Time series assignment
Venkatesh Inkollu
10 August 2017
Problem Description
Build time series models in using GE Power Stock in the BSE.Our goal is to forecast the closing price of the GEPower for future time period using time series algorithms and evaluating each of the algorithms
clearing the environment
rm(list = ls(all = TRUE))library call
library(Quandl)## Loading required package: xts## Warning: package 'xts' was built under R version 3.3.3## Loading required package: zoo## Warning: package 'zoo' was built under R version 3.3.3##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numericlibrary(forecast)
library(lubridate)## Warning: package 'lubridate' was built under R version 3.3.3##
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## datelibrary(DataCombine)## Warning: package 'DataCombine' was built under R version 3.3.3library(imputeTS)## Warning: package 'imputeTS' was built under R version 3.3.3##
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## na.locflibrary(dplyr)## Warning: package 'dplyr' was built under R version 3.3.3##
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## intersect, setdiff, union## The following objects are masked from 'package:xts':
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## first, last## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionlibrary(TTR)## Warning: package 'TTR' was built under R version 3.3.3library(graphics)
library(data.table)##
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## shift## The following objects are masked from 'package:lubridate':
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## hour, isoweek, mday, minute, month, quarter, second, wday,
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## first, lastlibrary(plyr)## Warning: package 'plyr' was built under R version 3.3.3## -------------------------------------------------------------------------## You have loaded plyr after dplyr - this is likely to cause problems.
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## library(plyr); library(dplyr)## -------------------------------------------------------------------------##
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## arrange, count, desc, failwith, id, mutate, rename, summarise,
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## herelibrary(zoo)Pulling the data from Quandl website and reading the data.
GE_data <-Quandl("BSE/BOM532309", api_key="Q2ek_939U_2zj-S9cMqN")basic understanding of the data
names(GE_data)## [1] "Date" "Open"
