diff --git a/docs/Stats_review.html b/docs/Stats_review.html
index 47ed22c..c90a20c 100644
--- a/docs/Stats_review.html
+++ b/docs/Stats_review.html
@@ -340,7 +340,7 @@ <h1>Random Variables (RVs)</h1>
 <li>discrete (numbers of items or successes)</li>
 <li>continuous (heights, times, weights)</li>
 </ul>
-<p>We usually use capital letters – e.g.&nbsp;<span class="math inline">X</span>, <span class="math inline">Y</span>, sometimes with bold or with subscripts – to denote the RVs. In contrast we use lower case letters, e.g.&nbsp;<span class="math inline">x</span>, <span class="math inline">y</span>, <span class="math inline">k</span>, to denote the values that the RV takes. For instance, lets say that the heights of the woman at Virginia Tech are the RV, <span class="math inline">X</span>, and <span class="math inline">X</span> has a normal distribution with mean 62 inches and variance 6<span class="math inline">^2</span>, i.e., <span class="math inline">X \sim \mathrm{N}(62,6^2)</span> distribution. Say we then observe the heights of 3 individuals drawn from this distribution – we would write this as: <span class="math inline">x=(</span> 63.5, 66.3, 62.1 <span class="math inline">)</span>.</p>
+<p>We usually use capital letters – e.g.&nbsp;<span class="math inline">X</span>, <span class="math inline">Y</span>, sometimes with bold or with subscripts – to denote the RVs. In contrast we use lower case letters, e.g.&nbsp;<span class="math inline">x</span>, <span class="math inline">y</span>, <span class="math inline">k</span>, to denote the values that the RV takes. For instance, lets say that the heights of the woman at Virginia Tech are the RV, <span class="math inline">X</span>, and <span class="math inline">X</span> has a normal distribution with mean 62 inches and variance 6<span class="math inline">^2</span>, i.e., <span class="math inline">X \sim \mathrm{N}(62,6^2)</span> distribution. Say we then observe the heights of 3 individuals drawn from this distribution – we would write this as: <span class="math inline">x=(</span> 58.8, 62.1, 64.2 <span class="math inline">)</span>.</p>
 <p><br> <br> </p>
 </section>
 <section id="probability-distributions" class="level1">
@@ -564,7 +564,7 @@ <h1>Probability Distributions in <code>R</code></h1>
 <div class="cell">
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">rnorm</span>(<span class="dv">3</span>, <span class="at">mean=</span><span class="dv">0</span>, <span class="at">sd=</span><span class="dv">1</span>) <span class="do">## random draws</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>[1] -0.2085685  0.1056252  1.8288412</code></pre>
+<pre><code>[1] 0.02318727 0.28406256 0.56712882</code></pre>
 </div>
 </div>
 <div class="cell">
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diff --git a/docs/VB_RegDiagTrans.html b/docs/VB_RegDiagTrans.html
index 7539570..8140768 100644
--- a/docs/VB_RegDiagTrans.html
+++ b/docs/VB_RegDiagTrans.html
@@ -455,7 +455,7 @@ <h2>SLR model assumptions</h2>
 </section>
 <section id="example-model-violations" class="slide level2">
 <h2>Example model violations</h2>
-<p><span style="color: red;">Anscombe’s quartet</span> comprises four datasets that have similar statistical properties ...</p>
+<p><span style="color: red;">Anscombe’s quartet</span> comprises four datasets that have similar statistical properties …</p>
 <table>
 <thead>
 <tr class="header">
@@ -505,7 +505,7 @@ <h2>Example model violations</h2>
 </section>
 <section class="slide level2">
 
-<p>...but vary considerably when graphed:</p>
+<p>…but vary considerably when graphed:</p>
 
 <img data-src="VB_RegDiagTrans_files/figure-revealjs/unnamed-chunk-2-1.png" class="quarto-figure quarto-figure-center r-stretch" width="672"></section>
 <section class="slide level2">
@@ -519,7 +519,7 @@ <h2>Example model violations</h2>
 <img data-src="VB_RegDiagTrans_files/figure-revealjs/unnamed-chunk-5-1.png" class="quarto-figure quarto-figure-center r-stretch" width="672"></section>
 <section class="slide level2">
 
-<p>The regression lines and <span class="math inline">\(R^2\)</span> values are the same...<br>
+<p>The regression lines and <span class="math inline">\(R^2\)</span> values are the same…<br>
 </p>
 <table>
 <thead>
@@ -560,7 +560,7 @@ <h2>Example model violations</h2>
 </section>
 <section class="slide level2">
 
