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ARIMA forecasting

*The author of this computation has been verified*

R Software Module: /rwasp_arimaforecasting.wasp (opens new window with default values)

Title produced by software: ARIMA Forecasting

Date of computation: Tue, 08 Dec 2009 15:29:39 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4.htm/, Retrieved Tue, 08 Dec 2009 23:30:44 +0100

BibTeX entries for LaTeX users:

@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4.htm/},
    year = {2009},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2009},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

216234 213587 209465 204045 200237 203666 241476 260307 243324 244460 233575 237217 235243 230354 227184 221678 217142 219452 256446 265845 248624 241114 229245 231805 219277 219313 212610 214771 211142 211457 240048 240636 230580 208795 197922 194596 194581 185686 178106 172608 167302 168053 202300 202388 182516 173476 166444 171297 169701 164182 161914 159612 151001 158114 186530 187069 174330 169362 166827 178037 186412 189226 191563 188906 186005 195309 223532 226899 214126

Output produced by software:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	9 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Univariate ARIMA Extrapolation Forecast
time	Y[t]	F[t]	95% LB	95% UB	p-value (H0: Y[t] = F[t])	P(F[t]>Y[t-1])	P(F[t]>Y[t-s])	P(F[t]>Y[57])
45	182516	-	-	-	-	-	-	-
46	173476	-	-	-	-	-	-	-
47	166444	-	-	-	-	-	-	-
48	171297	-	-	-	-	-	-	-
49	169701	-	-	-	-	-	-	-
50	164182	-	-	-	-	-	-	-
51	161914	-	-	-	-	-	-	-
52	159612	-	-	-	-	-	-	-
53	151001	-	-	-	-	-	-	-
54	158114	-	-	-	-	-	-	-
55	186530	-	-	-	-	-	-	-
56	187069	-	-	-	-	-	-	-
57	174330	-	-	-	-	-	-	-
58	169362	161911.6759	151768.8156	172054.5362	0.075	0.0082	0.0127	0.0082
59	166827	154160.358	140580.3447	167740.3713	0.0338	0.0141	0.0381	0.0018
60	178037	155959.5371	137709.9248	174209.1493	0.0089	0.1216	0.0498	0.0242
61	186412	153647.4941	132070.6024	175224.3857	0.0015	0.0134	0.0724	0.0301
62	189226	147966.2384	123019.6132	172912.8636	6e-04	0.0013	0.1013	0.0192
63	191563	143794.7633	116022.5982	171566.9285	4e-04	7e-04	0.1005	0.0156
64	188906	140812.0779	110337.6607	171286.4952	0.001	5e-04	0.1133	0.0156
65	186005	134244.3285	101338.549	167150.108	0.001	6e-04	0.1591	0.0085
66	195309	138336.7725	103126.0003	173547.5448	8e-04	0.004	0.1355	0.0226
67	223532	169308.4773	131952.0929	206664.8616	0.0022	0.0863	0.1831	0.3961
68	226899	171209.4508	131811.3649	210607.5367	0.0028	0.0046	0.2151	0.4383
69	214126	156510.2753	115177.2014	197843.3492	0.0031	4e-04	0.1991	0.1991

Univariate ARIMA Extrapolation Forecast Performance
time	% S.E.	PE	MAPE	Sq.E	MSE	RMSE
58	0.032	0.046	0.0038	55507329.9208	4625610.8267	2150.7233
59	0.0449	0.0822	0.0068	160443820.2407	13370318.3534	3656.5446
60	0.0597	0.1416	0.0118	487414370.0176	40617864.1681	6373.2146
61	0.0716	0.2132	0.0178	1073512849.4675	89459404.1223	9458.2982
62	0.086	0.2788	0.0232	1702367930.7292	141863994.2274	11910.6672
63	0.0985	0.3322	0.0277	2281804433.2195	190150369.435	13789.5021
64	0.1104	0.3415	0.0285	2313025339.851	192752111.6542	13883.5194
65	0.1251	0.3856	0.0321	2679167114.6497	223263926.2208	14942.0188
66	0.1299	0.4118	0.0343	3245834701.7672	270486225.1473	16446.4654
67	0.1126	0.3203	0.0267	2940190418.4688	245015868.2057	15652.9827
68	0.1174	0.3253	0.0271	3101325888.5226	258443824.0435	16076.1881
69	0.1347	0.3681	0.0307	3319571733.8306	276630977.8192	16632.2271

