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<li class="divider"></li> <li class="chapter" data-level="" data-path="index.html"><a href="index.html">Preface</a></li> <li class="chapter" data-level="1" data-path="intro.html"><a href="intro.html">1 Getting started</a> <ul> <li class="chapter" data-level="1.1" data-path="what-can-be-forecast.html"><a href="what-can-be-forecast.html">1.1 What can be forecast?</a></li> <li class="chapter" data-level="1.2" data-path="planning.html"><a href="planning.html">1.2 Forecasting, planning and goals</a></li> <li class="chapter" data-level="1.3" data-path="determining-what-to-forecast.html"><a href="determining-what-to-forecast.html">1.3 Determining what to forecast</a></li> <li class="chapter" data-level="1.4" data-path="data-methods.html"><a href="data-methods.html">1.4 Forecasting data and methods</a></li> <li class="chapter" data-level="1.5" data-path="case-studies.html"><a href="case-studies.html">1.5 Some case studies</a></li> <li class="chapter" data-level="1.6" data-path="basic-steps.html"><a href="basic-steps.html">1.6 The basic steps in a forecasting task</a></li> <li class="chapter" data-level="1.7" data-path="perspective.html"><a href="perspective.html">1.7 The statistical forecasting perspective</a></li> <li class="chapter" data-level="1.8" data-path="intro-exercises.html"><a href="intro-exercises.html">1.8 Exercises</a></li> <li class="chapter" data-level="1.9" data-path="intro-reading.html"><a href="intro-reading.html">1.9 Further reading</a></li> </ul></li> <li class="chapter" data-level="2" data-path="graphics.html"><a href="graphics.html">2 Time series graphics</a> <ul> <li class="chapter" data-level="2.1" data-path="ts-objects.html"><a href="ts-objects.html">2.1 <code>ts</code> objects</a></li> <li class="chapter" data-level="2.2" data-path="time-plots.html"><a href="time-plots.html">2.2 Time plots</a></li> <li class="chapter" data-level="2.3" data-path="tspatterns.html"><a href="tspatterns.html">2.3 Time series patterns</a></li> <li class="chapter" data-level="2.4" data-path="seasonal-plots.html"><a href="seasonal-plots.html">2.4 Seasonal plots</a></li> <li class="chapter" data-level="2.5" data-path="seasonal-subseries-plots.html"><a href="seasonal-subseries-plots.html">2.5 Seasonal subseries plots</a></li> <li class="chapter" data-level="2.6" data-path="scatterplots.html"><a href="scatterplots.html">2.6 Scatterplots</a></li> <li class="chapter" data-level="2.7" data-path="lag-plots.html"><a href="lag-plots.html">2.7 Lag plots</a></li> <li class="chapter" data-level="2.8" data-path="autocorrelation.html"><a href="autocorrelation.html">2.8 Autocorrelation</a></li> <li class="chapter" data-level="2.9" data-path="wn.html"><a href="wn.html">2.9 White noise</a></li> <li class="chapter" data-level="2.10" data-path="graphics-exercises.html"><a href="graphics-exercises.html">2.10 Exercises</a></li> <li class="chapter" data-level="2.11" data-path="graphics-reading.html"><a href="graphics-reading.html">2.11 Further reading</a></li> </ul></li> <li class="chapter" data-level="3" data-path="toolbox.html"><a href="toolbox.html">3 The forecaster’s toolbox</a> <ul> <li class="chapter" data-level="3.1" data-path="simple-methods.html"><a href="simple-methods.html">3.1 Some simple forecasting methods</a></li> <li class="chapter" data-level="3.2" data-path="transformations.html"><a href="transformations.html">3.2 Transformations and adjustments</a></li> <li class="chapter" data-level="3.3" data-path="residuals.html"><a href="residuals.html">3.3 Residual diagnostics</a></li> <li class="chapter" data-level="3.4" data-path="accuracy.html"><a href="accuracy.html">3.4 Evaluating forecast accuracy</a></li> <li class="chapter" data-level="3.5" data-path="prediction-intervals.html"><a href="prediction-intervals.html">3.5 Prediction intervals</a></li> <li class="chapter" data-level="3.6" data-path="the-forecast-package-in-r.html"><a href="the-forecast-package-in-r.html">3.6 The forecast package in R</a></li> <li class="chapter" data-level="3.7" data-path="toolbox-exercises.html"><a href="toolbox-exercises.html">3.7 Exercises</a></li> <li class="chapter" data-level="3.8" data-path="toolbox-reading.html"><a href="toolbox-reading.html">3.8 