-<p>...but the residuals, <span class="math inline">\(e\)</span>, <strong><em>(plotted vs.&nbsp;<span class="math inline">\(\hat{Y~}\)</span>)</em></strong> look totally different.</p>
+<p>…but the residuals, <span class="math inline">\(e\)</span>, <strong><em>(plotted vs.</em></strong> <span class="math inline">\(\hat{Y~}\)</span>) look totally different.</p>
 
 <img data-src="VB_RegDiagTrans_files/figure-revealjs/unnamed-chunk-6-1.png" class="quarto-figure quarto-figure-center r-stretch" width="672"></section>
 <section class="slide level2">
@@ -628,7 +628,7 @@ <h2>Understanding Leverage</h2>
 </ul>
 <p>Here is a nice online (interactive) illustration of leverage:</p>
 <center>
-<a href="https://omaymas.shinyapps.io/Influence_Analysis/">https://omaymas.shinyapps.io/Influence_Analysis/</a>
+<a href="https://omaymas.shinyapps.io/Influence_Analysis/" class="uri">https://omaymas.shinyapps.io/Influence_Analysis/</a>
 </center>
 <p><br></p>
 <p><span style="color: red;">Outliers do more damage if they have high leverage!</span></p>
@@ -647,7 +647,7 @@ <h2>Standardized residuals</h2>
 </section>
 <section class="slide level2">
 
-<p>About estimating <span class="math inline">\(s\)</span> under sketchy SLR assumptions ...</p>
+<p>About estimating <span class="math inline">\(s\)</span> under sketchy SLR assumptions …</p>
 <p>We want to see whether any particular <span class="math inline">\(e_i\)</span> is “too big”, but we don’t want a single outlier to make <span class="math inline">\(s\)</span> artificially large.</p>
 <p><br></p>
 <div class="columns">
@@ -715,7 +715,7 @@ <h2>How to deal with outliers</h2>
 <section id="outliers-leverage-and-residuals" class="slide level2">
 <h2>Outliers, leverage, and residuals</h2>
 <p><span style="color: red;">Warning:</span> <span style="color: dodgerblue;">Unfortunately, outliers with high leverage are hard to catch through</span> <span class="math inline">\(\color{dodgerblue}{r_i}\)</span> <span style="color: dodgerblue;">(since the line is pulled towards them)</span>.</p>
-<p>Means get distracted by outliers...<br>
+<p>Means get distracted by outliers…<br>
 </p>
 <center>
 <img data-src="graphics/distracted_mean_meme.jpg" style="height:4in">
@@ -870,7 +870,7 @@ <h2>Solution 1: Variance stabilizing transformations</h2>
 <p><span style="color: dodgerblue;">We have a multiplicative model now!</span></p>
 </div>
 <p><br></p>
-<p>Note: <strong>you <span style="color: red;">CANNOT</span> compare <span class="math inline">\(R^2\)</span> values for regressions corresponding to different transformations of the response.</strong></p>
+<p>Note: <strong>you <span style="color: red;">CANNOT</span> compare</strong> <span class="math inline">\(R^2\)</span> values for regressions corresponding to different transformations of the response.</p>
 <ul>
 <li><p><span class="math inline">\(Y\)</span> and <span class="math inline">\(f(Y)\)</span> may not be on the same scale,</p></li>
 <li><p>therefore <span class="math inline">\(\text{var}(Y)\)</span> and <span class="math inline">\(\text{var}(f(Y))\)</span> may not be either.</p></li>
@@ -912,7 +912,7 @@ <h2>Testing for nonlinearity</h2>
 <p>To see if you need more nonlinearity, try the regression which includes the next polynomial term, and see if it is significant.</p>
 <p>For example, to see if you need a <span style="color: dodgerblue;">quadratic term</span>,</p>
 <ul>
-<li><p>fit the model then run the regression <span class="math inline">\(\mathbb{E}[Y] = \beta_0 + \beta_1 X +  \beta_2 X^2\)</span>.</p></li>
+<li><p>fit the model then run the regression <span class="math inline">\(\mathbb{E}[Y] = \beta_0 + \beta_1 X + \beta_2 X^2\)</span>.</p></li>
 <li><p><span style="color: red;">If your test implies</span> <span class="math inline">\(\color{dodgerblue}{\beta_2 \neq 0}\)</span>, <span style="color: red;">you need</span> <span class="math inline">\(\color{dodgerblue}{X^2}\)</span> in your model.</p></li>
 </ul>
 <p>Note: <span class="math inline">\(p\)</span>-values are calculated “given the other <span class="math inline">\(\beta\)</span>’s are nonzero”; i.e., conditional on <span class="math inline">\(X\)</span> being in the model.</p>
@@ -953,7 +953,7 @@ <h2>The log-log model</h2>
 <p>In practice, this is often used in conjunction with a <span class="math inline">\(\log(Y)\)</span> response transformation. The log-log model is <span class="math display">\[
     \color{red}{\log(Y) = \beta_0 + \beta_1 \log(X) + \varepsilon}.
