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<h1>Time Series Forecasting</h1>
<blockquote>
<p>This is a follow-up to the introduction to <a href="Time-Series-Analysis-With-R">time series analysis</a>, but focused more on forecasting rather than analysis.</p>
</blockquote>
<h2>Simple Moving Average</h2>
<p>Simple moving average can be calculated using <code>ma()</code> from forecast</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">sm <-<span class="st"> </span><span class="kw">ma</span>(ts, <span class="dt">order=</span><span class="dv">12</span>) <span class="co"># 12 month moving average</span>
<span class="kw">lines</span>(sm, <span class="dt">col=</span><span class="st">"red"</span>) <span class="co"># plot</span></code></pre></div>
<h2>Exponential Smoothing</h2>
<p>Simple, Double and Triple exponential smoothing can be performed using the HoltWinters() function. There are multiple implementations of the Holt Winters method – <code>hw()</code> {forecast} and <code>ets()</code>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(forecast)
<span class="co"># Simple exponential smoothing: Level Only</span>
model <-<span class="st"> </span><span class="kw">hw</span>(trainingData, <span class="dt">initial =</span> “optimal”, <span class="dt">h=</span>(forecastPeriodLen), <span class="dt">beta=</span><span class="ot">NULL</span>, <span class="dt">gamma=</span><span class="ot">NULL</span>) <span class="co"># h is the no. periods to forecast</span>
<span class="co"># Double Exponential smoothing: Level and Trend components</span>
model <-<span class="st"> </span><span class="kw">hw</span>(trainingData, <span class="dt">initial =</span> “optimal”, <span class="dt">h=</span>(forecastPeriodLen), <span class="dt">gamma=</span><span class="ot">NULL</span>)
<span class="co"># Holt Winters: Level, Trend and Seasonality</span>
model <-<span class="st"> </span><span class="kw">hw</span>(trainingData, <span class="dt">initial =</span> “optimal”, <span class="dt">h=</span>(forecastPeriodLen))
<span class="kw">plot</span>(model)
<span class="kw">accuracy</span>(model) <span class="co"># calculate accuracy measures</span></code></pre></div>
<h2>ARIMA</h2>
<p>The forecast package offers auto.arima() function to fit ARIMA models. It can also be manually fit using Arima(). A caveat with ARIMA models in R is that it does not have the functionality to fit long seasonality of more than 350 periods eg: 365 days for daily data or 24 hours for 15 sec data.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Fit and forecast with auto.arima()</span>
autoArimaFit <-<span class="st"> </span><span class="kw">auto.arima</span>(tsData)
<span class="kw">plot</span>(<span class="kw">forecast</span>(autoArimaFit, <span class="dt">h=</span><span class="dv">20</span>))
<span class="co"># Fit and forecast with Arima()</span>
arimaFit <-<span class="st"> </span><span class="kw">Arima</span>(tsData,<span class="dt">order=</span><span class="kw">c</span>(<span class="dv">3</span>,<span class="dv">1</span>,<span class="dv">0</span>))
<span class="kw">plot</span>(<span class="kw">forecast</span>(arimafit,<span class="dt">h=</span><span class="dv">20</span>))</code></pre></div>
<h2>How To Forecast ARIMA Models With Long Seasonality (Greater Than 350 Periods)?</h2>
<p>Upon plotting your Arima() forecast, you find a more or less flat forecast, it could be because of long seasonality. In such case, you can feed in the seasonality as an external regressor through the ‘xreg’ argument.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">Fit <-<span class="st"> </span><span class="kw">Arima</span>(tsData,<span class="dt">order=</span><span class="kw">c</span>(<span class="dv">3</span>,<span class="dv">1</span>,<span class="dv">0</span>)) <span class="co"># fit Arima model</span>
Fit <-<span class="st"> </span><span class="kw">auto.arima</span>(tsData, <span class="dt">seasonal=</span><span class="ot">FALSE</span>, <span class="dt">xreg=</span><span class="kw">fourier</span>(tsData,<span class="dv">4</span>)) <span class="co"># fit auto.arima model</span>
<span class="kw">plot</span>(<span class="kw">forecast</span>(Fit,<span class="dt">h=</span><span class="dv">20</span>))
