ". pmW. Q}B |iO=YD. |w. = Ov} Qi l}. uORs o. w}UQoQwD=. AR(p). =YDN= x@. Q. xt = 1 xt 1 + 2xt 2 + ;. 1 3 2. =kDv= Qorta ?UL Q@ Q}N= xrO=at Qo = "OvDUy. p = 0 = 6. D t. w}UQoQwD= Ov} Qi. O. = QDt=Q=B. @ p t | y. i. w. O}iU xiwv. w Oy=wN Q} R. ;. ;. ;. ;. w}UQoQwD=. u. x@ ?@U u}ty x@. w. 1 = 1. CU= xDWPo. u QO. =. xm CU=. =. | yu tR QO. wt. f. w x. g u QO. Q. =} QUB. AR(1). x. =. X N. Cr=L |iO=YD. x. =y x@ C@Uv t uw}UQoQ `k=w. =mv x@ uwvm =. C. =. uORs o . O u}=. QO p t. Ov} wo. ". = CU= Q@=Q@. % @. x^t = 1xt 1 + 2xt 2 + ;. ;. xm. pB p)xt = wt O}vm xHwD Q} R. O = |t. Q. | U. CQ Y @ 'OO o u @ w. ". " W @. |v=tR. g. ;. "O @. p(B )xt = (1 1B 2B 2. xt. =Q f. + pxt p + wt. ;. p. p x@ Q. O. x W. 3 2. |=ypOt h} QaD. Qo = 'Ov} wo. x. R=. + t pxt ;. t. =. Q. u tR QO OQw @ .

(32) 1391 '. |vWwOv|wUwt. 24. 0.0. 0.2. 0.4. ACF. 0.6. 0.8. 1.0. Series diff(x). 0. 5. 10. 15. 20. 25. 30. Lag. p=iD p=ta= =@ |iO=YD. =. =ov|oDU@tyOwN V}=tv. uORs o Q. Q |t. "OO o. Q =]N. OQw @. =a@ Qt pk=OL. C. w}UQoQwD=. O Qo = "OW=@ \rDNt =}. Q k. = Ov} Qi. uORs o. u}}aD. w. |k}kL. = xW} Q. | y. CU= =DU}= Ov} Qi x=ov. QO ". Q OL=w. 'OO o. R=. Q Q} R p=Ft Q=yJ x@ uwvm = "CU= =DU}==v Ov} Qi. ;. 1 2. B=0x. QUB p=kDv= Qorta. w. w O W. t. xrtHOvJ u}= xW} Q "OwW|t xH}Dv QDoQR@. R=. w. w. = QDt=Q=B. ;. m. m. 2 3 2. |}=DU}=. p(B ) = 0 x. =Q. J | y. }Q s. ;. } i. w x. CQ Y @. }. }= v w. AR(1). CU= QDoQR@ OL=w. R=. }. t. ; ". R=. |m} =Q} R 'CU= =DU}==v t. ;. } ;. ;. U=. ". |va}. U=. | y. L=w R=. 1 + 41 B 2 = 0. } Q |=Q=O. oQ @. ;. rO. t. ;. w. ;. |=. t Q k = } R "OO o. xt =. B p. ;. t. x. }. 1 4 t;2. }= v. + wt. v=. H. @. p t 3. a R= xO. J "O W. U. Q=. w x. CQ Y @. }=. t. L. CQ Y @. ;. ;. = xW} Q =Q} R 'CU= =DU}=. | y. ;. }. O. x = 21 xt 1 + 12 xt 2 + wt w x AR(2) O 1 1)(B + 2)xt = wt |va 12 (B 2 + B 2)xt = wt QU =kD Qort =iD = 2 (B C B = 1 2 = xW. (B ) = 21 (B 1)(B + 2) xrt Ov w | xH}D C O QD R B = 2 jr] O Q Q | | =DU = ?@ B = 1 Ok \k CU= OL=w =yxW} Q. ". D. p t 1. rO. ;. iO. p t 2. ;. UO @. t. xm CU=. CQ Y @. ;. }. m. =at. rO. D. ;. U=. H D. ;. ;. CU=. | y. x = xt 1 41 xt 2 + wt w x AR(2) O 1 2)2 xt = wt |va 41 (B 2 4B + 4)xt = wt x =a 4 (B B = 2 x O | C x (B ) = 14 (B 2)2 x =a p =. =iDU= =@ =Q} R "CU= =DU}= t. |=. O = |t OL=w. H. x = 12 xt 1 + wt. =at xW} Q =Q} R 'CU= =DU}= t. rO. R= xO. " W @. |}=DU}==v. |=. ". w. x@ pOt. B ) xrt Ov = xW =t jr] | B = 1 x C = 1 B | =Y O}v x w AR Ov Q | =DU = | =DU. QDW}@ p (. ". @= @. VwQ. Ov=wD|t xm Ovt=v xYNWt xrO=at. |=Q=O. |= @. B=2QQ 1. QO. pmW. %62. U= @. t. t. AR(2). v. i. O. p t 4.

(33) x}=B |iO=YD. 25. 2QQ. = xW} Q. @= @ y. R=. l} Qy jr]t QOk "OW=@|t. i=. p ;. 1. w. CU= \rDNt. O. O= a=. xm OvDUy. CU= OL=w. ". xrO=at O. x W. = xW} Q. | y. xDio `@=D =@. w |t xm. u= D. =yJ. sQ. w. OQ=O O Hw. w. w. polyroot(). =F. w s U OQ= t 'p t u=. wvax@. = x@ |a@=D. s v. Q |UQQ@. "O m. =Q. R. QO |=. =. 2. pYi. B = 2i . QDoQR@ xm OW=@|t. R=. xrtHOvJ. =y|QU |}=DU}=. O. | yp t. = xW} Q x@U=Lt. | y. wtv x@U=Lt. w O. Q. |= @. xYNWt. =Q. w |t x@U=Lt. "O W. w. =F. %s U p t. > polyroot(c(-2,1,1)) 1] 1-0i -2+0i. %R=. OvDQ=@a =yxW} Q jr]t QOk. > Mod(polyroot(c(-2,1,1))) 1] 1 2. =yJ p=Ft. %sQ. > polyroot(c(1,0,1/4)) 1] 0+2i 0-2i. %R=. OvDQ=@a =yxW} Q jr]t QOk. > Mod(polyroot(c(1,0,1/4))) 1] 2 2. AR(1) p t swO O. ". w x. CU= Q} R. CQ Y @. AR(1). O. x@DQt. O. 3 3 2. X=wN. O =Wt ,q@k xm Qw]u=ty. p t ' W x y. xt = xt 1 + wt ;. xm. =Wv. O=O u. w |t "OW=@|t. u= D. 2 T =. v } Q=w w. QiY u}ov=}t =@ O}iU xiwv. wt. f. g u QO. xm. x = 0 k = k 2=(1 2) ;. xm CWwv. w |t. u= D. < 1 Ov Q. j. j. } i. |}=DU}=. Q. \ W w. B Qort. =iDU= =@. a R= xO. (1 ; B )xt = wt wtv x@U=Lt. "O. xt = (1 B ) 1wt ;. ;. = wt + wt 1 + wt 2 + ;. 2. ;. =. 1 X. i wt. i. ;. i=0. =Q. xt. w |t uwvm =. u= D.