## [3] "High" "Low"
## [5] "Close" "WAP"
## [7] "No. of Shares" "No. of Trades"
## [9] "Total Turnover" "Deliverable Quantity"
## [11] "% Deli. Qty to Traded Qty" "Spread H-L"
## [13] "Spread C-O"summary(GE_data)## Date Open High Low
## Min. :2000-03-21 Min. : 16.0 Min. : 16.4 Min. : 15.05
## 1st Qu.:2004-09-09 1st Qu.: 134.0 1st Qu.: 137.0 1st Qu.: 131.34
## Median :2009-01-10 Median : 379.3 Median : 385.1 Median : 371.00
## Mean :2009-01-03 Mean : 377.9 Mean : 385.4 Mean : 370.09
## 3rd Qu.:2013-05-09 3rd Qu.: 581.0 3rd Qu.: 590.0 3rd Qu.: 570.05
## Max. :2017-08-11 Max. :1099.0 Max. :1109.0 Max. :1060.00
##
## Close WAP No. of Shares No. of Trades
## Min. : 15.75 Min. : 15.93 Min. : 1 Min. : 1
## 1st Qu.: 133.20 1st Qu.: 134.26 1st Qu.: 5166 1st Qu.: 193
## Median : 377.68 Median : 378.95 Median : 17924 Median : 603
## Mean : 376.85 Mean : 378.01 Mean : 58081 Mean : 1294
## 3rd Qu.: 578.49 3rd Qu.: 580.31 3rd Qu.: 53533 3rd Qu.: 1466
## Max. :1082.10 Max. :1091.21 Max. :3067625 Max. :77660
##
## Total Turnover Deliverable Quantity % Deli. Qty to Traded Qty
## Min. :3.300e+01 Min. : 39 Min. : 1.57
## 1st Qu.:1.173e+06 1st Qu.: 3036 1st Qu.: 31.64
## Median :5.139e+06 Median : 9088 Median : 45.30
## Mean :2.220e+07 Mean : 20992 Mean : 47.45
## 3rd Qu.:1.782e+07 3rd Qu.: 23432 3rd Qu.: 61.70
## Max. :1.993e+09 Max. :441337 Max. :100.00
## NA's :444 NA's :444
## Spread H-L Spread C-O
## Min. : 0.00 Min. :-88.650
## 1st Qu.: 5.40 1st Qu.: -5.250
## Median : 12.07 Median : -0.700
## Mean : 15.29 Mean : -1.088
## 3rd Qu.: 20.25 3rd Qu.: 2.500
## Max. :250.00 Max. : 90.950
## dim(GE_data)## [1] 4162 13length(GE_data)## [1] 13str(GE_data)## 'data.frame': 4162 obs. of 13 variables:
## $ Date : Date, format: "2017-08-11" "2017-08-10" ...
## $ Open : num 650 675 685 704 710 ...
## $ High : num 673 678 695 704 712 ...
## $ Low : num 647 646 670 687 698 ...
## $ Close : num 669 656 676 694 706 ...
## $ WAP : num 659 662 683 696 705 ...
## $ No. of Shares : num 2079 2312 2128 479 2521 ...
## $ No. of Trades : num 110 166 104 43 155 97 77 220 750 220 ...
## $ Total Turnover : num 1369360 1530677 1452890 333396 1777251 ...
## $ Deliverable Quantity : num 1362 1461 1439 295 1796 ...
## $ % Deli. Qty to Traded Qty: num 65.5 63.2 67.6 61.6 71.2 ...
## $ Spread H-L : num 26.4 31.5 24.9 17.4 14.1 ...
## $ Spread C-O : num 19.45 -18.7 -9.05 -9.7 -4 ...
## - attr(*, "freq")= chr "daily"basic data view
head(GE_data,30)## Date Open High Low Close WAP No. of Shares
## 1 2017-08-11 650.00 673.35 646.90 669.45 658.66 2079
## 2 2017-08-10 674.95 677.85 646.35 656.25 662.06 2312
## 3 2017-08-09 685.10 695.05 670.10 676.05 682.75 2128
## 4 2017-08-08 704.00 704.00 686.60 694.30 696.03 479
## 5 2017-08-07 710.00 712.00 697.95 706.00 704.98 2521
## 6 2017-08-04 699.00 710.00 691.00 709.20 705.21 2537
## 7 2017-08-03 705.15 708.00 697.95 705.05 704.12 1804
## 8 2017-08-02 700.05 713.60 699.00 710.65 708.53 4491
## 9 2017-08-01 717.00 717.00 678.85 706.00 699.20 8257
## 10 2017-07-31 714.00 717.00 707.40 714.35 712.20 3870
## 11 2017-07-28 680.00 712.50 680.00 710.30 706.64 13180
## 12 2017-07-27 695.00 721.00 667.00 678.65 692.89 6484
## 13 2017-07-26 