DATA 624 - Homework2 - Forecast
Soumya Ghosh
September 07, 2020
Libraries
library(kableExtra)
library(tidyverse)
library(ggplot2)
library(dplyr)
library(corrplot)
library(RColorBrewer)
library(GGally)
library(fpp2)
library(grid)
library(gridExtra)
#library(ggpubr)Forecasting: Principles & Practice
Section 3.7 - Exercise 1
For the following series, find an appropriate Box-Cox transformation in order to stabilise the variance.
- usnetelec
- usgdp
- mcopper
- enplanements
DataSet: usnetelec
Description: Annual US net electricity generation (billion kwh) for 1949-2003
# Timeseries plot before Transformation:
plot <- autoplot(usnetelec,ylab="billion kwh",xlab="Year") + ggtitle("Annual US net electricity generation")
lambda <- BoxCox.lambda(usnetelec)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: 0.5167714# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(usnetelec,lambda),ylab="billion kwh",xlab="Year") + ggtitle("Annual US net electricity generation (w/ BoxCox Transformation)")
grid.arrange(plot,plot_boxcox, ncol=2) 
DataSet: usgdp
Description: Quarterly US GDP. 1947:1 - 2006.1.
# Timeseries plot before Transformation:
plot <- autoplot(usgdp,xlab="Year",ylab="US Dollars") + ggtitle("Quarterly US GDP")
lambda <- BoxCox.lambda(usgdp)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: 0.366352# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(usgdp,lambda),xlab="Year",ylab="US Dollars") + ggtitle("Quarterly US GDP (w/ BoxCox Transformation)")
grid.arrange(plot,plot_boxcox, ncol=2) 
DataSet: mcopper
Description: Monthly copper prices. Copper, grade A, electrolytic wire bars/cathodes,LME,cash (pounds/ton) Source: UNCTAD
# Timeseries plot before Transformation:
plot <- autoplot(mcopper,ylab="pounds per ton",xlab="Year") + ggtitle("Monthly copper price")
lambda <- BoxCox.lambda(mcopper)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: 0.1919047# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(mcopper,lambda),ylab="pounds per ton",xlab="Year") + ggtitle("Monthly copper price (w/ BoxCox Transformation)")
grid.arrange(plot,plot_boxcox, ncol=2) 
DataSet: enplanements
Description: “Domestic Revenue Enplanements (millions): 1996-2000. SOURCE: Department of Transportation, Bureau of Transportation Statistics, Air Carrier Traffic Statistic Monthly.
# Timeseries plot before Transformation:
plot <- autoplot(enplanements,ylab="millions",xlab="Year") + ggtitle("US domestic enplanements")
lambda <- BoxCox.lambda(enplanements)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: -0.2269461# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(enplanements,lambda),ylab="millions",xlab="Year") + ggtitle("US domestic enplanements (w/ BoxCox Transformation)")
grid.arrange(plot,plot_boxcox, ncol=2) 
Section 3.7 - Exercise 2
Q. Why is a Box-Cox transformation unhelpful for the cangas data?
DataSet: cangas
Description: Monthly Canadian gas production, billions of cubic metres, January 1960 - February 2005.
# Timeseries plot before Transformation:
plot <- autoplot(cangas,ylab="billion cubic metres",xlab="Year") + ggtitle("Monthly Canadian gas production")
lambda <- BoxCox.lambda(cangas)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: 0.5767759# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(cangas,lambda),ylab="billion cubic metres",xlab="Year") + ggtitle("Monthly Canadian gas production (w/ BoxCox Transformation)")
grid.arrange(plot,plot_boxcox, ncol=2) 
Analyzing the original plot for cangas data, below variability in seasonal behavior can be observed -
- Seasonal variability was smaller for years before 1975
- Increasing seasonal variability can be observed after 1975, and reached highest around 1985
- Seasonal variability began to decrease significantly after 1990
Comparing the two plots above for cangas data set, it doesn’t look like BoxCox transformation has helped in making the seasonal variation more uniform across time periods. Hence applying Boxcox transformation is not going to simplify the forecasting model for this data set.
Section 3.7 - Exercise 3
Q. What Box-Cox transformation would you select for your retail data (from Exercise 3 in Section 2.10)?
DataSet: Retail
retaildata <- readxl::read_excel("retail.xlsx", skip=1)
head(retaildata, 20) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")| Series ID | A3349335T | A3349627V | A3349338X | A3349398A | A3349468W | A3349336V | A3349337W | A3349397X | A3349399C | A3349874C | A3349871W | A3349790V | A3349556W | A3349791W | A3349401C | A3349873A | A3349872X | A3349709X | A3349792X | A3349789K | A3349555V | A3349565X | A3349414R | A3349799R | A3349642T | A3349413L | A3349564W | A3349416V | A3349643V | A3349483V | A3349722T | A3349727C | A3349641R | A3349639C | A3349415T | A3349349F | A3349563V | A3349350R | A3349640L | A3349566A | A3349417W | A3349352V | A3349882C | A3349561R | A3349883F | A3349721R | A3349478A | A3349637X | A3349479C | A3349797K | A3349477X | A3349719C | A3349884J | A3349562T | A3349348C | A3349480L | A3349476W | A3349881A | A3349410F | A3349481R | A3349718A | A3349411J | A3349638A | A3349654A | A3349499L | A3349902A | A3349432V | A3349656F | A3349361W | A3349501L | A3349503T | A3349360V | A3349903C | A3349905J | A3349658K | A3349575C | A3349428C | A3349500K | A3349577J | A3349433W | A3349576F | A3349574A | A3349816F | A3349815C | A3349744F | A3349823C | A3349508C | A3349742A | A3349661X | A3349660W | A3349909T | A3349824F | A3349507A | A3349580W | A3349825J | A3349434X | A3349822A | A3349821X | A3349581X | A3349908R | A3349743C | A3349910A | A3349435A | A3349365F | A3349746K | A3349370X | A3349754K | A3349670A | A3349764R | A3349916R | A3349589T | A3349590A | A3349765T | A3349371A | A3349588R | A3349763L | A3349372C | A3349442X | A3349591C | A3349671C | A3349669T | A3349521W | A3349443A | A3349835L | A3349520V | A3349841J | A3349925T | A3349450X | A3349679W | A3349527K | A3349526J | A3349598V | A3349766V | A3349600V | A3349680F | A3349378T | A3349767W | A3349451A | A3349924R | A3349843L | A3349844R | A3349376L | A3349599W | A3349377R | A3349779F | A3349379V | A3349842K | A3349532C | A3349931L | A3349605F | A3349688X | A3349456L | A3349774V | A3349848X | A3349457R | A3349851L | A3349604C | A3349608L | A3349609R | A3349773T | A3349852R | A3349775W | A3349776X | A3349607K | A3349849A | A3349850K | A3349606J | A3349932R | A3349862V | A3349462J | A3349463K | A3349334R | A3349863W | A3349781T | A3349861T | A3349626T | A3349617R | A3349546T | A3349787F | A3349333L | A3349860R | A3349464L | A3349389X | A3349461F | A3349788J | A3349547V | A3349388W | A3349870V | A3349396W |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1982-04-01 | 303.1 | 41.7 | 63.9 | 408.7 | 65.8 | 91.8 | 53.6 | 211.3 | 94.0 | 32.7 | 126.7 | 178.3 | 50.4 | 22.2 | 43.0 | 62.4 | 178.0 | 61.8 | 85.4 | 147.2 | 1250.2 | 257.9 | 17.3 | 34.9 | 310.2 | 58.2 | 55.8 | 59.1 | 173.1 | 93.6 | 26.3 | 119.9 | 104.2 | 42.2 | 15.6 | 31.6 | 34.4 | 123.7 | 36.4 | 48.7 | 85.1 | 916.2 | 139.3 | NA | NA | 161.8 | 31.8 | 46.6 | 13.3 | 91.6 | 28.9 | 13.9 | 42.8 | 67.5 | 18.4 | 11.1 | 22.0 | 25.8 | 77.3 | 18.7 | 26.7 | 45.4 | 486.3 | 83.5 | 6.0 | 11.3 | 100.8 | 15.2 | 16.0 | 8.6 | 39.7 | 19.1 | 6.6 | 25.7 | 48.9 | 8.1 | 6.1 | 7.2 | 12.9 | 34.2 | 14.3 | 15.8 | 30.1 | 279.4 | 96.6 | 12.3 | 13.1 | 122.0 | 19.2 | 22.5 | 8.6 | 50.4 | 21.4 | 7.4 | 28.8 | 36.5 | 9.7 | 6.5 | 14.6 | 11.3 | 42.1 | 8.0 | 10.4 | 18.4 | 298.3 | 26.0 | NA | NA | 28.4 | 6.1 | 5.1 | 2.4 | 13.6 | 6.7 | 1.9 | 8.7 | NA | 2.9 | 1.8 | 4.0 | NA | NA | 1.9 | 3.5 | 5.4 | 79.9 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.7 | 1.2 | 1.6 | 15.5 | 2.7 | 4.4 | 2.6 | 9.7 | 3.7 | 2.2 | 5.9 | 10.3 | 2.3 | 1.1 | 2.5 | 2.2 | 8.1 | 4.4 | 3.2 | 7.6 | 57.1 | 933.4 | 79.6 | 149.6 | 1162.6 | 200.3 | 243.4 | 148.6 | 592.3 | 268.5 | 91.4 | 359.9 | 460.1 | 135.1 | 64.9 | 125.6 | 153.5 | 479.1 | 146.3 | 196.1 | 342.4 | 3396.4 |
| 1982-05-01 | 297.8 | 43.1 | 64.0 | 404.9 | 65.8 | 102.6 | 55.4 | 223.8 | 105.7 | 35.6 | 141.3 | 202.8 | 49.9 | 23.1 | 45.3 | 63.1 | 181.5 | 60.8 | 84.8 | 145.6 | 1300.0 | 257.4 | 18.1 | 34.6 | 310.1 | 62.0 | 58.4 | 59.2 | 179.5 | 95.3 | 27.1 | 122.5 | 110.2 | 42.1 | 15.8 | 31.5 | 34.4 | 123.9 | 36.2 | 48.9 | 85.1 | 931.2 | 136.0 | NA | NA | 158.7 | 32.8 | 49.6 | 12.7 | 95.0 | 30.6 | 14.7 | 45.3 | 69.7 | 17.7 | 11.7 | 21.9 | 25.9 | 77.2 | 19.5 | 27.3 | 46.8 | 492.8 | 80.6 | 5.4 | 11.1 | 97.1 | 17.2 | 19.0 | 9.5 | 45.7 | 21.6 | 7.0 | 28.6 | 52.2 | 7.5 | 6.5 | 7.5 | 13.0 | 34.4 | 14.2 | 15.8 | 30.0 | 288.0 | 96.4 | 11.8 | 13.4 | 121.6 | 21.9 | 27.8 | 8.2 | 57.9 | 24.1 | 8.0 | 32.1 | 43.7 | 11.0 | 7.2 | 15.2 | 11.6 | 45.0 | 8.0 | 10.3 | 18.3 | 318.5 | 25.4 | NA | NA | 27.7 | 6.3 | 4.7 | 2.5 | 13.4 | 7.4 | 1.9 | 9.3 | NA | 2.9 | 1.9 | 4.0 | NA | NA | 2.0 | 3.5 | 5.5 | 78.9 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.1 | 1.4 | 1.6 | 15.1 | 3.0 | 4.9 | 3.3 | 11.1 | 3.8 | 2.1 | 5.9 | 10.6 | 2.5 | 1.0 | 2.5 | 2.0 | 8.0 | 3.4 | 3.3 | 6.7 | 57.3 | 920.5 | 80.8 | 149.7 | 1150.9 | 210.3 | 268.3 | 151.0 | 629.6 | 289.8 | 96.8 | 386.6 | 502.6 | 134.9 | 67.7 | 128.7 | 154.8 | 486.1 | 145.5 | 196.6 | 342.1 | 3497.9 |
| 1982-06-01 | 298.0 | 40.3 | 62.7 | 401.0 | 62.3 | 105.0 | 48.4 | 215.7 | 95.1 | 32.5 | 127.6 | 176.3 | 48.0 | 22.8 | 43.7 | 59.6 | 174.1 | 58.7 | 80.7 | 139.4 | 1234.2 | 261.2 | 18.1 | 34.6 | 313.9 | 53.8 | 53.7 | 59.8 | 167.3 | 85.2 | 24.3 | 109.6 | 96.7 | 38.5 | 15.2 | 29.6 | 33.5 | 116.8 | 35.7 | 47.1 | 82.8 | 887.0 | 143.5 | NA | NA | 166.6 | 34.9 | 51.4 | 12.9 | 99.2 | 30.5 | 14.5 | 45.1 | 60.7 | 17.7 | 11.5 | 22.7 | 25.9 | 77.7 | 18.6 | 26.2 | 44.8 | 494.1 | 82.3 | 5.2 | 11.2 | 98.7 | 17.4 | 18.1 | 8.4 | 43.9 | 18.3 | 6.0 | 24.3 | 48.9 | 6.7 | 6.1 | 7.5 | 12.5 | 32.7 | 13.4 | 15.3 | 28.7 | 277.2 | 95.6 | 11.3 | 13.5 | 120.4 | 19.9 | 26.7 | 7.9 | 54.4 | 21.4 | 7.0 | 28.5 | 38.0 | 10.7 | 6.6 | 14.5 | 10.9 | 42.5 | 7.3 | 10.4 | 17.7 | 301.5 | 25.3 | NA | NA | 27.7 | 6.4 | 5.2 | 2.1 | 13.7 | 6.7 | 1.8 | 8.6 | NA | 2.9 | 1.9 | 3.9 | NA | NA | 2.0 | 3.1 | 5.1 | 77.5 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.5 | 1.3 | 1.7 | 15.5 | 2.5 | 4.8 | 2.7 | 9.9 | 3.2 | 2.0 | 5.1 | 9.9 | 2.3 | 1.0 | 2.5 | 2.0 | 7.8 | 3.6 | 3.5 | 7.1 | 55.3 | 933.6 | 77.3 | 149.0 | 1160.0 | 198.7 | 266.1 | 142.6 | 607.4 | 261.9 | 88.6 | 350.5 | 443.8 | 128.2 | 65.5 | 125.0 | 148.8 | 467.5 | 140.2 | 188.5 | 328.7 | 3357.8 |