(34) 1391 '. |vWwOv|wUwt. 26. ". . 1 X. E (xt ) = E. !. i wt. =. i. ;. i=0. 1 X. O}vm xHwD u}ov=}t x@U=Lt x@ uwvm =. iE (wt i) = 0 ;. i=0. CU= Q} R KQW x@ Tv=} Q=wwmwD= !. ". k = Cov(xt xt+k ) = Cov =. 1 X i=0. X. j =k +i. = k 2. wt i i. 1 X. ;. j =0. O}vm s}UkD Tv=} Q=w Q@. =Q. ;. X1. i=0. QDW}@ CaQU =@ QDmJwm. |. O |t. 72. " }. pmW. QO. 2i = k 2=(1 2) ;. Ov}Qi. 4 3 2. Q=ov|oDU@ty. =. t w. Q. |= @. (k 0). | y. |iv. ;. Tv=} Q=wwmwD= xm CU= |i=m |oDU@tyOwN `@=D x@U=Lt. k = k p}t QiY CtU x@. j. ;. i j Cov(wt i wt+k j ). AR(1) ". j wt+k. |=R=. x@ Q=ov|oDU@ty u}=Q@=v@ "CU=. C@Ft Q}O=kt. Q =ov|oDU@ty. |= @ Q. j. <1 j. =F xt=O=. wO p t. u QO. xm. Ovm|t. QO ". > plot(0:10, rho(0:10, -0.7), type = "b", xlab="lag k") > rho <{ function(k, alpha) alpha^k > par(mfrow=c(2,1), mar=c(4,4,2,4)) > plot(0:10, rho(0:10, 0.7), type = "b", xlab="lag k",. + ylab=expression(rk]), main=expression(alpha==0.7)) > plot(0:10, rho(0:10, -0.7), type = "b", xlab="lag k", + ylab=expression(rk]), main=expression(alpha==-0.7)) > abline(h=0, lty=2) 9. x. x@ \ki t Qo = |DL 'OvDUy QiY hr=Nt =yQ}N-=D s=tD. |} RH |oDU@tyOwN `@=D. Q =y|oDU@tyOwN Q}O=kt. |= @. k. k Q}. '. N=. 5 3 2. xrO=at j@=]t. x. =y|oDU@ty QF= xm CU= |oDU@ty R= |awv ' Q}N-=D QO |} RH |oDU@tyOwN `@=D "OW=@ xDW=O |oDU@ t;1 w Oy=wN QiY. "O @. CU=. '. AR(k). k>1. s. O xDi=}. p t. =tD. AR(1) | R ? Q u} k =. Q. |= @. Q. VR= @. } H. } . t=. @. |oDU@tyOwN `@=D p=Ft. |oDU@tyOwN `@=D Q=ov|oDU@ty u}=Q@=v@ "CU= QiY w. x@U=Lt. Q. |= @. pacf(). = x@ |a@=D. s v. R. Q}o Q=Qk. QO "O. k Q} = QQ k>p. Q@=Q@. N-D QO. @= @. =iDU=. xO. w. OQ t. Q. P. QDmJwm Q}N-=D =@. |} RH |oDU@tyOwN `@=D '|rm Cr=L. O w | AR Ov Q |=R= v= D. QO. x@ k ?}=Q AR(p) Ov} Qi QO |va} t. w. "OQ=O O Hw. 9. Partial autocorrelation function. Q. |= @ "OO o h L. } i. x@DQt u}}aD. Q |} RH. |= @. |} RH |oDU@tyOwN `@=D s}UQD.

(35) x}=B |iO=YD. 27. =. O. | yp t. 2. pYi. 0.0. 0.4. rk. 0.8. α = 0.7. 0. 2. 4. 6. 8. 10. 8. 10. lag k. 0.0 −0.5. rk. 0.5. 1.0. α = − 0.7. 0. 2. 4. 6 lag k. = 0:7 ;0:7. Q. |= @. AR(1) Ov Q. =ov|oDU@ty V}=tv. } i Q. %72. pmW |R=Ux}@W. QO. |} RH |oDU@tyOwN. w. |oDU@tyOwN `@=wD. |oDU@tyOwN Q=ov|oDU@ty QO. k>1Q. =kt. }O. w. Ov} Qi. O. = x}@W Q} R. w x W |R U. w x. CQ Y @. R. QO. 6 3 2. AR(1). O. p t. Q O}vm|t x_Lqt xm Qw]u=ty "OOQo|t s}UQD 82 pmW. |= @. O. w. |vat |oDU@ty '|} RH. "OQ= v O Hw |Q=O. xOW |R=Ux}@W |=yxO=O Q@. x.ar. Q. Q. R= xO. =iDU= =@. | U |= @. u. w}UQoQwD= pOt 'Q} R |=yO)m. u. xDi=}. VR=Q@ |=ypOt. 7 3 2. xDi=}. VR=Q@ |=ypOt. 8 3 2. O = |t VR=Q@ =yxO=O Q@ ar() `@=D \UwD R QO. QO " @ }. 10 |@v=Ht Tv=} Q=w xm 'O@=}|t. Q O. VR= @ x W xO=O. QDt=Q=B. w. %95 u=v}t]=. w |t. "O W. > set.seed(1) > x <{ w <{ rnorm(100) > for (t in 2:100) xt] <- 0.7 xt - 1] + wt] > x.ar <{ ar(x, method = "mle") > x.ar$order pOt. x@DQ. pOt. QDt=Q=B. 1] 1 > x.ar$ar 10. asymtotic. AR(p). = = O. O. p t. = x}@W. xR @ @ x W |R U. QNDU=. G=. x.ar$asy.var.

(36) |vWwOv|wUwt. 28. −1. 0. x. 1. 2. 3. 1391 '. 0. 20. 40. 60. 80. 100. 0.6 0.2 −0.2. ACF. 1.0. Index. 0. 5. 10. 15. 20. 0.2 0.4 0.6 −0.2. Partial ACF. Lag. 5. 10. 15. 20. Lag. V}=y|oDU@tyOwN. w. AR(1) Ov Q. V}=tv. Q@ 'OW. =iDU= =yO)m. } i. %82. pmW. 1] 0.6009459 > x.ar$ar + c(-2, 2) sqrt(x.ar$asy.var) pOt. QDt=Q=B. u=v}t]= xR=@. 1] 0.4404031 0.7614886 ?. =NDv= "CU= Q=wDU= |}=tvDUQO QFm =OL `@=D. =. T U=. Q Ov} Qi. xO. QO VR= @. x]@= x@U=Lt "CU= pOt QDt=Q=B pk=OL Qov=}@ xm OwW|t s=Hv= 12 AIC x]@= ". QO. xm 11 mle. Ov} Qi x@ \w@ Qt. R= '. CU= Q} R. p. VwQ. Okt. Q=. w x O. CQ Y @ x W. xDio. AIC = ;2 log-likelihood + 2 number of parameters CUQO x@DQt Q=. Okt. R=. =. = O xm O}vm xHwD "Ovm|t Q=}DN=. 'q @ | y ) m. ^ = 0:6 Q Q AR(1). QDmJwm xm 'Ot CUOx@. O. w. "OQ= v O Hw. E}L u}=. @= @. Q. Q. | U. w |t O}OQD =@ "CU=. u= D. =@ =aO= u}=. = =. O QDt=Q=B. p t. AIC u QDm }. Q. wtv |@=} R=@. OQw @ "O. Q |t pOt QDt=Q=B Q=Okt pt=W %95 u=v}t]=. x =Nro QF=. |= v. R=. 11. Maximum likelihood estimation. AR p t %. |W=v xm OyO|t u=Wv. Q CU= xOW. |Q oR U |= @ ". VR=Q@. P. h L. w =. O. =. =Q tO. |v=yH. 12. Akaike Information Criterion. R=. =Q. = =t=. xR @. |W=v. 1970 |=x. @ D QO. } p t. ". U=. |=tO |QU. OvwQ V}=Ri=. |iO=YD xO}OB l} p=kDv=. ar() ` = p = 1 |va O C = 0:7 O. J m @ p t '. R= |O= }= w OO o. xDi=}. =aO=. Q u} QDy@. =Q VR= @. =. 9 3 2. Q. p U R= | U. u}=. Ov}=Ri OvwQ xm Owtv.