     \]</span></p>
-<p>It is super useful, and has some special properties ...</p>
+<p>It is super useful, and has some special properties …</p>
 </section>
 <section class="slide level2">
 
@@ -1012,7 +1012,7 @@ <h2>Elasticity and the log-log model</h2>
 </section>
 <section class="slide level2">
 
-<p><strong><em>Summary: <span style="color: red;">Plots of residuals v.s.</span> <span class="math inline">\(\color{red}X\)</span> <span style="color: red;">or</span> <span class="math inline">\(\color{red}{\hat{Y~}}\)</span> <span style="color: red;">are most important for diagnosing problems</span>.</em></strong></p>
+<p><strong><em>Summary: <span style="color: red;">Plots of residuals v.s.</span></em></strong> <span class="math inline">\(\color{red}X\)</span> <span style="color: red;">or</span> <span class="math inline">\(\color{red}{\hat{Y~}}\)</span> <span style="color: red;">are most important for diagnosing problems</span>.</p>
 <ul>
 <li><p>Log transform is your top tool (log(<span class="math inline">\(X\)</span>), log(<span class="math inline">\(Y\)</span>), or both).</p></li>
 <li><p>Add polynomial terms (e.g.&nbsp;<span class="math inline">\(X^2\)</span>) to get nonlinear mean patterns.</p>
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new file mode 100644
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+++ b/docs/VB_TimeDepData.html
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+
+<section id="title-slide" data-background-image="VectorByte-logo_lg.png" data-background-opacity="0.2" data-background-size="contain" class="quarto-title-block center">
+  <h1 class="title">VectorByte Methods Training</h1>
+  <p class="subtitle">Regression Methods for Time Dependent Data</p>
+
+<div class="quarto-title-authors">
+<div class="quarto-title-author">
+<div class="quarto-title-author-name">
+The VectorByte Team (Leah R. Johnson, Virginia Tech) 
+</div>
+</div>
+</div>
+
+</section>
+<section id="learning-objectives" class="slide level2">
+<h2>Learning Objectives</h2>
+<ol type="1">
+<li></li>
+</ol>
+</section>
+<section id="time-dependent-data" class="slide level2">
+<h2>Time Dependent Data</h2>
+<p>Much of the data we collect in VBD applications depends on <em><span style="color: red;">time</span></em> such as</p>
+<ul>
+<li>observed cases in a city or country</li>
+<li>number of mosquitoes over time</li>
+</ul>
+<p>Additionally, we often assume that the reasons these change over time may be because covariates (e.g., temperature, precipitation, insecticide spraying) change over time.</p>
+<p><strong><em><span style="color: ForestGreen;">How do we incorporate these time varying factors into our regression models?</span></em></strong></p>
+</section>
+<section id="types-of-time-dependent-data" class="slide level2">
+<h2>Types of time dependent data</h2>
+<p>The most common type of time-dependent data that statisticians talk about is <span style="color: dodgerblue;">time series data</span>. These are data where observations are evenly spaced with no (or very little) missing observations.</p>
+<p><br></p>
+<p>Although evenly spaced data are ideal (and the most common methods are designed for them), in VBD survey data we often don’t have evenly spaced observations. These data don’t have a specific name, and most time-series methods can’t be directly used with them.</p>
+</section>
+<section id="time-series-data-and-dependence" class="slide level2">
+<h2>Time series data and dependence</h2>
+<p>Time-series data are simply a collection of observations gathered over time. For example, suppose <span class="math inline">\(y_1, \ldots, y_T\)</span> are</p>
+<ul>
+<li>daily temperature,</li>
+<li>solar activity,</li>
+<li>CO<span class="math inline">\(_2\)</span> levels,</li>
+<li>yearly population size.</li>
+</ul>
+<p><span style="color: dodgerblue;">In each case, we might expect what happens at time</span> <span class="math inline">\(t\)</span> <span style="color: dodgerblue;">to be correlated with time</span> <span class="math inline">\(t-1\)</span>.</p>
+</section>
+<section class="slide level2">
+
+<p>Suppose we measure temperatures, daily, for several years.</p>
+<p>Which would work better as an estimate for today’s temp:</p>
+<ul>
+<li>The average of the temperatures from the previous year?</li>
+<li>The temperature on the previous day?</li>
+</ul>
+<p><br></p>
+<p>How would this change if the readings were iid <span class="math inline">\(\mathcal{N}(\mu, \sigma^2)\)</span>?</p>
+<p><br></p>
+<p><span style="color: red;">Correlated errors require fundamentally different techniques.</span></p>
+</section>
+<section class="slide level2">
+
+<p><span style="color: ForestGreen;">Example</span>: <span class="math inline">\(Y_t =\)</span> average daily temp. at ROA, Feb-Mar 2018.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a></a>weather <span class="ot">&lt;-</span> <span class="fu">read.csv</span>(<span class="st">"data/Roanoke_weather2018.csv"</span>)</span>
+<span id="cb1-2"><a></a>days<span class="ot">&lt;-</span><span class="dv">32</span><span class="sc">:</span><span class="dv">90</span> <span class="do">## This picks out February and March</span></span>
+<span id="cb1-3"><a></a>weather<span class="ot">&lt;-</span>weather[days,<span class="fu">c</span>(<span class="dv">1</span>,<span class="dv">3</span>,<span class="dv">7</span>,<span class="dv">4</span>)]</span>
+<span id="cb1-4"><a></a><span class="fu">names</span>(weather)[<span class="dv">3</span>]<span class="ot">&lt;-</span><span class="st">"temp"</span></span>