660.00 746.50 660.00 691.35 709.36 30980
## 14 2017-07-25 651.35 663.00 650.00 660.70 654.42 2397
## 15 2017-07-24 640.10 660.00 640.10 651.50 649.78 430
## 16 2017-07-21 645.05 655.00 644.95 648.95 649.04 757
## 17 2017-07-20 650.00 655.00 640.10 649.00 652.00 1490
## 18 2017-07-19 673.95 673.95 632.90 649.95 650.05 1249
## 19 2017-07-18 625.50 647.70 625.00 637.00 634.63 1354
## 20 2017-07-17 626.60 637.05 621.00 632.80 631.38 1699
## 21 2017-07-14 651.00 651.00 631.60 641.80 638.19 1239
## 22 2017-07-13 666.00 669.90 645.25 647.10 658.49 1722
## 23 2017-07-12 655.05 679.00 655.05 669.85 670.55 4912
## 24 2017-07-11 662.00 666.60 645.20 653.20 660.75 2218
## 25 2017-07-10 658.00 669.00 650.00 664.45 660.08 7645
## 26 2017-07-07 622.25 661.00 621.65 655.20 652.57 5552
## 27 2017-07-06 632.05 632.55 623.05 625.00 628.56 493
## 28 2017-07-05 624.00 641.70 623.80 630.30 630.95 1128
## 29 2017-07-04 628.00 630.85 617.00 620.20 621.21 1070
## 30 2017-07-03 615.05 634.00 615.05 632.15 628.37 146
## No. of Trades Total Turnover Deliverable Quantity
## 1 110 1369360 1362
## 2 166 1530677 1461
## 3 104 1452890 1439
## 4 43 333396 295
## 5 155 1777251 1796
## 6 97 1789108 2213
## 7 77 1270228 1338
## 8 220 3181986 3136
## 9 750 5773259 4596
## 10 220 2756203 2308
## 11 637 9313504 7115
## 12 487 4492699 2307
## 13 2052 21975976 9473
## 14 102 1568645 1708
## 15 43 279406 258
## 16 54 491322 564
## 17 34 971477 1446
## 18 76 811912 826
## 19 77 859286 1135
## 20 92 1072722 880
## 21 58 790715 1009
## 22 160 1133926 998
## 23 157 3293741 4199
## 24 120 1465554 1458
## 25 244 5046322 6572
## 26 236 3623058 3228
## 27 23 309881 306
## 28 77 711712 596
## 29 42 664692 679
## 30 25 91742 128
## % Deli. Qty to Traded Qty Spread H-L Spread C-O
## 1 65.51 26.45 19.45
## 2 63.19 31.50 -18.70
## 3 67.62 24.95 -9.05
## 4 61.59 17.40 -9.70
## 5 71.24 14.05 -4.00
## 6 87.23 19.00 10.20
## 7 74.17 10.05 -0.10
## 8 69.83 14.60 10.60
## 9 55.66 38.15 -11.00
## 10 59.64 9.60 0.35
## 11 53.98 32.50 30.30
## 12 35.58 54.00 -16.35
## 13 30.58 86.50 31.35
## 14 71.26 13.00 9.35
## 15 60.00 19.90 11.40
## 16 74.50 10.05 3.90
## 17 97.05 14.90 -1.00
## 18 66.13 41.05 -24.00
## 19 83.83 22.70 11.50
## 20 51.80 16.05 6.20
## 21 81.44 19.40 -9.20
## 22 57.96 24.65 -18.90
## 23 85.48 23.95 14.80
## 24 65.73 21.40 -8.80
## 25 85.96 19.00 6.45
## 26 58.14 39.35 32.95
## 27 62.07 9.50 -7.05
## 28 52.84 17.90 6.30
## 29 63.46 13.85 -7.80
## 30 87.67 18.95 17.10tail(GE_data,30)## Date Open High Low Close WAP No. of Shares
## 4133 2000-05-08 39.10 39.10 38.90 39.00 38.93077 650
## 4134 2000-05-05 36.25 42.50 36.25 36.25 37.30066 18100
## 4135 2000-05-04 39.40 39.40 39.40 39.40 39.40000 250
## 4136 2000-05-03 43.00 43.00 42.80 42.80 42.83333 300
## 4137 2000-04-28 46.50 46.50 46.50 46.50 46.50000 50
## 4138 2000-04-27 47.10 50.00 47.10 50.00 49.45875 800
## 4139 2000-04-26 54.00 54.00 49.15 50.30 50.25952 2100
## 4140 2000-04-25 56.50 56.50 53.40 53.40 53.98889 900
## 4141 2000-04-24 57.75 58.00 57.75 58.00 57.92500 600
## 4142 2000-04-20 55.50 56.05 55.50 55.50 55.70769 650
## 4143 2000-04-19 60.00 60.00 60.00 60.00 60.00000 100
## 4144 2000-04-18 58.50 61.00 56.65 