(37) x}=B |iO=YD. 29. "O. wtv. =iDU= u}at `@=wD. xO. R=. xm. O. wtv. u uw @ xO. O}vm xHwD xvq=U. ". =. =. O. |R Up t =Q. Q. | tO | U. =. O. | yp t. 2. pYi. OvwQ xirw-t CU= umtt '=yxO=O. u}ov=}t x@ xDi=}. Q. VR= @. AR. O x@. p t. > www <{ "http://www.massey.ac.nz/~pscowper/ts/global.dat" > Global <{ scan(www) > Global.ts <{ ts(Global, st = c(1856, 1), end = c(2005, 12), fr = 12) > Global.ar <{ ar(aggregate(Global.ts, FUN = mean), method = "mle") > mean(aggregate(Global.ts, FUN = mean)). −0.2. 0.0. 0.2. 0.4. ACF. 0.6. 0.8. 1.0. 1] -0.1382628 > Global.ar$order 1] 4 > Global.ar$ar 1] 0.58762026 0.01260253 0.11116731 0.26763656 > acf(Global.ar$res-(1:Global.ar$order)], lag = 50, main=""). 0. 10. 20. 30. 40. 50. Lag. =yxOv=t}k=@ Q=ov|oDU@ty V}=tv. Q}N-=D. QO. rk O. Okt RH x@ =Q} R "OyO|t u=Wv. Q=. |t Q_v x@ ?U=vt '=yxO=O Q@. " UQ xO=O. CUOx@. O. = QDt=Q=B. =Q p t | y. w. =Q. O}iU xiwv. AR(4). xvq=U \UwDt. =. O. Q. | U. %92. pmW. l} '=yxOv=t}k=@ Q P O. Q. 92. p t VR= @ = r " vQ=O Q= k u. | tO. u}ov=}t Q=Okt xm Q}N=. pmW Q=ov|oDU@ty. =v}t]=. =. xR @ QO. =yv x}k@. 27. = O G}=Dv x@ xHwD =@ uwvm =. | y ) m. CWwv. ". w |t 'CU=. u= D. x^t = 0:14+0:59(xt 1 +0:14)+0:013(xt 2+0:14)+0:11(xt 3 +0:15)+0:27(xt 4 +0:15) ;. ;. ;. ;. ;.

(38) swU pYi. =DU}= |=ypOt O}m =. ,=. xt. =Q f. g. |v=tR. Q. w |t. | U "O W. |. Q}o}B EL@. xt=O=. u. |va} 'Ovmv Q}}eD u=tR. f (xt1 xt2 Cov(xt xs) T =. w. u. wvm = "OW KQ]t |@r=]t |}=DU}= x@ `H=Q ,q@k. =kDv= QF=. QO p. xt ) = f (xt1 +h xt2 +h. xt +h). k. w |t pt=W R}v =Q u=tR. v } Q=w m w O W. QO. w `} RwD `@=D Qo = Ov} wo 1 =DU}=. Q. QO | U s= D. k. C@=F Tv=} Q=w w u}ov=}t 'Omw-t |}=DU}= xm O}vm xHwD O = xDW=O |oDU@. " W @. lQLDt X=wN w. xrtH. quQ. } N. u}vJty. w. O}iU xiwv. =. |Q H. xrtH. R=. q x@ Q. CU= Q} R. w x. ". 1. Strictly stationary. xt = wt +

(39) 1 wt 1 + ;. 30. u}ov=}t. h} QaD. |]N ?}mQD. D t R=. CQ Y @ u. +

(40) q wt. k= t s. q. ;. ;. j. Q}N-=D x@ \ki. |=ypOt. MA(q). %. (MA). j. Ov}Qi. 1 3 1 1 3. QLDt u}ov=}t Ov} Qi. l. x]@=Q "OW=@|t O}iU xiwv |r@k.

(41) =DU}=. 31. =kDv= Qorta ?UL Q@. p. w |t. u= D. . 2 = xrO=at "CU= w Tv=} Q=w. =Q q @. =. QiY u}ov=}t =@ O}iU xiwv. w. O. | yp t. wt. f. g u QO. wtv |U} wvR=@. "O. xt = (1 +

(42) 1B +

(43) 2B 2 +. =DU}= CqtH R= |y=vDt `tH pt=W MA |=yOv} Qi xm u}= x@ Q_v "CU= w. =. x@ C@Uv Tv=} Q=wwmwD=. w. B. '. w. xm. QUB. +

(44) q B q )wt = q (B )wt. |. "OQ=O O Hw u tR. pYi. 3. u}ov=}t. =@. QO C F w. qx. xrtHOvJ q. HQO R= |=. u QO. xm. CU= Q=QkQ@ |}=DU}= =Pr 'CU= O}iU xiwv. CU= x@U=Lt p@=k |oO=U x@. ". xt. f. g. Tv=} Q=w. w. u}ov=}t. E (xt ) = E (wt +

(45) 1 wt 1 + +

(46) q wt q ) = E (wt) +

(47) 1 E (wt 1) + +

(48) q E (wt q ) =0 ;. ;. ;. ;. OvDUy QiY Q@=Q@ =yu}ov=}t. ". R=. l} Qy =Q} R. Var(xt ) = Var(wt +

(49) 1wt 1 + +

(50) q wt q ) = Var(wt) +

(51) 12 Var(wt 1) + +

(52) q2 Var(wt q ) = w2 (1 +

(53) 12 + +

(54) q2) ;. ;. ;. ". OvDUy pkDUt Qo}Om} ". 8 > > > 1 > > > q ;k > < i i+k > > > > > :. C. x. qtH pt=W t;1. w. xt. Q. =Um} Tv=} Q=w. CU= Q} R QwYx@. k 0. qtH s=tD =Q} R. |=Q=O C. Q |oDU@tyOwN `@=D. |= @. i. i=0. = }R '. u. k = 1 : : : q. q. k>q. 0. O}iU xiwv pkDUt. w2. R= w. k=0. P (k) = > P 2 > i=0. ;. k>q. CU= QiY Q@=Q@ `@=D. Q CU=. |= @ ".

(55) 0 = 1. CU= QiY =yv Tv=} Q=wwm u}vJty. ". xrtH. O |y=vDt=v x@DQt =@ =DU}= w}UQoQwD= Ov} Qi ?UL Q@. uw @. w. =} Q} R. "O W u @. wD@ Qo = Ov} wo 2 Q}PBuwQ=w. =Q u u=. x = (1

(56) B )wt x]. w x Ov=wD|t t. CQ Y @. ;. @=Q. =@. MA(1) Ov Q. =F. =Q. u QO w. xm. OvDUy. MA Ov Q. } i. Q CWwv =]N. } i 'p t |= @ ". wt = (1

(57) B ) 1xt = xt +

(58) xt 1 +

(59) 2xt 2 + ;. ;. ;. Ovm. ". OL=w. R=. QDoQR@ |oty. 2. invertible. (B ). O. j Y. ;. |}=Qoty. Q. \ W. = xW} Q jr]t QOk Qo = CU= Q}PBuwQ=w Ov} Qi l}. | y. =D CU=.

(60) <1. j. j. MA(q) Ov Q. u QO. xm. |rm Qw]x@. } i '. OvW=@. ".

(61) 1391 '. |vWwOv|wUwt. 32. R | yp. |R=Ux}@W w Q=ov|oDU@ty %. w. xkrL. =iDU= =@ |a@=D Q=m u}=. R= xO. Q. wtv. |= @ "O. R. =}. xO B. w |t. QO u= D. =Q. MA(q) Ov Q. =. 2 1 3. =Ft. Q |oDU@tyOwN `@=D. } i |= @. w |t h} QaD. "O W. Q. \ W. > rho <{ function(k, beta) f + q <{ length(beta) - 1 + if (k > q) ACF <{ 0 else f + s1 <{ 0 s2 <{ 0 + for (i in 1:(q-k+1)) s1 <- s1 + betai] betai+k] + for (i in 1:(q+1)) s2 <{ s2 + betai]^2 + ACF <{ s1 / s2g + ACFg Ov Q Q Q = O C x@ =L p = MA(q) Ov Q Q |oDU@t w ` = = = O =iD = C pm w x

(62) 3 = 0:2

(63) = 0:5

(64) 1 = 0:7 = QD = = MA(3) > beta <{ c(1, 0.7, 0.5, 0.2) > rho.k <{ rep(1, 10) > for (k in 1:10) rho.kk] <{ rho(k, beta) > plot(0:10, c(1, rho.k), pch = 4, ylab = expression(rhok]), xlab="lag k") > abline(0, 0) }R | y ) m ". U. U= 13. t. @ k. } i |= @. W CQ Y. @. yO N. w. @ D 'q @ | y ) m R= xO. '. | y. 0.6. 0.8. 1.0. ". U=. 0.0. 0.2. 0.4. ρk. } i |= @. 0. 2. 4. 6. 8. lag k. MA(3). O. =ov|oDU@ty V}=tv. p t Q. %13. pmW. 10. t=Q B. @. U=. @.