| 1982-07-01 | 307.9 | 40.9 | 65.6 | 414.4 | 68.2 | 106.0 | 52.1 | 226.3 | 95.3 | 33.5 | 128.8 | 172.6 | 48.6 | 23.2 | 46.5 | 61.9 | 180.2 | 60.3 | 82.4 | 142.7 | 1265.0 | 266.1 | 18.9 | 35.2 | 320.2 | 57.9 | 56.9 | 59.8 | 174.5 | 91.6 | 25.6 | 117.2 | 104.6 | 38.9 | 15.2 | 35.2 | 33.4 | 122.7 | 34.6 | 47.5 | 82.1 | 921.3 | 150.2 | NA | NA | 172.9 | 34.6 | 50.9 | 13.9 | 99.4 | 27.9 | 15.2 | 43.1 | 67.9 | 18.4 | 13.1 | 24.3 | 28.7 | 84.4 | 22.6 | 25.2 | 47.8 | 515.6 | 88.2 | 5.6 | 12.1 | 105.9 | 18.7 | 20.3 | 10.3 | 49.3 | 18.6 | 6.4 | 25.0 | 48.3 | 7.8 | 6.6 | 7.9 | 13.9 | 36.2 | 14.5 | 17.0 | 31.4 | 296.1 | 103.3 | 12.1 | 13.8 | 129.2 | 19.3 | 28.2 | 8.7 | 56.2 | 21.8 | 7.2 | 29.0 | 42.0 | 9.0 | 7.0 | 14.6 | 11.4 | 42.0 | 7.8 | 10.3 | 18.1 | 316.4 | 27.8 | NA | NA | 30.3 | 5.9 | 5.2 | 2.7 | 13.7 | 7.1 | 1.8 | 8.9 | NA | 3.1 | 1.8 | 4.4 | NA | NA | 1.9 | 3.6 | 5.5 | 82.7 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.2 | 1.4 | 1.6 | 16.1 | 2.8 | 5.1 | 2.4 | 10.2 | 3.4 | 2.1 | 5.4 | 8.8 | 2.6 | 1.1 | 2.6 | 2.0 | 8.3 | 4.0 | 3.5 | 7.5 | 56.3 | 972.6 | 80.4 | 153.5 | 1206.4 | 208.7 | 273.5 | 150.1 | 632.4 | 267.2 | 92.1 | 359.3 | 459.1 | 129.9 | 68.5 | 136.6 | 156.1 | 491.1 | 146.5 | 192.0 | 338.5 | 3486.8 |
| 1982-08-01 | 299.2 | 42.1 | 62.6 | 403.8 | 66.0 | 96.9 | 54.2 | 217.1 | 82.8 | 29.4 | 112.3 | 169.6 | 51.3 | 21.4 | 44.8 | 60.7 | 178.1 | 56.1 | 80.7 | 136.8 | 1217.6 | 247.2 | 19.0 | 33.8 | 300.1 | 59.2 | 56.7 | 62.2 | 178.1 | 85.2 | 23.5 | 108.7 | 92.5 | 39.5 | 14.5 | 34.7 | 33.2 | 122.0 | 32.5 | 49.3 | 81.8 | 883.2 | 144.0 | NA | NA | 165.9 | 32.9 | 51.6 | 12.8 | 97.3 | 27.4 | 14.1 | 41.5 | 66.5 | 17.8 | 13.0 | 23.6 | 27.7 | 82.1 | 22.6 | 25.6 | 48.2 | 501.4 | 82.3 | 5.7 | 11.7 | 99.7 | 18.6 | 19.6 | 10.6 | 48.9 | 17.1 | 6.0 | 23.1 | 49.4 | 7.9 | 6.3 | 8.3 | 13.7 | 36.1 | 13.6 | 17.5 | 31.1 | 288.4 | 96.6 | 12.0 | 13.3 | 121.9 | 19.6 | 27.4 | 7.9 | 55.0 | 18.7 | 6.6 | 25.3 | 38.5 | 9.1 | 6.8 | 15.3 | 10.9 | 42.1 | 7.6 | 10.1 | 17.7 | 300.5 | 26.6 | NA | NA | 29.0 | 5.7 | 4.8 | 2.9 | 13.4 | 5.8 | 1.7 | 7.5 | NA | 3.1 | 1.8 | 4.2 | NA | NA | 1.9 | 3.6 | 5.5 | 78.1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.7 | 1.6 | 1.6 | 15.8 | 2.8 | 4.6 | 2.7 | 10.1 | 3.1 | 2.0 | 5.0 | 8.8 | 2.6 | 0.9 | 2.8 | 2.0 | 8.4 | 3.6 | 3.7 | 7.3 | 55.4 | 923.5 | 81.6 | 147.3 | 1152.5 | 206.2 | 262.7 | 153.7 | 622.6 | 241.5 | 83.7 | 325.2 | 438.4 | 133.0 | 65.2 | 134.7 | 152.8 | 485.7 | 138.8 | 192.7 | 331.5 | 3355.9 |
| 1982-09-01 | 305.4 | 42.0 | 64.4 | 411.8 | 62.3 | 97.5 | 53.6 | 213.4 | 89.4 | 32.2 | 121.6 | 181.4 | 49.6 | 21.8 | 43.9 | 61.2 | 176.5 | 58.1 | 82.1 | 140.2 | 1244.9 | 262.4 | 18.4 | 35.4 | 316.2 | 57.1 | 58.9 | 63.6 | 179.6 | 89.5 | 24.3 | 113.8 | 98.3 | 41.7 | 15.1 | 34.2 | 34.5 | 125.5 | 33.9 | 50.7 | 84.6 | 917.9 | 146.9 | NA | NA | 169.5 | 33.7 | 49.6 | 14.5 | 97.9 | 29.1 | 15.5 | 44.5 | 73.4 | 18.8 | 13.0 | 21.8 | 29.0 | 82.6 | 23.2 | 26.7 | 49.8 | 517.7 | 84.2 | 5.8 | 12.0 | 102.0 | 18.8 | 19.9 | 11.5 | 50.2 | 18.2 | 6.4 | 24.6 | 48.5 | 7.8 | 6.4 | 7.8 | 14.1 | 36.0 | 13.9 | 17.8 | 31.7 | 293.0 | 101.4 | 12.3 | 13.4 | 127.1 | 19.9 | 27.0 | 8.7 | 55.6 | 19.5 | 7.4 | 26.9 | 40.2 | 10.0 | 7.1 | 15.1 | 11.7 | 43.9 | 8.2 | 10.3 | 18.5 | 312.3 | 27.1 | NA | NA | 29.6 | 5.3 | 4.8 | 2.6 | 12.8 | 5.8 | 1.7 | 7.5 | NA | 3.2 | 1.8 | 4.0 | NA | NA | 1.9 | 3.8 | 5.7 | 79.1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.9 | 1.4 | 1.8 | 16.0 | 2.6 | 4.3 | 3.1 | 10.0 | 3.4 | 2.2 | 5.6 | 9.2 | 2.6 | 1.0 | 2.8 | 2.2 | 8.6 | 4.2 | 3.9 | 8.1 | 57.5 | 955.9 | 81.4 | 151.8 | 1189.1 | 200.9 | 263.1 | 157.9 | 622.0 | 256.2 | 90.1 | 346.3 | 465.1 | 135.5 | 66.8 | 130.4 | 157.2 | 489.9 | 144.3 | 197.6 | 341.9 | 3454.3 |
| 1982-10-01 | 318.0 | 46.1 | 66.0 | 430.1 | 66.2 | 99.3 | 58.0 | 223.5 | 83.3 | 31.9 | 115.2 | 173.9 | 51.6 | 21.0 | 45.6 | 62.1 | 180.3 | 53.9 | 87.3 | 141.2 | 1264.2 | 285.4 | 20.9 | 38.0 | 344.3 | 66.9 | 59.6 | 64.1 | 190.5 | 93.0 | 25.8 | 118.7 | 102.8 | 46.2 | 16.3 | 35.9 | 36.7 | 135.2 | 37.7 | 54.1 | 91.7 | 983.3 | 143.7 | NA | NA | 166.2 | 31.7 | 49.1 | 13.1 | 93.8 | 33.4 | 15.2 | 48.6 | 68.3 | 20.2 | 12.0 | 19.3 | 27.0 | 78.5 | 20.8 | 28.1 | 48.8 | 504.2 | 88.9 | 6.6 | 12.7 | 108.2 | 18.7 | 19.7 | 10.8 | 49.3 | 20.7 | 7.4 | 28.1 | 46.1 | 7.6 | 7.4 | 8.4 | 15.0 | 38.4 | 17.2 | 20.6 | 37.8 | 307.9 | 107.0 | 14.2 | 14.1 | 135.4 | 18.0 | 25.5 | 10.2 | 53.6 | 20.8 | 8.3 | 29.1 | 37.4 | 7.7 | 7.5 | 15.0 | 12.6 | 42.8 | 9.3 | 11.0 | 20.3 | 318.7 | 27.0 | NA | NA | 29.5 | 5.5 | 4.2 | 2.6 | 12.3 | 5.3 | 1.6 | 7.0 | NA | 2.9 | 1.8 | 4.2 | NA | NA | 2.0 | 3.9 | 5.9 | 78.7 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.5 | 1.5 | 1.7 | 16.6 | 3.7 | 4.7 | 3.5 | 11.9 | 3.4 | 2.3 | 5.8 | 9.7 | 2.7 | 1.2 | 2.6 | 2.5 | 9.0 | 4.8 | 4.0 | 8.9 | 61.9 | 999.3 | 90.8 | 157.3 | 1247.4 | 211.9 | 263.3 | 162.6 | 637.8 | 261.3 | 92.9 | 354.2 | 452.7 | 140.6 | 67.7 | 132.0 | 160.6 | 500.9 | 146.6 | 211.9 | 358.4 | 3551.5 |
| 1982-11-01 | 334.4 | 46.5 | 65.3 | 446.2 | 68.9 | 107.8 | 67.2 | 243.9 | 99.3 | 35.0 | 134.3 | 206.6 | 55.8 | 23.5 | 45.3 | 68.3 | 192.9 | 61.2 | 87.4 | 148.7 | 1372.6 | 291.9 | 22.4 | 38.2 | 352.5 | 78.1 | 63.2 | 82.5 | 223.8 | 107.9 | 29.0 | 136.9 | 114.6 | 43.5 | 17.5 | 38.0 | 40.7 | 139.7 | 40.3 | 57.3 | 97.7 | 1065.2 | 152.7 | NA | NA | 175.4 | 33.8 | 53.2 | 14.9 | 101.9 | 35.5 | 15.9 | 51.4 | 73.4 | 21.5 | 13.2 | 19.2 | 29.7 | 83.6 | 22.7 | 27.6 | 50.4 | 536.0 | 87.0 | 6.5 | 12.2 | 105.7 | 21.0 | 22.7 | 13.1 | 56.8 | 23.6 | 8.0 | 31.6 | 58.5 | 8.8 | 7.8 | 8.8 | 15.8 | 41.2 | 17.3 | 20.9 | 38.2 | 332.1 | 108.7 | 14.2 | 13.8 | 136.7 | 19.0 | 27.4 | 13.2 | 59.6 | 23.8 | 8.8 | 32.6 | 42.4 | 8.4 | 7.9 | 15.7 | 13.9 | 45.9 | 9.6 | 11.1 | 20.8 | 337.9 | 28.0 | NA | NA | 30.6 | 6.0 | 5.3 | 3.2 | 14.5 | 7.1 | 1.9 | 9.0 | NA | 3.1 | 2.0 | 4.7 | NA | NA | 2.0 | 3.9 | 5.9 | 86.5 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 14.1 | 1.5 | 1.7 | 17.2 | 3.9 | 5.1 | 4.6 | 13.6 | 3.6 | 2.6 | 6.2 | 11.3 | 3.0 | 1.3 | 3.1 | 2.9 | 10.3 | 5.4 | 4.3 | 9.6 | 68.3 | 1031.9 | 92.3 | 156.5 | 1280.7 | 232.2 | 285.9 | 199.0 | 717.2 | 302.4 | 101.5 | 403.9 | 522.9 | 145.7 | 73.6 | 135.7 | 176.1 | 531.1 | 159.3 | 215.4 | 374.7 | 3830.5 |