+<span id="cb1-5"><a></a><span class="fu">plot</span>(days, weather<span class="sc">$</span>temp, <span class="at">xlab=</span><span class="st">"day of year"</span>, <span class="at">ylab=</span><span class="st">"temp (F)"</span>, <span class="at">type=</span><span class="st">"l"</span>, <span class="at">col=</span><span class="dv">2</span>, <span class="at">lwd=</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-1-1.png" class="quarto-figure quarto-figure-center r-stretch" width="768"><ul>
+<li>“sticky” sequence: today tends to be close to yesterday.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p><span style="color: ForestGreen;">Example</span>: <span class="math inline">\(Y_t =\)</span> monthly UK deaths due to lung infections</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a></a>ld<span class="ot">&lt;-</span><span class="fu">as.vector</span>(ldeaths)</span>
+<span id="cb2-2"><a></a><span class="fu">plot</span>(ld, <span class="at">xlab=</span><span class="st">"month"</span>, <span class="at">ylab=</span><span class="st">"deaths"</span>, <span class="at">type=</span><span class="st">"l"</span>, </span>
+<span id="cb2-3"><a></a>     <span class="at">col=</span><span class="dv">4</span>, <span class="at">lwd=</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-2-1.png" class="quarto-figure quarto-figure-center r-stretch" width="768"><ul>
+<li>The same pattern repeats itself year after year.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p><span style="color: ForestGreen;">Example</span>: uncorrelated samples from a normal random number generator.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a></a><span class="fu">plot</span>(<span class="fu">rnorm</span>(<span class="dv">200</span>), <span class="at">xlab=</span><span class="st">"t"</span>, <span class="at">ylab=</span><span class="st">"Y_t"</span>, <span class="at">type=</span><span class="st">"l"</span>, </span>
+<span id="cb3-2"><a></a>     <span class="at">col=</span><span class="dv">6</span>, <span class="at">lwd=</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-3-1.png" class="quarto-figure quarto-figure-center r-stretch" width="768"><ul>
+<li>It is tempting to see patterns even where they don’t exist.</li>
+</ul>
+</section>
+<section id="checking-for-dependence" class="slide level2">
+<h2>Checking for dependence</h2>
+<p>To see if <span class="math inline">\(Y_{t-1}\)</span> would be useful for predicting <span class="math inline">\(Y_t\)</span>, just plot them together and see if there is a relationship.</p>
+<div class="columns">
+<div class="column" style="width:50%;">
+<div class="cell" data-layout-align="center">
+<div class="cell-output-display">
+<div class="quarto-figure quarto-figure-center">
+<figure>
+<p><img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-4-1.png" class="quarto-figure quarto-figure-center" width="480"></p>
+</figure>
+</div>
+</div>
+</div>
+</div><div class="column" style="width:50%;">
+<p><br></p>
+<ul>
+<li>Correlation between <span class="math inline">\(Y_t\)</span> and <span class="math inline">\(Y_{t-1}\)</span> is called <span style="color: red;">autocorrelation</span>.</li>
+</ul>
+</div>
+</div>
+</section>
+<section class="slide level2">
+
+<p>We can plot <span class="math inline">\(Y_t\)</span> against <span class="math inline">\(Y_{t-\ell}\)</span> to see <span class="math inline">\(\color{red}\ell\)</span><span style="color: red;">-period lagged relationships</span>.</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-5-1.png" class="quarto-figure quarto-figure-center r-stretch" width="816"><ul>
+<li>It appears that the correlation is getting weaker with increasing <span class="math inline">\(\ell\)</span>.</li>
+</ul>
+</section>
+<section id="autocorrelation-for-time-series-data" class="slide level2">
+<h2>Autocorrelation (for time series data)</h2>
+<p>To summarize the time-varying dependence, compute lag-<span class="math inline">\(\ell\)</span> correlations for <span class="math inline">\(\ell=1,2,3,\ldots\)</span></p>
+<p>In general, the autocorrelation function (ACF) for <span class="math inline">\(Y\)</span> is <span class="math display">\[\color{red}{r(\ell) = \mathrm{cor}(Y_t, Y_{t-\ell})}\]</span></p>
+<p>For our Roanoke temperature data:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a></a><span class="fu">print</span>(<span class="fu">acf</span>(weather<span class="sc">$</span>temp))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<pre><code>     0      1      2      3      4      5      6      7      8    
+ 1.000  0.658  0.298  0.263  0.297  0.177  0.111  0.008 -0.099   
+    9     10    11     12     13     14     15     16     17 
+-0.045 0.071 -0.020 -0.157 -0.156 -0.146 -0.278 -0.346 -0.314 </code></pre>
+</section>
+<section class="slide level2">
+
+<p>R’s <code>acf</code> function shows the ACF visually.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a></a><span class="fu">acf</span>(weather<span class="sc">$</span>temp)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-7-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><p>It provides a visual summary of our data dependence.</p>
+<p>(<span style="color: dodgerblue;">Blue lines mark “statistical significance” for the acf values.</span>)</p>
+</section>
+<section class="slide level2">
+
+<p>The lung infection data shows an alternating dependence structure which causes time series oscillations.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a></a><span class="fu">acf</span>(ld, <span class="at">lag.max=</span><span class="dv">30</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-8-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section class="slide level2">