(65) =DU}=. 33. w. |v=tR. Q. | U "O. Q}o|t Q=Qk. =iDU=. xO. w. =ov|oDU@ty s}UQD. OQ t Q. w. MA(3) Ov Q. =. O. | yp t. = x}@W. } i |R U. 3. Q Q} R. |= @. pYi. = O. | y ) m. |oDU@tyOwN `@=D Q=Okt xU u}rw= 'CiQ|t Q=_Dv= xm Qw]u=ty "CU= xOW s}UQD 23 pmW QO Q=ov|oDU@ty O. Q. " vQ=O Q= k u. =v}t]=. =. xR @ QO. CU=. ". |. r. 3. =y k Q}O=kt. Q}oxvwtv. '. Q}}eD. C=. R= R=. QDW}@. = Q}N-=D. | y. Q. |= @ w. OvDUy QiY. R= |Q=O. 4 2 0 −2. x. −4. 200. 400. 600. 800. 1000. 0.4 0.0. ACF. 0.8. time t. 0. 5. 10. 15. 20. 25. 30. Lag. u Q. =ov|oDU@ty. w. Cw. =iD. |=Q=O. |W=v xm OW=@ xDW=O OwHw |oORuwQ}@ CU= umtt sy %5 xD@r=. > set.seed(1) > b <{ c(0.8, 0.6, 0.4) > x <{ w <{ rnorm(1000) > for (t in 4:1000) f + for (j in 1:3) xt] <{ xt] + bj] wt - j] +g > par(mfrow=c(2,1), mar=c(4,4,2,4)) > plot(x, type = "l", xlab="time t") > acf(x, main=""). 0. |vat. MA(3). = x}@W Ov}Qi V}=tv. |R U. %23. pmW.

(66) 1391 '. |vWwOv|wUwt. 34. xDi=}. VR=Q@. xOW |R=Ux}@W |QU. = w. =@ pOt x@DQt xm. u t oQ. QUm. u}ov=}t. =Q. w. OQ=O O Hw. Q V}B. Z i. arima(). w x. O. CQ Y @ x W. R. = x@ |a@=D. s v. xDio `@=D. '. O=. x@ xDi=}. Q =. ar() ` =. qN Q@. Q |t. "O W. Q 3 <=O@t. MA(q) O order=c(0,0,q). p t. Q xrtH. OQw @. 1 2 3. VR=Q@ pOt. w |t s}_vD. @ D h. 2 3. |=ypOt. Q@ Ov=wD|t. QO " @ } VR= @ yxO=O. "OO o C. MA. R= Z a. Ovm|tv. QO w. =a@ Qt `wtHt '=yQDt=Q=B OQw Q@ |=Q@ arima() `@=D "OvwW|t OQw Q@ |OOa sD} Qwor= l} \UwD pOt |=yQDt=Q=B O =. " W @. = QDt=Q=B. ' y. Q} R. |=R=. w x. CQ Y @. x@ xm CU=. method=c("CSS") = w x | w w u x MA(q) Ov Q Q Q | Q Q | x@ =L Qm Q wt = O =t} = u t oQ. CQ Y. w^t |va. }=. @. DQ Y QO. } i VR= @ |= @. yx v. } VOQw @ w. m. k @ "OO o. t. U. xD@r=. = |t pk=OL. OR U. =Q. |]QW. =a@ Qt `wtHt sD} Qwor= K} wD. ] W C. t Q. w x =yxOv=t}k=@. =a@ Qt `wtHt. t Q ] @. C. CU=. ". S (

(67) ^1 : : :

(68) ^q ) = Q =kt. R= | }O. O. '|O a |. n X t=1. w^t = 2. n n X. wHDUH "OvW=@ QiY =@ Q@=Q@ '=yv Q=QmD. pt=W Q. . Q Q}N= xOW. VR= @. = x}@W. |R U. coeff: 2 s.e. of coeff: =@ Qk , }. . D O. =. | yxO=O. Q}o|t Q=Qk. ;. w^0 : : : w^t. Q. u. Q@. \w. =a@ Qt `wtHt xm Ovm|t u}at. =Q. =yQDt=Q=B. `w W QO. Q pk=OL q=@. = QDt=Q=B "O@=}|t. +

(69) ^q w^t q ). ;. t=1. "OO o | y. o2. xt ; (

(70) ^1w^t 1 +. q Q}O=kt xm. ;. C. x.ma %95 =v}t x@. u. QLDt u}ov=}t. 'l. =. ]= xR @ QO. O. Q. QWt. = O. p t ' }R | y ) m QO. xt=vQ@ |HwQN. O. Q. QO x W OQw @. =_Dv= xm u=vJty ,=vt "CU= xOW xO=iDU= =yv R= |R=Ux}@W QO xm OwW|t (0.8,0.6,0.4) |=yQDt=Q=B Q}O=kt O QiY =@. "OQ= v. include.mean=FALSE = w = u Q C include.mean=TRUE > set.seed(1) > b <{ c(0.8, 0.6, 0.4) > x <{ w <{ rnorm(1000) > for (t in 4:1000) f + for (j in 1:3) xt] <{ xt] + bj] wt - j] +g > x.ma <{ arima(x, order = c(0, 0, 3)) > x.ma `@=D. QO. u t oQ R= Q m. ". U=. Call: arima(x = x, order = c(0, 0, 3)). 3. Intercept. |Q=O. |vat. Cw. =iD. wtv s}_vD QiY Q@=Q@. }= |= @ "O. Q V}B u}=Q@=v@. Z i. <= =Q. O@t. u}ov=}t Q=Okt. w |t. "O W. Q CiQ|t. R= Z a. =iDU=. xO. w |t. u= D. arima().

(71) =DU}=. 35. =. O. | yp t. pYi. 3. Coefficients: ma1. ma2. ma3. 0.7898. 0.5665. 0.3959. s.e.. 0.0307. intercept. 0.0351. -0.0322. 0.0320. sigma^2 estimated as 1.068:. 0.0898. log likelihood = -1452.41,. ARMA. Ov}Qi. aic = 2914.83. 3 3. %|@}mQD |=ypOt. 1 3 3. h} QaD. CU= Q} R. ". w x. CQ Y @. p x@ Q. D t R=. w}UQoQwD= Ov} Qi. xt = 1 xt 1 + 2xt 2 + ;. MA AR Ov Q x |D = O (ARMA) QLD u}o =} w} Q w. } i. l. m. v t. CU= Q} R. w Ov} Qi |=Q=O. w x. ". CQ Y @ u. xt = 1xt 1 + 2xt 2 + ;. ". ;. x]@=Q. xt. f. w |t. =Wv. "O W. = QDt=Q=B =y i. | y. |v=tR |QU. g. xO=O u. w. w. O}iU xiwv f t g. Q |t pY=L 'OvwW|t xOwRi= sy x@. ARMA(p,q) =. @ w. ;. ;. w x. w |t. CQ Y @ u= D. xm. u QO. "OO o. CU=. + pxt p + wt +

(72) 1wt 1 +

(73) 2 wt 2 +. CWwv wQUB p=kDv= Qorta. xm Qw]u=ty. ;. O}it xDUO "OvDUy pOt. U oQ D=. g |QU 'OW xOy=Wt ,q@k. + pxt p + wt. ;. kw yp t R= |. t. xt. f. (p,q) x@ Q. D t R=. +

(74) q wt. ;. w x]@=Q "CU= O}iU xiwv. =Q j i. wt. f. ;. q. xm. g u QO. p(B )xt = q (B )wt ". ". OvW=@ OL=w. OvW=@ OL=w. ". R=. QDoQR@. R=. QDoQR@. wYN. X. QDW}@ |} Q=m ,=@r=e. QO |. ARMA(p,q). w. OQ t QO. Q} R. 1. = xW} Q s=tD jr]t QOk Qo = CU= Q}PBuwQ=w Ov} Qi. 2. | y. CU=. ARMA(p,0). p t X N. CU=. ARMA(0,q). p t X N. ARMA. w. OQ= t. = xW} Q s=tD jr]t QOk Qo = CU= =DU}= Ov} Qi. | y. ". ". =yQDt=Q=B. CU= xHwD QwNQO. O. =. Cr=L. AR(p). p t 3. O. =. Cr=L. MA(q). p t 4. w |t s=Hv=. p t 'O W. "OQ=O. O. |}=yvD x@. O O. Q |Dkw %QDt=Q=B pk=OL. VR= @. AR = MA } w. =. O x@ C@Uv. | yp t. 5.

