1. Time Series Analysis & fpp
• Used to predict wind speeds at forecast horizons of 10
minutes to 1 day ahead, based on historical wind speeds (10
minute averages).
• Forecast package: Rob Hyndman
• https://www.otexts.org/fpp - good explanations and worked
examples.
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Fiona McGroarty
Dublin R 24/03/2016
2. Time Series Data
• Forecasting extrapolates trend and seasonal patterns.
• Trend : long term increased/decrease.
• Seasonal: e.g. daily
weekly, yearly (fixed
& known length).
• Cycle: rises/falls that
are not a fixed period
(variable & unknown
length).
Trend: Generally increasing
Seasonal Pattern: Sharp rise at the
end of each year (stockpiling)
• Observations (at regular intervals) sequentially over time.
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Fiona McGroarty
Dublin R 24/03/2016
3. Autocorrelations
• Correlation: lags 4 & 8 –
seasonal pattern, peaks are
4 quarters apart.
• Linear relationship between
lagged values of a time series
(quarterly beer production):
Lag.plot(beer2, lags=9).
• Corr. Coefficient -1 ≤ r ≤ 1.
• Correlation: lags 2 & 6 –
troughs are 2 quarters
behind peaks.
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Fiona McGroarty
Dublin R 24/03/2016
4. Autocorrelations
• Plot ACF (acf(beer2)):
• White Noise: 95% of spikes
are within ±2/T (T = length
of time series=50).
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Fiona McGroarty
Dublin R 24/03/2016
5. Autocorrelations
• Plot ACF (acf(beer2)):
• White Noise: 95% of spikes
are within ±2/T (T = length
of time series=50).
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Fiona McGroarty
Dublin R 24/03/2016
6. Simple Forecasts (Benchmarks)
• Naïve method: forecast = value of last observation:
naive(beer2, h=11) or rwf(beer2, h=11).
• Average method: forecast = mean of historical data:
meanf(beer2, h=11).
• Seasonal naïve: forecast = value of last observation from the
previous season
(month/quarter/year ...)
snaive(beer2, h=11).
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Fiona McGroarty
Dublin R 24/03/2016
7. Simple Forecasts (Benchmarks)
• Drift: variation on naïve, allow forecast to increase/decrease
over time: rwf (dj2, h=42, drift=TRUE)
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Fiona McGroarty
Dublin R 24/03/2016
8. Time Series Decomposition
• fit <- stl (elecequip,
t.window=15,
s.window=“periodic”,
robust=TRUE).
Plot(fit).
• Time series decomposition: yt = St + Tt + Et or yt = St × Tt × Et
Electrical equipment orders. (Seasonal, Trend, Error)
• Set trend window and seasonal
window sizes – small values
allow more rapid changes.
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Fiona McGroarty
Dublin R 24/03/2016
9. Forecast: ETS
• Effects of components can be additive (A), multiplicative (M) or
ignored (N, none). EG ETS (MAN) forecasts by multiplying errors
(M), adding trends (A) and ignoring seasonal effects (N).
SES: Simple Exponential Smoothing
Holts Linear Method
Additive Holts-Winter Method
Multiplicative Holts-Winter Method
Holts-Winter damped method
• ETS (ANN) SES with + errors; ETS (MNN) SES with × errors.
• Some are numerically unstable e.g. if data contains 0 or neg. values.
• ETS(ZZZ) – runs all (stable) ETS models and returns the optimal one.
• ETS (Error, Trend, Seasonal OR ExponenTial Smoothing)
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Fiona McGroarty
Dublin R 24/03/2016
11. Forecast: ETS
• oildata <- window(oil, start=1996,end=2007)
fit <- ets(oildata, model=“ANN”)
plot(forecast(fit, h=3), ylab=“Oil (millions of tonnes)”)
• Point forecasts and 80% and 95% prediction intervals.
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Fiona McGroarty
Dublin R 24/03/2016
12. ARIMA Models
• ARIMA (p, d, q) e.g. ARIMA101 model uses one past time-
lagged wind speed autocorrelation term (p = 1), is not
differentiated (d = 0) and uses one past forecast error (q = 1).
• ARIMA = Auto Regression (detects similarity in the data using
timelagged values of the variable) Integrated (the data may
have to be differentiated a number of times to make it
stationary) Moving Average (using weighted moving average
of the past few forecast errors).
• Can specify the ARIMA (p, d, q) model to use, or use
auto.arima
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Fiona McGroarty
Dublin R 24/03/2016
13. ARIMA Models
• fit <- auto.arima (usconsumption[,1], seasonal=FALSE
plot(forecast(fit, h=10), include=80)
• Best ARIMA model is returned, in this case it’s ARIMA(0,0,3).
• fit <- Arima(usconsumption[,1], order=c(0,0,3))
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Fiona McGroarty
Dublin R 24/03/2016
14. Lots of other things to consider….
• Stationarity of data
• Transforming data before forecasting – differentiating…
• Error Metrics! Calculate MAE, RMSE,…..
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Fiona McGroarty
Dublin R 24/03/2016
