HW 6

Load packages and data

library(fpp3)
library(readxl)
library(cowplot)

My Dataset

Milky_boy <- read_excel("~/Downloads/Milky boy.xlsx", 
                        col_types = c("date", "numeric", "text"))

Cleaning and OG Plot

milk <- Milky_boy %>%
  mutate(Month = yearmonth(Year)) %>%
  select(Month, Price) %>%
  as_tsibble(index = Month)

mplot <- autoplot(milk) +
  labs(y = "Price in $", x = "Months", title = "Monthly Gallon Of Milk Prices")

## Plot variable not specified, automatically selected `.vars = Price`

mplot

STL Decomp

milk %>%
  model(STL(Price ~ season(window = 13), robust = TRUE)) %>%
  components() %>%
  autoplot() +
  labs(title = "STL Decomp")

Data not stationary and needs 1 diff

milk %>%
  features(Price, unitroot_kpss)

## # A tibble: 1 × 2
##   kpss_stat kpss_pvalue
##       <dbl>       <dbl>
## 1      1.29        0.01

milk %>%
  features(Price, unitroot_ndiffs)

## # A tibble: 1 × 1
##   ndiffs
##    <int>
## 1      1

milk %>%
  gg_tsdisplay(Price, plot_type = 'partial')

Diff check and partial resid of diff

Milkdiff2 <- milk %>%
  mutate(Price = difference(Price)) %>%
  na.omit()

Milkdiff2 %>%
  ggplot(aes(x = Month, y = Price)) +
  geom_line(aes(y = Price))

Milkdiff2 %>%
  features(Price, unitroot_kpss)

## # A tibble: 1 × 2
##   kpss_stat kpss_pvalue
##       <dbl>       <dbl>
## 1    0.0914         0.1

Milkdiff2 %>%
  features(Price, unitroot_ndiffs)

## # A tibble: 1 × 1
##   ndiffs
##    <int>
## 1      0

Milkdiff2 %>%
  gg_tsdisplay(Price, plot_type = 'partial')

ARIMA auto

Milkauto <- milk %>%
  model('Auto' = ARIMA(Price))

report(Milkauto)

## Series: Price 
## Model: ARIMA(0,1,2) 
## 
## Coefficients:
##          ma1     ma2
##       0.3441  0.1289
## s.e.  0.0613  0.0714
## 
## sigma^2 estimated as 0.004806:  log likelihood=342.2
## AIC=-678.41   AICc=-678.32   BIC=-667.58

ARIMA Compare

Milkcomp <- milk %>% 
  model(
    '012' = ARIMA(Price ~ pdq(0,1,2)),
    '212' = ARIMA(Price ~ pdq(2,1,2)),
    '110' = ARIMA(Price ~ pdq(1,1,0)),
    "011" = ARIMA(Price ~ pdq(0,1,1)),
    "111" = ARIMA(Price ~ pdq(1,1,1))) 

Milkcomp %>%
  glance()

## # A tibble: 5 × 8
##   .model  sigma2 log_lik   AIC  AICc   BIC ar_roots  ma_roots 
##   <chr>    <dbl>   <dbl> <dbl> <dbl> <dbl> <list>    <list>   
## 1 012    0.00481    342. -678. -678. -668. <cpl [0]> <cpl [2]>
## 2 212    0.00474    345. -680. -680. -662. <cpl [2]> <cpl [2]>
## 3 110    0.00483    341. -678. -678. -671. <cpl [1]> <cpl [0]>
## 4 011    0.00485    341. -677. -677. -670. <cpl [0]> <cpl [1]>
## 5 111    0.00483    341. -677. -677. -666. <cpl [1]> <cpl [1]>

#012 and 212

ETS Auto

MilkETSauto <- milk %>%
  model('Auto' = ETS(Price))

report(MilkETSauto)

## Series: Price 
## Model: ETS(M,N,N) 
##   Smoothing parameters:
##     alpha = 0.9999 
## 
##   Initial states:
##      l[0]
##  2.783582
## 
##   sigma^2:  5e-04
## 
##      AIC     AICc      BIC 
## 107.6908 107.7796 118.5301

ETS Comp

MilkETS <- milk %>% 
  model(
    'Auto' = ETS(Price),
    'Simple' = ETS(Price ~ error("A") + trend("N") + season("N")),
    'Holt' = ETS(Price ~ error("A") + trend("A") + season("N")),
    'Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N"))) 

MilkETS %>%
  glance()