(75) 1391 '. |vWwOv|wUwt. 36. pOt swO. C. qtH ?UL Q@ =Q O Q}o@ Q_v. " }. x. OD@= QO xm CU= QDy@ 'f t g |v=tR. u =. QO =Q. ARMA(1,1). O. =F. ARMA(p,q). Q. | U. Q OvDUy pkDUt sy. p t p t |= @ ". xt = xt 1 + wt +

(76) wt =Q. Q}N= xrO=at "CU=. Var(wt) = w2 E (wt ) w. O. " } QO. R=. 2 3 3. X=wN. swO. x@DQt X=wN u}}aD |=Q@. =yv =Q} R "O}U} wv@ O}iU xiwv. 1. ;. ;. Q@. x@DQt. u. O}iU xiwv. Tv=} Q=w. w. wt. u}ov=}t "CU= O}iU xiwv. = xirw-t ?UL Q@ xm. | y. xO. wtv ?DQt. w. |Q ]. u QO. xt. xm. =. T U=. xt = (1 B ) 1(1 +

(77) B )wt ;. ;. CU= Q} R. w x. ". xt = (1 + B + 2B 2 + =. 1 X . i=0. = 1+. w. CU=. E (wt i) = 0 = i ;. y. Q \U@. h ]. )(1 +

(78) B )wt. !. iB i (1 +

(79) B )wt 1 X. i=0. i+1B i+1 +. = wt + ( +

(80) ) Tv=} Q=w. = x]@=Q CU=Q. CQ Y @ q @. =tD. s. |=R=. 1 X. 1 X. i=0. i 1wt ;. ;. i=0. !. i

(81) B i+1 wt. i. E (xt ) u}o =}. x@ =Q} R 'CU= QiY Q@=Q@. ". Var(xt ) = Var wt + ( +

(82) ). 1 X. = x]@=Q x@ xHwD =@. v t 'q @. #. i 1wt ;. i. ;. i=1. = w2 + w2 ( +

(83) )(1 + 2). ;. 1. w |t xH}Dv. "O W. k>0. Cov(xt xt+k ) = ( +

(84) ) k 1w2 + ( +

(85) )2w2 k ;. Q. |= @. 1 X. i=0. 2i. k T = 2. 1. ;. . w |t xH}Dv k |oDU@tyOwN `@=D. "O W. k = k =0 = Cov(xt xt+k )= Var(xt ) k 1 = ( +

(86) )(1 +2

(87) ) 1 +

(88) +

(89) ;. Q. ;. = ( +

(90) ) k 1w2 + ( +

(91) )2w2 k (1 ; 2) ;. ww. v } Q=w m D= |= @. Q. |= @.

(92) =DU}=. 37. k = k |@ QHDp}rLD. ;. CiQo xH}Dv. 1. ARMA. %. =. O. | yp t. 3. w |t Q}N= x]@=Q. u= D. w |t. w |t EL@. u= D. c(p,0,q) x@ Q. =. D t @. Oa@ pYi. =Q O W. arima() ` =. @ D. |. ARIMA ARMA(p,q). QO. \UwD Ov=wD|t. xm. = Ov} Qi. | y. O. wtv. p t "O. u R=. R=. 4 3. |=ypOt. 1 4 3. VR=Q@ w |R=Ux}@W. `@=D \UwD. pYi. ARMA Ov Q R arima.sim(). QD|twta. = x}@W. |R U. w. } i. QO. Q. "O=O VR= @. O TBU. p t. w |t. 'O W. l}ORv xvwtv. R=. = x}@W. |R U.

(93) w.

(94) = 0:5 = 0:6 = ARMA(1,1) Ov Q ;. w. = QDt=Q=B. | y. @. Q =. Q} R. } i |= @ yxO=O. Q CiQ|t Q=_Dv= xm Qw]u=ty "O@=}|t. OQw @. Q =yv Q@. VR= @. = O. | y ) m QO. ARMA(1,1). CU= q=@. ". QO. =yv Q}O=kt. > set.seed(1) > x <{ arima.sim(n = 10000, list(ar = -0.6, ma = 0.5)) > coef(arima(x, order = c(1, 0, 1))) ar1. ma1. -0.596966371. intercept. 0.502703368 -0.006571345. p}O@D. x]@= 33. w. O@=}|t. pmW "OW=@|t =yxO=O. CU= lJwm. w '. Q p}O@D. VR= @. Q. Q. |=Q=O. ARMA(1,1) AR(1) MA(1) =v O ARMA(1,1) Ov Q O Q | x O | =W ARMA(1,1) O. Q@. Q ?U. |= @ | D. = |oDU@tyOwN. | y. Q. Mv | U. w. t p t m. yO. } i " vO o. t u. Ovm|t O}}=D. ". '. v =Q. O u}=. =Q p t. t. 2 4 3. =. = O. O. Q. m QO | yp t ' } R | y ). xU}=kt sy =@ =yv. AIC. O =t}k=@ Q=ov|oDU@ty. p t x v. R= xO. =iDU=. w. OvDUy O}iU xiwv =@ Q=oR=U. > www <{ "http://www.massey.ac.nz/~pscowper/ts/pounds_nz.dat" > x <{ read.table(www, header = T) > x.ts <{ ts(x, st = 1991, fr = 4) > x.ma <{ arima(x.ts, order = c(0, 0, 1)) > x.ar <{ arima(x.ts, order = c(1, 0, 0)) > x.arma <{ arima(x.ts, order = c(1, 0, 1)) > AIC(x.ma) 1] -3.526895 > AIC(x.ar) 1] -37.40417 > AIC(x.arma) 1] -42.27357 > x.arma. MQv |QU.

(95) 1391 '. |vWwOv|wUwt. 38. Call: arima(x = x.ts, order = c(1, 0, 1)) Coefficients: ar1 ma1 intercept 0.892 0.532 2.960 s.e. 0.076 0.202 0.244 sigma^2 estimated as 0.0151: log likelihood = 25.1, aic = -42.3. 0.4 −0.2. 0.0. 0.2. ACF. 0.6. 0.8. 1.0. > acf(resid(x.arma)). 0. 1. 2. 3. Lag. p}O@D MQv. Q. Q. | U |= @. ARMA(1,1). Q. O =t}k=@ Q=ov|oDU@ty. | U x v. %33. pmW.

(96) sQ=yJ pYi. =DU}==v |=ypOt OvwQ. w. |rYi. Q}eD. C=. QF-=Dt =Q} R 'OvDUy =DU}==v =y|QU. R=. u}= QO "OW p}O@D =DU}= |QU x@ p =iD Q=@ l} =@ xm CU=. ". QLDt u}ov=}t. l. w. w}UQoQwD=. w =yv. "O @. =}U@ Ot xDWPo. R= |Q R=. |m} |iO=YD uORs=o p=Ft. qtH pt=W xm O@=}|t \U@ |iO=YD. C. w |t xO}t=v 3 ARIMA Q=YDN= x@ xm 'CU= 2 |oJQ=Bm} =}. "O W. SARIMA. w |t. w x. CQ Y @ =Q u u= D. O = |t |v=tR. " W @. `}=W |r=t |v=tR. =. =. Q. = |t =DU}==v. w OR U. =}U@ p}rLD. | y| U R= |Q. Q. | y| U QO. = pYi. | y. =Q u. QO |. xm CU= |rYi. w. 1 `}tHD. wvax@ "OvW=@|t =. O pYi. uORs o p t. =DLt |r =iD. qtH pt=W. ARIMA. ?r]t u}= "Ovm|t Q}eD Kw w Qw]x@ Tv=} Q=w =DU}==v. xmQw]u=ty. G. C. Ovtv=wv Q=R@= |rYi. u=. QO. =. Q. | U. ARIMA Ov Q. } i. =. O. | yp t "O=O u. Q. | y| U R=. =Wv. |a@. QO. w}UQoQwD= pOt '=y|QU `wv u}= |R=UpOt |=Q@ =yOQm} wQ R= |m} "OQ=O RwQ@ u=mt= R}v |t}rk= |=yOQwmQ QO "CU= 5. GARCH. =YDN= x@. Q. u. xDi=} s}taD `wv. w |t xO}t=v 4 ARCH pOt Q=YDN= x@. "O W. w. CU= Tv=} Q=w. Q. |= @. w |t xO}t=v. "O W. 1. aggregated 2. integrated 3. AutoRegressive Integreted Moving Avrage 4. AutoRegressive Conditional Heteroskedastic 5. Generalised AutoRegressive Conditional Heteroskedastic 39.

