Time series Analysis & fpp package

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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.

Fiona McGroarty Dublin R 24/03/2016

https://www.otexts.org/fpp

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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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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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Autocorrelations• Plot ACF (acf(beer2)):

• White Noise: 95% of spikes are within ±2/T (T = length of time series=50).


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Autocorrelations• Plot ACF (acf(beer2)):

• White Noise: 95% of spikes are within ±2/T (T = length of time series=50).


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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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Simple Forecasts (Benchmarks)• Drift: variation on naïve, allow forecast to increase/decrease

over time: rwf (dj2, h=42, drift=TRUE)


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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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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 SmoothingHolts Linear Method

Additive Holts-Winter MethodMultiplicative Holts-Winter MethodHolts-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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EG: Forecasts using Holt’s method


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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.Fiona McGroarty

Dublin R 24/03/2016

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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 time lagged 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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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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Lots of other things to consider….

• Stationarity of data• Transforming data before forecasting – differentiating…• Error Metrics! Calculate MAE, RMSE,…..


Data & Analytics

Time series Analysis & fpp package