pred <-<span class="st"> </span><span class="kw">predict</span> (Fit, <span class="dt">newxreg=</span>newXregVar) <span class="co"># alternate way to forecast</span>
<span class="kw">plot</span>(<span class="kw">forecast</span>(fit, <span class="dt">h=</span>h, <span class="dt">xreg=</span><span class="kw">fourierf</span>(tsData,<span class="dv">4</span>,h))) <span class="co"># h is number of forecasts</span></code></pre></div>
<p>If you are using a numeric vector as an external regressor (xreg), make sure you change it to a data.frame() before feeding it as an xreg parameter to auto.arima(). You can also use multiple external regressors by binding them together as a data.frame().</p>
<h2>Some Useful External Regressors For Arima() and auto.arima()</h2>
<p>Any dataframe with as many rows as length of ts data can be used as ‘xreg’ argument. A couple of common ‘xreg’s that are used to model seasonal effects are below.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">Xreg1 <-<span class="st"> </span><span class="kw">seasonaldummy</span>(tsData) <span class="co"># creates dummy binary variable for each period in a season.</span>
Xreg2 <-<span class="st"> </span><span class="kw">model.matrix</span>(~<span class="st"> </span><span class="kw">as.factor</span>(weekday) +<span class="st"> </span><span class="dv">0</span>)<span class="er">)</span> <span class="co"># weekday could be a monthday, hour-of-day, holiday indicator etc .. </span></code></pre></div>
<h2>How To Model Time Series With Complex Seasonality Pattern?</h2>
<p>Use the tbats() in forecast package. Time series with multiple-seasonality can be modelled with this method. Since this is a computationally intensive procedure, the in-built parallel processing facility may be leveraged.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">tbatsFit <-<span class="st"> </span><span class="kw">tbats</span>(tsData, <span class="dt">use.parallel=</span><span class="ot">TRUE</span>, <span class="dt">num.cores =</span> <span class="dv">2</span>) <span class="co"># fit tbats model</span>
<span class="kw">plot</span>(<span class="kw">forecast</span>(fit)) <span class="co"># plot</span>
components <-<span class="st"> </span><span class="kw">tbats.components</span>(tbatsFit)
<span class="kw">plot</span>(components)</code></pre></div>
<h2>How To Find Confidence Intervals For My Forecasts?</h2>
<p>The predict() function has the facility. By providing the argument ‘prediction.interval=TRUE’ and ‘level = n’, the prediction intervals for a given confidence is calculated. Below is a general format of the code.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">model <-<span class="st"> </span><span class="kw">HoltWinters</span>(TS) <span class="kw">predict</span>(model, <span class="dv">50</span>, <span class="dt">prediction.interval =</span> <span class="ot">TRUE</span>, <span class="dt">level=</span> <span class="fl">0.99</span>) <span class="co"># prediction.interval = TRUE</span></code></pre></div>
<h2>More Useful Functions Related To Time Series</h2>
<table>
<thead>
<tr class="header">
<th align="left">Functions</th>
<th align="left">Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">accuracy()</td>
<td align="left">accuracy measures of forecast</td>
</tr>
<tr class="even">
<td align="left">BoxCox, invBoxCox()</td>
<td align="left">Box-Cox transformation</td>
</tr>
<tr class="odd">
<td align="left">decompose()</td>
<td align="left">Decompose time series data into components</td>
</tr>
<tr class="even">
<td align="left">dm.test()</td>
<td align="left">Diebold-Mariano test compares the forecast accuracy</td>
</tr>
<tr class="odd">
<td align="left">monthdays()</td>
<td align="left">number of days in seasonal series</td>
</tr>
<tr class="even">
<td align="left">na.interp()</td>
<td align="left">interpolate missing values</td>
</tr>
<tr class="odd">
<td align="left">seasadj()</td>
<td align="left">Remove the seasonal components from a time series</td>
</tr>
<tr class="even">
<td align="left">seasonaldummy()</td>
<td align="left">create matrix of seasonal indicator variables</td>
</tr>
<tr class="odd">
<td align="left">seasonplot()</td>
<td align="left">Plot seasonal effects</td>
</tr>
</tbody>
</table>
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