## # A tibble: 4 × 9
##   .model                 sigma2 log_lik   AIC  AICc   BIC     MSE   AMSE    MAE
##   <chr>                   <dbl>   <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>  <dbl>
## 1 Auto                 0.000508   -50.8  108.  108.  119. 0.00532 0.0143 0.0150
## 2 Simple               0.00536    -51.7  109.  110.  120. 0.00532 0.0143 0.0493
## 3 Holt                 0.00542    -52.3  115.  115.  133. 0.00535 0.0142 0.0494
## 4 Multiplicative Trend 0.00539    -51.4  113.  113.  131. 0.00531 0.0142 0.0493

#Auto

Test and Training sets

choochoo <- milk %>%
  filter(year(Month) <= 2015)
choochoo

## # A tsibble: 192 x 2 [1M]
##       Month Price
##       <mth> <dbl>
##  1 2000 Jan  2.78
##  2 2000 Feb  2.78
##  3 2000 Mar  2.75
##  4 2000 Apr  2.77
##  5 2000 May  2.78
##  6 2000 Jun  2.76
##  7 2000 Jul  2.78
##  8 2000 Aug  2.81
##  9 2000 Sep  2.81
## 10 2000 Oct  2.80
## # … with 182 more rows

testies <- milk %>%
  filter(year(Month) > 2015)
testies

## # A tsibble: 82 x 2 [1M]
##       Month Price
##       <mth> <dbl>
##  1 2016 Jan  3.31
##  2 2016 Feb  3.23
##  3 2016 Mar  3.19
##  4 2016 Apr  3.16
##  5 2016 May  3.16
##  6 2016 Jun  3.12
##  7 2016 Jul  3.06
##  8 2016 Aug  3.14
##  9 2016 Sep  3.23
## 10 2016 Oct  3.29
## # … with 72 more rows

RMSE of first 4 models

Milkfour <- choochoo %>%
  model(
    'Mean' = MEAN(Price),
    'Naive' = NAIVE(Price),
    'Seasonal_naive' = SNAIVE(Price),
    'Drift' = RW(Price ~ drift()))

Milkacc <- Milkfour %>%
  forecast(h = 82)

accuracy(Milkacc, milk)

## # A tibble: 4 × 10
##   .model         .type        ME  RMSE   MAE    MPE  MAPE  MASE RMSSE  ACF1
##   <chr>          <chr>     <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Drift          Test  -0.115    0.334 0.271 -4.44   8.32 1.03  0.953 0.945
## 2 Mean           Test   0.0647   0.361 0.258  0.916  7.46 0.979 1.03  0.949
## 3 Naive          Test  -0.000817 0.355 0.270 -1.08   7.97 1.02  1.01  0.949
## 4 Seasonal_naive Test  -0.112    0.393 0.326 -4.50   9.89 1.24  1.12  0.867

#Drift lowest RMSE
#Mean has lowest MAPE

RMSE for ARIMA

Milktrain <- choochoo %>% 
  model(
    '012' = ARIMA(Price ~ pdq(0,1,2)),
    '212' = ARIMA(Price ~ pdq(2,1,2)),
    '110' = ARIMA(Price ~ pdq(1,1,0)),
    "011" = ARIMA(Price ~ pdq(0,1,1)),
    "111" = ARIMA(Price ~ pdq(1,1,1))) 

Milktrain %>%
  glance()

## # A tibble: 5 × 8
##   .model  sigma2 log_lik   AIC  AICc   BIC ar_roots  ma_roots 
##   <chr>    <dbl>   <dbl> <dbl> <dbl> <dbl> <list>    <list>   
## 1 012    0.00531    230. -454. -454. -445. <cpl [0]> <cpl [2]>
## 2 212    0.00515    234. -458. -457. -442. <cpl [2]> <cpl [2]>
## 3 110    0.00539    228. -452. -452. -446. <cpl [1]> <cpl [0]>
## 4 011    0.00545    227. -450. -450. -444. <cpl [0]> <cpl [1]>
## 5 111    0.00541    228. -451. -451. -441. <cpl [1]> <cpl [1]>

#012 and 212

best <- choochoo %>% 
  model(
    '012' = ARIMA(Price ~ pdq(0,1,2)),
    '212' = ARIMA(Price ~ pdq(2,1,2)),
    '110' = ARIMA(Price ~ pdq(1,1,0)),
    "011" = ARIMA(Price ~ pdq(0,1,1)),
    "111" = ARIMA(Price ~ pdq(1,1,1))) %>%
  forecast(h = 82)


accuracy(best, milk)

