Analytics
Dan Wigodsky
Data 624 Homework 3
February 21, 2019
Time Series Decomposition
Question 6:2
Our time series comes from 5 years worth of monthly values relating to production of a plastic product.
From a basic graph of our time series, a seasonal and positive trend component appear to be fairly strong.
## Time-Series [1:60] from 1 to 5.92: 742 697 776 898 1030 ...
A multiplicative decomposition is applied to our dataset. The result confirms the visual inspection. The overall trend is positive, until the end. The seasonal component is fairly regular.
## $x
## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 1 742 697 776 898 1030 1107 1165 1216 1208 1131 971 783
## 2 741 700 774 932 1099 1223 1290 1349 1341 1296 1066 901
## 3 896 793 885 1055 1204 1326 1303 1436 1473 1453 1170 1023
## 4 951 861 938 1109 1274 1422 1486 1555 1604 1600 1403 1209
## 5 1030 1032 1126 1285 1468 1637 1611 1608 1528 1420 1119 1013
##
## $seasonal
## Jan Feb Mar Apr May Jun Jul
## 1 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## 2 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## 3 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## 4 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## 5 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## Aug Sep Oct Nov Dec
## 1 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
## 2 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
## 3 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
## 4 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
## 5 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
##
## $trend
## Jan Feb Mar Apr May Jun Jul
## 1 NA NA NA NA NA NA 976.9583
## 2 1000.4583 1011.2083 1022.2917 1034.7083 1045.5417 1054.4167 1065.7917
## 3 1117.3750 1121.5417 1130.6667 1142.7083 1153.5833 1163.0000 1170.3750
## 4 1208.7083 1221.2917 1231.7083 1243.2917 1259.1250 1276.5833 1287.6250
## 5 1374.7917 1382.2083 1381.2500 1370.5833 1351.2500 1331.2500 NA
## Aug Sep Oct Nov Dec
## 1 977.0417 977.0833 978.4167 982.7083 990.4167
## 2 1076.1250 1084.6250 1094.3750 1103.8750 1112.5417
## 3 1175.5000 1180.5417 1185.0000 1190.1667 1197.0833
## 4 1298.0417 1313.0000 1328.1667 1343.5833 1360.6250
## 5 NA NA NA NA NA
##
## $random
## Jan Feb Mar Apr May Jun Jul
## 1 NA NA NA NA NA NA 1.0247887
## 2 0.9656005 0.9745267 0.9750081 0.9894824 1.0061175 1.0024895 1.0401641
## 3 1.0454117 0.9953920 1.0079773 1.0142083 0.9990100 0.9854384 0.9567618
## 4 1.0257400 0.9924762 0.9807020 0.9798704 0.9684851 0.9627557 0.9917766
## 5 0.9767392 1.0510964 1.0498039 1.0299302 1.0398787 1.0628077 NA
## Aug Sep Oct Nov Dec
## 1 1.0157335 1.0040354 0.9724119 0.9961368 0.9489762
## 2 1.0230774 1.0040674 0.9962088 0.9735577 0.9721203
## 3 0.9969907 1.0132932 1.0314752 0.9910657 1.0258002
## 4 0.9776897 0.9920952 1.0133954 1.0527311 1.0665946
## 5 NA NA NA NA NA
##
## $figure
## [1] 0.7670466 0.7103357 0.7765294 0.9103112 1.0447386 1.1570026 1.1636317
## [8] 1.2252952 1.2313635 1.1887444 0.9919176 0.8330834
##
## $type
## [1] "multiplicative"
##
## attr(,"class")
## [1] "decomposed.ts"
When we change a point to turn it into an outlier, it changes the seasonally adjusted series in the area where the change is made. It creates a dramatic spike, but the effect is local. If we look at the decomposition, however, we see more of an effect. If it brings a high value up, the seasonal pattern becomes more spiked throughout. If, however, it is changed opposite the seasonal direction, it turns a smooth curve into a set of dual peaks. The seasonal series is changed, by a fraction of the outlier’s distance, throughout the entire series.
