U3_SARIMA_Seasonal ARIMA time series model.pptx

Seasonal ARIMA

1. Unit 3 : SARIMA Ravindra Nath Shukla

2. Types of Seasonality

3. SARIMA Model

4. s

5. s

6. SARIMA in R

7. [RANKING

instal caca nal
Spar

ITY GWALIOR, MR INDIA
ersiy.acin

SARIMA Model

SARIMA stands for “Seasonal Auto-regressive Integrated Moving
Average”.

We can convert trends to a stationary series by taking differences.

We can convert seasonality to stationarity by taking seasonal
differences.

8. RANKING

Let us make a transformation:
Z, = (1-B°)P(1-B)@Y,,

Where Dis the number of seasonal differences (usually O or 1)
And d is the number of regular differences (usually < 3).

Idea of SARIMA is to model regular and seasonal impacts separately
and then combine into a model multiplicatively.

9. 1 RANKING

ya
M UNIVERS

ITY GWALIOR, MR INDIA
ersiy.acin

The SARIMA model takes the following structure:

PB (BA — BS)PC1 — B)*¥, = 6,(B°)0,(B)e,

Where,

(BS) = 1—0,BS 一 … 一 の pg is the seasonal AR operator of order P
HB) =1— $B — dB is the regular AR operator of order p
O9(BS) = 1— 0, BS 一 … 一 OpBS* is the seasonal MA operator of order Q
6,(B) =1 一 98 一 … 一 6 有 B is the regular MA operator of order q

This is defined as: SARIMA(p, d, q) x (P,D,Q)s

10. Seasonal ARIMA

ㆍ Aseasonal ARIMA model is formed by including additional seasonal terms in the
ARIMA models we have seen so far. It is written as follows:

ARIMA (p,d,q) (P,D,Q 5
t 个

of the model of the model

( Non-seasonal part ) ( Seasonal part

+ Where m = number of periods per season.

ㆍ We use uppercase notation for the seasonal parts of the model, and lowercase notation
for the non-seasonal parts of the model.

12. RANKING

GWALIOR, MP, INDIA
iyacin

ACF & PACF

+ The seasonal part of an AR or MA model will be seen in the seasonal lags of the PACF and
ACF.
+ For example, an ARIMA(0,0,0)(0,0,1),, model will show:
・ Aspike at lag 12 in the ACF but no other significant spikes.
+ The PACF will show exponential decay in the seasonal lags; that is, at lags 12, 24, 36, ....
+ Similarly, an ARIMA(0,0,0)(1,0,0),, model will show:
Exponential decay in the seasonal lags of the ACF.
A single significant spike at lag 12 in the PACF.

・ In considering the appropriate seasonal orders for an ARIMA model, restrict attention to the
seasonal lags.

+ The modelling procedure is almost the same as for non-seasonal data, except that we need
to select seasonal AR and MA terms as well as the non-seasonal components of the model.

13. GWALIOR, MP, INDIA
iyacin

RANKING
SARIMA in R

European Quarterly Retail Trade

+ We will describe the seasonal ARIMA modelling procedure using quarterly European
retail trade data from 1996 to 2011.

+ We will use Forecast Library in R studio.

Retail index

0 2 9 9% 98 100

2000 2005 2010
Year

plot(euretail, ylab="Retail index", xlab="Year")

14. GWALIOR, MP, INDIA
iyacin

Make It Stationary

・ The data are clearly non-stationary, with some seasonality, so we will first take a
seasonal difference.

32-10

2000 2005 200

er

94 00 04 08
PAOF

94 oo 04 08

La 0
| tsdisplay( diff(euretail,4) )

15. ITY GWALIOR, MR INDIA
iyacin

+ These also appear to be non-stationary, and so we take an additional first difference.

oo to

0

20

ep 02 04

a

tsdisplay( diff( diff(euretail,4) ) )

16. GWALIOR, MP, INDIA
iyacin

Find Appropriate ARIMA Model

+ Based on the ACF and PACF shown:

+ The significant spike at lag 1 in the ACF
suggests a non-seasonal MA(1)
component.

- The significant spike at lag 4 in the ACF
suggests a seasonal MA(1) component.

・ Consequently, we begin with an
ARIMA(0,1,1)(0,1,1), model, indicating
a first and seasonal difference, and
non-seasonal and seasonal MA(1)
components.

AM Iw J

206 am

17. ITY GWALIOR, MR INDIA
acin

Find Appropriate ARIMA Model
ARIMA(0,1,1)(0,1,1),

spikes at lag 3, indicating some
additional non-seasonal terms need to
be included in the model.

Both the ACF and PACF show significant
i A u

spikes at lag 2, and almost significant
200

The AlCc of:
ㆍ ARIMA(0,1,2)(0,1,1), model is 74.36,
ㆍ ARIMA(0,1,3)(0,1,1), model is 68.53.

ac

22 oo oe 04

PACE

We tried other models with AR terms
as well, but none that gave a smaller E To ms
AlCc value. us wm

fit <- Arima(euretail, order=c(0,1,1), seasonal=c(0,1,1))
tsdisplay(residuals(fit))

18. 1 RANKING

GWALIOR, MP, INDIA
versity ain

Find Appropriate ARIMA Model
ARIMA(0,1,3)(0,1,1),
ㆍ All the spikes are now within the

significance limits, and so the residuals
appear to be white noise.

| "a MM My Hi

ㆍ ALjung-Box test also shows that the Al 1
residuals have no remaining 더 더 에
autocorrelations.

ser

up in
fit3 <- Arima(euretail, order=c(0,1,3), seasonal=c(0,1,1)) |
res <- tsdisplay{residuals(fit3))

Box.test(res, lag=16, fitdf=4, type="Ljung")

19. GWALIOR, MP, INDIA
iyacin

Forecast Model

+ Forecasts from the model for the next Forecasts from ARIMA(O,1,3)(0,4, 114]
six years as shown.

+ Notice how the forecasts follow the
recent trend in the data (this occurs
because of the double differencing).

e

+ The large and rapidly increasing - gt っ プ |
prediction intervals show that the retail
trade index could start increasing or 7
decreasing at any time while the point p

forecasts trend downwards.

2000 2005 2010 2015

plot( forecast(fit3, h=24) )

20. GWALIOR, MP, INDIA
iyacin

Forecast Without Seasonality

fit <- Arima(euretail, order=c(1,2,0)) Forecasts from ARIMA(1,2,0)

tsdisplay(residuals(fit))

160

plot(forecast(fit, h=24))

21. ITY GWALIOR, MP INDIA
in

Find Appropriate ARIMA Model

Other Method
+ We could have used auto.arima() to do > auto.arima(euretail, stepwise=FALSE,
most of this work for us. It would have approximation=FALSE)

given the following result.
ARIMA(0,1,3)(0,1,1)[4]

Coefficients:
mal ma2 ma3 smal
0.2625 0.3697 0.4194 -0.6615
s.e. 0.1239 0.1260 0.1296 0.1555

sigma^2 estimated as 0.1451: log likeliho
od=-28.7
AIC=67.4 AlCc=68.53 BIC=77.78