| 1982-12-01 | 389.6 | 53.8 | 77.9 | 521.3 | 90.8 | 155.5 | 146.3 | 392.6 | 142.9 | 51.7 | 194.6 | 346.6 | 69.9 | 31.4 | 55.0 | 104.0 | 260.3 | 75.7 | 97.2 | 172.9 | 1888.3 | 334.6 | 29.7 | 43.9 | 408.2 | 87.5 | 90.3 | 143.0 | 320.8 | 148.2 | 39.8 | 188.0 | 208.5 | 57.2 | 21.5 | 56.5 | 57.3 | 192.5 | 45.2 | 64.1 | 109.3 | 1427.3 | 172.8 | NA | NA | 198.0 | 42.6 | 79.0 | 29.4 | 151.0 | 48.8 | 22.1 | 70.9 | 127.9 | 30.9 | 16.2 | 23.8 | 41.5 | 112.4 | 24.5 | 31.1 | 55.7 | 715.9 | 99.1 | 8.6 | 14.5 | 122.1 | 23.8 | 30.3 | 25.4 | 79.6 | 33.4 | 11.7 | 45.1 | 88.9 | 12.9 | 10.5 | 11.1 | 23.1 | 57.6 | 22.8 | 24.8 | 47.6 | 440.9 | 128.5 | 16.2 | 16.0 | 160.7 | 23.0 | 37.6 | 26.6 | 87.2 | 34.8 | 13.1 | 47.9 | 71.9 | 11.8 | 11.0 | 19.6 | 21.5 | 63.9 | 13.4 | 12.4 | 25.7 | 457.4 | 32.7 | NA | NA | 35.7 | 7.7 | 7.9 | 6.0 | 21.7 | 11.1 | 2.6 | 13.8 | NA | 4.6 | 2.5 | 5.8 | NA | NA | 2.4 | 4.3 | 6.7 | 118.6 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 16.5 | 1.6 | 1.9 | 20.0 | 4.2 | 8.0 | 7.4 | 19.7 | 4.7 | 3.5 | 8.2 | 18.5 | 4.9 | 1.8 | 3.9 | 4.1 | 14.6 | 6.9 | 4.3 | 11.2 | 92.2 | 1190.4 | 111.0 | 182.3 | 1483.7 | 281.2 | 410.7 | 385.0 | 1077.0 | 426.1 | 145.2 | 571.4 | 889.3 | 194.0 | 95.8 | 176.7 | 258.7 | 725.2 | 192.6 | 240.5 | 433.1 | 5179.7 |
| 1983-01-01 | 311.4 | 43.8 | 65.1 | 420.3 | 58.0 | 95.1 | 66.6 | 219.7 | 78.5 | 31.4 | 109.8 | 135.3 | 50.1 | 20.7 | 47.4 | 63.9 | 182.1 | 54.2 | 93.0 | 147.2 | 1214.5 | 270.7 | 22.9 | 36.0 | 329.6 | 58.8 | 55.5 | 64.3 | 178.6 | 81.6 | 25.0 | 106.6 | 81.5 | 43.7 | 15.6 | 34.1 | 35.8 | 129.3 | 36.9 | 57.7 | 94.6 | 920.3 | 146.9 | NA | NA | 169.3 | 28.8 | 50.1 | 14.1 | 92.9 | 29.7 | 14.9 | 44.6 | 64.0 | 22.8 | 12.0 | 17.7 | 27.8 | 80.4 | 20.5 | 30.7 | 51.2 | 502.4 | 82.7 | 7.1 | 12.5 | 102.3 | 19.7 | 18.8 | 9.2 | 47.7 | 20.0 | 6.4 | 26.4 | 43.5 | 8.0 | 6.7 | 8.1 | 13.9 | 36.6 | 15.3 | 24.2 | 39.5 | 295.9 | 94.6 | 15.7 | 12.1 | 122.3 | 16.6 | 25.8 | 9.6 | 52.0 | 18.8 | 7.2 | 26.0 | 35.6 | 7.4 | 6.7 | 14.3 | 11.4 | 39.8 | 8.0 | 11.6 | 19.6 | 295.4 | 26.8 | NA | NA | 29.3 | 4.7 | 4.7 | 2.6 | 12.0 | 5.3 | 1.5 | 6.8 | NA | 2.9 | 1.7 | 3.9 | NA | NA | 1.9 | 3.6 | 5.5 | 75.2 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.0 | 1.0 | 1.6 | 14.6 | 3.0 | 4.3 | 3.3 | 10.6 | 2.7 | 1.9 | 4.6 | 7.4 | 2.5 | 1.0 | 2.5 | 2.1 | 8.1 | 3.8 | 3.9 | 7.7 | 53.0 | 959.3 | 91.7 | 151.9 | 1202.8 | 190.7 | 255.4 | 169.9 | 615.9 | 237.7 | 88.8 | 326.5 | 379.2 | 138.6 | 64.9 | 128.5 | 159.3 | 491.4 | 141.8 | 226.9 | 368.6 | 3384.5 |
| 1983-02-01 | 327.2 | 39.3 | 62.3 | 428.8 | 63.7 | 105.1 | 59.2 | 228.0 | 72.9 | 29.4 | 102.3 | 144.2 | 64.7 | 22.1 | 44.0 | 64.8 | 195.5 | 56.7 | 85.1 | 141.8 | 1240.6 | 278.4 | 20.8 | 35.4 | 334.6 | 59.7 | 60.2 | 64.6 | 184.5 | 73.5 | 23.4 | 96.9 | 86.6 | 44.3 | 16.3 | 34.0 | 36.4 | 130.9 | 38.0 | 50.2 | 88.2 | 921.7 | 149.3 | NA | NA | 170.5 | 26.2 | 47.5 | 12.3 | 86.0 | 25.2 | 12.6 | 37.9 | 53.5 | 20.2 | 11.5 | 17.0 | 25.8 | 74.5 | 19.7 | 27.9 | 47.6 | 470.0 | 85.3 | 6.4 | 11.7 | 103.5 | 18.9 | 19.8 | 8.5 | 47.2 | 17.3 | 5.9 | 23.2 | 39.7 | 8.9 | 6.4 | 7.1 | 13.0 | 35.4 | 13.9 | 21.2 | 35.1 | 284.1 | 100.6 | 13.3 | 12.3 | 126.2 | 16.7 | 24.9 | 9.6 | 51.1 | 18.0 | 7.0 | 25.0 | 33.2 | 7.4 | 6.6 | 13.2 | 11.2 | 38.4 | 7.9 | 10.7 | 18.6 | 292.6 | 26.9 | NA | NA | 29.3 | 5.0 | 4.5 | 2.4 | 11.9 | 5.6 | 1.7 | 7.3 | NA | 3.2 | 1.9 | 3.8 | NA | NA | 2.0 | 3.3 | 5.3 | 76.5 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 12.8 | 1.1 | 1.6 | 15.5 | 3.3 | 4.4 | 2.6 | 10.3 | 2.7 | 1.9 | 4.6 | 8.0 | 3.0 | 1.0 | 2.5 | 2.1 | 8.6 | 4.2 | 3.9 | 8.2 | 55.1 | 995.5 | 82.0 | 146.7 | 1224.2 | 194.8 | 267.5 | 159.4 | 621.7 | 216.4 | 82.3 | 298.7 | 378.0 | 152.8 | 66.4 | 122.1 | 157.9 | 499.1 | 143.7 | 204.4 | 348.1 | 3369.8 |
| 1983-03-01 | 350.9 | 43.4 | 65.7 | 460.0 | 66.0 | 124.1 | 67.3 | 257.5 | 93.3 | 34.2 | 127.5 | 180.5 | 63.1 | 24.9 | 47.7 | 70.0 | 205.7 | 60.9 | 83.7 | 144.6 | 1375.7 | 303.8 | 23.5 | 39.1 | 366.4 | 71.6 | 67.6 | 73.9 | 213.0 | 100.6 | 28.2 | 128.8 | 108.0 | 48.3 | 16.8 | 36.7 | 39.1 | 140.9 | 37.0 | 55.0 | 92.0 | 1049.2 | 162.4 | NA | NA | 185.8 | 30.1 | 58.6 | 16.6 | 105.3 | 31.1 | 15.2 | 46.3 | 64.4 | 20.9 | 13.3 | 18.9 | 30.4 | 83.4 | 21.8 | 28.8 | 50.5 | 535.7 | 95.9 | 6.9 | 14.0 | 116.8 | 22.9 | 24.1 | 9.9 | 56.8 | 23.5 | 7.6 | 31.2 | 54.4 | 9.8 | 7.7 | 7.8 | 15.3 | 40.5 | 16.2 | 24.6 | 40.8 | 340.5 | 107.6 | 15.4 | 13.7 | 136.7 | 18.0 | 28.2 | 10.1 | 56.3 | 19.7 | 7.5 | 27.2 | 37.6 | 7.3 | 7.3 | 14.8 | 12.2 | 41.6 | 8.7 | 11.6 | 20.3 | 319.6 | 29.8 | NA | NA | 32.6 | 6.0 | 5.7 | 3.0 | 14.7 | 6.5 | 1.9 | 8.5 | NA | 3.5 | 2.1 | 4.2 | NA | NA | 2.3 | 3.4 | 5.7 | 89.1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.8 | 1.1 | 1.8 | 16.7 | 3.6 | 5.3 | 3.1 | 12.0 | 3.8 | 2.5 | 6.3 | 10.6 | 3.1 | 1.1 | 2.6 | 2.2 | 9.1 | 4.0 | 4.4 | 8.5 | 63.1 | 1080.8 | 91.4 | 160.3 | 1332.4 | 219.8 | 315.1 | 184.2 | 719.1 | 279.7 | 97.5 | 377.2 | 472.1 | 157.3 | 73.7 | 133.2 | 174.4 | 538.7 | 151.9 | 213.9 | 365.8 | 3805.3 |
