非平稳序列的确定性分析.

1、应用时间序列分析实验报告实验名称非平稳序列的确定性分析姓名学号班级实验地点实验日期指导教师实验目的：1、掌握时间序列线性、非线性趋势拟合分析方法；2、掌握X-11过程季节调整方法；3、掌握Forecast过程预测方法。涉及实验的相关情况介绍（包含使用软件或实验设备等情况）：SAS软件、excel表格实验内容：5、我国1949年2008年年末人口总数 仲位:万人）序列如表4 8所示（行数据） 表4 8541675519656300574825879660266614656282864653659946720766207658596729569172704997253874542763687853

2、4806718299285229871778921190859924209371794974962599754298705100072 101654 103008 104357 105851 107507 109300 111026 112704 114333 115823 117171 118517 119850 121121 122389 123626 124761 125786 126743 127627 128453 129227 129988 130756 131448 132129 132802选择适当的模型拟合该序列的长期数据，并作5期预测7、某地区1962 1970年平均每头奶

3、牛的月度产奶量数据（单位：磅）如表 410所 示（行数据）。表 4 10589561640656727697640599568577553582600566653673742716660617583587565598628618688705770736678639604611594634658622709722782756702653615621602635677635736755811798735697661667645688713667762784837817767722681687660698717696775796858826783740701706677711734690785805

4、871845801764725723690734750707807824886859819783740747711751（1）绘制该序列时序图，直观考察该序列的特点。（2）使用因素分解法，拟合该序列的发展，并预测下一年该地区奶牛的月度产奶量（3）使用X 11方法，确定该序列的趋势。&某城市1980年1月至1995年8月每月屠宰生猪的数量（单位：头）如表 411 所示（行数据）。表 41176378719473387396428105084957411106479059510145776889812919164396228102736952409168010125910956476892

5、857739521010030694089102680779199356111706281225104114109959978801053869647997580109490107188941771150971136961145321201109360710306910335111133110616111159099447101987895438926582719794987484673819770296957175722641827735763292593807833285472701337912585805817788685269069722587344576131860827544373

6、969781398003470694818237564075540822297534575982780747758884100979668905193503816358979781022782657727185043954189129710124411452510113993866951711001839707790901903368873283759992677329290130910551060621035601040751017839379110901196499102430103002918159906711006788905899361067238430711489610674987

7、892选择适当的模型拟合该序列的发展， 并预测1995年9月至1997年9月该城市 的生猪屠宰量。实验过程记录（含程序、数据记录及分析和实验结果等）：5、时序图如下：通过时序图，我可以发现我国1949年一2008年年末人口总数（随时间的变化呈现 出线性变化。故此时我可以用线性模型拟合序列的发展。Xt 二a b lt,t =1,2,3,111,60! ” ”2E lt =0,var It 仝其中，It为随机波动；Xt二a b就是消除随机波动的影响之后该序列的长期趋势。由进行线性拟合后的输出结果可知，两个参数的p值明显小于0.05,即这两个参数都是具有显著非零。The AUTOREQ Proced

8、ureDependent 恂irisiblexOrdinar? Least Squares EstlyatesSSE20229G6O7USE4522355SEC1095鞠9舲证1714.04232NAPEL85950D8Durbi r-fatson0.0724VariableDFE&t imeiteIntercept1-2770820lime11445DFERoot M8E MC41CCRegress RSquar Total RjSquare58 2127 109IJ1I141091e.921G70.30310.9931Standard耶 P rosErrortValuePr >

9、; It31&66S0-34<.0001IE 蜩 39丨舗<,0001又因为Regress R-square=total R-square=0.9931即拟合度达到99.31%所以用这个模型 拟合的非常好。因此，拟合模型为：Xt =-2770828 1449t lt,t =1,2,3,| 山602E lt =0,var lt作5期预测:proc forecast data=example1 method=stepar tren d=2 lead=5 out=out outfull outest=est; id t;var x;proc gplot data=out;plot

10、 x*time=_type_/href=2008;symbol1 i=join v=none c=red;symbol2 i=join v=none c=black l=2;symbol3 i=join v=none c=black l=2; run;data example1;n putx;time=1949 +_n_- 1;SAS程序如下:cards ;54167 55196 56300 57482 58796 60266 61465 6282864653 65994 67207 66207 65859 67295 69172 7049972538 74542 76368 78534 80

11、671 82992 85229 87177 89211 90859 92420 93717 94974 96259 97542 98705 100072 101654 103008 104357 105851 107507 109300 111026 112704 114333 115823 117171 118517 119850 121121 122389 123626 124761 125786 126743 127627 128453 129227 129988 130756 131448 132129 132802 proc gplot data =example1;plot x*t