+
+<p>An acf plot for <span class="math inline">\(iid\)</span> normal data shows no significant correlation.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a></a><span class="fu">acf</span>(<span class="fu">rnorm</span>(<span class="dv">40</span>), <span class="at">lag.max=</span><span class="dv">40</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-9-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section id="autoregression" class="slide level2">
+<h2>Autoregression</h2>
+<p>How do we model data that exhibits autocorrelation?</p>
+<p>Suppose <span class="math inline">\(Y_1 = \varepsilon_1\)</span>, <span class="math inline">\(Y_2 = \varepsilon_{1} + \varepsilon_{2}\)</span>, <span class="math inline">\(Y_3 = \varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3}\)</span>, etc.</p>
+<p><br></p>
+<p><strong><em>Then</em></strong> <span class="math inline">\(\color{dodgerblue}{Y_t =  \sum_{i=1}^{t}\varepsilon_i = Y_{t-1} + \varepsilon_t}\)</span> and <span class="math inline">\(\color{dodgerblue}{ \mathbb{E}[Y_t] = Y_{t-1}}\)</span>.</p>
+<p><br></p>
+<p>This is called a random walk model for <span class="math inline">\(Y_t\)</span>:</p>
+<ul>
+<li>the expectation of what will happen is always what happened most recently.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>Random walks are just a version of a more general model ...</p>
+<p>The autoregressive model of order one holds that <span class="math display">\[
+\color{red}{AR(1): Y_t = \beta_0 + \beta_1Y_{t-1} + \varepsilon_t,
+\;\;\; \varepsilon_t \stackrel{\mathrm{iid}}{\sim}\mathcal{N}(0, \sigma^2).}
+\]</span> This is just a SLR model of <span class="math inline">\(Y_t\)</span> regressed onto lagged <span class="math inline">\(Y_{t-1}\)</span>.</p>
+<p>It assumes all of our standard regression model conditions.</p>
+<ul>
+<li><p>The residuals should look <span class="math inline">\(iid\)</span> and be uncorrelated with <span class="math inline">\(\hat{Y~}_t\)</span>.</p></li>
+<li><p>All of our previous diagnostics and transforms still apply.</p></li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>For the Roanoke temperatures, there is a clear autocorrelation:</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a></a>tempreg <span class="ot">&lt;-</span> <span class="fu">lm</span>(weather<span class="sc">$</span>temp[<span class="dv">2</span><span class="sc">:</span><span class="dv">59</span>] <span class="sc">~</span> weather<span class="sc">$</span>temp[<span class="dv">1</span><span class="sc">:</span><span class="dv">58</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a></a><span class="fu">summary</span>(tempreg)  <span class="do">## abbreviated output</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<p><br></p>
+<pre><code>Coefficients:
+                   Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept)         15.2420     4.5729   3.333  0.00153 ** 
+weather$temp[1:58]   0.6584     0.1007   6.541 1.98e-08 ***
+---
+Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
+
+Residual standard error: 7.328 on 56 degrees of freedom
+Multiple R-squared:  0.4331,    Adjusted R-squared:  0.423 
+F-statistic: 42.78 on 1 and 56 DF,  p-value: 1.98e-08</code></pre>
+<p><br></p>
+<ul>
+<li>The autoregressive term (<span class="math inline">\(b_1 \approx 0.66\)</span>) is highly significant!</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>We can check residuals for any “left-over” correlation.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a></a><span class="fu">acf</span>(tempreg<span class="sc">$</span>residuals)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-12-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>Looks like we’ve got a good fit.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>For the lung infection data, the autoregressive term is also highly significant.</p>
+<p><br></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a></a>lungreg <span class="ot">&lt;-</span> <span class="fu">lm</span>(ld[<span class="dv">2</span><span class="sc">:</span><span class="dv">72</span>] <span class="sc">~</span>  ld[<span class="dv">1</span><span class="sc">:</span><span class="dv">71</span>])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a></a><span class="fu">summary</span>(lungreg)  <span class="do">## abbreviated output</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<p><br></p>
+<pre><code>Coefficients:
+             Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept) 487.13140  162.00888   3.007  0.00368 ** 
+ld[1:71]      0.75571    0.07546  10.015 4.39e-15 ***
+---
+Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
+
+Residual standard error: 387.6 on 69 degrees of freedom
+Multiple R-squared:  0.5924,    Adjusted R-squared:  0.5865 
+F-statistic: 100.3 on 1 and 69 DF,  p-value: 4.39e-15</code></pre>
+</section>
+<section class="slide level2">
+
+<p>But residuals show a clear pattern of left-over autocorrelation.</p>
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a></a><span class="fu">acf</span>(lungreg<span class="sc">$</span>residuals)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+
+</div>
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-15-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>We’ll talk shortly about how to model this type of pattern ...</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>Many different types of series may be written as an AR<span class="math inline">\((1)\)</span>. <span class="math display">\[
+AR(1): Y_t = \beta_0 + \beta_1Y_{t-1} + \varepsilon_t
+\]</span></p>