57.00 57.14570 22100
## 4145 2000-04-17 61.60 61.60 61.55 61.55 61.57500 200
## 4146 2000-04-13 72.00 72.00 66.80 66.80 66.97222 2700
## 4147 2000-04-12 73.70 73.80 72.55 72.60 72.92701 11550
## 4148 2000-04-11 72.00 76.00 72.00 76.00 72.83471 17612
## 4149 2000-04-10 77.00 77.00 72.50 72.60 72.56287 102750
## 4150 2000-04-07 73.00 75.50 72.00 75.00 74.48571 350
## 4151 2000-04-06 71.75 74.90 70.00 72.65 71.56308 1300
## 4152 2000-04-05 67.50 72.00 65.25 72.00 68.21958 2400
## 4153 2000-04-04 72.45 72.45 67.00 67.00 68.11879 2315
## 4154 2000-04-03 67.10 67.10 67.10 67.10 67.10000 8200
## 4155 2000-03-31 62.00 69.25 62.00 62.15 63.14929 4200
## 4156 2000-03-29 74.10 76.95 73.15 73.20 74.53115 24400
## 4157 2000-03-28 78.70 83.90 78.70 79.50 79.49109 24700
## 4158 2000-03-27 95.00 95.00 85.50 85.50 91.41200 2250
## 4159 2000-03-24 92.70 93.00 92.70 92.90 92.83200 1250
## 4160 2000-03-23 100.75 100.75 100.75 100.75 100.75000 400
## 4161 2000-03-22 109.50 109.50 109.50 109.50 109.50000 600
## 4162 2000-03-21 111.00 119.00 101.00 119.00 108.38000 500
## No. of Trades Total Turnover Deliverable Quantity
## 4133 3 25305 NA
## 4134 56 675142 NA
## 4135 2 9850 NA
## 4136 3 12850 NA
## 4137 1 2325 NA
## 4138 10 39567 NA
## 4139 20 105545 NA
## 4140 6 48590 NA
## 4141 6 34755 NA
## 4142 11 36210 NA
## 4143 1 6000 NA
## 4144 66 1262920 NA
## 4145 2 12315 NA
## 4146 15 180825 NA
## 4147 92 842307 NA
## 4148 38 1282765 NA
## 4149 49 7455835 NA
## 4150 5 26070 NA
## 4151 11 93032 NA
## 4152 14 163727 NA
## 4153 17 157695 NA
## 4154 9 550220 NA
## 4155 33 265227 NA
## 4156 133 1818560 NA
## 4157 118 1963430 NA
## 4158 9 205677 NA
## 4159 10 116040 NA
## 4160 6 40300 NA
## 4161 4 65700 NA
## 4162 6 54190 NA
## % Deli. Qty to Traded Qty Spread H-L Spread C-O
## 4133 NA 0.20 -0.10
## 4134 NA 6.25 0.00
## 4135 NA 0.00 0.00
## 4136 NA 0.20 -0.20
## 4137 NA 0.00 0.00
## 4138 NA 2.90 2.90
## 4139 NA 4.85 -3.70
## 4140 NA 3.10 -3.10
## 4141 NA 0.25 0.25
## 4142 NA 0.55 0.00
## 4143 NA 0.00 0.00
## 4144 NA 4.35 -1.50
## 4145 NA 0.05 -0.05
## 4146 NA 5.20 -5.20
## 4147 NA 1.25 -1.10
## 4148 NA 4.00 4.00
## 4149 NA 4.50 -4.40
## 4150 NA 3.50 2.00
## 4151 NA 4.90 0.90
## 4152 NA 6.75 4.50
## 4153 NA 5.45 -5.45
## 4154 NA 0.00 0.00
## 4155 NA 7.25 0.15
## 4156 NA 3.80 -0.90
## 4157 NA 5.20 0.80
## 4158 NA 9.50 -9.50
## 4159 NA 0.30 0.20
## 4160 NA 0.00 0.00
## 4161 NA 0.00 0.00
## 4162 NA 18.00 8.00Observation & Analysis
GE_data.Day = ddply(GE_data,.(Date),summarize,close_price = mean(Close))
GE_data.Day$Date=as.Date(GE_data.Day$Date,format="%Y-%m-%d")
minDate=min(as.Date(GE_data.Day$Date,format="%Y-%m-%d"))
maxDate =max(as.Date(GE_data.Day$Date,format="%Y-%m-%d"))Creating a data frame for sequence of Days.
seq <- data.frame("dateRange"=seq(minDate,maxDate,by="days"))Merge the data frames of seq of dates with the dates of sale.
GE_data1 = merge(seq,GE_data.Day,by.x = "dateRange",by.y = "Date",all.x = T)
head(GE_data1)## dateRange close_price
## 1 2000-03-21 119.00
## 2 2000-03-22 109.50
## 3 2000-03-23 100.75
## 4 2000-03-24 92.90
## 5 2000-03-25 NA
## 6 2000-03-26 NAImpuation of Missing Values.
GE_data1$close_price<-na.locf(GE_data1$close_price)