(97) 1391 '. |vWwOv|wUwt. 40. |rYi Q}e. jQ@. w. x= N w. |iO=YD. p =iD u}rw=. = pFt OW=@ |iO=YD x=wN 'Ovm. uORs o. xt = Xt 1 + wt |va. '. }. ;. |iO=YD. P. h L =Q. =. ARIMA. O}rwD. OvwQ Ov=wD|t. |QU w. f. xt. Q@ p =iD. =ta=. p. w pFt OW=@ u}at. uORs o QO ". |OQ t. xt = xt xt 1 = wt xt = a + bt + wt Ov Q p =k. O = |t =DU}= xm CU=. " W @. 1 1 4. p=iD. g |QU. CU= |]N OvwQ xm. 1 4. pOt. r. ;. ;. xt = xt xt 1 =o C O}i x w = =] = |] O l x Q R arima() ` = =W MA(1) = w | x C =DU QLD u}o =} Ov Q l O} Q R \ w u}a O Q |r =i O O} wN =t Q =t |t =H C = Ok = = O =iD = |r =i Q Q ARMA O w | xD@ w | s}_v Qi u}o =} = |r =i Q =o Q d=0 include.mean=FALSE arima() ` = xt = yt = b + wt + wt 1 u Q =v O = xt = a + bt + wt x O}v Q ARIMA O = =@ Pt Q MA(1) Q O | C x xt = x0 + i=1 yi x] yt = xt wma h Qa =iD = C O = x}@ xt T = u Q =v C -1 Q Q wt 1 ? Q O = Q yt |r =i | = Q. r. ;. VR= @ '. v '. QO. U= @. U=. U. @ D "O=O u. yO VR= @. xO. x. ;. U D. D | U. t. vwQ. N @. v. @. @ =Q. i v| y. N. @ =Q u u= D. D p t. p t u= D. y=. t. @. @. x W |R U. W f. t. W. t. UO @. g. v } Q=w. }= l. W. D. v R. Y. t. t. @= @. v t. @ F Q=. v t @. w. t. @. t QO. } i. @ =Q. m Z i. r. T. D | U x. v. @ D R=. p t @ \ DQ= QO. t. }. } ' @ } VR= @ f. ;. }. yp t. w. m. U=. } i. U=. t. @=Q. }= @ @ ". m. m. o = "OQ. }= @ @ ' W @. ;. o = " }. } QO. r= "O W. "O=O VR= @. r. vwQ. D R= xO. g. U= @ ". D. U=. v tR | U. Ci=} Oy=wN V}=Ri= s}kDUt \N pwL. ". O}vm xOy=Wt Q} R. ". yt = xt = xt xt 1 = a + bt + wt = b + wt + wt. QO. O}v=wD|t. =Q. Q}N=. r. ;. ;. a + b(t 1) + wt. ; f. ;. 1. ;. ). x0 +. t X i=1. yi = x0 +. t X. (b + wi ; wt 1) ;. i=1. = x0 + bt + wt = xt. ;. 1g. Q. =. =} R. h= o =Q B C } H.

(98) =DU}==v. 41. Q |t. "OO o. x0 = a. w0 = 0 x xH}D xt 1 Q. CU=. w. w |t. "O W. xt = xt = xt = xt = xt .... =. ;. ;. ;. ;. ;. ;. ;. ;. ;. = x0 + bt + wt + (1 +

(99) ). t X i=1. ;. ;. 3. ;. wt CUDQ=@a Tv=} Q=w. %R=. Q@=Q@ Tv=} Q=w. w. CQ Y. u}=. QO. QUB |v} Ro}=H =@. |= @ w. ;. + b + wt +

(100) wt 1 2 + b + wt 1 +

(101) wt 2 + b + wt +

(102) wt 1 2 + 2b + wt + (1 +

(103) )wt 1 +

(104) wt 2 3 + 3b + wt + (1 +

(105) )wt 1 + (1 +

(106) )wt 2 +

(107) wt 1. ;. pYi. 4. O}W=@ xDW=O xHwD. m. v. O. | yp t. xm. Var(xt ) = w2 f1 + (1 +

(108) )2(t ; 1)g O =

(109) = ;1 x u Qo O = | V R R} W @. m. }=. t ' @ }. t. }= i=. v. Vv=} Q=w. t. Q=. Okt V}=Ri= =@ xm. . 2 CU= w. ". sD} Q=or Q. | U '. =. =F. Q. Q}o|t. w. | yxO=O p t |= @ "O. |v=tR. =F |r =iD. p t. Q. | U 14. Q. pmW. CQ Y. di \. R. w. U D. wtv xU Q} R. QO. p =iD u}rw= 'OW x_Lqt ,q@k xm Qw]u=ty. = O O Q}o@ Q_v. | y ) m " }. Q=O. xDU} QDmr= O}rwD x@. QO =Q. w |t xOy=Wt xm Qw]u=ty "OyO|t u=Wv. | U wO QO O W. sD} Q=or |r =iD. =Q. w Q |a}@]. \ @ t. Q. | U w. |r =iD. w |tv Qy=_ Qo}O OvwQ xirw-t 'xOW xDio. "O W. > www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat" > CBE <{ read.table(www, he = T) > Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12) > par(mfrow=c(3,1), mar=c(4,4,2,4)) > plot(Elec.ts) > plot(di(Elec.ts)) > plot(di(log(Elec.ts))) 6 |k}irD pOt. =@. w. Q. Ov} wo. d x@ Q. D t R=. |k}irD pOt. O = |t wQUB p=kDv= Qorta. | U " W @. =ov. =Q u x. B. u QO. wt. Q. 'OO o f. xm CU=. g. d. r. O}iU xiwv p =iD u}t=d. (1 ; B )d x. m. s}O}O. R=. TB. w |t. "O W. Qo = 'CU=. xt. f. 2 1 4 g |QU. xO=O u. d x@ Q. |k}irD. D t R=. (1 ; B ) = wt di() ` = w | C I (1) = C = | =Y C Q p = di(x, d=2) = di(di(x)) p. =Wv. Qo =. I (d) xt. f. g. d. =F. wvax@ "CW=O R}v q=@ x@DQt =@. p t u=. wva Q} R `@=D u}=. u=. Qo}O u=twoQ. QO |. 6. Integrated model. =Q. ". @ D u= D. U=. mP. @ k. t ". U=. X N. } w. r L. iO. = O =. D uORs o " W @. =iD Q=@ wO. Q. |= @.