+<p><span style="color: dodgerblue;">The value of</span> <span class="math inline">\(\color{dodgerblue}{\beta_1}\)</span> <span style="color: dodgerblue;">is key!</span></p>
+<ul>
+<li><p>If <span class="math inline">\(|\beta_1| = 1\)</span>, we have a <em>random walk</em>.</p></li>
+<li><p>If <span class="math inline">\(|\beta_1| &gt; 1\)</span>, the series <em>explodes</em>.</p></li>
+<li><p>If <span class="math inline">\(|\beta_1| &lt; 1\)</span>, the values are <em>mean reverting</em>.</p></li>
+</ul>
+</section>
+<section id="random-walk" class="slide level2">
+<h2>Random walk</h2>
+<p>In a random walk, the series just wanders around.</p>
+<center>
+<span class="math inline">\(\beta_1 = 1\)</span>
+</center>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-16-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section class="slide level2">
+
+<p>Autocorrelation of a random walk stays high for a long time.</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-17-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section class="slide level2">
+
+<p>The random walk has some special properties ...</p>
+<p><span class="math inline">\(\color{dodgerblue}{Y_t - Y_{t-1} = \beta_0 + \varepsilon_t}\)</span>, and <span class="math inline">\(\beta_0\)</span> is called the “drift parameter”.</p>
+<p>The series is <span style="color: dodgerblue;">nonstationary</span>:</p>
+<ul>
+<li>it has no average level that it wants to be near, but rather just wanders off into space.</li>
+</ul>
+<p>The random walk <span style="color: red;">without drift</span> (<span class="math inline">\(\color{red}{\beta_0 = 0}\)</span>) is a common model for simple processes</p>
+<ul>
+<li>since <span class="math inline">\(\mathbb{E}[Y_t] =  \mathbb{E}[Y_{t-1}]\)</span>, e.g., tomorrow <span class="math inline">\(\approx\)</span> today</li>
+</ul>
+</section>
+<section id="exploding-series" class="slide level2">
+<h2>Exploding series</h2>
+<p>For AR term <span class="math inline">\(&gt;1\)</span>, the <span class="math inline">\(Y_t\)</span>’s move exponentially far from <span class="math inline">\(Y_1\)</span>.</p>
+<center>
+<span class="math inline">\(\beta_1 = 1.02\)</span>
+</center>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-18-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>Useless for modeling and prediction.</li>
+</ul>
+</section>
+<section id="stationary-series" class="slide level2">
+<h2>Stationary series</h2>
+<p>For <span class="math inline">\(\beta_1&lt;1\)</span>, <span class="math inline">\(Y_t\)</span> is always pulled back towards the mean.</p>
+<center>
+<span class="math inline">\(\beta_1 = 0.8\)</span>
+</center>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-19-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>These are the most common and useful type of AR series.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>Autocorrelation for the stationary series drops off right away.</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-20-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>The past matters, but with limited horizon.</li>
+</ul>
+</section>
+<section id="mean-reversion" class="slide level2">
+<h2>Mean reversion</h2>
+<p>An important property of stationary series is <span style="color: red;">mean reversion</span>.</p>
+<p>Think about shifting both <span class="math inline">\(Y_t\)</span> and <span class="math inline">\(Y_{t-1}\)</span> by their mean <span class="math inline">\(\mu\)</span>. <span class="math display">\[
+\color{dodgerblue}{Y_t - \mu = \beta_1 (Y_{t-1} - \mu) +\varepsilon_t}
+\]</span> Since <span class="math inline">\(|\beta_1| &lt; 1\)</span>, <span class="math inline">\(Y_t\)</span> is expected to be closer to <span class="math inline">\(\mu\)</span> than <span class="math inline">\(Y_{t-1}\)</span>.</p>
+<p>Mean reversion is all over, and helps predict future behavior:</p>
+<ul>
+<li>weekly sales numbers,</li>
+<li>daily temperature.</li>
+</ul>
+</section>
+<section id="negative-correlation" class="slide level2">
+<h2>Negative correlation</h2>
+<p>It is also possible to have negatively correlated AR(1) series.</p>
+<center>
+<span class="math inline">\(\beta_1 = -0.8\)</span>
+</center>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-21-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>But you see these far less often in practice.</li>
+</ul>
+</section>
+<section id="summary-of-ar1-behavior" class="slide level2">
+<h2>Summary of AR(1) behavior</h2>
+<ul>
+<li><p><span class="math inline">\(\color{dodgerblue}{|\beta_1|&lt;1|}\)</span>: The series has a mean level to which it reverts over time (stationary). For <span class="math inline">\(+\beta_1\)</span>, the series tends to wander above or below the mean level for a while. For <span class="math inline">\(-\beta_1\)</span>, the series tends to flip back and forth around the mean.</p></li>
+<li><p><span class="math inline">\(\color{dodgerblue}{|\beta_1|=1|}\)</span>: A random walk series. The series has no mean level and, thus, is called nonstationary. The drift parameter <span class="math inline">\(\beta_0\)</span> is the direction in which the series wanders.</p></li>
+<li><p><span class="math inline">\(\color{dodgerblue}{|\beta_1|&gt;1|}\)</span>: The series explodes, is nonstationary, and pretty much useless for prediction.</p></li>
+</ul>
+</section>
+<section id="arp-models" class="slide level2">
+<h2>AR(<span class="math inline">\(p\)</span>) models</h2>
+<p>It is possible to expand the AR idea to higher lags <span class="math display">\[
+\color{red}{AR(p): Y_t = \beta_0 + \beta_1Y_{t-1} + \cdots + \beta_pY_{t-p} + \varepsilon}.