## # A tibble: 5 × 10
##   .model .type       ME  RMSE   MAE   MPE  MAPE  MASE RMSSE  ACF1
##   <chr>  <chr>    <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 011    Test  -0.00622 0.355 0.272 -1.25  8.02  1.03  1.01 0.949
## 2 012    Test  -0.0143  0.355 0.274 -1.50  8.12  1.04  1.01 0.949
## 3 110    Test  -0.00665 0.355 0.272 -1.26  8.03  1.03  1.01 0.949
## 4 111    Test  -0.00789 0.355 0.272 -1.30  8.04  1.03  1.01 0.949
## 5 212    Test  -0.00396 0.355 0.271 -1.18  8.00  1.03  1.01 0.949

#212 and 011 have lowest RMSE
#All RMSE are extremely close
#212 and 011 have lowest MAPE
#All are extremely close except 012

RMSE for ETS

Milkband <- choochoo %>% 
  model(
    'Auto' = ETS(Price),
    'Simple' = ETS(Price ~ error("A") + trend("N") + season("N")),
    'Holt' = ETS(Price ~ error("A") + trend("A") + season("N")),
    'Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N"))) 

Milkband %>%
  glance()

## # A tibble: 4 × 9
##   .model                 sigma2 log_lik   AIC  AICc   BIC     MSE   AMSE    MAE
##   <chr>                   <dbl>   <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>  <dbl>
## 1 Auto                 0.000573   -10.3  32.6  33.0  52.1 0.00582 0.0177 0.0156
## 2 Simple               0.00615    -14.9  35.8  35.9  45.6 0.00608 0.0168 0.0506
## 3 Holt                 0.00648    -19.0  48.0  48.3  64.3 0.00635 0.0204 0.0537
## 4 Multiplicative Trend 0.00628    -15.9  41.8  42.1  58.1 0.00615 0.0168 0.0511

#Auto

MilkETS2 <- choochoo %>% 
  model(
    'Auto' = ETS(Price),
    'Simple' = ETS(Price ~ error("A") + trend("N") + season("N")),
    'Holt' = ETS(Price ~ error("A") + trend("A") + season("N")),
    'Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N"))) %>%
  forecast(h = 82)


accuracy(MilkETS2, milk)

## # A tibble: 4 × 10
##   .model              .type       ME  RMSE   MAE     MPE  MAPE  MASE RMSSE  ACF1
##   <chr>               <chr>    <dbl> <dbl> <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Auto                Test   3.30e-2 0.357 0.263 -0.0530  7.66 0.996 1.02  0.949
## 2 Holt                Test   8.20e-1 1.12  0.831 23.0    23.4  3.15  3.19  0.960
## 3 Multiplicative Tre… Test  -6.83e-2 0.337 0.268 -3.07    8.09 1.01  0.962 0.947
## 4 Simple              Test  -8.16e-4 0.355 0.270 -1.08    7.97 1.02  1.01  0.949

#Multi has lowest RMSE

#Auto has lowest MAPE

Plots for all the best models

Milkdrift <- choochoo %>%
  model('Drift' = RW(Price ~ drift())) %>%
  forecast(h = 82)

plot1 <- Milkdrift %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "Drift")

## Plot variable not specified, automatically selected `.vars = Price`

bestfr <- choochoo %>% 
  model('012' = ARIMA(Price ~ pdq(0,1,2))) %>%
  forecast(h = 82)

plot2 <- bestfr %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (0,1,2)")

## Plot variable not specified, automatically selected `.vars = Price`

bestfr3 <- choochoo %>% 
  model('212' = ARIMA(Price ~ pdq(2,1,2))) %>%
  forecast(h = 82)

plot3 <- bestfr3 %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (2,1,2)")

## Plot variable not specified, automatically selected `.vars = Price`

bestfr4 <- choochoo %>% 
  model('011' = ARIMA(Price ~ pdq(0,1,1))) %>%
  forecast(h = 82)

plot4 <- bestfr4 %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (0,1,1)")

## Plot variable not specified, automatically selected `.vars = Price`

BestETS <- choochoo %>%
   model('Auto' = ETS(Price)) %>%
   forecast(h = 82)

plot5 <- BestETS %>%
   autoplot() +
   autolayer(milk) +
  labs(title = "ETS Auto")

## Plot variable not specified, automatically selected `.vars = Price`

BestETS2 <- choochoo %>%
   model('Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N"))) %>%
   forecast(h = 82)
 
plot6 <- BestETS2 %>%
   autoplot() +
   autolayer(milk) +
  labs(title = "ETS Multi")