Further reading</a></li> </ul></li> <li class="chapter" data-level="4" data-path="judgmental.html"><a href="judgmental.html">4 Judgmental forecasts</a> <ul> <li class="chapter" data-level="4.1" data-path="judgmental-limitations.html"><a href="judgmental-limitations.html">4.1 Beware of limitations</a></li> <li class="chapter" data-level="4.2" data-path="judgmental-principles.html"><a href="judgmental-principles.html">4.2 Key principles</a></li> <li class="chapter" data-level="4.3" data-path="delphimethod.html"><a href="delphimethod.html">4.3 The Delphi method</a></li> <li class="chapter" data-level="4.4" data-path="analogies.html"><a href="analogies.html">4.4 Forecasting by analogy</a></li> <li class="chapter" data-level="4.5" data-path="scenarios.html"><a href="scenarios.html">4.5 Scenario forecasting</a></li> <li class="chapter" data-level="4.6" data-path="new-products.html"><a href="new-products.html">4.6 New product forecasting</a></li> <li class="chapter" data-level="4.7" data-path="judgmental-adjustments.html"><a href="judgmental-adjustments.html">4.7 Judgmental adjustments</a></li> <li class="chapter" data-level="4.8" data-path="judgmental-reading.html"><a href="judgmental-reading.html">4.8 Further reading</a></li> </ul></li> <li class="chapter" data-level="5" data-path="regression.html"><a href="regression.html">5 Time series regression models</a> <ul> <li class="chapter" data-level="5.1" data-path="regression-intro.html"><a href="regression-intro.html">5.1 The linear model</a></li> <li class="chapter" data-level="5.2" data-path="least-squares.html"><a href="least-squares.html">5.2 Least squares estimation</a></li> <li class="chapter" data-level="5.3" data-path="regression-evaluation.html"><a href="regression-evaluation.html">5.3 Evaluating the regression model</a></li> <li class="chapter" data-level="5.4" data-path="useful-predictors.html"><a href="useful-predictors.html">5.4 Some useful predictors</a></li> <li class="chapter" data-level="5.5" data-path="selecting-predictors.html"><a href="selecting-predictors.html">5.5 Selecting predictors</a></li> <li class="chapter" data-level="5.6" data-path="forecasting-regression.html"><a href="forecasting-regression.html">5.6 Forecasting with regression</a></li> <li class="chapter" data-level="5.7" data-path="regression-matrices.html"><a href="regression-matrices.html">5.7 Matrix formulation</a></li> <li class="chapter" data-level="5.8" data-path="nonlinear-regression.html"><a href="nonlinear-regression.html">5.8 Nonlinear regression</a></li> <li class="chapter" data-level="5.9" data-path="causality.html"><a href="causality.html">5.9 Correlation, causation and forecasting</a></li> <li class="chapter" data-level="5.10" data-path="regression-exercises.html"><a href="regression-exercises.html">5.10 Exercises</a></li> <li class="chapter" data-level="5.11" data-path="regression-reading.html"><a href="regression-reading.html">5.11 Further reading</a></li> </ul></li> <li class="chapter" data-level="6" data-path="decomposition.html"><a href="decomposition.html">6 Time series decomposition</a> <ul> <li class="chapter" data-level="6.1" data-path="components.html"><a href="components.html">6.1 Time series components</a></li> <li class="chapter" data-level="6.2" data-path="moving-averages.html"><a href="moving-averages.html">6.2 Moving averages</a></li> <li class="chapter" data-level="6.3" data-path="classical-decomposition.html"><a href="classical-decomposition.html">6.3 Classical decomposition</a></li> <li class="chapter" data-level="6.4" data-path="x11.html"><a href="x11.html">6.4 X11 decomposition</a></li> <li class="chapter" data-level="6.5" data-path="seats.html"><a href="seats.html">6.5 SEATS decomposition</a></li> <li class="chapter" data-level="6.6" data-path="stl.html"><a href="stl.html">6.6 STL decomposition</a></li> <li class="chapter" data-level="6.7" data-path="seasonal-strength.html"><a href="seasonal-strength.html">6.7 Measuring strength of trend and seasonality</a></li> <li class="chapter" data-level="6.8" data-path="forecasting-decomposition.html"><a href="forecasting-decomposition.html">6.8 Forecasting with decomposition</a></li> <li class="chapter" data-level="6.9" data-path="decomposition-exercises.html"><a