head(GE_data1)## dateRange close_price
## 1 2000-03-21 119.00
## 2 2000-03-22 109.50
## 3 2000-03-23 100.75
## 4 2000-03-24 92.90
## 5 2000-03-25 92.90
## 6 2000-03-26 92.90Observation on close_price on weekly basis.
GE_data1$WEEK <-as.numeric(format(GE_data1$dateRange,format="%Y.%W"))
GE_data1.Week<-ddply(GE_data1, .(WEEK),summarize,close_price=mean(close_price))
head(GE_data1.Week)## WEEK close_price
## 1 2000.12 101.32500
## 2 2000.13 71.12143
## 3 2000.14 71.96429
## 4 2000.15 69.77143
## 5 2000.16 57.22143
## 6 2000.17 50.17143splitting of the data
Train=GE_data1.Week[which(GE_data1.Week$WEEK<=2013.52),]
Test=GE_data1.Week[which(GE_data1.Week$WEEK>2013.52),]converting into time series
Price <- ts(Train$close_price,frequency =52)visualize the time series data
plot(Price,
type="l",
lwd=3,
col="red",
xlab="yearly",
ylab="Price",
main="Time series plot for Book-xyzabc")
Decomposed time series
Pricedecomposed=decompose(Price)
plot(Pricedecomposed,col="Red")
ACF,PACF
par(mfrow=c(2,2))
acf(Price,lag=30)
pacf(Price,lag=30)
Price1 <- ts(Train$close_price, frequency =1)
plot(Price1,xlab = "Weekly",ylab="Price")
acf(Price1,lag=30)
pacf(Price1,lag=30)
Stationarize by differencing & taking log transform
par(mfrow=c(2,3))
plot(diff(Price1,lag = 1),type="l"); acf(diff(Price1,lag = 1),lag=30) ;pacf(diff(Price1,lag = 1),lag=30)
plot(diff(Price1,lag=2),type="l"); acf(diff(Price1,lag = 2),lag=30); pacf(diff(Price1,lag = 2),lag=30)
plot(diff(log10(Price1),type="l"));acf(diff(log10(Price1),lag=30));pacf(diff(log10(Price1),lag=30))
ndiffs(Price)## [1] 1Modelling the time series using simple moving averages
fitsma <- SMA(Price,n=2)
pred<-forecast(fitsma,h=8)## Warning in ets(object, lambda = lambda, allow.multiplicative.trend =
## allow.multiplicative.trend, : Missing values encountered. Using longest
## contiguous portion of time seriesplot(pred)
lines(fitsma,col="red")
Define the metric MAPE
smaMape <- mean(abs((Price[2:length(Price)]-fitsma[2:length(Price)])/Price[2:length(Price)]))
smaMape## [1] 0.02292155Weighted Moving Averages
fitwma<- WMA(Price,n=2,1:2)
wmaMape <- mean(abs((Price[2:length(Price)]-fitwma[2:length(Price)])/Price[2:length(Price)]))
wmaMape## [1] 0.01528103pred<-forecast(fitwma,h=8)## Warning in ets(object, lambda = lambda, allow.multiplicative.trend =
## allow.multiplicative.trend, : Missing values encountered. Using longest
## contiguous portion of time seriesplot(pred)
lines(fitwma,col="yellow")
Exponential Moving Averages
fitEma <- EMA(Price, n = 2)
emaMape <- mean(abs((Price[2:length(Price)]-fitEma[2:length(Price)])/Price[2:length(Price)]))
emaMape## [1] 0.01810755pred<-forecast(fitEma,h=8)## Warning in ets(object, lambda = lambda, allow.multiplicative.trend =
## allow.multiplicative.trend, : Missing values encountered. Using longest
## contiguous portion of time seriesplot(pred)
lines(fitEma,col="purple")
Build a HoltWinters model with trend
holtpriceforecast <- HoltWinters(Train$close_price,gamma=FALSE)
head(holtpriceforecast$fitted)## Time Series:
## Start = 3
## End = 8
## Frequency = 1
## xhat level trend
## 3 40.91786 71.12143 -30.20357
## 4 44.59372 71.96429 -27.37057
## 5 44.69834 69.77143 -25.07309
## 6 33.29108 57.22143 -23.93035
## 7 27.78142 50.17143 -22.39001
## 8 19.34072 40.56429 -21.22357HoltWinters model with trend and Seasonality
priceholtforecast = HoltWinters(Price, beta=TRUE, gamma=TRUE, seasonal="additive")
