時間序列ARIMA模型

1、文檔實(shí)驗(yàn)容：給出實(shí)際問題的非平穩(wěn)時間序列，要求學(xué)生利用R統(tǒng)計(jì)軟件，對該序列進(jìn)行分析，通過平穩(wěn)性檢驗(yàn)、差分運(yùn)算、白噪聲檢驗(yàn)、擬合ARMA模型，建立ARIMA模型，在此基礎(chǔ)上進(jìn)行預(yù)測。實(shí)驗(yàn)要求：處理數(shù)據(jù)，掌握非平穩(wěn)時間序列的ARIMA建模方法，并根據(jù)具體的實(shí)驗(yàn)題目要求完成 實(shí)驗(yàn)報(bào)告，并及時上傳到給定的FTP和課程。實(shí)驗(yàn)步驟：第一步：編程建立R數(shù)據(jù)集；第二步：調(diào)用plot.ts程序?qū)?shù)據(jù)繪制時序圖。第三步：從時序圖中利用平穩(wěn)時間序列的定義判斷是否平穩(wěn)?第四步：若不滿足平穩(wěn)性，則可利用差分運(yùn)算是否能使序列平穩(wěn)？重復(fù)第三步步驟第五步：根據(jù)Box.test純隨機(jī)檢驗(yàn)結(jié)果，利用LB統(tǒng)計(jì)量和白噪聲特性檢驗(yàn)最后

2、處理的 時間序列是否為26.66323.59826.93124.74025.80624.36424.47723.90123.17523.22721.67221.87021.43921. 23.70921.66921.75220.76123.47923.82423.10523.11021.75922.21.93720. 23.59021.67222.22222.12323.95023.50422.23823.14221. 21.57321.54820.00022.42420.61521.76122.87424.10423.74823.26222.90721.51922.02522.60420.8

3、9424.67723.67325.32023.58324.67124.45424.12224.25222.08422.99123.28723.04925.07624.24.43024.66726.45125.61825.01425.11022.96423.98123.79822.27024.77522.64623.98824.73726.27625.81625.21025. 23.16224.70724.36422.64425.56524.06225.43124.63527.00926.60626.26826.46225.24625.18024.65723.30426.98226. 27.21

4、026.12226.70626.87826.15226.37924.71225.68824.99024.23926.72123.47524.76726.21928.36128.59927.91427.78425.69326.88126.21724.21827.91426.97528.52727. 28.98228.169實(shí)驗(yàn)：建立ARIMA模型(綜合性實(shí)驗(yàn))實(shí)驗(yàn)題目：某城市連續(xù)14年的月度嬰兒出生率數(shù)據(jù)如下表所示:28. 29.13626.29126.98726.58924.84828.87827.39028.06528.14127.13224.92428.96326.58928.40527.

5、94525.91226.61926.39825.56528.86530.00029. 28.48426.63427.73527.93128.00929.22928.75926.07625.28627.66025.95129.26129.01226.99227.897(1)選擇適當(dāng)模型擬和該序列的發(fā)展(2)使用擬合模型預(yù)測下一年度該城市月度嬰兒出生率27.54326.896文檔純隨機(jī)序列？第六步：在序列判斷為平穩(wěn)非白噪聲序列后，求出該觀察值序列的樣本自相關(guān)系數(shù)(ACF和樣本偏自相關(guān)系數(shù)(PACF的值，選擇階數(shù)適當(dāng)?shù)腁RIMA( p,d,q)模型進(jìn)行 擬合，并估計(jì)模型中未知參數(shù)的值。第七步：檢驗(yàn)?zāi)?/p>

6、型的有效性。如果擬合模型通不過檢驗(yàn)，轉(zhuǎn)向步驟6,重新選擇模型再擬合。第八步：模型優(yōu)化。如果擬合模型通過檢驗(yàn)，仍然轉(zhuǎn)向步驟6,充分考慮各種可能建立 多個擬合模型，從所有通過檢驗(yàn)的擬合模型中選擇最優(yōu)模型。第九步：利用最優(yōu)擬合模型，預(yù)測下一年度該城市月度嬰兒出生率。ex5.2=ts(sca n(ex5.2.txt), freque ncy=4)Read 168 itemsplot.ts(ex5.2)Time從圖中看出序列一開始有下降趨勢，后面有明顯上升趨勢，所以序列不平穩(wěn)。d12ex5.2 = diff(ex5.2,lag=12)acf(d12ex5.2,48)plot(d12ex5.2)文檔Box

7、-Lj ung testdata: d12ex5.2X-squared = 147.9254, df = 17, p-value 2.2e-16P值小于0.05，可以認(rèn)為是非白噪聲序列。par(mfrow=c(2,1); acf(d12ex5.2, 48); pacf(d12ex5.2, 48)文檔Senes d12ex5,210 12文檔Series d12ex5.2J .1In II 11 ii1 11i 1 11 1II.T |i11 11 11CMO024c8110112LagARIMA( 0,0,3)、ARIMA( 0,0,4)、ARIMA( 1,0,3)、ARIMA( 1,0,4)