href="decomposition-exercises.html">6.9 Exercises</a></li> <li class="chapter" data-level="6.10" data-path="decomposition-reading.html"><a href="decomposition-reading.html">6.10 Further reading</a></li> </ul></li> <li class="chapter" data-level="7" data-path="expsmooth.html"><a href="expsmooth.html">7 Exponential smoothing</a> <ul> <li class="chapter" data-level="7.1" data-path="ses.html"><a href="ses.html">7.1 Simple exponential smoothing</a></li> <li class="chapter" data-level="7.2" data-path="holt.html"><a href="holt.html">7.2 Trend methods</a></li> <li class="chapter" data-level="7.3" data-path="holt-winters.html"><a href="holt-winters.html">7.3 Holt-Winters’ seasonal method</a></li> <li class="chapter" data-level="7.4" data-path="taxonomy.html"><a href="taxonomy.html">7.4 A taxonomy of exponential smoothing methods</a></li> <li class="chapter" data-level="7.5" data-path="ets.html"><a href="ets.html">7.5 Innovations state space models for exponential smoothing</a></li> <li class="chapter" data-level="7.6" data-path="estimation-and-model-selection.html"><a href="estimation-and-model-selection.html">7.6 Estimation and model selection</a></li> <li class="chapter" data-level="7.7" data-path="ets-forecasting.html"><a href="ets-forecasting.html">7.7 Forecasting with ETS models</a></li> <li class="chapter" data-level="7.8" data-path="expsmooth-exercises.html"><a href="expsmooth-exercises.html">7.8 Exercises</a></li> <li class="chapter" data-level="7.9" data-path="expsmooth-reading.html"><a href="expsmooth-reading.html">7.9 Further reading</a></li> </ul></li> <li class="chapter" data-level="8" data-path="arima.html"><a href="arima.html">8 ARIMA models</a> <ul> <li class="chapter" data-level="8.1" data-path="stationarity.html"><a href="stationarity.html">8.1 Stationarity and differencing</a></li> <li class="chapter" data-level="8.2" data-path="backshift.html"><a href="backshift.html">8.2 Backshift notation</a></li> <li class="chapter" data-level="8.3" data-path="AR.html"><a href="AR.html">8.3 Autoregressive models</a></li> <li class="chapter" data-level="8.4" data-path="MA.html"><a href="MA.html">8.4 Moving average models</a></li> <li class="chapter" data-level="8.5" data-path="non-seasonal-arima.html"><a href="non-seasonal-arima.html">8.5 Non-seasonal ARIMA models</a></li> <li class="chapter" data-level="8.6" data-path="arima-estimation.html"><a href="arima-estimation.html">8.6 Estimation and order selection</a></li> <li class="chapter" data-level="8.7" data-path="arima-r.html"><a href="arima-r.html">8.7 ARIMA modelling in R</a></li> <li class="chapter" data-level="8.8" data-path="arima-forecasting.html"><a href="arima-forecasting.html">8.8 Forecasting</a></li> <li class="chapter" data-level="8.9" data-path="seasonal-arima.html"><a href="seasonal-arima.html">8.9 Seasonal ARIMA models</a></li> <li class="chapter" data-level="8.10" data-path="arima-ets.html"><a href="arima-ets.html">8.10 ARIMA vs ETS</a></li> <li class="chapter" data-level="8.11" data-path="arima-exercises.html"><a href="arima-exercises.html">8.11 Exercises</a></li> <li class="chapter" data-level="8.12" data-path="arima-reading.html"><a href="arima-reading.html">8.12 Further reading</a></li> </ul></li> <li class="chapter" data-level="9" data-path="dynamic.html"><a href="dynamic.html">9 Dynamic regression models</a> <ul> <li class="chapter" data-level="9.1" data-path="estimation.html"><a href="estimation.html">9.1 Estimation</a></li> <li class="chapter" data-level="9.2" data-path="regarima.html"><a href="regarima.html">9.2 Regression with ARIMA errors in R</a></li> <li class="chapter" data-level="9.3" data-path="forecasting.html"><a href="forecasting.html">9.3 Forecasting</a></li> <li class="chapter" data-level="9.4" data-path="stochastic-and-deterministic-trends.html"><a href="stochastic-and-deterministic-trends.html">9.4 Stochastic and deterministic trends</a></li> <li class="chapter" data-level="9.5" data-path="dhr.html"><a href="dhr.html">9.5 Dynamic harmonic regression</a></li> <li class="chapter" data-level="9.6" data-path="lagged-predictors.html"><a href="lagged-predictors.html">9.6 Lagged predictors</a></li> <li class="chapter" data-level="9.7" data-path="dynamic-exercises.html"><a