| 1983-04-01 | 323.4 | 43.7 | 61.9 | 429.0 | 58.3 | 112.3 | 57.7 | 228.2 | 111.2 | 39.4 | 150.6 | 199.4 | 51.1 | 24.5 | 52.9 | 65.3 | 193.7 | 63.5 | 79.7 | 143.2 | 1344.2 | 301.9 | 21.7 | 35.6 | 359.2 | 56.2 | 62.9 | 61.5 | 180.7 | 105.6 | 28.6 | 134.1 | 115.3 | 37.0 | 16.0 | 33.6 | 33.8 | 120.5 | 35.1 | 50.2 | 85.2 | 994.9 | 156.8 | NA | NA | 177.8 | 29.3 | 51.3 | 11.1 | 91.7 | 33.1 | 14.8 | 47.8 | 69.3 | 18.3 | 12.5 | 17.4 | 25.9 | 74.1 | 21.3 | 27.0 | 48.3 | 509.0 | 91.0 | 6.2 | 12.9 | 110.1 | 23.0 | 20.7 | 9.3 | 53.0 | 23.3 | 8.2 | 31.5 | 53.0 | 10.5 | 7.5 | 7.3 | 14.8 | 40.1 | 16.7 | 21.6 | 38.3 | 326.0 | 105.2 | 12.4 | 12.8 | 130.3 | 16.4 | 26.3 | 10.1 | 52.9 | 22.2 | 8.2 | 30.4 | 39.7 | 7.4 | 7.3 | 13.7 | 12.1 | 40.5 | 8.8 | 10.4 | 19.2 | 313.0 | 28.0 | NA | NA | 30.6 | 5.6 | 5.4 | 2.5 | 13.5 | 6.9 | 2.0 | 8.9 | NA | 3.1 | 2.1 | 3.9 | NA | NA | 2.4 | 3.1 | 5.5 | 83.5 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 14.3 | 1.3 | 1.7 | 17.2 | 3.4 | 4.0 | 3.1 | 10.6 | 4.7 | 2.7 | 7.4 | 11.7 | 2.6 | 1.1 | 2.2 | 2.3 | 8.2 | 4.4 | 3.7 | 8.2 | 63.3 | 1036.4 | 86.4 | 148.1 | 1270.9 | 193.5 | 284.2 | 155.7 | 633.4 | 308.3 | 104.2 | 412.5 | 503.4 | 131.2 | 71.5 | 131.8 | 159.3 | 493.8 | 153.0 | 198.1 | 351.1 | 3665.1 |
| 1983-05-01 | 316.6 | 42.3 | 63.7 | 422.6 | 67.8 | 120.5 | 64.9 | 253.2 | 112.5 | 41.4 | 153.9 | 200.5 | 54.8 | 25.4 | 55.0 | 68.9 | 204.1 | 64.5 | 81.1 | 145.6 | 1379.9 | 281.5 | 21.4 | 36.4 | 339.2 | 62.0 | 67.0 | 65.2 | 194.2 | 101.9 | 28.4 | 130.3 | 112.1 | 40.1 | 16.1 | 36.6 | 35.0 | 127.8 | 34.1 | 52.7 | 86.8 | 990.4 | 159.8 | NA | NA | 181.3 | 35.1 | 53.6 | 12.0 | 100.7 | 33.9 | 15.6 | 49.5 | 69.3 | 20.2 | 12.7 | 18.0 | 26.9 | 77.8 | 21.3 | 27.5 | 48.9 | 527.5 | 91.6 | 6.1 | 13.1 | 110.8 | 26.8 | 22.5 | 10.5 | 59.8 | 24.5 | 8.1 | 32.6 | 56.0 | 11.4 | 7.7 | 8.1 | 15.3 | 42.4 | 16.3 | 23.2 | 39.5 | 341.1 | 106.9 | 12.7 | 13.2 | 132.8 | 19.6 | 29.4 | 11.1 | 60.2 | 25.0 | 9.1 | 34.0 | 46.0 | 8.3 | 7.8 | 14.2 | 12.9 | 43.2 | 9.1 | 11.4 | 20.5 | 336.8 | 27.5 | NA | NA | 30.2 | 6.2 | 5.6 | 3.0 | 14.7 | 7.0 | 1.9 | 8.9 | NA | 3.1 | 2.1 | 3.9 | NA | NA | 2.2 | 3.6 | 5.8 | 85.1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 14.1 | 1.4 | 1.8 | 17.3 | 3.7 | 4.8 | 3.1 | 11.6 | 4.6 | 2.8 | 7.3 | 11.5 | 2.8 | 1.1 | 2.3 | 2.3 | 8.5 | 4.3 | 5.0 | 9.3 | 65.6 | 1014.2 | 85.0 | 152.4 | 1251.7 | 222.9 | 304.9 | 170.1 | 697.9 | 310.8 | 107.7 | 418.5 | 510.6 | 142.0 | 73.5 | 138.9 | 166.5 | 520.8 | 153.2 | 207.4 | 360.5 | 3760.0 |
| 1983-06-01 | 325.4 | 40.4 | 64.9 | 430.6 | 64.2 | 115.0 | 58.6 | 237.8 | 103.6 | 37.1 | 140.7 | 175.2 | 52.3 | 24.6 | 56.2 | 65.7 | 198.8 | 63.0 | 79.7 | 142.8 | 1325.8 | 290.6 | 20.8 | 34.2 | 345.6 | 57.0 | 66.2 | 60.2 | 183.3 | 90.3 | 25.6 | 115.9 | 100.1 | 38.2 | 16.1 | 35.9 | 33.7 | 123.8 | 34.9 | 46.4 | 81.3 | 950.0 | 158.8 | NA | NA | 180.2 | 30.9 | 53.6 | 12.0 | 96.5 | 34.0 | 15.5 | 49.5 | 72.6 | 19.8 | 12.6 | 18.7 | 26.8 | 77.9 | 21.0 | 26.5 | 47.5 | 524.2 | 94.0 | 6.2 | 13.1 | 113.2 | 28.5 | 22.9 | 9.8 | 61.2 | 22.4 | 7.4 | 29.8 | 51.9 | 11.3 | 7.4 | 7.7 | 14.9 | 41.3 | 15.7 | 21.9 | 37.6 | 335.0 | 106.9 | 13.7 | 13.4 | 134.0 | 18.4 | 25.8 | 11.0 | 55.2 | 22.2 | 8.1 | 30.3 | 37.8 | 7.2 | 7.2 | 14.1 | 12.2 | 40.6 | 8.6 | 10.4 | 19.0 | 316.9 | 27.3 | NA | NA | 30.2 | 6.4 | 5.2 | 2.5 | 14.1 | 6.7 | 1.9 | 8.6 | NA | 2.9 | 2.0 | 4.2 | NA | NA | 2.2 | 3.5 | 5.7 | 83.0 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 14.2 | 1.4 | 2.0 | 17.6 | 3.4 | 4.3 | 2.6 | 10.3 | 3.9 | 2.3 | 6.2 | 10.1 | 2.8 | 1.0 | 2.2 | 2.1 | 8.2 | 4.3 | 5.6 | 9.9 | 62.3 | 1033.9 | 83.7 | 151.6 | 1269.3 | 210.5 | 294.4 | 157.0 | 661.8 | 284.6 | 98.3 | 383.0 | 462.4 | 136.0 | 71.3 | 139.7 | 160.3 | 507.3 | 150.7 | 196.4 | 347.1 | 3630.8 |
| 1983-07-01 | 323.1 | 41.6 | 69.5 | 434.2 | 60.8 | 111.7 | 58.8 | 231.3 | 97.4 | 34.1 | 131.5 | 181.4 | 57.7 | 23.9 | 54.6 | 66.9 | 203.0 | 61.9 | 84.7 | 146.6 | 1328.1 | 297.6 | 21.3 | 36.2 | 355.2 | 54.9 | 64.0 | 59.9 | 178.8 | 95.1 | 26.6 | 121.7 | 103.4 | 39.0 | 16.2 | 36.9 | 34.2 | 126.4 | 35.8 | 49.8 | 85.6 | 971.0 | 162.9 | NA | NA | 185.1 | 32.9 | 55.0 | 14.4 | 102.2 | 33.5 | 16.0 | 49.5 | 65.9 | 20.8 | 13.0 | 19.5 | 29.0 | 82.2 | 22.1 | 27.8 | 49.9 | 534.8 | 98.3 | 6.2 | 13.5 | 118.0 | 25.7 | 22.2 | 11.1 | 58.9 | 24.0 | 7.9 | 31.9 | 51.7 | 8.3 | 8.1 | 8.3 | 15.8 | 40.5 | 18.2 | 23.1 | 41.3 | 342.3 | 106.2 | 13.9 | 13.1 | 133.2 | 18.4 | 30.2 | 9.7 | 58.3 | 24.7 | 8.4 | 33.0 | 40.3 | 7.6 | 7.7 | 14.3 | 12.3 | 41.8 | 8.9 | 10.9 | 19.8 | 326.5 | 28.3 | NA | NA | 31.3 | 5.9 | 5.1 | 2.7 | 13.7 | 6.0 | 1.8 | 7.8 | NA | 2.9 | 2.0 | 4.0 | NA | NA | 2.2 | 4.4 | 6.7 | 83.9 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.6 | 1.5 | 2.1 | 17.2 | 3.8 | 4.2 | 2.8 | 10.8 | 4.0 | 2.5 | 6.5 | 10.4 | 3.0 | 1.1 | 2.3 | 2.3 | 8.7 | 4.6 | 6.3 | 10.8 | 64.5 | 1047.4 | 85.9 | 159.5 | 1292.8 | 203.9 | 293.4 | 159.6 | 656.9 | 286.2 | 97.7 | 384.0 | 468.3 | 141.0 | 72.5 | 140.9 | 165.6 | 519.9 | 154.7 | 209.8 | 364.5 | 3686.5 |
