paper 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: Sat, 18 Dec 2010 13:46:38 +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/18/t1292680059ej5ru7ua8j8hm4c.htm/, Retrieved Sat, 18 Dec 2010 14:47:40 +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/18/t1292680059ej5ru7ua8j8hm4c.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:

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Dataseries X:

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100.00 100.42 100.50 101.14 101.98 102.31 103.27 103.80 103.46 105.06 106.08 106.74 107.35 108.96 109.85 109.81 109.99 111.60 112.74 112.78 113.66 115.37 116.26 116.24 116.73 118.76 119.78 120.23 121.48 124.07 125.82 126.92 128.48 131.44 133.51 134.58 136.68 140.10 142.45 143.91 146.19 149.84 152.31 153.62 155.79 159.89 163.21 165.32 167.68 171.79 175.38 177.81 181.09 186.48 191.07 194.23 197.82 204.41 209.26 212.24 214.88 218.87 219.86 219.75 220.89 224.02 222.27 217.27 213.23 212.44 207.87 199.46 198.19 199.77 200.10 195,76 191,27 195,79 192,7

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 'RServer@AstonUniversity' @ vre.aston.ac.uk 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[53]) 49 167.68 - - - - - - - 50 171.79 - - - - - - - 51 175.38 - - - - - - - 52 177.81 - - - - - - - 53 181.09 - - - - - - - 54 186.48 186.892 184.983 188.8409 0.3393 1 1 1 55 191.07 191.4897 188.3195 194.7684 0.401 0.9986 1 1 56 194.23 194.6576 190.7363 198.7434 0.4187 0.9574 1 1 57 197.82 198.9423 193.829 204.3327 0.3416 0.9567 1 1 58 204.41 206.2659 199.3941 213.6283 0.3106 0.9877 1 1 59 209.26 212.1223 203.7265 221.24 0.2692 0.9513 1 1 60 212.24 216.3157 206.4339 227.1911 0.2313 0.8982 1 1 61 214.88 221.9319 210.1671 235.092 0.1468 0.9256 0.9998 1 62 218.87 231.3613 217.0983 247.6301 0.0662 0.9765 0.9994 1 63 219.86 239.0685 222.3498 258.5058 0.0264 0.9792 0.9987 1 64 219.75 244.7631 225.65 267.4137 0.0152 0.9844 0.9976 1 65 220.89 252.324 230.3361 278.9529 0.0103 0.9917 0.9971 1 66 224.02 264.9483 238.8523 297.4459 0.0068 0.9961 0.9973 1 67 222.27 275.526 245.3162 314.2213 0.0035 0.9955 0.9976 1 68 217.27 283.5625 249.496 328.4032 0.0019 0.9963 0.9974 1 69 213.23 294.2234 255.3991 346.9672 0.0013 0.9979 0.9968 1 70 212.44 312.0485 266.026 377.326 0.0014 0.9985 0.9959 1 71 207.87 327.4071 274.2064 406.2209 0.0015 0.9979 0.9955 0.9999 72 199.46 339.4439 279.6009 431.8791 0.0015 0.9974 0.9952 0.9996 73 198.19 355.518 287.1729 466.5549 0.0027 0.9971 0.994 0.999 74 199.77 382.649 300.8143 525.6487 0.0061 0.9943 0.9902 0.9971 75 200.1 406.8882 311.481 586.5492 0.012 0.9881 0.985 0.9931 76 195.76 426.6367 318.6256 645.4321 0.0193 0.9788 0.9791 0.9861 77 191.27 453.4538 328.6547 731.055 0.0321 0.9656 0.9642 0.9728 78 195.79 499.8461 346.8164 894.5656 0.0655 0.9373 0.9319 0.9433 79 192.7 543.5993 361.263 1097.557 0.1072 0.8908 0.8879 0.9002 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 54 0.0053 -0.0022 0 0.1698 0 0 55 0.0087 -0.0022 0.0022 0.1761 0.173 0.4159 56 0.0107 -0.0022 0.0022 0.1828 0.1762 0.4198 57 0.0138 -0.0056 0.0031 1.2596 0.4471 0.6686 58 0.0182 -0.009 0.0042 3.4445 1.0466 1.023 59 0.0219 -0.0135 0.0058 8.193 2.2376 1.4959 60 0.0257 -0.0188 0.0077 16.6111 4.291 2.0715 61 0.0303 -0.0318 0.0107 49.7294 9.9708 3.1577 62 0.0359 -0.054 0.0155 156.0316 26.1998 5.1186 63 0.0415 -0.0803 0.022 368.9678 60.4766 7.7767 64 0.0472 -0.1022 0.0293 625.6562 111.8565 10.5762 65 0.0538 -0.1246 0.0372 988.096 184.8765 13.5969 66 0.0626 -0.1545 0.0462 1675.124 299.5109 17.3064 67 0.0717 -0.1933 0.0567 2836.2056 480.7034 21.9249 68 0.0807 -0.2338 0.0685 4394.6991 741.6364 27.233 69 0.0915 -0.2753 0.0815 6559.9242 1105.2794 33.2457 70 0.1067 -0.3192 0.0954 9921.8618 1623.9019 40.2977 71 0.1228 -0.3651 0.1104 14289.1166 2327.525 48.2444 72 0.1389 -0.4124 0.1263 19595.5012 3236.3658 56.8891 73 0.1593 -0.4425 0.1421 24752.1128 4312.1532 65.667 74 0.1907 -0.4779 0.1581 33444.7463 5699.4195 75.4945 75 0.2253 -0.5082 0.174 42761.341 7384.0523 85.9305 76 0.2617 -0.5412 0.19 53304.0435 9380.5737 96.8534 77 0.3123 -0.5782 0.2062 68740.3296 11853.8968 108.8756 78 0.4029 -0.6083 0.2223 92450.1401 15077.7466 122.7915 79 0.5199 -0.6455 0.2385 123130.3052 19233.6142 138.6853

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/18/t1292680059ej5ru7ua8j8hm4c/10g9n1292679994.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292680059ej5ru7ua8j8hm4c/10g9n1292679994.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292680059ej5ru7ua8j8hm4c/2wppv1292679994.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/18/t1292680059ej5ru7ua8j8hm4c/2wppv1292679994.ps (open in new window)

Parameters (Session):

par1 = 0 ; par2 = -1.0 ; par3 = 2 ; par4 = 0 ; par5 = 4 ; par6 = 3 ; par7 = 1 ; par8 = 1 ; par9 = 1 ; par10 = FALSE ;

Parameters (R input):

par1 = 0 ; par2 = -1.0 ; par3 = 2 ; par4 = 0 ; par5 = 4 ; par6 = 3 ; par7 = 1 ; par8 = 1 ; par9 = 1 ; par10 = FALSE ;

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

par1 <- 26 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')

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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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