head(priceholtforecast$fitted)## Time Series:
## Start = c(2, 1)
## End = c(2, 6)
## Frequency = 52
## xhat level trend season
## 2.000000 32.51193 27.52150 0.1115645 4.878858
## 2.019231 34.38084 28.29413 0.7726234 5.314091
## 2.038462 32.82178 28.51791 0.2237776 4.080094
## 2.057692 31.59478 28.71331 0.1954041 2.686069
## 2.076923 32.46066 28.92084 0.2075306 3.332292
## 2.096154 30.74751 29.15996 0.2391171 1.348432Prediction on the Train
holtforecastTrain = data.frame(priceholtforecast$fitted)
holtforecastTrainpredictions = holtforecastTrain$xhat
head(holtforecastTrainpredictions)## [1] 32.51193 34.38084 32.82178 31.59478 32.46066 30.74751Prediction on the Test
priceforecast<-forecast(priceholtforecast,h = 8)
plot(priceforecast,ylim = c(-1000,1000))
Arima Models
model1 = arima(Price,c(0,0,0))
model1##
## Call:
## arima(x = Price, order = c(0, 0, 0))
##
## Coefficients:
## intercept
## 316.1615
## s.e. 8.8724
##
## sigma^2 estimated as 57544: log likelihood = -5043.23, aic = 10090.47acf(Price) 
pacf(Price)
plot(Price)
Considering the difference from the graph as d=1 to stationarize
model2 = arima(Price,c(0,1,0))
model2##
## Call:
## arima(x = Price, order = c(0, 1, 0))
##
##
## sigma^2 estimated as 457.7: log likelihood = -3271.9, aic = 6545.8acf(diff(Price,lag = 1))
pacf(diff(Price,lag = 1))
plot(diff(Price))
plot taking log transform
acf(diff(log10(Price)))
pacf(diff(log10(Price)))
plot(diff(log10(Price)))
Considering the difference from the graph as d=2 to stationarize
model3 = arima(Price,c(0,2,0))
model3##
## Call:
## arima(x = Price, order = c(0, 2, 0))
##
##
## sigma^2 estimated as 667.2: log likelihood = -3404.78, aic = 6811.55plot(diff(Price,differences = 2))
acf(diff(Price,differences = 2))
pacf(diff(Price,differences = 2))
Observing the acf and pacf
model4 = arima(Price,c(1,1,1))
model4##
## Call:
## arima(x = Price, order = c(1, 1, 1))
##
## Coefficients:
## ar1 ma1
## 0.1096 0.1729
## s.e. 0.1593 0.1598
##
## sigma^2 estimated as 423.5: log likelihood = -3243.58, aic = 6493.16Plots of the models
par(mfrow=c(2,2))
plot(model1$residuals,ylim=c(-1000,1000))
plot(model2$residuals,ylim=c(-1000,1000))
plot(model3$residuals,ylim=c(-1000,1000))
plot(model4$residuals,ylim=c(-1000,1000))
Auto ARIMA model
MODEL_ARIMA <- auto.arima(Price, ic='aic')
summary(MODEL_ARIMA)## Series: Price
## ARIMA(0,1,1)(0,0,1)[52]
##
## Coefficients:
## ma1 sma1
## 0.2757 -0.0566
## s.e. 0.0352 0.0367
##
## sigma^2 estimated as 423.5: log likelihood=-3242.67
## AIC=6491.35 AICc=6491.38 BIC=6505.13
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.3162176 20.53697 12.46338 -0.009567683 4.453795 0.08373938
## ACF1
## Training set 0.004351009Forecast on the models
pricearimaforecasts1 <- forecast(model1, h=8)
plot(pricearimaforecasts1)
pricearimaforecast3 <- forecast(model3, h=8)
plot(pricearimaforecast3)
pricearimaforecasts_autoArima<- forecast(MODEL_ARIMA,h=8)
plot(pricearimaforecasts_autoArima,flwd = 4)
summary(MODEL_ARIMA)## Series: Price
## ARIMA(0,1,1)(0,0,1)[52]
##
## Coefficients:
## ma1 sma1
## 0.2757 -0.0566
## s.e. 0.0352 0.0367
##
## sigma^2 estimated as 423.5: log likelihood=-3242.67
## AIC=6491.35 AICc=6491.38 BIC=6505.13
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 0.3162176 20.53697 12.46338 -0.009567683 4.453795 0.08373938
## ACF1
## Training set 0.004351009