## Plot variable not specified, automatically selected `.vars = Price`

plot_grid(plot1, plot2, plot3, plot4, plot5, plot6)

RMSE values

# Drift = 0.3337870 (Best) # ARIMA (012) = 0.3551251 (5th) # ARIMA (212) = 0.3548632 (3rd) # ARIMA (011) = 0.3548885 (4th) # ETS Auto = 0.3568725 (Last) # ETS Multi = 0.3358933 (2nd)

MAPE values

# Drift = 8.321966 (Last) # ARIMA (012) = 8.124391 (5th) # ARIMA (212) = 8.000688 (2nd) # ARIMA (011) = 8.024482 (3rd) # ETS Auto = 7.662227 (Best) # ETS Multi = 8.036521 (4th)

Resid check for models

Drift

Milkdrift2 <- milk %>%
  model('Drift' = RW(Price ~ drift()))

Milkdrift2 %>%
  gg_tsresiduals() +
  labs(title = "Drift resid")

## Warning: Removed 1 row(s) containing missing values (geom_path).

## Warning: Removed 1 rows containing missing values (geom_point).

## Warning: Removed 1 rows containing non-finite values (stat_bin).

augment(Milkdrift2) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model lb_stat lb_pvalue
##   <chr>    <dbl>     <dbl>
## 1 Drift     35.4  0.000108

ARIMA (0,1,2)

AR012 <- milk %>% 
  model('012' = ARIMA(Price ~ pdq(0,1,2)))

AR012 %>%
  gg_tsresiduals() +
  labs(title = "ARIMA (0,1,2) resid")

augment(AR012) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model lb_stat lb_pvalue
##   <chr>    <dbl>     <dbl>
## 1 012       9.72     0.465

ARIMA (0,1,1)

AR011 <- milk %>% 
  model('011' = ARIMA(Price ~ pdq(0,1,1)))

AR011 %>%
  gg_tsresiduals() +
  labs(title = "ARIMA (0,1,1) resid")

augment(AR011) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model lb_stat lb_pvalue
##   <chr>    <dbl>     <dbl>
## 1 011       10.9     0.369

ARIMA (2,1,2)

AR212 <- milk %>% 
  model('212' = ARIMA(Price ~ pdq(2,1,2)))

AR212 %>%
  gg_tsresiduals() +
  labs(title = "ARIMA (2,1,2) resid")

augment(AR212) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model lb_stat lb_pvalue
##   <chr>    <dbl>     <dbl>
## 1 212       4.20     0.938

ETS Auto

AutoETS <- milk %>%
  model('Auto' = ETS(Price))

AutoETS %>%
  gg_tsresiduals() +
  labs(title = "ETS Auto")

augment(AutoETS) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model lb_stat lb_pvalue
##   <chr>    <dbl>     <dbl>
## 1 Auto      35.5  0.000101

ETS Multi

MultiETS <- milk %>%
  model('Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N")))

MultiETS %>%
  gg_tsresiduals() +
  labs(title = "ETS Multi")

augment(MultiETS) %>%
  features(.resid, ljung_box, lag = 10, dof = 0)

## # A tibble: 1 × 3
##   .model               lb_stat lb_pvalue
##   <chr>                  <dbl>     <dbl>
## 1 Multiplicative Trend    35.7 0.0000939

Ljung box p-values

# Drift = 0.0001077292 (Not WN) # ARIMA (012) = 0.4652559 (Is WN) # ARIMA (212) = 0.9376757 (Is WN) # ARIMA (011) = 0.3686044 (Is WN) # ETS Auto = 0.000101258 (Not WN)
# ETS Multi = 0.00009387677 (Not WN)

Plot of all Forecasts

Milkdriftcast <- milk %>%
  model('Drift' = RW(Price ~ drift())) %>%
  forecast(h = 12)

plot11 <- Milkdriftcast %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "Drift")

## Plot variable not specified, automatically selected `.vars = Price`

bestfrcast <- milk %>% 
  model('012' = ARIMA(Price ~ pdq(0,1,2))) %>%
  forecast(h = 12)

plot22 <- bestfrcast %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (0,1,2)")

## Plot variable not specified, automatically selected `.vars = Price`

bestfr3cast <- milk %>% 
  model('212' = ARIMA(Price ~ pdq(2,1,2))) %>%
  forecast(h = 12)

plot33 <- bestfr3cast %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (2,1,2)")