12、ime= 1;symbol1 c=black v=star i =joi n; run ;proc autoreg data =example1;model x=time;output out =out p=example1_cup; run ;procgplotdata =out;plot x*time= 1 example1_cup*time=2/ overlaysymbol2c=red v=none i-jo in w=2 1=3;run ;7、（ 1）时序图如下:time从图中可看出，该序列具有明显的周期性及递增趋势。 时序图：X 11拟合序列图:0IMO2000300040OJt剔除

13、季节效应后的时序图有非常显著的线性递增趋势。SAS程序如下:data xt4_7; input x; time=_n_; cards;58956164065672769764059956857755358260056665367374271666061758358756559862861868870577073667863960461159463465862270972278275670265361562160263567763573675581179873569766166764568871366776278483781776772268168766069871769677579685882

14、6783740701706677711734690785805871845801764725723690734750707807824886859819783740747711751proc gplot data=xt4_7; plot x*time=1;symbol c二black i二join v二star; run;(2)data example2;in putx;time=intnx ('month' , '01jan1962'd,_n_- 1);format time date;cards ;589561640656727697640599568577

15、553582600566653673742716660617583587565598628618688705770736678639604611594634658622709722782756702653615621602635677635736755811798735697661667645688713667762784837817767722681687660698717696775796858826783740701706677711734690785805871845801764725723690734750707807824886859819783740747711751 proc

16、gplot data =example2;plot x*time= 1;symbol1 c=red I =joi n v=star;run ;(3)data example2;in putx;t=intnx ('monthly' , '1jan1962'd,_n_- 1);cards ;58956164065672769764059956857755358260056665367374271666061758358756559862861868870577073667863960461159463465862270972278275670265361562160

17、2635677635736755811798735697661667645688713667762784837817767722681687660698717696775796858826783740701706677711734690785805871845801764725723690734750707807824886859819783740747711751;proc x11 data=example2;mon thly date=t;var x;output out=out b1=x d10=seas on d11=adiusted d12=tre nd d13=irr; data

18、out;set out;estimate=trend*season/100 ;proc gplot data =out;plot x*t= 1 estimate*t= 2/ overlay ;plot adjusted*t= 1tren d*t= 1irr*t= 1;symbol1 c=black i =joi n v =star;symbol2 c=red i =joinv = none w=2 l =3;run ;(4)data example4_7;n putx ;t=intnx ( 'quarter' , '1jan1962'd,_n_-1);forma

19、t t aa4.;cards ;589 561 640 656 727 697 640 599 568 577 553 582600566653673742716660617583587565598628618688705770736678639604611594634658622709722782756702653615621602635677635736755811798735697661667645688713667762784837817767722681687660698717696775796858826783740701706677711734690785805871845801

20、764725723690734750707807824886859819783740747711751 proc x11 data=example4_7;quarterly date=t;var x;output out=out b1=x d10=seas on d11=adjusted d12=tre nd d13=irr; data out;set out;estimate=trend*sesson/100 ;proc gplot data =out;plot seas on *t=2 adjusted*t= 2 tren d*t=2 irr*t= 2;plot x*t= 1 estima

21、te*t= 2/ overlay ;symbol1i =spli nev =starh=1cv =blue ci =red;symbol2i =spli nev =starh=1cv =redci =gree n;run ;8、时序图如下:TOOQT30OQC-12WOO*110W0：TCKKlOOIO«CWW：woo1. r从图中曲线可以看出，数据并没有周期性或者趋向性规律， 并且每月的生猪的屠 宰量大约在80000上下波动。所以由该时序图我可以认为它是平稳序列。即可以用 AR模型或者MA模型或者ARMA模型进行拟合。并且还需利用自相关图进一步辅 助识别。自相关图如下：The 翊】

22、WA ProcedureN&m of- v.Wean 口F Workinc Seri曲8Q64Qd44Siftndard ewlaftoffiISflfiS,9CNumber of CbMrYatisns18Cafar;anceGorrslat ion019290 msLD000O1I1462S4830.534212I03IB9M0e.ssm3lt：3lO61Zo.584an49799(0590.曲価5沖即Q皿制G8873J4I20 45 鹑 g191 EC ewe.4?!BE冃737C1M20.418E8990E?m7O.41ID0沖斛鵠阕期伽幹11759125950.393531&

