library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ ggplot2 3.3.6     ✔ purrr   0.3.4
## ✔ tibble  3.1.8     ✔ dplyr   1.0.9
## ✔ tidyr   1.2.0     ✔ stringr 1.4.1
## ✔ readr   2.1.2     ✔ forcats 0.5.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
library(tsibbledata)
library(tsibble)
## 
## Attaching package: 'tsibble'
## 
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, union
library(fpp3)
## ── Attaching packages ──────────────────────────────────────────── fpp3 0.4.0 ──
## ✔ lubridate 1.8.0     ✔ fable     0.3.1
## ✔ feasts    0.2.2     
## ── Conflicts ───────────────────────────────────────────────── fpp3_conflicts ──
## ✖ lubridate::date()     masks base::date()
## ✖ dplyr::filter()       masks stats::filter()
## ✖ tsibble::intersect()  masks base::intersect()
## ✖ lubridate::interval() masks tsibble::interval()
## ✖ dplyr::lag()          masks stats::lag()
## ✖ tsibble::setdiff()    masks base::setdiff()
## ✖ tsibble::union()      masks base::union()
proof<-c(0.067, 0.133, 0.200, 0.200, 0.200, 0.133, 0.067)
gas <- tail(aus_production, 5*4) %>% dplyr::select(Gas)
head(gas)
## # A tsibble: 6 x 2 [1Q]
##     Gas Quarter
##   <dbl>   <qtr>
## 1   221 2005 Q3
## 2   180 2005 Q4
## 3   171 2006 Q1
## 4   224 2006 Q2
## 5   233 2006 Q3
## 6   192 2006 Q4
gas %>%
  autoplot()+
  labs(title = " Gas Data")
## Plot variable not specified, automatically selected `.vars = Gas`

gas %>%
  model(classical_decomposition(Gas,type = "multiplicative")) %>%
  components() %>%
  autoplot() + 
  ggtitle(" Gas Data")
## Warning: Removed 2 row(s) containing missing values (geom_path).

descom<-gas %>%
  model(classical_decomposition(Gas,type = "multiplicative")) %>%
  components()

a=mean(descom$seasonal)

b =mean(descom$trend,na.rm=TRUE)
  
cat(" The calculated seasonality is ", a, " and the trend is", b)
##  The calculated seasonality is  1  and the trend is 216.3203

part 3: yes

gas_decom <- gas %>%
             model(classical_decomposition(Gas,type = "multiplicative")) %>%
             components()
gas_decom %>%
  ggplot(aes(x = Quarter)) +
  geom_line(aes(y = season_adjust,)) +
  labs(y = "",
       title = "Gas Data")

gas2 <- gas
gas2$Gas[15] <- gas2$Gas[11]+300

gas2 %>%
  model(classical_decomposition(Gas,type = "multiplicative")) %>%
  components() %>%
  autoplot() + 
  ggtitle("Gas Data with outlier")
## Warning: Removed 2 row(s) containing missing values (geom_path).

gas2 %>%
  model(classical_decomposition(Gas,type = "multiplicative")) %>%
  components() %>%
  ggplot(aes(x = Quarter)) +
  geom_line(aes(y = season_adjust,))

       title = ("Gas Data with 300 added ") +
  scale_colour_manual(
    values = c("gray"),
    breaks = c( "Seasonally Adjusted"))
       
gas3 <- gas
gas3$Gas[20] <- gas3$Gas[10]+300

gas3 %>%
  model(classical_decomposition(Gas,type = "multiplicative")) %>%
  components() %>%
  autoplot() + 
  ggtitle("Gas Data with outlier ")
## Warning: Removed 2 row(s) containing missing values (geom_path).

set.seed(23456789)
myseries <- aus_retail %>%
  filter(`Series ID` == sample(aus_retail$`Series ID`,1))

myseries %>%
  model(classical_decomposition(Turnover,type = "multiplicative")) %>%
  components() %>%
  autoplot() + 
  ggtitle("Multiplicative decomposition of my time series")
## Warning: Removed 6 row(s) containing missing values (geom_path).