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4/1ey3s1260311370.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4/1ey3s1260311370.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4/27tba1260311370.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/08/t1260311441a8n9b3ztxii3na4/27tba1260311370.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 2 ; par7 = 0 ; par8 = 0 ; par9 = 1 ; par10 = FALSE ;

Parameters (R input):

par1 = 12 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 2 ; par7 = 0 ; par8 = 0 ; par9 = 1 ; par10 = FALSE ;

R code (references can be found in the software module):

par1 <- as.numeric(par1) #cut off periods

par2 <- as.numeric(par2) #lambda

par3 <- as.numeric(par3) #degree of non-seasonal differencing

par4 <- as.numeric(par4) #degree of seasonal differencing

par5 <- as.numeric(par5) #seasonal period

par6 <- as.numeric(par6) #p

par7 <- as.numeric(par7) #q

par8 <- as.numeric(par8) #P

par9 <- as.numeric(par9) #Q

if (par10 == 'TRUE') par10 <- TRUE

if (par10 == 'FALSE') par10 <- FALSE

if (par2 == 0) x <- log(x)

if (par2 != 0) x <- x^par2

lx <- length(x)

first <- lx - 2*par1

nx <- lx - par1

nx1 <- nx + 1

fx <- lx - nx

if (fx < 1) {

fx <- par5

nx1 <- lx + fx - 1

first <- lx - 2*fx

}

first <- 1

if (fx < 3) fx <- round(lx/10,0)

(arima.out <- arima(x[1:nx], order=c(par6,par3,par7), seasonal=list(order=c(par8,par4,par9), period=par5), include.mean=par10, method='ML'))

(forecast <- predict(arima.out,fx))

(lb <- forecast$pred - 1.96 * forecast$se)

(ub <- forecast$pred + 1.96 * forecast$se)

if (par2 == 0) {

x <- exp(x)

forecast$pred <- exp(forecast$pred)

lb <- exp(lb)

ub <- exp(ub)

}

if (par2 != 0) {

x <- x^(1/par2)

forecast$pred <- forecast$pred^(1/par2)

lb <- lb^(1/par2)

ub <- ub^(1/par2)

}

(actandfor <- c(x[1:nx], forecast$pred))

(perc.se <- (ub-forecast$pred)/1.96/forecast$pred)

bitmap(file='test1.png')

opar <- par(mar=c(4,4,2,2),las=1)

ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub))

plot(x,ylim=ylim,type='n',xlim=c(first,lx))

usr <- par('usr')

rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon')

rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender')

abline(h= (-3:3)*2 , col ='gray', lty =3)

polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA)

lines(nx1:lx, lb , lty=2)

lines(nx1:lx, ub , lty=2)

lines(x, lwd=2)

lines(nx1:lx, forecast$pred , lwd=2 , col ='white')

box()

par(opar)

dev.off()

prob.dec <- array(NA, dim=fx)

prob.sdec <- array(NA, dim=fx)

prob.ldec <- array(NA, dim=fx)

prob.pval <- array(NA, dim=fx)

perf.pe <- array(0, dim=fx)

perf.mape <- array(0, dim=fx)

perf.se <- array(0, dim=fx)

perf.mse <- array(0, dim=fx)

perf.rmse <- array(0, dim=fx)

for (i in 1:fx) {

locSD <- (ub[i] - forecast$pred[i]) / 1.96

perf.pe[i] = (x[nx+i] - forecast$pred[i]) / forecast$pred[i]

perf.mape[i] = perf.mape[i] + abs(perf.pe[i])

perf.se[i] = (x[nx+i] - forecast$pred[i])^2

perf.mse[i] = perf.mse[i] + perf.se[i]

prob.dec[i] = pnorm((x[nx+i-1] - forecast$pred[i]) / locSD)

prob.sdec[i] = pnorm((x[nx+i-par5] - forecast$pred[i]) / locSD)

prob.ldec[i] = pnorm((x[nx] - forecast$pred[i]) / locSD)

prob.pval[i] = pnorm(abs(x[nx+i] - forecast$pred[i]) / locSD)