## Plot variable not specified, automatically selected `.vars = Price`

bestfr4cast <- milk %>% 
  model('011' = ARIMA(Price ~ pdq(0,1,1))) %>%
  forecast(h = 12)

plot44 <- bestfr4cast %>%
  autoplot() +
  autolayer(milk) +
  labs(title = "ARIMA (0,1,1)")

## Plot variable not specified, automatically selected `.vars = Price`

BestETScast <- milk %>%
   model('Auto' = ETS(Price)) %>%
   forecast(h = 12)

plot55 <- BestETScast %>%
   autoplot() +
   autolayer(milk) +
  labs(title = "ETS Auto")

## Plot variable not specified, automatically selected `.vars = Price`

BestETS2cast <- milk %>%
   model('Multiplicative Trend' = ETS(Price ~ error("A") + trend("M") + season("N"))) %>%
   forecast(h = 12)
 
plot66 <- BestETS2cast %>%
   autoplot() +
   autolayer(milk) +
  labs(title = "ETS Multi")

## Plot variable not specified, automatically selected `.vars = Price`

plot_grid(plot11, plot22, plot33, plot44, plot55, plot66)

Cross vally

ModCompfr <- milk %>%
  stretch_tsibble(.init = 10) %>%
  model(
    "Multiplicative Trend" = ETS(Price ~ error("A") + trend("M") + season("N")),
    'Auto' = ETS(Price),
    '212' = ARIMA(Price ~ pdq(2,1,2)),
    '012' = ARIMA(Price ~ pdq(0,1,2)),
    '011' = ARIMA(Price ~ pdq(0,1,1)),
    'Drift' = RW(Price ~ drift())) %>%
  forecast(h = 1)

## Warning: It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.

## Warning in sqrt(diag(best$var.coef)): NaNs produced

## Warning: It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.
## It looks like you're trying to fully specify your ARIMA model but have not said if a constant should be included.
## You can include a constant using `ARIMA(y~1)` to the formula or exclude it by adding `ARIMA(y~0)`.

## Warning: 11 errors (1 unique) encountered for 212
## [11] Could not find an appropriate ARIMA model.
## This is likely because automatic selection does not select models with characteristic roots that may be numerically unstable.
## For more details, refer to https://otexts.com/fpp3/arima-r.html#plotting-the-characteristic-roots

## Warning: 4 errors (1 unique) encountered for 012
## [4] Could not find an appropriate ARIMA model.
## This is likely because automatic selection does not select models with characteristic roots that may be numerically unstable.
## For more details, refer to https://otexts.com/fpp3/arima-r.html#plotting-the-characteristic-roots

accuracy(ModCompfr, milk)

## Warning: The future dataset is incomplete, incomplete out-of-sample data will be treated as missing. 
## 1 observation is missing at 2022 Nov

## # A tibble: 6 × 10
##   .model            .type      ME   RMSE    MAE    MPE  MAPE  MASE RMSSE    ACF1
##   <chr>             <chr>   <dbl>  <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl>   <dbl>
## 1 011               Test  0.00448 0.0722 0.0512 0.115   1.55 0.195 0.215 0.0563 
## 2 012               Test  0.00366 0.0729 0.0512 0.0935  1.55 0.196 0.217 0.0327 
## 3 212               Test  0.00401 0.0681 0.0505 0.102   1.52 0.193 0.203 0.00678
## 4 Auto              Test  0.00526 0.0745 0.0510 0.138   1.55 0.195 0.222 0.299  
## 5 Drift             Test  0.00145 0.0746 0.0508 0.0169  1.53 0.194 0.222 0.313  
## 6 Multiplicative T… Test  0.00109 0.0770 0.0530 0.0118  1.61 0.203 0.229 0.320

HW 6

Hunter Patenaude

Nov 20

Load packages and data

My Dataset

Cleaning and OG Plot

STL Decomp

Data not stationary and needs 1 diff

Diff check and partial resid of diff

ARIMA auto

ARIMA Compare

ETS Auto

ETS Comp

Test and Training sets

RMSE of first 4 models

RMSE for ARIMA

RMSE for ETS

Plots for all the best models

RMSE values

MAPE values

Resid check for models

Drift

ARIMA (0,1,2)

ARIMA (0,1,1)

ARIMA (2,1,2)

ETS Auto

ETS Multi

Ljung box p-values

Plot of all Forecasts

Cross vally

Best Model:

HiLo PI for best Model

Price of Milk in 12 periods

E

F

G

H