| 1983-08-01 | 338.1 | 42.2 | 67.9 | 448.2 | 64.8 | 117.2 | 64.8 | 246.9 | 96.3 | 34.0 | 130.2 | 179.7 | 61.5 | 25.0 | 54.6 | 70.4 | 211.5 | 64.7 | 85.2 | 149.9 | 1366.3 | 309.6 | 22.6 | 37.1 | 369.3 | 58.8 | 72.4 | 65.2 | 196.4 | 91.3 | 25.7 | 117.0 | 101.4 | 47.1 | 17.2 | 39.3 | 37.3 | 140.9 | 37.1 | 53.3 | 90.5 | 1015.5 | 167.3 | NA | NA | 189.4 | 35.1 | 61.0 | 14.0 | 110.1 | 36.6 | 16.4 | 52.9 | 60.4 | 21.2 | 13.9 | 22.1 | 29.5 | 86.7 | 22.8 | 28.7 | 51.5 | 551.0 | 101.7 | 6.7 | 13.8 | 122.1 | 27.8 | 24.9 | 11.2 | 63.9 | 23.0 | 7.9 | 30.9 | 54.0 | 9.0 | 8.5 | 8.5 | 16.3 | 42.3 | 18.6 | 24.6 | 43.2 | 356.4 | 111.9 | 14.2 | 13.5 | 139.6 | 19.4 | 34.2 | 11.0 | 64.6 | 24.1 | 8.4 | 32.4 | 38.0 | 8.9 | 8.2 | 15.3 | 13.1 | 45.5 | 9.2 | 11.7 | 20.8 | 340.9 | 29.6 | NA | NA | 32.6 | 6.4 | 5.8 | 3.1 | 15.3 | 6.5 | 1.9 | 8.3 | NA | 2.9 | 2.2 | 4.0 | NA | NA | 2.5 | 4.7 | 7.1 | 88.1 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.8 | 1.5 | 1.8 | 17.1 | 3.8 | 4.1 | 2.8 | 10.7 | 3.6 | 2.4 | 6.0 | 10.0 | 3.2 | 1.1 | 2.5 | 2.4 | 9.2 | 4.9 | 4.5 | 9.4 | 62.5 | 1089.4 | 88.5 | 159.1 | 1337.0 | 217.7 | 320.9 | 172.4 | 711.0 | 283.0 | 96.9 | 379.9 | 458.2 | 155.8 | 76.5 | 147.3 | 174.6 | 554.2 | 160.8 | 215.2 | 376.0 | 3816.3 |
| 1983-09-01 | 330.6 | 42.5 | 67.5 | 440.6 | 65.1 | 106.9 | 68.7 | 240.7 | 105.6 | 37.2 | 142.9 | 185.0 | 61.0 | 24.5 | 53.8 | 71.6 | 210.9 | 66.3 | 84.3 | 150.6 | 1370.8 | 310.2 | 22.4 | 37.4 | 370.0 | 57.4 | 69.7 | 66.4 | 193.6 | 94.7 | 26.5 | 121.3 | 105.2 | 46.1 | 16.9 | 38.3 | 37.2 | 138.5 | 36.8 | 54.0 | 90.8 | 1019.2 | 163.9 | NA | NA | 185.1 | 34.6 | 55.0 | 15.1 | 104.7 | 37.0 | 17.5 | 54.5 | 73.9 | 20.5 | 13.4 | 21.5 | 29.6 | 85.1 | 22.8 | 27.7 | 50.5 | 553.9 | 99.1 | 7.0 | 13.4 | 119.5 | 25.8 | 22.8 | 12.3 | 61.0 | 24.4 | 8.2 | 32.6 | 52.3 | 9.1 | 8.3 | 8.2 | 16.4 | 42.0 | 18.4 | 23.8 | 42.3 | 349.7 | 111.3 | 14.8 | 13.2 | 139.3 | 19.6 | 30.1 | 12.1 | 61.9 | 25.6 | 9.2 | 34.8 | 40.3 | 7.6 | 8.2 | 15.2 | 13.6 | 44.5 | 9.8 | 11.7 | 21.5 | 342.3 | 29.2 | NA | NA | 32.1 | 6.4 | 5.3 | 3.2 | 14.9 | 6.0 | 1.9 | 7.9 | NA | 2.9 | 2.0 | 4.2 | NA | NA | 2.3 | 5.0 | 7.4 | 88.0 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 13.5 | 1.5 | 2.1 | 17.1 | 3.8 | 4.0 | 3.1 | 10.9 | 3.6 | 2.5 | 6.1 | 10.3 | 3.2 | 1.0 | 2.3 | 2.4 | 8.9 | 4.5 | 6.4 | 10.9 | 64.1 | 1075.6 | 89.6 | 157.7 | 1322.8 | 213.9 | 295.1 | 181.4 | 690.3 | 298.5 | 103.5 | 402.0 | 482.7 | 152.4 | 74.9 | 144.6 | 176.0 | 547.9 | 162.0 | 215.6 | 377.6 | 3823.4 |
| 1983-10-01 | 351.1 | 45.0 | 66.0 | 462.1 | 66.3 | 114.4 | 84.1 | 264.8 | 97.9 | 37.3 | 135.2 | 194.4 | 56.9 | 24.6 | 55.6 | 74.9 | 212.0 | 63.7 | 80.1 | 143.8 | 1412.3 | 314.5 | 22.9 | 37.0 | 374.4 | 59.9 | 73.5 | 71.3 | 204.8 | 102.9 | 29.1 | 132.0 | 106.4 | 46.9 | 18.2 | 38.4 | 39.1 | 142.7 | 39.6 | 53.1 | 92.7 | 1053.0 | 167.2 | NA | NA | 189.6 | 36.4 | 52.6 | 14.7 | 103.7 | 33.1 | 16.2 | 49.3 | 65.5 | 21.1 | 13.2 | 20.9 | 29.5 | 84.7 | 22.9 | 29.4 | 52.4 | 545.1 | 96.7 | 7.2 | 12.7 | 116.6 | 21.9 | 22.7 | 10.2 | 54.8 | 22.5 | 7.6 | 30.0 | 51.5 | 8.4 | 8.0 | 8.7 | 15.2 | 40.3 | 17.8 | 21.6 | 39.4 | 332.8 | 112.3 | 15.1 | 13.0 | 140.4 | 17.8 | 26.8 | 12.8 | 57.4 | 24.2 | 9.1 | 33.3 | 41.5 | 8.5 | 7.9 | 15.9 | 13.9 | 46.2 | 10.1 | 11.6 | 21.7 | 340.5 | 29.9 | NA | NA | 33.0 | 6.3 | 4.9 | 3.3 | 14.5 | 6.4 | 1.9 | 8.4 | NA | 3.2 | 2.1 | 4.0 | NA | NA | 2.5 | 5.1 | 7.5 | 88.7 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 16.6 | 1.5 | 2.4 | 20.5 | 3.4 | 5.0 | 3.3 | 11.7 | 3.4 | 2.5 | 5.9 | 11.2 | 3.1 | 1.3 | 2.3 | 2.7 | 9.4 | 5.5 | 7.2 | 12.8 | 71.4 | 1105.9 | 93.1 | 156.4 | 1355.4 | 213.3 | 301.1 | 200.2 | 714.6 | 291.8 | 104.0 | 395.8 | 485.3 | 149.9 | 75.8 | 146.9 | 180.8 | 553.5 | 163.1 | 211.0 | 374.1 | 3878.7 |
| 1983-11-01 | 361.5 | 45.8 | 67.2 | 474.5 | 72.8 | 136.5 | 101.2 | 310.4 | 110.2 | 41.0 | 151.2 | 224.9 | 59.3 | 27.8 | 57.7 | 83.4 | 228.2 | 69.4 | 82.9 | 152.3 | 1541.6 | 336.8 | 24.0 | 38.4 | 399.1 | 64.3 | 80.3 | 82.8 | 227.4 | 109.7 | 30.0 | 139.6 | 123.1 | 48.9 | 19.4 | 40.7 | 42.6 | 151.7 | 42.0 | 53.9 | 95.8 | 1136.8 | 175.6 | NA | NA | 198.2 | 37.2 | 61.7 | 16.7 | 115.6 | 37.6 | 17.5 | 55.1 | 77.6 | 23.3 | 14.6 | 22.1 | 32.3 | 92.2 | 24.6 | 29.9 | 54.5 | 593.3 | 101.2 | 7.6 | 12.8 | 121.6 | 24.2 | 27.0 | 11.8 | 63.0 | 24.6 | 7.9 | 32.5 | 64.3 | 9.2 | 8.6 | 8.6 | 16.1 | 42.5 | 18.2 | 21.8 | 40.1 | 363.9 | 115.0 | 15.4 | 13.2 | 143.7 | 18.8 | 31.4 | 15.5 | 65.7 | 26.0 | 9.7 | 35.7 | 47.9 | 9.2 | 8.8 | 16.4 | 15.5 | 49.8 | 10.8 | 11.7 | 22.5 | 365.3 | 31.5 | NA | NA | 34.7 | 7.2 | 6.2 | 3.4 | 16.8 | 7.0 | 2.1 | 9.1 | NA | 3.4 | 2.3 | 4.0 | NA | NA | 2.8 | 5.2 | 8.0 | 97.7 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | 17.3 | 1.5 | 2.4 | 21.1 | 4.0 | 5.5 | 3.6 | 13.1 | 3.7 | 2.8 | 6.5 | 12.6 | 3.3 | 1.5 | 2.8 | 3.2 | 10.8 | 6.7 | 7.1 | 13.8 | 78.0 | 1155.9 | 95.4 | 159.6 | 1410.9 | 230.1 | 350.1 | 235.3 | 815.5 | 320.3 | 111.4 | 431.7 | 568.7 | 158.3 | 83.7 | 153.3 | 198.8 | 594.1 | 175.3 | 215.3 | 390.6 | 4211.5 |
Select one of the time series as follows (but replace the column name with your own chosen column):
myts <- ts(retaildata[,“A3349873A”],frequency=12, start=c(1982,4))
I have selected “A3349337W” as the timeseries from the retail data set for this exercise.
myts <- ts(retaildata[,"A3349337W"],frequency=12, start=c(1982,4))
myts## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov
## 1982 53.6 55.4 48.4 52.1 54.2 53.6 58.0 67.2
## 1983 66.6 59.2 67.3 57.7 64.9 58.6 58.8 64.8 68.7 84.1 101.2
## 1984 73.7 69.6 77.7 68.5 70.0 60.5 60.2 70.0 69.5 81.5 96.5
## 1985 69.4 69.8 74.1 71.9 83.6 68.8 71.8 79.4 76.0 97.0 126.8
## 1986 90.3 89.8 89.6 91.9 96.0 89.3 79.4 89.1 88.1 116.8 128.6
## 1987 103.9 97.3 97.9 97.2 106.5 88.2 97.7 100.2 110.8 137.3 150.5
## 1988 126.6 119.4 123.6 108.8 121.0 113.9 110.9 124.3 118.5 143.9 172.1
## 1989 160.7 155.2 161.0 149.3 165.6 140.1 128.2 140.4 130.2 143.3 185.3
## 1990 96.4 95.0 103.8 97.1 104.6 100.7 98.2 106.6 96.7 113.3 126.2
## 1991 89.1 99.6 129.0 125.6 127.3 111.7 114.1 118.0 119.6 121.5 128.5
## 1992 100.1 108.2 113.2 108.0 98.2 95.2 101.4 93.5 112.0 118.9 125.7
## 1993 100.7 102.8 113.5 99.2 95.4 89.3 84.4 91.1 102.2 101.4 108.5
## 1994 111.0 121.4 125.6 116.2 125.1 119.1 117.5 123.8 134.5 141.0 145.2
## 1995 120.8 121.0 132.6 116.3 113.2 120.2 124.3 134.0 140.6 163.7 176.2
## 1996 157.5 147.7 158.1 152.4 171.0 158.0 174.0 157.5 167.0 181.0 189.6
## 1997 168.0 154.9 169.9 159.8 172.7 154.1 144.9 141.3 164.3 162.7 172.8
## 1998 157.0 145.0 158.6 145.9 146.8 140.2 135.8 141.7 158.7 148.4 148.0
## 1999 133.1 120.5 132.2 126.0 141.0 135.0 143.7 144.4 171.7 185.5 167.9
## 2000 169.7 163.2 167.6 148.7 161.4 188.5 158.3 174.5 193.2 194.5 209.7
## 2001 209.6 185.2 202.2 200.0 200.3 200.3 193.6 211.4 218.2 236.3 230.6
## 2002 219.9 196.6 218.7 216.8 205.5 198.2 233.9 246.2 259.8 277.3 294.3
## 2003 247.0 229.3 250.3 241.6 247.0 258.7 271.3 291.1 312.7 324.6 315.2
## 2004 258.9 246.5 260.9 249.0 256.5 257.4 275.4 269.8 279.8 307.3 323.9
## 2005 281.8 250.6 274.1 270.3 268.2 264.0 266.9 298.6 303.1 329.4 345.6
## 2006 288.0 277.3 302.8 288.5 290.4 275.4 262.4 272.9 279.7 299.3 313.3
## 2007 286.4 268.4 286.6 260.0 273.0 248.5 259.7 272.2 293.6 294.9 294.3
## 2008 263.0 246.2 255.2 240.2 239.6 226.9 238.7 253.1 271.3 283.1 299.0
## 2009 289.3 249.6 272.1 272.9 279.4 267.8 273.1 307.7 318.2 334.0 325.0
## 2010 309.2 272.6 311.1 298.2 313.1 305.8 307.3 330.9 362.8 361.7 364.2
## 2011 311.6 283.7 322.2 310.8 319.5 305.1 308.9 355.6 384.9 401.1 382.1
## 2012 334.0 292.1 309.6 305.8 325.0 314.2 327.2 363.7 406.9 397.1 379.6
## 2013 340.0 293.9 330.7 290.7 291.8 281.1 309.8 344.6 360.7 384.7 367.9
## Dec
## 1982 146.3
## 1983 192.3
## 1984 179.4
## 1985 221.2
## 1986 235.4
## 1987 248.8
## 1988 307.4
## 1989 228.9
## 1990 159.5
## 1991 151.4
## 1992 154.7
## 1993 179.0
## 1994 180.7
## 1995 225.4
## 1996 249.8
## 1997 248.7
## 1998 183.0
## 1999 200.7
## 2000 266.3
## 2001 291.0
## 2002 341.9
## 2003 360.8
## 2004 361.1
## 2005 395.2
## 2006 341.6
## 2007 339.3
## 2008 360.2
## 2009 348.9
## 2010 395.4
## 2011 409.0
## 2012 428.0
## 2013 430.7title <- 'Retail Sales for Category = A3349337W'
# Timeseries plot before Transformation:
plot <- autoplot(myts,ylab="$ Sales Turnover",xlab="Year") + ggtitle(title)
lambda <- BoxCox.lambda(myts)
cat('BoxCox Transofrmation Parameter, Lambda:', lambda)## BoxCox Transofrmation Parameter, Lambda: 0.9165544# Timeseries plot after applying BoxCox Transformation:
plot_boxcox <- autoplot(BoxCox(myts,lambda),ylab="$ Sales Turnover",xlab="Year") + ggtitle(paste(title," (w/ BoxCox Transformation)"))
grid.arrange(plot,plot_boxcox, ncol=2) 
Section 3.7 - Exercise 8
For your retail time series (from Exercise 3 in Section 2.10):
- Split the data into two parts using - myts.train <- window(myts, end=c(2010,12)) myts.test <- window(myts, start=2011)
myts.train <- window(myts, end=c(2010,12))
myts.test <- window(myts, start=2011)- Check that your data have been split appropriately by producing the following plot - autoplot(myts) + autolayer(myts.train, series=“Training”) + autolayer(myts.test, series=“Test”)
autoplot(myts) +
autolayer(myts.train, series="Training") +
autolayer(myts.test, series="Test")
- Calculate forecasts using snaive applied to myts.train - fc <- snaive(myts.train)
fc <- snaive(myts.train)
fc %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")| Point Forecast | Lo 80 | Hi 80 | Lo 95 | Hi 95 | |
|---|---|---|---|---|---|
| Jan 2011 | 309.2 | 275.4855 | 342.9145 | 257.6381 | 360.7619 |
| Feb 2011 | 272.6 | 238.8855 | 306.3145 | 221.0381 | 324.1619 |
| Mar 2011 | 311.1 | 277.3855 | 344.8145 | 259.5381 | 362.6619 |
| Apr 2011 | 298.2 | 264.4855 | 331.9145 | 246.6381 | 349.7619 |
| May 2011 | 313.1 | 279.3855 | 346.8145 | 261.5381 | 364.6619 |
| Jun 2011 | 305.8 | 272.0855 | 339.5145 | 254.2381 | 357.3619 |
| Jul 2011 | 307.3 | 273.5855 | 341.0145 | 255.7381 | 358.8619 |
| Aug 2011 | 330.9 | 297.1855 | 364.6145 | 279.3381 | 382.4619 |
| Sep 2011 | 362.8 | 329.0855 | 396.5145 | 311.2381 | 414.3619 |
| Oct 2011 | 361.7 | 327.9855 | 395.4145 | 310.1381 | 413.2619 |
| Nov 2011 | 364.2 | 330.4855 | 397.9145 | 312.6381 | 415.7619 |
| Dec 2011 | 395.4 | 361.6855 | 429.1145 | 343.8381 | 446.9619 |
| Jan 2012 | 309.2 | 261.5205 | 356.8795 | 236.2804 | 382.1196 |
| Feb 2012 | 272.6 | 224.9205 | 320.2795 | 199.6804 | 345.5196 |
| Mar 2012 | 311.1 | 263.4205 | 358.7795 | 238.1804 | 384.0196 |
| Apr 2012 | 298.2 | 250.5205 | 345.8795 | 225.2804 | 371.1196 |
| May 2012 | 313.1 | 265.4205 | 360.7795 | 240.1804 | 386.0196 |
| Jun 2012 | 305.8 | 258.1205 | 353.4795 | 232.8804 | 378.7196 |