}

perf.mape = perf.mape / fx

perf.mse = perf.mse / fx

perf.rmse = sqrt(perf.mse)

bitmap(file='test2.png')

plot(forecast$pred, pch=19, type='b',main='ARIMA Extrapolation Forecast', ylab='Forecast and 95% CI', xlab='time',ylim=c(min(lb),max(ub)))

dum <- forecast$pred

dum[1:12] <- x[(nx+1):lx]

lines(dum, lty=1)

lines(ub,lty=3)

lines(lb,lty=3)

dev.off()

load(file='createtable')

a<-table.start()

a<-table.row.start(a)

a<-table.element(a,'Univariate ARIMA Extrapolation Forecast',9,TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'time',1,header=TRUE)

a<-table.element(a,'Y[t]',1,header=TRUE)

a<-table.element(a,'F[t]',1,header=TRUE)

a<-table.element(a,'95% LB',1,header=TRUE)

a<-table.element(a,'95% UB',1,header=TRUE)

a<-table.element(a,'p-value<br />(H0: Y[t] = F[t])',1,header=TRUE)

a<-table.element(a,'P(F[t]>Y[t-1])',1,header=TRUE)

a<-table.element(a,'P(F[t]>Y[t-s])',1,header=TRUE)

mylab <- paste('P(F[t]>Y[',nx,sep='')

mylab <- paste(mylab,'])',sep='')

a<-table.element(a,mylab,1,header=TRUE)

a<-table.row.end(a)

for (i in (nx-par5):nx) {

a<-table.row.start(a)

a<-table.element(a,i,header=TRUE)

a<-table.element(a,x[i])

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.element(a,'-')

a<-table.row.end(a)

}

for (i in 1:fx) {

a<-table.row.start(a)

a<-table.element(a,nx+i,header=TRUE)

a<-table.element(a,round(x[nx+i],4))

a<-table.element(a,round(forecast$pred[i],4))

a<-table.element(a,round(lb[i],4))

a<-table.element(a,round(ub[i],4))

a<-table.element(a,round((1-prob.pval[i]),4))

a<-table.element(a,round((1-prob.dec[i]),4))

a<-table.element(a,round((1-prob.sdec[i]),4))

a<-table.element(a,round((1-prob.ldec[i]),4))

a<-table.row.end(a)

}

a<-table.end(a)

table.save(a,file='mytable.tab')

a<-table.start()

a<-table.row.start(a)

a<-table.element(a,'Univariate ARIMA Extrapolation Forecast Performance',7,TRUE)

a<-table.row.end(a)

a<-table.row.start(a)

a<-table.element(a,'time',1,header=TRUE)

a<-table.element(a,'% S.E.',1,header=TRUE)

a<-table.element(a,'PE',1,header=TRUE)

a<-table.element(a,'MAPE',1,header=TRUE)

a<-table.element(a,'Sq.E',1,header=TRUE)

a<-table.element(a,'MSE',1,header=TRUE)

a<-table.element(a,'RMSE',1,header=TRUE)

a<-table.row.end(a)

for (i in 1:fx) {

a<-table.row.start(a)

a<-table.element(a,nx+i,header=TRUE)

a<-table.element(a,round(perc.se[i],4))

a<-table.element(a,round(perf.pe[i],4))

a<-table.element(a,round(perf.mape[i],4))

a<-table.element(a,round(perf.se[i],4))

a<-table.element(a,round(perf.mse[i],4))

a<-table.element(a,round(perf.rmse[i],4))

a<-table.row.end(a)

}

a<-table.end(a)

table.save(a,file='mytable1.tab')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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