href="dynamic-exercises.html">9.7 Exercises</a></li> <li class="chapter" data-level="9.8" data-path="dynamic-reading.html"><a href="dynamic-reading.html">9.8 Further reading</a></li> </ul></li> <li class="chapter" data-level="10" data-path="hierarchical.html"><a href="hierarchical.html">10 Forecasting hierarchical or grouped time series</a> <ul> <li class="chapter" data-level="10.1" data-path="hts.html"><a href="hts.html">10.1 Hierarchical time series</a></li> <li class="chapter" data-level="10.2" data-path="gts.html"><a href="gts.html">10.2 Grouped time series</a></li> <li class="chapter" data-level="10.3" data-path="bottom-up.html"><a href="bottom-up.html">10.3 The bottom-up approach</a></li> <li class="chapter" data-level="10.4" data-path="top-down.html"><a href="top-down.html">10.4 Top-down approaches</a></li> <li class="chapter" data-level="10.5" data-path="middle-out.html"><a href="middle-out.html">10.5 Middle-out approach</a></li> <li class="chapter" data-level="10.6" data-path="mapping-matrices.html"><a href="mapping-matrices.html">10.6 Mapping matrices</a></li> <li class="chapter" data-level="10.7" data-path="reconciliation.html"><a href="reconciliation.html">10.7 The optimal reconciliation approach</a></li> <li class="chapter" data-level="10.8" data-path="hierarchical-exercises.html"><a href="hierarchical-exercises.html">10.8 Exercises</a></li> <li class="chapter" data-level="10.9" data-path="hierarchical-reading.html"><a href="hierarchical-reading.html">10.9 Further reading</a></li> </ul></li> <li class="chapter" data-level="11" data-path="advanced.html"><a href="advanced.html">11 Advanced forecasting methods</a> <ul> <li class="chapter" data-level="11.1" data-path="complexseasonality.html"><a href="complexseasonality.html">11.1 Complex seasonality</a></li> <li class="chapter" data-level="11.2" data-path="VAR.html"><a href="VAR.html">11.2 Vector autoregressions</a></li> <li class="chapter" data-level="11.3" data-path="nnetar.html"><a href="nnetar.html">11.3 Neural network models</a></li> <li class="chapter" data-level="11.4" data-path="bootstrap.html"><a href="bootstrap.html">11.4 Bootstrapping and bagging</a></li> <li class="chapter" data-level="11.5" data-path="advanced-exercises.html"><a href="advanced-exercises.html">11.5 Exercises</a></li> <li class="chapter" data-level="11.6" data-path="advanced-reading.html"><a href="advanced-reading.html">11.6 Further reading</a></li> </ul></li> <li class="chapter" data-level="12" data-path="practical.html"><a href="practical.html">12 Some practical forecasting issues</a> <ul> <li class="chapter" data-level="12.1" data-path="weekly.html"><a href="weekly.html">12.1 Weekly, daily and sub-daily data</a></li> <li class="chapter" data-level="12.2" data-path="counts.html"><a href="counts.html">12.2 Time series of counts</a></li> <li class="chapter" data-level="12.3" data-path="limits.html"><a href="limits.html">12.3 Ensuring forecasts stay within limits</a></li> <li class="chapter" data-level="12.4" data-path="combinations.html"><a href="combinations.html">12.4 Forecast combinations</a></li> <li class="chapter" data-level="12.5" data-path="aggregates.html"><a href="aggregates.html">12.5 Prediction intervals for aggregates</a></li> <li class="chapter" data-level="12.6" data-path="backcasting.html"><a href="backcasting.html">12.6 Backcasting</a></li> <li class="chapter" data-level="12.7" data-path="long-short-ts.html"><a href="long-short-ts.html">12.7 Very long and very short time series</a></li> <li class="chapter" data-level="12.8" data-path="forecasting-on-training-and-test-sets.html"><a href="forecasting-on-training-and-test-sets.html">12.8 Forecasting on training and test sets</a></li> <li class="chapter" data-level="12.9" data-path="missing-outliers.html"><a href="missing-outliers.html">12.9 Dealing with missing values and outliers</a></li> <li class="chapter" data-level="12.10" data-path="further-reading.html"><a href="further-reading.html">12.10 Further reading</a></li> </ul></li> <li class="chapter" data-level="" data-path="appendix-using-r.html"><a href="appendix-using-r.html">Appendix: Using R</a></li> <li class="chapter" data-level="" data-path="appendix-for-instructors.html"><a href="appendix-for-instructors.html">Appendix: For instructors</a></li> <li class="chapter" data-level="" data-path="appendix-reviews.html"><a href="appendix-reviews.html">Appendix: Reviews</a></li> <li class="chapter" data-level="" data-path="translations.html"><a href="translations.html">Translations</a></li> <li class="chapter" data-level="" data-path="about-the-authors.html"><a href="about-the-authors.html">About the authors</a></li> <li class="chapter" data-level="" data-path="buy-a-print-or-downloadable-version.html"><a href="buy-a-print-or-downloadable-version.html">Buy a print or downloadable version</a></li> <li class="chapter" data-level="" data-path="report-an-error.html"><a href="report-an-error.html">Report an error</a></li> <li class="chapter" data-level="" data-path="bibliography.html"><a href="bibliography.html">Bibliography</a></li> <li class="divider"></li> <li><a href="https://OTexts.com" target="blank">Published by OTexts™ with bookdown</a></li> </ul> </nav> </div> <div class="book-body"> <div class="body-inner"> <div class="book-header" role="navigation"> <h1> <a href="./">Forecasting: Principles and Practice (2nd ed)</a> </h1> </div> <div class="page-wrapper" tabindex="-1" role="main"> <div class="page-inner"> <section class="normal" id="section-"> <div id="stl" class="section level2 hasAnchor" number="6.6"> <h2>6.6 STL decomposition<a href="stl.html#stl" class="anchor-section" aria-label="Anchor link to header"></a></h2> STL is a versatile and robust method for decomposing time series. STL is an acronym for “Seasonal and Trend decomposition using Loess”, while Loess is a method for estimating nonlinear relationships. The STL method was developed by R. B. Cleveland, Cleveland, McRae, & Terpenning (<a href="#ref-Cleveland1990">1990</a>). STL has several advantages over the classical, SEATS and X11 decomposition methods: <ul> <li>Unlike SEATS and X11, STL will handle any type of seasonality, not only monthly and quarterly data.</li> <li>The seasonal component is allowed to change over time, and the rate of change can be controlled by the user.</li> <li>The smoothness of the trend-cycle can also be controlled by the user.</li> <li>It can be robust to outliers (i.e., the user can specify a robust decomposition), so that occasional unusual observations will not affect the estimates of the trend-cycle and seasonal components. They will, however, affect the remainder component.</li> </ul> On the other hand, STL has some disadvantages. In particular, it does not handle trading day or calendar variation automatically, and it only provides facilities for additive decompositions. It is possible to obtain a multiplicative decomposition by first taking logs of the data, then back-transforming the components. Decompositions between additive and multiplicative can be obtained using a Box-Cox transformation of the data with \(0<\lambda<1\). A value of \(\lambda=0\) corresponds to the multiplicative decomposition while \(\lambda=1\) is equivalent to an additive decomposition. The best way to begin learning how to use STL is to see some examples and experiment with the settings. Figure <a href="components.html#fig:elecequip-stl">6.2</a> showed an example of STL applied to the electrical equipment orders data. Figure <a href="stl.html#fig:elecequip-stl2">6.13</a> shows an alternative STL decomposition where the trend-cycle is more flexible, the seasonal component does not change over time, and the robust option has been used. Here, it is more obvious that there has been a down-turn at the end of the series, and that the orders in 2009 were unusually low (corresponding to some large negative values in the remainder component). <div class="sourceCode" id="cb102"><pre class="sourceCode r"><code class="sourceCode r"><a href="stl.html#cb102-1" tabindex="-1"></a>elecequip %>% <a href="stl.html#cb102-2" tabindex="-1"></a> stl(t.window=13, s.window="periodic", robust=TRUE) %>% <a href="stl.html#cb102-3" tabindex="-1"></a> autoplot()</code></pre></div> <div class="figure" style="text-align: center"> <img src="fpp_files/figure-html/elecequip-stl2-1.png" alt="The electrical equipment orders (top) and its three additive components obtained from a robust STL decomposition with flexible trend-cycle and