+\]</span></p>
+<p>However, it is seldom necessary to fit AR lags for <span class="math inline">\(p&gt;1\)</span>.</p>
+<ul>
+<li><p>Like having polynomial terms higher than 2, this just isn’t usually required in practice.</p></li>
+<li><p>You lose all of the stationary/nonstationary intuition.</p></li>
+<li><p>Often, the need for higher lags is symptomatic of (missing) a more persistent trend or periodicity in the data, or needing predictors ...</p></li>
+</ul>
+</section>
+<section id="trending-series" class="slide level2">
+<h2>Trending series</h2>
+<p>Often, you’ll have a linear trend in your time series.</p>
+<p><span class="math inline">\(\Rightarrow\)</span> AR structure, sloping up or down in time.</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-22-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section class="slide level2">
+
+<p>This is easy to deal with: just put “time” in the model.</p>
+<p>AR with linear trend: <span class="math display">\[
+\color{red}{Y_t =  \beta_0 + \beta_1Y_{t-1} +
+\beta_2t + \varepsilon_t}
+\]</span></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a></a>t <span class="ot">&lt;-</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">199</span></span>
+<span id="cb17-2"><a></a>sst.fit <span class="ot">&lt;-</span> <span class="fu">lm</span>(sst[<span class="dv">2</span><span class="sc">:</span><span class="dv">200</span>] <span class="sc">~</span> sst[<span class="dv">1</span><span class="sc">:</span><span class="dv">199</span>] <span class="sc">+</span> t)</span>
+<span id="cb17-3"><a></a><span class="fu">summary</span>(sst.fit)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<p><br></p>
+<pre><code>Coefficients:
+             Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept) -0.571525   0.178110  -3.209  0.00156 ** 
+sst[1:199]   0.735840   0.048062  15.310  &lt; 2e-16 ***
+t           -0.009179   0.002160  -4.249 3.32e-05 ***</code></pre>
+</section>
+<section id="periodic-models" class="slide level2">
+<h2>Periodic models</h2>
+<p>It is very common to see seasonality or periodicity in series.</p>
+<ul>
+<li><p>Temperature goes up in Summer and down in Winter.</p></li>
+<li><p>Gas consumption in Blacksburg would do the opposite.</p></li>
+</ul>
+<p>Recall the monthly lung infection data:</p>
+<div class="columns">
+<div class="column" style="width:70%;">
+<div class="cell" data-layout-align="center">
+<div class="cell-output-display">
+<div class="quarto-figure quarto-figure-center">
+<figure>
+<p><img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-24-1.png" class="quarto-figure quarto-figure-center" width="768"></p>
+</figure>
+</div>
+</div>
+</div>
+</div><div class="column" style="width:30%;">
+<p><br></p>
+<ul>
+<li>Appears to oscillate on a 12-month cycle.</li>
+</ul>
+</div>
+</div>
+</section>
+<section class="slide level2">
+
+<p>The straightforward solution: Add periodic predictors.</p>
+<p><span class="math inline">\(\mathrm{Period}\!-\!k~\mbox{model}\!:\)</span> <span class="math display">\[
+\color{red}{Y_t = \beta_0 + \beta_1\sin(2\pi t /k) + \beta_2\cos(2\pi t /k)
++ \varepsilon_t}
+\]</span></p>
+<p>Remember your <span style="color: red;">sine</span> and <span style="color: dodgerblue;">cosine</span>! (Repeating every <span class="math inline">\(2\pi\)</span>)</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-25-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"></section>
+<section class="slide level2">
+
+<p><span class="math inline">\(\mathrm{Period}\!-\!k~\mbox{model}\!:\)</span> <span class="math display">\[
+\color{red}{Y_t = \beta_0 + \beta_1\sin(2\pi t /k) + \beta_2\cos(2\pi t /k)
++ \varepsilon_t}
+\]</span></p>
+<p>It turns out that you can represent any smooth periodic function as a sum of sines and cosines.</p>
+<p>You choose <span class="math inline">\(k\)</span> to be the number of “times” in a single period.</p>
+<ul>
+<li><p>For monthly data, <span class="math inline">\(k = 12\)</span> implies an annual cycle.</p></li>
+<li><p>For quarterly data, usually <span class="math inline">\(k=4\)</span>.</p></li>
+<li><p>For hourly data, <span class="math inline">\(k=24\)</span> gives you a daily cycle.</p></li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>Let’s fit an AR with sine/cosine predictors: <span class="math display">\[
+\color{red}{Y_t =  \beta_0 + \beta_1Y_{t-1} +
+\beta_2\sin(2\pi t /k) + \beta_3\cos(2\pi t /k)  + \varepsilon_t}
+\]</span></p>
+<p>We want to make new predictors/data , much like when we add polynomial terms, and then fit.</p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a></a>tmax<span class="ot">&lt;-</span><span class="fu">length</span>(ld)</span>
+<span id="cb19-2"><a></a>t <span class="ot">&lt;-</span> <span class="dv">2</span><span class="sc">:</span>tmax</span>
+<span id="cb19-3"><a></a>YX <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(<span class="at">ld=</span>ld[<span class="dv">2</span><span class="sc">:</span>tmax], <span class="at">ldpast=</span>ld[<span class="dv">1</span><span class="sc">:</span>(tmax<span class="dv">-1</span>)], <span class="at">t=</span>t, </span>
+<span id="cb19-4"><a></a>                 <span class="at">sin12=</span><span class="fu">sin</span>(<span class="dv">2</span><span class="sc">*</span>pi<span class="sc">*</span>t<span class="sc">/</span><span class="dv">12</span>), <span class="at">cos12=</span><span class="fu">cos</span>(<span class="dv">2</span><span class="sc">*</span>pi<span class="sc">*</span>t<span class="sc">/</span><span class="dv">12</span>))</span>
+<span id="cb19-5"><a></a>lunglm <span class="ot">&lt;-</span> <span class="fu">lm</span>(ld <span class="sc">~</span> ldpast <span class="sc">+</span> sin12 <span class="sc">+</span> cos12, <span class="at">data=</span>YX)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a></a><span class="fu">summary</span>(lunglm) <span class="do">## abbreviated output</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