(110) |vWwOv|wUwt. 42. 8000 2000. Elec.ts. 14000. 1391 '. 1960. 1965. 1970. 1975. 1980. 1985. 1990. 1980. 1985. 1990. 1980. 1985. 1990. 1000 0 −1500. diff(Elec.ts). Time. 1960. 1965. 1970. 1975. 0.00 0.10 0.20 −0.15. diff(log(Elec.ts)). Time. 1960. 1965. 1970. 1975 Time. |tD} Q=or |r=iD. Q. | U w. u}= "CU= OL=w Q@=Q@ ZQiV}B CQwYx@ xirw-t. di(x, lag=12). =F. Q. |r=iD. Q. |v=tR. Q. | U '. Okt "Ovm XNWt. u Q=. Q |atH pOt. p t |= @ "OO o. QO. | U. =Q. |rYi. V}=tv. %14. pmW. p =iD Q}N-=D Ov=wD|t xm CU= OwHwt lag Q}}eD xirw-t. C=. P. h L. ?@U Ov=wD|t u=twoQ. =iD CU= umtt =Hv}= QO "O}=tv|t hPL xv=y=t |=y|QU QO =Q |atH pOt |rYi C=Q}}eD QF= w |]N OvwQ. Cw u. O)m TBU 'O}vm xHwD. wt = vt vt ;. ;. 3. =F x@ s=y@= `iQ. p t. lag d. Q O = xDW=O s=y@=. |= @ " W @. ". w. CU= Q} R. w x. CQ Y @. = w. u t oQ wO. u}@. di() ` =. @ D QO. > w <{ di(v, lag=2, d=1) =yp=Ft. Ov} Qi x@ p}O@D. xt. f. g |QU. B )yt = q (B )wt. = x Qo = uwvm = "CU= p (. | H @. p =iD u}t=d Qo = 'CU= x. =ov. w. ARIMA(p,d,q) Ov Q O = yt = (1 ; B )d xt Q. xt | = Q Q ARMA(p,q) Q} Q xt Q}eD yt. } i |=Q=O f. ' W @. 3 1 4. h} QaD. g. v tR | U. o = "OO o. xm OwW|t xH}Dv. 'O. o Q= k. t. p(B )(1 B )dxt = q (B )wt ;. ARIMA. =. O. . =F OvJ x@ uwvm = "OvDUy q w p x@DQt R= ?}DQD x@ |}=y|=xrtH OvJ q w p. | yp t R= p t. u QO. xm. O}vm xHwD. ".

(111) =DU}==v. 43. QUB p=kDv= Qorta. w x. w. CQ Y @ u.

(112). wvm = "CU= pOt QDt=Q=B. =. O. 4. 1. Ov} Qi. | yp t. x = xt 1 + wt +

(113) wt. xm t. u QO. ;. ;. pYi. w |t xDWwv. "O W. xt xt 1 = wt +

(114) wt ;. ;. QLDt u}ov=}t |k}irD pOt. l. 1 ). ;. u. x@. CU=. w '. ARIMA(0,d,q) IMA(d,q) |r. m. (1 ; B )xt = (1 +

(115) B )wt. ARIMA(0,1,1). Cr=L. w |t. =Wv. QO "O W. xt Q IMA(1,1). w x. g | U 'xU}=kt =@. CQ Y @ f. xO=O u. u}=Q@=v@. w x xm Ov} wo. CQ Y @. CU=. ". Qorta. w x. CQ Y @ u. wvm = "CU= pOt QDt=Q=B. u QO. x = xt 1 + xt. xm t. ;. 1 ;. ;. xt 2 + wt Ov Q. } i . ;. w |t xDWwv wQUB p=kDv=. "O W. xt xt ;. |rm Cr=L. 1;. ;. w |t. QO "O W. xt 1 + xt 2 = wt ;. xO=O u. ;. =Wv. ARI(1,1). ). (1 ; B )(1 ; B )xt = wt. w x xm Ov} wo |k}irD w}UQoQwD=. CQ Y @. CU=. ". O. p t u. x@. ARI(p,d) ARIMA(p,d,0) 4 1 4. VR=Q@ w |R=Ux}@W. u}}aD =. |R U. c(p,d,q) = O x@ Q x O = | Q R arima() ` = x}@ xt = 0:5xt 1 + xt 1 + wt + 0:3wt 1 x w Q = @ p t. W. D t. ;. m. @ }. t VR= @. QO. ;. ;. w |t. "O W. w. @ D. =@ =yxO=O Q@. @ \ @ t | yxO=O. Q =yv Q@. xO=O VR= @. ARIMA(p,d,q) Ov Q. } i. Q} R |=yO)m. =F. Q. Q |t. QO p t |= @ "OO o. ARIMA(1,1,1). O TBU. p t. O. w x W. > set.seed(1) > x <{ w <{ rnorm(1000) > for (i in 3:1000) xi] <{ 0.5 xi - 1] + xi - 1] - 0.5 + xi - 2] + wi] + 0.3 wi - 1] > arima(x, order = c(1, 1, 1)) Call: arima(x = x, order = c(1, 1, 1)) Coecients: ar1 ma1 0.4235 0.3308 s.e. 0.0433 0.0450 sigma^2 estimated as 1.067: log likelihood = -1450.13, aic = 2906.26 =. "OQ=O s v. arima.sim() ` =. @ D. u}= "OyO|t s=Hv=. = O)m u=ty Q=m xm. =Q q @. CU= Q} R. ". w. x =N@=Dm `@=D. OQ=O O Hw |= v. w x O. QO. xD@r=. = x}@W sDU}U uwvm =. CQ Y @ x W |R U. > x <{ arima.sim(model = list(order = c(1, 1, 1), ar = 0.5,. R.

(116) 1391 '. |vWwOv|wUwt. 44. + ma = 0.3), n = 1000) > arima(x, order = c(1, 1, 1)) Call: arima(x = x, order = c(1, 1, 1)) Coecients: ar1 ma1 0.5567 0.2502 s.e. 0.0372 0.0437 sigma^2 estimated as 1.079: log likelihood = -1457.45, aic = 2920.91. ARIMA. |rYi. 2 4. |=ypOt. 1 2 4. h} QaD. |atH pOt |rYi C=Q}}eD QF= =D Ovm|t xO=iDU= (s) pYi xR=Ov= x@ |Q}N-=D =@ p =iD R= ARIMA |rYi pOt. s Q}. u}ov=}t xrtH Qov=}@ p =iD O. s Q}. O = |t. p t " W @. CWwv Q} R. = =@. N-D. l. =. N-D w. Ovm|t. QLDt u}ov=}t. w. P. OvwQ xirw-t p =iD Qorta OL=w Q}N-=D =@. h L =Q. w}UQoQwD=. w x QUB p=kDv= Qorta ?UL Q@. w |t. CQ Y @ w. u= D. =Q. P. ARIMA |rY O C QLD ARIMA(p d q)(P D Q)s |rY. qtH pt=W. C. Q. "OO o h L. i p t ". U= l. t i. P (B s) p(B )(1 ; B s)D (1 ; B )d xt = Q(B s) q (B )wt |twta Cr=L xrtH OvJ. |=. OvW=@|t. QO ". q Qp P w. '. qtH xYNWt xrO=at. = xrtH OvJ ?}DQD x@. R= | y. = xW} Q jr]t QOk. C. |rYi pOt p=Ft OvJ x@ uwvm =. x@DQt. '. | y. w. q Q p P. D=d=0Q. o=. w Oy=wN =DU}= x=ov 'OvW=@ OL=w. "O @. R=. '. w. '. QDoQR@ |oty Q}N= xrO=at AJ O}vm xHwD. =. =. p U x t |wQ. w x xm. CQ Y @. ARIMA(0 0 0)(1 0 0)12 12 |rY '. xDWPo p=U x=t xm. u. Q@. O =. \w. QWt CU= ?U=vt xv=y=t. 1=12. " W @ j. ;. j. >1Q. o=. =v =@. i ?w D. =. AR. =. O. xO U p t . Q |rOt u}vJ "CU=. | yxO=O |= @. CU= =DU}= pOt u}= "OW=@ xDW=Po QF=. |}=H@=H =@. ' }. t. "O W. '. U=. \U@ t. ;. ;. ;. ;. UO @. t. t. "O=O. } w. Y L. }=. @. }= v p t " W @. ;. t. W v u= D. }= "OQ=O. t. v r. r. W o p U QO x t } Q |=Q=O. m. U=. i. m. @=Q. CQ Y @ =Q p t. ;. } u tR u. ;. t =Q q @. ;. } i '. H D @ ". CQ Y @ u= D. ;. =. |Q H. y QO H. D. t W. @. @. oQ. @. t. @. m. H D. }=. m. u tR QO. J h ] |=. @ p t . @=Q |. }. H. Q. h ]. ARIMA. x = xt 1 + xt 12 xt 13 + wt w x w | = Ov O O | C x 1 (B 12 )(1 B )xt = wt = (1 B 12 )(1 B )xt = wt x] Q} wD = w | p = ARIMA(0 1 0)(1 0 0)12 Ov Q |rY ARIMA |r x] = xU =k = x ?r] u x x w = C w w | R} xt = xt 12 + wt w x O u x O}v x w C =DU = O u xD P = = |va = =t Q}}e x |oDU t = Q}}e x O = | B = 1 xW x C (1 B ) xrt p = A Q xrt Ov Q. w. xm. =t= "CU= =DU}==v pOt u}=. ". xt = x12 + wt. u QO. m i. D. m. m. J = }R.

