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paper - arima forecast

*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: Wed, 22 Dec 2010 19:37:44 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/22/t1293046623yrof9pcuke4q4br.htm/, Retrieved Wed, 22 Dec 2010 20:37:03 +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/2010/Dec/22/t1293046623yrof9pcuke4q4br.htm/},
    year = {2010},
}
@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 = {2010},
    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:

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631923 654294 671833 586840 600969 625568 558110 630577 628654 603184 656255 600730 670326 678423 641502 625311 628177 589767 582471 636248 599885 621694 637406 595994 696308 674201 648861 649605 672392 598396 613177 638104 615632 634465 638686 604243 706669 677185 644328 644825 605707 600136 612166 599659 634210 618234 613576 627200 668973 651479 619661 644260 579936 601752 595376 588902 634341 594305 606200 610926 633685 639696 659451 593248 606677 599434 569578 629873 613438 604172 658328 612633 707372 739770 777535 685030 730234 714154 630872 719492 677023 679272 718317 645672

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	2 seconds
R Server	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

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[72])
60	610926	-	-	-	-	-	-	-
61	633685	-	-	-	-	-	-	-
62	639696	-	-	-	-	-	-	-
63	659451	-	-	-	-	-	-	-
64	593248	-	-	-	-	-	-	-
65	606677	-	-	-	-	-	-	-
66	599434	-	-	-	-	-	-	-
67	569578	-	-	-	-	-	-	-
68	629873	-	-	-	-	-	-	-
69	613438	-	-	-	-	-	-	-
70	604172	-	-	-	-	-	-	-
71	658328	-	-	-	-	-	-	-
72	612633	-	-	-	-	-	-	-
73	707372	658733.0943	616761.9495	700704.2391	0.0116	0.9843	0.8789	0.9843
74	739770	658216.9497	615906.2839	700527.6155	1e-04	0.0114	0.8045	0.9826
75	777535	663157.4901	618803.0194	707511.9609	0	4e-04	0.5651	0.9872
76	685030	614676.6212	569942.8479	659410.3946	0.001	0	0.8261	0.5357
77	730234	618480.8485	572888.4619	664073.235	0	0.0021	0.6941	0.5992
78	714154	610371.782	564147.9225	656595.6416	0	0	0.6786	0.4618
79	630872	585474.6421	538547.9162	632401.3679	0.029	0	0.7466	0.1283
80	719492	633810.7987	586224.2784	681397.3191	2e-04	0.5482	0.5644	0.8085
81	677023	625711.9868	577460.1469	673963.8267	0.0186	1e-04	0.691	0.7024
82	679272	616195.6431	567298.4653	665092.8209	0.0057	0.0074	0.6851	0.5568
83	718317	655313.6232	605767.2787	704859.9677	0.0063	0.1716	0.4525	0.9543
84	645672	619433.2405	569301.3122	669565.1688	0.1525	1e-04	0.6048	0.6048

Univariate ARIMA Extrapolation Forecast Performance
time	% S.E.	PE	MAPE	Sq.E	MSE	RMSE
73	0.0325	0.0738	0	2365743149.7767	0	0
74	0.0328	0.1239	0.0989	6650900008.6574	4508321579.217	67144.0361
75	0.0341	0.1725	0.1234	13082214768.166	7366285975.5334	85827.0702
76	0.0371	0.1145	0.1212	4949597903.1231	6762113957.4308	82232.0738
77	0.0376	0.1807	0.1331	12488766877.3144	7907444541.4075	88923.8131
78	0.0386	0.17	0.1392	10770748763.071	8384661911.6848	91567.7995
79	0.0409	0.0775	0.1304	2060920108.1769	7481270225.4694	86494.3364
80	0.0383	0.1352	0.131	7341268247.7116	7463769978.2496	86393.113
81	0.0393	0.082	0.1256	2632820075.4157	6926997766.8236	83228.5874
82	0.0405	0.1024	0.1232	3978626802.3832	6632160670.3796	81438.0787
83	0.0386	0.0961	0.1208	3969425486.2076	6390093835.4549	79938.0625
84	0.0413	0.0424	0.1142	688472500.5178	5914958724.2101	76908.7688

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/22/t1293046623yrof9pcuke4q4br/1uvtv1293046660.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/22/t1293046623yrof9pcuke4q4br/1uvtv1293046660.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/22/t1293046623yrof9pcuke4q4br/285q41293046660.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/22/t1293046623yrof9pcuke4q4br/285q41293046660.ps (open in new window)

Parameters (Session):

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

Parameters (R input):

par1 = 12 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 1 ; par7 = 1 ; par8 = 1 ; 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,par1))

(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)

}

if (par2 < 0) {

olb <- lb

lb <- ub

ub <- olb

}

(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.mape1 <- array(0, dim=fx)

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

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

perf.mse1 <- 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.se[i] = (x[nx+i] - forecast$pred[i])^2

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[1] = abs(perf.pe[1])

perf.mse[1] = abs(perf.se[1])

for (i in 2:fx) {

perf.mape[i] = perf.mape[i-1] + abs(perf.pe[i])

perf.mape1[i] = perf.mape[i] / i

perf.mse[i] = perf.mse[i-1] + perf.se[i]

perf.mse1[i] = perf.mse[i] / i

}

perf.rmse = sqrt(perf.mse1)

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:par1] <- 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.mape1[i],4))

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

a<-table.element(a,round(perf.mse1[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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