23、#163;33C37tS60.4ft!41185?1392170.2362114E72H165O.C5415&127$EI29 曲确1阳371210280.192431752D8«?60.70001934341OGG193439520.1Z92510汕祁0.14176211312520.1139S222G5«mi0.13783sa333082450.1715724盯曲馆21o.ieeoaAutocorrel-itiorf T 1 卜百 ”十劇當«a14414IBA*阳1*«軋At*3|(.IMW,W*#,«WV»* di* *

24、1Std Erroro.v»at O.0952BS 0.1 W083 ILQBS25 0J3$i3S 0.1422EI 0.1499E4 O.I5816» 0J619S9 0.16753?0J7602A O.I8300B QJ8553B OjAbw 0.19003t 0.191085 0.13083 onaiBS 0. IMS* 0.195734 OJ9G124 0.186S8* 0F197 Ui由图可知，样本自相关系数在延迟4阶之后几乎全部落入2倍标准差范围内，并 且向零衰减的速度较快，所以可认为该序列是平稳序列。由时序图和自相关图结合可看出该序列为平稳序列，因此可用AR

25、模型或者MA模型或者ARMA模型进行拟合。The AR1M4 ProcedureAutocorralet ion Check for White Moi stTo L*事Chi"SquareDFPr >ChiSqAutocorrelat ions-E3H.92g<.00010.5940.636l.6«60.4300.4950.45012513.05t2<.00010.4230.4140.4!»0.348034038518593 J 7t8（m0.29S0.2370.266CJ9Z0,2700 J73£4的 4. 5Ncmi0J79D.l

26、ft2#1200J380J730J8S由上图数据可知，由于p值显著小于0.05,故可以否定原假设（H。），接受备选 假设（HJ,即可以认为该序列是平稳的非纯随机序列。这说明可以根据历史信息预 测未来的生猪屠宰量。Part i a I Autocorre I i oinsLag Correlation -1 9 0 7 6 S 4 9 2 I 0 1 2 3 4 5 B 7 S 9 11L 59421:山.L.lial t. J driJ,* L JaiL *T>ST *P?L2BQ93出狀Mi*讯粘$L317?e用狀W*讯狀4-().08324'*5&9.086S3出7桝

27、1.8-0.015493：.102 23桝,IQ-(1,0913311h13765Hi轴1.30162H44«*13-0.2033514TJ.091E915-1J.11645IEL 00452170.05740汁-0.03715Hi .J.007S720-0-03671*21'0-31754U0.04722讯23Q.059S9H!£40.03541观察自相关图和偏自相关图，从中可以看出，偏自相关图是拖尾的，而自相关系 数是拖尾的，因此可以用ARMA模型进行拟合，并且还需利用计算机进行最优定阶 相对最优定阶输出结果：Minimuh Informat i on Crit

28、er ionL昨HA 0祸1MA 2MA 3MA 4MA 5AR018,3367719.8368218,9876818,7770618,7886318,74368AR116.448751E.3703718.3786518.3083616.2840618.26745AR218.360118.3755718.2404918.2065618.2281816.25191AR318.17&4318.1757916.TS14718J346718.T610118.17063AR418JC0418.1779518 J85911GJ53S818.147671S, 10855AR5W. 1932118.1

29、86171S.210611GJ994616.1130418.13422Error series modek AR(10)上图中可以看出，在众多模型中， ARMA模型的BIC信息量最小的是 ARMA (4,5),因此采用ARMA模型进行拟合。参数估计输出结果：The 酉RI郦 ProcedureCond i t ionaI Least Squares Est i mationParameterEctihiateStandard Errort ValueApproxPr > HILagMU78431J7064.110.82<.00010MA1J0J44110,239143.59O.OO

30、Q51船1,2-0.46tUB0.238482.02C.U4522MAL3-0.4242S0,19939-2.190.03479h1A1,40.663050.095636.93<.00014MA1t5-0.204870,11111-L84O.OB895AR1JL214540.237585J1<.00011ARL2I0J022B0.35150-2.DO0,04722ARL30.049930.300960J6G.871A3AR1t40.412350,189582,ie0,030S4Constant IEstIkate1944*897Variance EstI mate99166958S

31、id ErrorEst inate9958.261AIC4004.78SBC4037.126Number ofResi due 1 s188律 AIC and SBC donot includeIok determinant.TheProcedureCix wist icr ofRariMtar'PlrabeterMUMIJMAt,3MAI34MI.5AM,朋IJARI ,41.0004U89-L0J60.011&J9S血1-).0l«0.007".他NM1wI.OflO-0.4O.?54&.03?-4J31Q,)50«.78SM.601M

32、t 2Ml1.000'0.BS5UM-o.sran.934-OJ?I0J3SHI.90,0110J54-I0JI61.M0(M12I.7U<U977朋NM4-(.OS?0$弼-I.EIEKOOD-V4V1-0.1S?-D.Oft0.55&-0.3EJ-G,IIIi.m-0,7t7»41-0.51,0.ARL1-a.oj：QJ5I-0J7E0.754-0JS?-4J17t.aaa-K9MC.«40«Ai.i-H.Q1C-n.asi0.334SM酣4.641-051WO-0.9340.U7ARL3U.0U7VJOSQ.S?60.897-a血-0.