Question 6:3
When we use an x-11 decomposition for our retail series from previous homeworks, some features that were not clear before pop out. Some large spikes show up in the residuals early in the series. Our trend rises in 2009 after a short dip. As it turns out, this rise is quite interesting. A combination of population growth, trading partners and banking conservatism allowed Australia to escape ill effects of a recession that most of the world felt much more strongly. (http://www.cbsnews.com/8301-505125_162-51352693/how-australia-ducked-the-crisis/)
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_______________________Appendix____________________________________________
str(plastics) autoplot(plastics) +xlab(‘year’)+ theme(text = element_text(family = “corben”,color=‘#249382’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’))
#compute multiplicative decomposition traditional_multiplicative<-decompose(plastics,type=“multiplicative”) traditional_multiplicative plastics %>% decompose(type=“multiplicative”) %>% autoplot(color=‘#249382’) + xlab(“Year”) + ggtitle(“classical multiplicative decomposition of product a, plastics”)+ theme(text = element_text(family = “corben”,color=‘#249382’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’)) #seasonally adjust the plastics set autoplot(plastics, series=“Data”) + autolayer(seasadj(traditional_multiplicative), series=“Seasadj”,size=1) + xlab(“Year”) + ylab(“Sales”) + ggtitle(“Plastics, Product a (seasonally adjusted)”) + scale_colour_manual(values=c(“Data”=“#baa9f9”,“Seasadj”=“#2267d8”), breaks=c(“Data”,“Seasadj”))+ theme(panel.background = element_rect(fill = ‘#f4f4ef’),panel.grid.major = element_blank(), panel.grid.minor = element_blank(),text = element_text(family = “corben”,color=‘#2267d8’,size=12))
plastics[26]<-1100
traditional_multiplicative<-decompose(plastics,type=“multiplicative”) autoplot(plastics, series=“Data”) + autolayer(seasadj(traditional_multiplicative), series=“Seasadj”,size=1) + xlab(“Year”) + ylab(“Sales”) + ggtitle(“Plastics, high outlier at low of cycle”) + scale_colour_manual(values=c(“Data”=“#baa9f9”,“Seasadj”=“#55a9d1”), breaks=c(“Data”,“Seasadj”))+ theme(panel.background = element_rect(fill = ‘#f4f4ef’),panel.grid.major = element_blank(), panel.grid.minor = element_blank(),text = element_text(family = “corben”,color=‘#55a9d1’,size=12))
plastics[26]<-793 plastics[29]<-1600
traditional_multiplicative<-decompose(plastics,type=“multiplicative”) autoplot(plastics, series=“Data”) + autolayer(seasadj(traditional_multiplicative), series=“Seasadj”,size=1) + xlab(“Year”) + ylab(“Sales”) + ggtitle(“Plastics, high outlier at middle of cycle”) + scale_colour_manual(values=c(“Data”=“#baa9f9”,“Seasadj”=“#55a9d1”), breaks=c(“Data”,“Seasadj”))+ theme(panel.background = element_rect(fill = ‘#f4f4ef’),panel.grid.major = element_blank(), panel.grid.minor = element_blank(),text = element_text(family = “corben”,color=‘#55a9d1’,size=12))
plastics[29]<-1204 plastics[33]<-2000
traditional_multiplicative<-decompose(plastics,type=“multiplicative”) autoplot(plastics, series=“Data”) + autolayer(seasadj(traditional_multiplicative), series=“Seasadj”,size=1) + xlab(“Year”) + ylab(“Sales”) + ggtitle(“Plastics, high outlier at top of cycle”) + scale_colour_manual(values=c(“Data”=“#baa9f9”,“Seasadj”=“#55a9d1”), breaks=c(“Data”,“Seasadj”))+ theme(panel.background = element_rect(fill = ‘#f4f4ef’),panel.grid.major = element_blank(), panel.grid.minor = element_blank(),text = element_text(family = “corben”,color=‘#55a9d1’,size=12))
plastics %>% decompose(type=“multiplicative”) %>% autoplot(color=‘#249382’) + xlab(“Year”) + ggtitle(“multiplicative decomposition, high outlier at top of cycle”)+ theme(text = element_text(family = “corben”,color=‘#5574d1’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’))
plastics[33]<-1000
traditional_multiplicative<-decompose(plastics,type=“multiplicative”) autoplot(plastics, series=“Data”) + autolayer(seasadj(traditional_multiplicative), series=“Seasadj”,size=1) + xlab(“Year”) + ylab(“Sales”) + ggtitle(“multiplicative decomposition, low outlier at top of cycle”) + scale_colour_manual(values=c(“Data”=“#baa9f9”,“Seasadj”=“#55a9d1”), breaks=c(“Data”,“Seasadj”))+ theme(panel.background = element_rect(fill = ‘#f4f4ef’),panel.grid.major = element_blank(), panel.grid.minor = element_blank(),text = element_text(family = “corben”,color=‘#55a9d1’,size=12))
plastics %>% decompose(type=“multiplicative”) %>% autoplot(color=‘#249382’) + xlab(“Year”) + ggtitle(“classical multiplicative decomposition of product a, plastics”)+ theme(text = element_text(family = “corben”,color=‘#5574d1’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’))
retaildata <- readxl::read_excel(“C:/Users/dawig/Desktop/Data624/retail.xlsx”, skip=1) turnover <- ts(retaildata[,“A3349608L”], frequency=12, start=c(1982,4)) autoplot(turnover, ylab=“turnover”, xlab=“”)+ theme(panel.background = element_rect(fill = ‘#efeae8’))+ ggtitle(“clothing, footwear and personal accessory turnover series”)+ theme(text = element_text(family = “corben”,color=‘#249382’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’)) turnover %>% seas(x11=“”) -> fit autoplot(fit) + ggtitle(“X11 decomposition of clothing, footwear and personal accessory turnover”)+ theme(text = element_text(family = “corben”,color=‘#249382’,size=12),panel.background = element_rect(fill = ‘#f4f4ef’))