| Jul 2012 | 307.3 | 259.6205 | 354.9795 | 234.3804 | 380.2196 |
| Aug 2012 | 330.9 | 283.2205 | 378.5795 | 257.9804 | 403.8196 |
| Sep 2012 | 362.8 | 315.1205 | 410.4795 | 289.8804 | 435.7196 |
| Oct 2012 | 361.7 | 314.0205 | 409.3795 | 288.7804 | 434.6196 |
| Nov 2012 | 364.2 | 316.5205 | 411.8795 | 291.2804 | 437.1196 |
| Dec 2012 | 395.4 | 347.7205 | 443.0795 | 322.4804 | 468.3196 |
# Calculating Forecast applying BoxCox transformation
fc1 <- snaive(myts.train, lambda = lambda)
#fc1 %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
# Calculating Forecast applying BoxCox transformation
fc2 <- snaive(myts.train, lambda = lambda, biasadj = TRUE)
autoplot(myts.train) +
autolayer(fc, series="Without Transformation") +
autolayer(fc1, series="Simple Back Transformation", PI=FALSE) +
autolayer(fc2, series="Bias Adjusted", PI=FALSE) +
guides(colour=guide_legend(title="Forecast"))
- Compare the accuracy of your forecasts against the actual values stored in myts.test - accuracy(fc,myts.test)
accuracy(fc,myts.test)## ME RMSE MAE MPE MAPE MASE
## Training set 9.460661 26.30758 21.23363 4.655690 12.762886 1.0000000
## Test set 17.212500 21.26067 17.39583 4.748234 4.807728 0.8192584
## ACF1 Theil's U
## Training set 0.8070166 NA
## Test set 0.4843871 0.6934111# With BoxCox Transformation
accuracy(fc1,myts.test)## ME RMSE MAE MPE MAPE MASE
## Training set 9.460661 26.30758 21.23363 4.655690 12.762886 1.0000000
## Test set 17.212500 21.26067 17.39583 4.748234 4.807728 0.8192584
## ACF1 Theil's U
## Training set 0.8070166 NA
## Test set 0.4843871 0.6934111- Check the residuals - checkresiduals(fc)
Q. Do the residuals appear to be uncorrelated and normally distributed?
checkresiduals(fc)
##
## Ljung-Box test
##
## data: Residuals from Seasonal naive method
## Q* = 856.11, df = 24, p-value < 2.2e-16
##
## Model df: 0. Total lags used: 24Analysis of Residuals:
- Time plot shows that residuals show variations, residual values are not mostly zeros and does not show any seasonal patter.
- From the ACF plot, residuals seem to have high positive autocorrelation between r1 to r10. So based on Box-Pierce test, Q statistic (\(Q=T\sum _{ k=1 }^{ h }{ { { r }_{ k } }^{ 2 } }\)) using h=10 for non-seasonal residual data, seems to have a relatively high value showing strong correlation amongst residuals.
- From the Histogram, it is clear that the mean of the residuals is not zero and residuals are also not normally distributed.
- Based on the outpout of Ljung-Box test, \({ Q }^{ * }\) statistic (\({ Q }^{ * }=T(T+2)\sum _{ k=1 }^{ h }{ { { { (T-k) }^{ -1 }r }_{ k } }^{ 2 } }\)) shows significantly high value with very small p-value. This suggests that the residual autocorrelations DO NOT come from white noise series. Hence it can be concluded that the residuals show sufficient correlation.
- Q. How sensitive are the accuracy measures to the training/test split?
The approach to gauge sensitivity of accuracy measures to the training/test split would be to iterate over multiple years as splitting point and verify the impact on the measures.
calcAccuracy <- function(year){
train <- window(myts, end=c(year, 12))
test <- window(myts, start=year+1)
model_accuracy <- accuracy(snaive(train), test)
return(model_accuracy)
}
splitYears <- c(2005:2011)
modelAccuracyDF <- data.frame()
for (year in splitYears){
currentAccuracy <- calcAccuracy(year)
testRow <- data.frame(t(currentAccuracy[2,]))
modelAccuracyDF <- rbind(modelAccuracyDF, testRow)
}
row.names(modelAccuracyDF) <- paste('Dec',splitYears,sep = '-')
modelAccuracyDF <- tibble::rownames_to_column(modelAccuracyDF, "Split_Period")
modelAccuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")| Split_Period | ME | RMSE | MAE | MPE | MAPE | MASE | ACF1 | Theil.s.U |
|---|---|---|---|---|---|---|---|---|
| Dec-2005 | -9.4625000 | 26.79323 | 22.22083 | -3.0565350 | 7.532356 | 1.0803141 | 0.6458613 | 1.2908423 |
| Dec-2006 | -17.9083333 | 25.60143 | 20.61667 | -7.1891576 | 8.014002 | 0.9961769 | 0.5989406 | 1.1544668 |
| Dec-2007 | -1.6375000 | 22.04366 | 19.97083 | -1.3680734 | 7.258093 | 0.9818795 | 0.7868247 | 0.9620150 |
| Dec-2008 | 46.5541667 | 52.76536 | 47.49583 | 14.7565145 | 15.026410 | 2.3323712 | 0.6436880 | 2.2893119 |
| Dec-2009 | 39.6166667 | 41.75055 | 39.61667 | 11.6729690 | 11.672969 | 1.9047330 | 0.4997840 | 1.4503696 |
| Dec-2010 | 17.2125000 | 21.26067 | 17.39583 | 4.7482337 | 4.807728 | 0.8192584 | 0.4843871 | 0.6934111 |
| Dec-2011 | 0.8666667 | 16.47723 | 14.34167 | 0.0508195 | 4.245806 | 0.6839372 | 0.4736556 | 0.4855843 |
ggplot(modelAccuracyDF,aes(x=Split_Period)) +
geom_line(aes(y = RMSE, color = "blue"),group=1,size=2 ) +
geom_line(aes(y = MAPE, color = "red"),group=1,size=2) +
geom_line(aes(y = MAE, color = "green"),group=1,size=2) +
geom_line(aes(y = MASE, color = "orange"),group=1,size=2) +
xlab('Train/Test Split Period') +
ylab('Measure Value') +
scale_color_discrete(name = "Accuracy Measures", labels = c("RMSE","MAPE","MAE","MASE")) +
ggtitle("Train/Test Split Accuracy Measures Sensitivity")
From the table and chart above, we can clearly see that choice of Training/Test split point impacts the accuracy measures for the test data sets. Also, it appears that Dec’2011 is the best choice for the split with lowest RMSE, MAE, MAPE and MASE etc. values.