data<-ts(myseries$Turnover,start=c(1982,4), end=c(2000,12), frequency=12)

x11_dcmp <- myseries %>%
  model(x11 = X_13ARIMA_SEATS(Turnover ~ x11())) %>%
  components()
autoplot(x11_dcmp) +
  labs(title =
    "myseries using X-11.")

canadian_gas %>%
  model(
    STL(Volume ~ trend(window = 25) +
                   season(window = 10),
    robust = TRUE)) %>%
  components() %>%
  autoplot()+
  labs(title = "STL decomposition of Canadian Gas Production")

  1. The trend shows that the labor force is overall increasing. Seasonally, the highest employment seems to be at the end of the year, and the lowest in January. The trend is increasing at a slower rate starting in 1991.
  1. you can see the recession in every part but the seasonal plot. The value had its largest drop, the trend began increasing slower, and there is a large outlier in the remainder
canadian_gas %>%
  autoplot(Volume)+
  labs(title = "Monthly Canadian Gas Production",
       subtitle = "autoplot()",
       y = "billionsr")

canadian_gas %>%
  gg_subseries(Volume)+
  labs(title = "Monthly Canadian Gas Production",
       subtitle = "gg_subseries()",
       y = "billions")

canadian_gas %>%
  gg_season(Volume)+
  labs(title = "Monthly Canadian Gas Production",
       subtitle = "gg_season()",
       y = "billions")

canadian_gas %>%
  model(
    STL(Volume ~ trend(window = 25) +
                   season(window = 10),
    robust = TRUE)) %>%
  components() %>%
  autoplot()+
  labs(title = "STL decomposition of Canadian Gas Production")

  1. it increases
canadian_gas %>%
 model(
    STL(Volume ~ trend(window = 25) +
                   season(window = 10),
    robust = TRUE)) %>%
  components() %>%
  ggplot(aes(x = Month)) +
  geom_line(aes(y = season_adjust))

labs(title = "STL decomposition of Canadian Gas Production") +
  scale_colour_manual(
    values = c("gray"),
    breaks = c( "Seasonally Adjusted"))
## NULL
canadian_gas %>%
  model(x11 = X_13ARIMA_SEATS(Volume ~ x11())) %>%
  components() %>%
  autoplot()+
  labs(title = "X-11 decomposition")

canadian_gas %>%
  model(seats = X_13ARIMA_SEATS(Volume ~ seats())) %>%
  components() %>%
autoplot() +
  labs(title ="SEATS Decomposition of Canadian Gas Production")

they look almost the exact same except for more variation with SEATS

liquor<-aus_retail%>%
  filter(Industry == "Liquor retailing" & year(Month)>= 2000)%>%
  summarise(Turnover = sum(Turnover))

ggplot(data= liquor)+
  geom_line(aes(x=Month, y=Turnover))

  1. 2). a moving average of a moving average
liquor <-liquor%>%
  mutate(`12-MA` = slider::slide_dbl(Turnover, mean,
                                    .before = 5, .after = 6, .complete = TRUE))%>%
  mutate(`2x12-MA` = slider::slide_dbl(`12-MA`, mean,
                                    .before = 1, .after = 0, .complete = TRUE))
liquor%>%
  autoplot(Turnover) +
  geom_line(aes(y =`2x12-MA`), colour = "#0072B2") +
  labs(y = "Liquor retailing Turnover",
       title = "Total AUS Liquor retailing Turnover")+
  guides(colour = guide_legend(title = "series"))
## Warning: Removed 12 row(s) containing missing values (geom_path).

10.

Additive: 1. compute the trend cycle component 2. calculate the detrended series with yt-Tt 3. average the detrended values for the season to estimate seasonal component 4. make sure seasonal effects are normalized 5. subtract estimated season and trend components to find remainder

Multiplicative: 1. Compute the trend cycle compenent 2. calculate detrended series with yt/Tt 3.estimate seasonal compenent by averaging detrended values 4. make sure seasonal effects are normanlized 5. calculate the raminder component by dividing the estimated seasonal and trend cycle components