Stat 565 Fitting ARMA Models Jan 30 2014 Charlotte Wickham Saturday, February 1, 14 stat565.cwick.co.nz 1 HW#2 Be careful smoothing things that aren’t smooth Log transform helped a bit A sudden break at Saturday, February 1, 14 2 Procedure for ARMA modeling We'll assume the primary goal is getting a forecast. 1. Plot the data. Transform? Outliers? 2. Examine acf and pacf and pick p and q. Or try a few. acf and pacf 3. Fit ARMA(p, 0, q) model to original data. Or fit a few. AIC might be useful. arima 4. Check model diagnostics 5. Forecast (back transform?) Saturday, February 1, 14 3 Diagnostics If the correct model is fit, then the residuals (our estimates of the white noise process), should be roughly i.i.d Normal variates. Check: Time plot of residuals Autocorrelation (and partial) plot of residuals Normal qq plot Saturday, February 1, 14 4 Pair data analysis Each pair gets a time series Try to identify and estimate the model. Once you are satisfied, find the other pair(s), with the same series, and see if you agree. Explain your series to a pair with a different one. Saturday, February 1, 14 5 Time series load(url('http://stat565.cwick.co.nz/data/series-1.rda')) load(url('http://stat565.cwick.co.nz/data/series-2.rda')) load(url('http://stat565.cwick.co.nz/data/series-3.rda')) load(url('http://stat565.cwick.co.nz/data/series-4.rda')) load(url('http://stat565.cwick.co.nz/data/series-5.rda')) load(url('http://stat565.cwick.co.nz/data/series-6.rda')) load(url('http://stat565.cwick.co.nz/data/series-7.rda')) load(url('http://stat565.cwick.co.nz/data/series-8.rda')) Saturday, February 1, 14 6 0 x x 2 −2 −4 0 25 50 75 4 2 0 −2 100 0 50 100 t x x 0 −5 25 50 75 100 0 50 100 200 t 2 2 0 0 x x 150 4 2 0 −2 t −2 −2 0 25 50 75 100 0 100 t 200 300 t 5.0 2.5 0.0 −2.5 −5.0 4 x x 200 t 5 0 150 0 −4 0 25 50 t Saturday, February 1, 14 75 100 0 50 100 150 200 t 7 Solutions 1. ARMA(1, 1) list(ar = 0.9, ma = -0.5) 2. MA(2) list(ma = c(0.9, 0.3) 3. ARMA(2, 1) list(ar = c(1.2, -0.4), ma = 0.8) 4. ARMA(1, 1) list(ar = -0.8, ma = 0.2) 5. AR(2) list(ar = c(0.4, 0.4) 6. MA(3) list(ma = c(0.2, -0.8, 0.3)) 7. ARMA(1, 2) list(ar = 0.7, ma = c(0.6, 0.7)) 8. AR(3) list(ar = c(0.8, -1.1, 0.3)) This is how I simulated the datasets. It doesn’t necessarily mean you should be able to pick the correct process. Choosing the order of mixed ARMA processes is hard! Saturday, February 1, 14 8 Stationarity by differencing HW#2.2 you saw once differencing a process with a linear trend results in a stationary process. In general "d" times differencing a process with a polynomial trend of order d results in a stationary process. Suggests the approach: 1. Difference the data until it is stationary 2. Fit a stationary ARMA model to capture the correlation 3. Work backwards (by summing) to get original series The alternative is to explicitly model the trend and seasonality...more next week Saturday, February 1, 14 9 HW #2 example xt = β0 + β1t + wt a linear trend ∇xt = xt - xt-1 = β1 + wt - wt-1 an MA(1) process with constant mean β1 xt is called ARIMA(0, 1, 1) Saturday, February 1, 14 10 ARIMA(p, d, q) Autoregressive Integrated Moving Average A process xt is ARIMA(p, d, q) if xt d differenced d times (∇ xt) is an ARMA(p, q) process. I.e. xt is defined by ɸ(B) d ∇ xt ɸ(B) (1 - = θ(B) wt d B) xt = θ(B) wt forces constant in 1st differenced series arima(x, order = c(p, 1, q), xreg = 1:length(x)) Saturday, February 1, 14 11 Procedure for ARIMA modeling We'll assume the primary goal is getting a forecast. diff 1. Plot the data. Transform? Outliers? Differencing? 2. Difference until series is stationary, i.e. find d. 3. Examine differenced series and pick p and q. 4. Fit ARIMA(p, d, q) model to original data. 5. Check model diagnostics 6. Forecast (back transform?) Saturday, February 1, 14 12 Pick one: Oil prices install.packages('TSA') data(oil.price, package = 'TSA') Global temperature load(url("http://www.stat.pitt.edu/stoffer/tsa3/tsa3.rda")) gtemp US GNP load(url("http://www.stat.pitt.edu/stoffer/tsa3/tsa3.rda")) gnp Sulphur Dioxide (LA county) load(url("http://www.stat.pitt.edu/stoffer/tsa3/tsa3.rda")) so2 Saturday, February 1, 14 13 My code to turn a ts into a data.frame library(ggplot2) source(url("http://stat565.cwick.co.nz/code/fortify-ts.r")) fortify(gtemp) Saturday, February 1, 14 14 SARIMA models I haven't shown you any data with seasonality. The idea is very similar, if one seasonal cycle lasts for s measurements, then if we difference at lag s, yt = ∇sxt = xt - xt-s = (1 - s B )xt, we will remove the seasonality. Differencing seasonally D times is denoted, ∇Dsxt = (1 - Bs)Dxt, Saturday, February 1, 14 15 SARIMA A multiplicative seasonal autoregressive integrated moving average model, SARIMA(p, d, q) x (P, D, Q)s is given by s ɸ(B )ɸ(B) D d ∇ s∇ xt = s ϴ(B )θ(B)wt ∇Ds∇dxt is just an ARMA model with lots of coefficients set to zero. Have to specify s, then choose p, d, q, P, D and Q Saturday, February 1, 14 16 Procedure for SARIMA modeling We'll assume the primary goal is getting a forecast. 1. Plot the data. Transform? Outliers? Differencing? 2. Difference to remove trend, find d. Then difference to remove seasonality, find D. 3. Examine acf and pacf of differenced series. Find P and Q first, by examining just at lags s, 2s, 3s, etc. Find p and q by examining between seasonal lags. 4. Fit SARIMA(p, d, q)x(P, D, Q)s model to original data. 5. Check model diagnostics 6. Forecast (back transform?) Saturday, February 1, 14 17 Saturday, February 1, 14 18 library(forecast) Has some "automatic" ways to select arima models (and seasonal ARIMA models). auto.arima(log(oil$price)) Finds the model (up to a certain order) with the lowest AIC. You should still check if a simpler model has almost the same AIC. Saturday, February 1, 14 19

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