(117) =DU}==v. 45. x = (1

(118) B 4)wt = wt

(119) wt. u}= "CU= t. ;. OvW=@ |iO=YD OvwQ. '. w x. ;. x = xt 1 + wt

(120) wt. w |t w 'CU= t. =v. ;. ;. 4. ;. O. O. pYi. 4. QLDt u}ov=}t pOt l}. CQ Y @. =yxO=O Qo = "CU= ?U=vt OvwQ. |=Q=O. CQ Y @ u= D ?w D QO. w x pYi Q=yJ. 4. ;. =. | yp t. l. =. Q \ki. uw @ | yxO=O |= @. CU= =DU}= pOt. w. xm O}=}|t \U@ pw= x@DQt p =iD CQwYx@ pOt. 1 0)(0 0 1)4. p =iD 'OW=@ |iO=YD OvwQ |=Q=O |rYi CqtH Qo = "CWwv R}v ARIMA(0. w x. O. CQ Y @ =Q u w ' yO. |t CUOx@. xt = xt 4 + wt

(121) wt. =Q. ;. ;. ". CW=O l} Q}N-=D =@ p =iD. w |t u}=Q@=v@ 'Ovm|t. u= D. CWwv P. 4. ;. x]@=Q. w |t R}v. u= D. |]N OvwQ. h L =Q. Q |t p=ta= R}v |rYi. w OO o. ARIMA(0 0 0)(0 1 1)4. s Q}. = = p =iD xm CW=O xHwD O}=@. N-D @. '. QLDt u}ov=}t CqtH x=ov 'OQ=O OwHw |]N OvwQ xm |r=L QO OOQo p=ta= p =iD QO l} Q}N-=D Qo = #Q}N =}. l. Q |t |iQat O}iU xiwv. CQ Y @. w x. =F. wvax@. w. |atH. O =t}k=@. t] x. "OO o. O l} CQwYx@ O}iU xiwv w |rYi C=Q}}eD w |]N OvwQ. p t. u QO. xm. 4. =v = |v=tR |QU x@. 'p t u=. ?w D @. ". O}vm xHwD. OQ=O O Hw. xt = a + bt + s t] + wt O}vm xHwD. ". 4 Q}. = = p =iD u}rw= x@ uwvm =. N-D @. s 4] = s t. 4]. ;. u}=Q@=v@ "OyO|t u=Wv. (1 ; B 4 )xt = xt ; xt 4 = a + bt ; (at + b(t ; 4)) + s t] ; s t = 4b + wt ; wt 4. =Q. 4Q t @. x v. m. ;. 4]. ;. + wt ; wt. 4. ;. ;. u}rw= xm O}vm. Q. Z i u. wvm =. wtv u=}@. "O. 4b C =. @ F. =ov. x. ARIMA(0,0,0)(0,1,1) w x =t 4 Q} = = p =i p@ l Q} =. xrtH =@ Q. 'OO o p. w | t xm. CQ Y @ =Q u u= D. N-D @. a=. D R=. k. }. = p =iD x@DQt. N-D @. (1 ; B 4 )(1 ; B )xt = (1 ; B 4)(b + s t] ; s t 1] + wt ; wt 1) = wt ; wt 1 ; wt 4 + wt 5 ;. ;. ;. CU= C@=F Q=Okt Ok=i xm CWwv. ". ;. ;. ARIMA(0 1 1)(0 1 1)4. w x. w |t xm. CQ Y @ =Q u u= D. VR=Q@. 2 2 4. x} wQ. u}=Q@=v@ "OW=@ CqtH R= |@}mQD w OOaDt |=yQDt=Q=B pt=W Ov=wD|t xwkr=@ CQwYx@ '|rYi ARIMA |=ypOt R=. QD?U=vt pOt. =NDv=. ?. Q. w. =LDt= =yxO=O Q@. |= @ w O W u. Q. Q. O. VR= @ |= @ p t R=. |a}Uw. xOw. OLt xm CU= xDU}=W. xiwv CQwYx@ xOv=t}k=@ Q=ov|oDU@ty |=Q=O ,=k]vt 'xOW xDi=} VR=Q@ ?U=vt pOt "OOQo xO=iDU= AIC x]@= ". xm O. w. OQ=O O Hw. x W. seasonal. xDiQo Q_v. QO. wva Q} R |v=twoQ `@=D u}=. u=. xDU} QDmr= O}rwD. =. | yxO=O. sD} Q=or. w |t. QO "O W. Q. | U. =iDU=. xO. Q} R p=Ft. Q. Q. VR= @ |= @. Q |t. QO "OO o. CU= O}iU. arima() ` =. GQO u QO. R. @ D R= '. |rYi. QO. = xirw-t. | y.

(122) 1391 '. |vWwOv|wUwt. QDy@. =Q |. Q. VR= @. ARI. 46. O. Ovm|t. p t ". P. h L =Q. |]N OvwQ xm CU=. d=1. w. OQ t wO. Qy. QDmJwm. "OQ=O |. QO. p =iD QDt=Q=B "CU=. AIC. Q. O. = } R ' yO. |t u=Wv. > www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat" > CBE <{ read.table(www, he = T) > Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12) > AIC (arima(log(Elec.ts), order = c(1,1,0),. + seas = list(order = c(1,0,0), 12))) 1] -1764.741 > AIC (arima(log(Elec.ts), order = c(0,1,1), + seas = list(order = c(0,0,1), 12))) 1] -1361.586 `@=D Qt= u}= VwQ. QO. CrwyU. Q. w |t. |= @ "O W. |Dkw OQm} wQ u}= "OyO CUOx@ O}vm xHwD Q} R. ". xO =Q. =iDU= =]N. w. |aU. QD?U=vt pOt. = O x@ uwvm = "CU=. | y ) m. R=. QD?U=vt pOt uDi=}. AIC x] =. =. @ T U=. Q wDU= sD} Qwor=. | DQ=. w. > www <{ "http://www.massey.ac.nz/~pscowper/ts/cbe.dat" > CBE <{ read.table(www, he = T) > Elec.ts <{ ts(CBE, 3], start = 1958, freq = 12) > get.best.arima <{ function(x.ts, maxord = c(1,1,1,1,1,1)) +f + best.aic <{ 1e8 + n <{ length(x.ts). w. =ypOt xvt=O pQDvm. Q@ =D CWwv OyO|t. + for (p in 0:maxord1]) for(d in 0:maxord2]) for(q in 0:maxord3]) + for (P in 0:maxord4]) for(D in 0:maxord5]) for(Q in 0:maxord6]) +f + t <{ arima(x.ts, order = c(p,d,q), + seas = list(order = c(P,D,Q), + frequency(x.ts)), method = "CSS") + t.aic <{ -2 t$loglik + (log(n) + 1) length(t$coef) + if (t.aic < best.aic) +f + best.aic <{ t.aic + best.t <{ t + best.model <{ c(p,d,q,P,D,Q) +g +g + list(best.aic, best.t, best.model) +g > best.arima.elec <{ get.best.arima( log(Elec.ts), + maxord = c(2,2,2,2,2,2)) > best.t.elec <- best.arima.elec2]] > acf( resid(best.t.elec) ) > best.arima.elec 3]] 1] 0 1 1 2 0 2. w. w |t. u= D. ?= H. =Q. Q. |= @. |mJwm. QDy@ OW=@. CSS.