8、四個模型分別進(jìn)行擬合檢驗(yàn)(rec.ols = arima(d12ex5.2,order=c(0,0,3)Call:arima(x = d12ex5.2, order = c(0, 0, 3)Coefficie nts:ma1 ma2 ma3 in tercept0.79490.44800.11560.2150s.e. 0.08390.08320.08850.1744sigmaA2 estimated as 0.8621: log likelihood = -210.12,rec.pr = predict(rec.ols, n. ahead=5)U = rec.pr$pred + 1.96*re

9、c.pr$seL = rec.pr$pred - 1.96*rec.pr$seminx = min( d12ex5.2,L)maxx = max(d12ex5.2,U)ts.plot(d12ex5.2, rec.pr$pred, ylim=c( min x,maxx)aic = 430.25oo文檔lin es(rec.pr$pred, col=red, type=o)lin es(U, col=blue, lty=dashed)lin es(L, col=blue, lty=dashed)qqno rm(rec.ols$resid)qqli ne(rec.ols$resid)文檔Normal

10、 Q-Q Plot-2-1012Theoreticial Quarvtilesshapiro.test(rec.ols$resid)Shapiro-Wilk no rmality testdata: rec.ols$residW = 0.9777, p-value = 0.0125用shapiro檢驗(yàn)，發(fā)現(xiàn)p值為0.0125，在5%的顯著性水平下顯著，所以為ARIMA( 0,0,3)模型不合理。(rec.ols = arima(d12ex5.2,order=c(0,0,4)Call:arima(x = d12ex5.2, order = c(0, 0, 4)Coefficie nts:ma1

11、ma2 ma3 ma4 in tercept0.83060.49430.22540.20700.2041s.e. 0.09020.11580.09250.08890.1994文檔sigmaA2 estimated as 0.828: log likelihood = -207.07, aic = 426.15 rec.pr = predict(rec.ols, n. ahead=5)U = rec.pr$pred + 1.96*rec.pr$seL = rec.pr$pred - 1.96*rec.pr$seminx = min( d12ex5.2,L)maxx = max(d12ex5.2,

12、U)ts.plot(d12ex5.2, rec.pr$pred, ylim=c( min x,maxx) lin es(rec.pr$pred, col=red, type=o) lin es(U,col=blue, lty=dashed) lin es(L, col=blue, lty=dashed)Timeqqno rm(rec.ols$resid) qqli ne(rec.ols$resid)文檔shapiro.test(rec.ols$resid)Shapiro-Wilk no rmality testdata: rec.ols$residW = 0.9689, p-value = 0

13、.001363用shapiro檢驗(yàn)，發(fā)現(xiàn)p值為0.001363，在5%的顯著性水平下顯著，所以為ARIMA(0,0,4)模型不合理。(rec.ols = arima(d12ex5.2,order=c(1,0,3)Call:Normal 6Q Plot/a_ctu/a_ctununu a_a_文檔arima(x = d12ex5.2, order = c(1, 0, 3)Coefficie nts:ar1ma1ma2ma3intercept0.9288-0.1369-0.2156-0.15860.0240s.e. 0.06690.10650.09210.08790.4984sigmaA2 est

14、imated as 0.7986: log likelihood = -204.35, aic = 420.7 rec.pr = predict(rec.ols, n. ahead=5)U = rec.pr$pred + 1.96*rec.pr$seL = rec.pr$pred - 1.96*rec.pr$seminx = min( d12ex5.2,L)maxx = max(d12ex5.2,U)ts.plot(d12ex5.2, rec.pr$pred, ylim=c( min x,maxx) lin es(rec.pr$pred, col=red, type=o) lin es(U,c

15、ol=blue, lty=dashed) lin es(L, col=blue, lty=dashed)Ti meqqno rm(rec.ols$resid)qqli ne(rec.ols$resid)文檔Normal Q-Q Plot-2 -1 0 1 Theoretical Quantilesshapiro.test(rec.ols$resid)Shapiro-Wilk no rmality testdata: rec.ols$residW = 0.9783, p-value = 0.01454(rec.ols = arima(d12ex5.2,order=c(1,0,4)Call:ari

16、ma(x = d12ex5.2, order = c(1, 0, 4)Coefficie nts:ar1ma1ma2ma3ma4 in tercept0.9084-0.1288-0.2457-0.15110.13090.0493s.e. 0.07250.10780.10210.07780.09600.4684U)QU)Q=C=Ceneno oo o一dlue文檔sigmaA2 estimated as 0.7891: log likelihood = -203.47,rec.pr = predict(rec.ols, n. ahead=5)U = rec.pr$pred + 1.96*rec.pr$seL = rec.pr$pred - 1.96*rec.pr$seminx = min( d12ex5.2,L)maxx = max(d12ex5.2,U)ts.plot(d12ex5.2, rec.pr$pred, ylim=c( min x,maxx)lin es(rec.pr$pred, col=red, type=o)lin es(U, col=blue, lty=dashed)li