Pure Seasonal Models ARIMA Modeling with R Pure Seasonal Models O - PowerPoint PPT Presentation

ARIMA MODELING WITH R Pure Seasonal Models

ARIMA Modeling with R Pure Seasonal Models ● O � en collect data with a known seasonal component ● Air Passengers (1 cycle every S = 12 months) ● Johnson & Johnson Earnings (1 cycle every S = 4 quarters)

ARIMA Modeling with R Pure Seasonal Models ● Consider pure seasonal models such as an SAR( P = 1) s = 12 X t = Φ X t − 12 + W t

ARIMA Modeling with R ACF and PACF of Pure Seasonal Models SAR(P) s SMA(Q) s SARMA(P, Q) s ACF* Tails o ff Cuts o ff lag QS Tails o ff PACF* Cuts o ff lag PS Tails o ff Tails o ff * The values at the nonseasonal lags are zero SAR(1) 1 SMA(1) 1

ARIMA MODELING WITH R Let’s practice!

ARIMA MODELING WITH R Mixed Seasonal Models

ARIMA Modeling with R Mixed Seasonal Model ● Mixed model: SARIMA(p, d, q) x (P, D, Q) s model ● Consider a SARIMA(0, 0, 1) x (1, 0, 0) 12 model X t = Φ X t − 12 + W t + θ W t − 1 ● SAR(1): Value this month is related to 
 X t − 12 last year’s value ● MA(1): This month’s value related to last 
 W t − 1 month’s shock

ARIMA Modeling with R ACF and PACF of SARIMA(0,0,1) x (1,0,0) s=12 ● The ACF and PACF for this mixed model: X t = . 8 X t − 12 + W t − . 5 W t − 1 Seasonal Non-seasonal

ARIMA Modeling with R Seasonal Persistence Hawaiian Quarterly Occupancy Rate 85 3 1 1 3 1 3 1 80 1 1 1 3 3 1 3 3 2 2 4 2 3 1 3 4 1 Quarterly Occupancy Rate: 
 3 2 75 3 4 4 1 4 2 x 2 2 4 3 4 2 4 4 1 % rooms filled 4 3 1 70 4 2 2 2 3 2 4 2 1 65 4 2 4 2002 2004 2006 2008 2010 2012 2014 2016 Time Seasonal Component Seasonal Component 1 1 4 1 1 3 1 1 1 3 3 1 1 3 3 3 1 3 1 1 Seasonal Component: 
 1 3 1 3 3 2 3 3 3 3 this year vs. last year Qx 0 Q1 ≈ Q1, Q2 ≈ Q2, 
 2 -2 2 2 2 2 2 2 2 2 2 4 4 2 2 4 4 4 4 4 4 4 2 4 2 Q3 ≈ Q3, Q4 ≈ Q4 -4 4 4 4 4 2002 2004 2006 2008 2010 2012 2014 2016 Time Seasonal Difference 2 3 4 1 1 2 Remove seasonal persistence 5 3 4 3 1 2 3 4 1 2 2 4 1 4 3 4 1 2 1 4 4 2 3 1 3 diff(x, 4) 3 4 1 2 3 by a seasonal di ff erence: 
 0 1 2 4 2 1 4 3 4 2 3 3 X t - X t-4 or D = 1, S = 4 -5 2 1 2 3 4 for quarterly data -10 1 2004 2006 2008 2010 2012 2014 2016 Time

ARIMA Modeling with R Air Passengers ● Monthly totals of international airline passengers, 1949-1960 x: AirPassengers lx: log(x) dlx: diff(lx) ddlx: diff(dlx, 12)

ARIMA Modeling with R Air Passengers: ACF and PACF of ddlx ● Seasonal: ACF cu � ing o ff at lag 1s (s = 12); PACF tailing o ff at lags 1s, 2s, 3s… ● Non-Seasonal: ACF and PACF both tailing o ff

ARIMA Modeling with R Air Passengers > airpass_fit1 <- sarima(log(AirPassengers), p = 1, d = 1, q = 1, P = 0, D = 1, Q = 1, S = 12) > airpass_fit1$ttable Estimate SE t.value p.value ar1 0.1960 0.2475 0.7921 0.4296 ma1 -0.5784 0.2132 -2.7127 0.0075 sma1 -0.5643 0.0747 -7.5544 0.0000 > airpass_fit2 <- sarima(log(AirPassengers), 0, 1, 1, 0, 1, 1, 12) > airpass_fit2$ttable Estimate SE t.value p.value ma1 -0.4018 0.0896 -4.4825 0 sma1 -0.5569 0.0731 -7.6190 0

ARIMA Modeling with R Air Passengers Model: (0,1,1) (0,1,1) [12] Standardized Residuals 3 2 1 − 1 − 3 1950 1952 1954 1956 1958 1960 Time ACF of Residuals Normal Q − Q Plot of Std Residuals 3 ● ● 0.4 Sample Quantiles ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ●● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ACF ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● − 1 ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● − 0.2 ● − 3 ● 0.5 1.0 1.5 − 2 − 1 0 1 2 LAG Theoretical Quantiles p values for Ljung − Box statistic 0.8 ● ● p value ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 5 10 15 20 25 30 35 lag

ARIMA MODELING WITH R Let’s practice!

ARIMA MODELING WITH R Forecasting Seasonal ARIMA

ARIMA Modeling with R Forecasting ARIMA Processes ● Once model is chosen, forecasting is easy because the model describes how the dynamics of the time series behave over time ● Simply continue the model dynamics into the future ● In the astsa package, use sarima.for() for forecasting

ARIMA Modeling with R Forecasting Air Passengers ● In the previous video, we decided that a SARIMA(0,1,1)x(0,1,1) 12 model was appropriate > sarima.for(log(AirPassengers), n.ahead = 24, 0, 1, 1, 0, 1, 1, 12) 6.5 log(AirPassengers) 6.0 5.5 1954 1956 1958 1960 1962 Time

ARIMA MODELING WITH R Let’s practice!

ARIMA MODELING WITH R Congratulations!

ARIMA Modeling with R What you’ve learned ● How to identify an ARMA model from data looking at ACF and PACF ● How to use integrated ARMA (ARIMA) models for nonstationary time series ● How to cope with seasonality

ARIMA Modeling with R Don’t stop here! ● astsa- package ● Other DataCamp courses in Time Series Analysis

ARIMA MODELING WITH R Thank you!