33、5 !30,9404.934-o.wARI.4-0,O?4-0.1010.799-0,975n.m曲-E*M4d皿丁-0.34?1.004The 4RIHA ProcedureCorrell at i)nsofParameter Est1natesParameterARI.1ARI,2ARI , 3Afil dflRit2Q.641-n.3511,000-0-034D”断7ARI,3-LSI 90.340一 CL52dmo-.942ARI.40lS69-o.3440.BS?-0.342iTuonAutocorrelat ion Check of Residue IsToChi-Pr >$

34、<4udir«DFChi凶t*= AuiACi)rtit 1 Ait i ©ns * * g¥0-0.009(L(W80.033-0.0230.018-0.0341217,8130.0005-0JO3-0.0310.032-0.02S0.0260*2641823,3790.0044-0,0?90280.0200,0840.啊-O.lflfi243k13150.0084-0.027U28-0.063Q血0,0600.1543040.35?l0.0067-0.023-0.043-0.0960.037-0,1663G5U9270.0030,0840曲-0J3C0.

35、0970,0410.030Model for variable xEstimated Me«n 7643LOG知1口esrsssiYe FaclQrsFedor 1: 1 ' 141457+ 0.7W；l 6«(灯-啊9捕 B«O) - Q.d1"3 B«(4)该输出形式即为：人二 1 -1.21457B 0.70228B2 -0.04985B3 -0.41243B4 壮 因此该模型为：xt -v B厂B ；t序列预测(1995年9月至1997年9月)结果：forecast lead=24 id=time out=example3yc;

36、run;The ARJhIA ProcedureForecasts for variable xObsForecflist Std Error951 Conf idence Limits9012945678$OH23456 oo Qu 4U 4U OJ rt3 flv o- fl- o- u 111111111112 2 r ? 2 .? 2?9 90 oo O1 ? ? t ?9(316.90349953.1493SS4fiO.1+5210619.608S£S9£.SS1&IOSgO.94957636 35411313.8789fi79e.S9S912096.90

37、S05486,9ftS0T2386.880S4e53.?6iS12755.493951&2.133413342.99155E92.066013818.296953J2.2504U115.92154435,7554U3.4S0S3732.4Z6J14S79.44653641J&53I443L42E8386S.2a415284.300碉£12 疋冊15567.4Q29 觀 90.OMD1S763.22282667.5S815354.38192369.504616193.22102385.6516459.5?92590.4CS916699.080S2t)aa .836716

38、873.88991615.72(417099.535N*9I253.945S17208.90421291133,303217406.1IG76798.3013 7SS46.2941 75709.OTftS 74286.0282 79047t19?9 71209.06S1 C9B53.4547 63090.3517 6S508.2189 67845.5530 林3&氣酋12 C51S7.2an «4976,1474 63890.7559 63301*3?60 £94 .7215 £1336.8513 G0G91.9744 60136.7663 59622.0

39、729 53021.5629(1.8116 57525.1140 57017.9119115833.S05& II?2?4.0043 II£202.3247 1098?.9625 10426.5989 11S?64.?4O9 1IS654.070? 121333.9162 122675.4110 1?Z378.34?8 1J25W.259& 122307.S157 12290G.2$9l 123B45.S62& 124324.4716 124195.4167 12:3338. 7286 124107.634B 124655.324? 126tO0CeS 125

40、166.0504 125000.S429 1?4582.77?6 125248.684&上图为对数据进行两年二十四期的预测结果， 其预测数据均可从上图得出。其中 数据由左到右分别表示序列值的序号、预测值、预测值的标准差、 95%的置信下限和 95%的置信上限。下图为上述五中数据的图形表达：proc gplot data=example3yc;plot x*time=1 forecast*time=2 195*time=3 u95*time=3/overlay; symboll c=black i=spli ne v=star;symbol2 c=red i=join v=dot;symbol3 c=gree n i=join v=dot; run;SGIMlSAS程序如下:data example3; in putx;time=_n_;cards :76378719473387396428105084 95741110647 100331 9413310305590595101457 76889812919164396228102736 100264 103491 970279524091680101259 109564 76892857739