fixed seasonality." width="100%" /> Figure 6.13: The electrical equipment orders (top) and its three additive components obtained from a robust STL decomposition with flexible trend-cycle and fixed seasonality. </div> The two main parameters to be chosen when using STL are the trend-cycle window (<code>t.window</code>) and the seasonal window (<code>s.window</code>). These control how rapidly the trend-cycle and seasonal components can change. Smaller values allow for more rapid changes. Both <code>t.window</code> and <code>s.window</code> should be odd numbers; <code>t.window</code> is the number of consecutive observations to be used when estimating the trend-cycle; <code>s.window</code> is the number of consecutive years to be used in estimating each value in the seasonal component. The user must specify <code>s.window</code> as there is no default. Setting it to be infinite is equivalent to forcing the seasonal component to be periodic (i.e., identical across years). Specifying <code>t.window</code> is optional, and a default value will be used if it is omitted. The <code>mstl()</code> function provides a convenient automated STL decomposition using <code>s.window=13</code>, and <code>t.window</code> also chosen automatically. This usually gives a good balance between overfitting the seasonality and allowing it to slowly change over time. But, as with any automated procedure, the default settings will need adjusting for some time series. As with the other decomposition methods discussed in this book, to obtain the separate components plotted in Figure <a href="classical-decomposition.html#fig:classical-elecequip">6.8</a>, use the <code>seasonal()</code> function for the seasonal component, the <code>trendcycle()</code> function for trend-cycle component, and the <code>remainder()</code> function for the remainder component. The <code>seasadj()</code> function can be used to compute the seasonally adjusted series. </div> <h3>Bibliography<a href="bibliography.html#bibliography" class="anchor-section" aria-label="Anchor link to header"></a></h3> <div id="refs" class="references csl-bib-body hanging-indent" entry-spacing="0"> <div id="ref-Cleveland1990" class="csl-entry"> Cleveland, R. B., Cleveland, W. S., McRae, J. E., & Terpenning, I. J. (1990). STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics, 6(1), 3–33. <a href="http://bit.ly/stl1990">http://bit.ly/stl1990</a> </div> </div> </section> </div> </div> </div> <a href="seats.html" class="navigation navigation-prev " aria-label="Previous page"></a> <a href="seasonal-strength.html" class="navigation navigation-next " aria-label="Next page"></a> </div> </div> <script src="libs/gitbook-2.6.7/js/app.min.js"></script> <script src="libs/gitbook-2.6.7/js/clipboard.min.js"></script> <script src="libs/gitbook-2.6.7/js/plugin-search.js"></script> <script src="libs/gitbook-2.6.7/js/plugin-sharing.js"></script> <script src="libs/gitbook-2.6.7/js/plugin-fontsettings.js"></script> <script src="libs/gitbook-2.6.7/js/plugin-bookdown.js"></script> <script src="libs/gitbook-2.6.7/js/jquery.highlight.js"></script> <script src="libs/gitbook-2.6.7/js/plugin-clipboard.js"></script> <script> gitbook.require(["gitbook"], function(gitbook) { gitbook.start({ "sharing": { "github": false, "facebook": true, "twitter": true, "linkedin": false, "weibo": false, "instapaper": false, "vk": false, "whatsapp": false, "all": ["facebook", "twitter", "linkedin", "weibo", "instapaper"] }, "fontsettings": { "theme": "white", "family": "sans", "size": 2 }, "edit": { "link": null, "text": null }, "history": { "link": null, "text": null }, "view": { "link": null, "text": null }, "download": null, "search": { "engine": "fuse", "options": null }, "toc": { "collapse": "section" } }); }); </script>  <script> (function () { var script = document.createElement("script"); script.type = "text/javascript"; var src = "true"; if (src === "" || src === "true") src = "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/latest.js?config=TeX-MML-AM_CHTML"; if (location.protocol !== "file:") if (/^https?:/.test(src)) src = src.replace(/^https?:/, ''); script.src = src; document.getElementsByTagName("head")[0].appendChild(script); })(); </script> </body> </html>