+<pre><code>Coefficients:
+             Estimate Std. Error t value Pr(&gt;|t|)    
+(Intercept) 1379.3598   247.5536   5.572 4.84e-07 ***
+ldpast         0.3260     0.1189   2.741 0.007853 ** 
+sin12        377.5492   101.9206   3.704 0.000431 ***
+cos12        402.7845    44.8015   8.990 3.98e-13 ***</code></pre>
+</section>
+<section class="slide level2">
+
+<p>The model predictions look pretty good!</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-28-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>Sine and cosine trends seem to capture the periodicity.</li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>The residuals look pretty good.</p>
+
+<img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-29-1.png" class="quarto-figure quarto-figure-center r-stretch" width="960"><ul>
+<li>Maybe a bit of downward linear trend?</li>
+</ul>
+</section>
+<section id="alternative-periodicity" class="slide level2">
+<h2>Alternative Periodicity</h2>
+<p>An alternative way to add periodicity would be to simply add a dummy variable for each month (<code>feb, mar, apr, ...</code>).</p>
+<ul>
+<li><p>This achieves basically the same fit as above, without requiring you to add sine or cosine.</p></li>
+<li><p>However, this takes 11 periodic parameters while we use only 2.</p></li>
+</ul>
+</section>
+<section class="slide level2">
+
+<p>I like to think of the periodicity as a smooth oscillation, with sharp day/month effects added for special circumstances.</p>
+<ul>
+<li><p>Requires more thought, but leads to better models.</p></li>
+<li><p>The <span class="math inline">\(\sin+\cos\)</span> technique works regardless of the number of increments in a period (e.g.&nbsp;365 days).</p></li>
+</ul>
+<p><span style="color: red;">The exception:</span></p>
+<ul>
+<li>Since quarterly data has a period of only 4, it is often fine to just add “quarter” effects.</li>
+</ul>
+</section>
+<section id="non-time-series-data" class="slide level2">
+<h2>Non-time series data</h2>
+<center>
+<strong><em><span style="color: ForestGreen;">What happens if data aren’t evenly sampled?</span></em></strong>
+</center>
+<p><br></p>
+<p>All of the models/tools we explored that incorporate auto-correlation <span style="color: red;">are not valid if data are not evenly spaced</span>.</p>
+<ul>
+<li>You can’t calculate an auto-correlation if the gap between data points and the earlier points aren’t all the same because we don’t expect all lags to have the same correlation.</li>
+</ul>
+<center>
+<strong><span style="color: dodgerblue;">So what can we do?</span></strong>
+</center>
+</section>
+<section class="slide level2">
+
+<p>The simplest approach is to just go back to treating it like normal regression (without AR terms) – acknowledging that there could be residual auto-correlation that you can’t get rid of because you can’t quantify it properly.</p>
+<p><span style="color: ForestGreen;">So we might:</span></p>
+<ul>
+<li>include <code>time</code> as a predictor – not simple counting, but incorporating gaps. E.g., the iso-week of a sample.</li>
+<li>include in <code>sine</code> and <code>cosine</code> as before, but again, with the appropriate value corresponding to the time of observation instead of number of observation</li>
+<li><span style="color: dodgerblue;">add in time dependent predictors</span></li>
+</ul>
+</section>
+<section id="time-dependent-predictors" class="slide level2">
+<h2>Time Dependent Predictors</h2>
+<p>Often we have additional measurements of possible covariates that might impact the time-dependent responses that we want to model. E.g. in VBD systems:</p>
+<ul>
+<li><span style="color: red;">weather variables</span>: temperature, rainfall, humidity</li>
+<li><span style="color: dodgerblue;">habitat/climate variables</span>: greenness, ENSO, land use, container densities</li>
+<li><span style="color: ForestGreen;">socio-economic variables</span>: bed net coverage, insecticide spraying</li>
+</ul>
+<p>These may all depend on time, and can be incorporated into a model for all time dependent data (<span style="color: red;">including time series!</span>).</p>
+</section>
+<section id="time-lagged-predictors" class="slide level2">
+<h2>Time-Lagged Predictors</h2>
+<p>Additionally, sometimes there may be a lag between an observed covariate and the response.</p>
+<p><br></p>
+<p><span style="color: ForestGreen;">Example</span>: The number of people being hospitalized for dengue on a particular day reflect the number of people infected days before, and potentially mosquitoes infected days before that!</p>
+<p>Thus, proxies of mosquito abundance, like temperature or humidity, weeks earlier may be appropriate predictors.</p>
+<p><br></p>
+<p><strong><span style="color: dodgerblue;">How can we determine an appropriate lag for a predictor?</span></strong></p>
+</section>
+<section id="two-strategies" class="slide level2">
+<h2>Two Strategies</h2>
+<p>The first is what we might call a <span style="color: dodgerblue;">scientific approach</span>:</p>
+<ul>
+<li>Using our system knowledge, we can define what might be feasible time lags to include in a model, given evenly sampled predictor data. We decide and include just those <em>a priori</em> lags, and maybe do model/feature selection to narrow down.</li>
+</ul>
+<p>This approach may miss a <em>best</em> lag for time series data, but is often the main way we can try to find appropriate lags for unevenly sampled data.</p>
+<p>(Note, we almost always assume a lag of at least 1.)</p>
+</section>
+<section class="slide level2">
+
+<p>Alternatively, for time series data we might take a <span style="color: dodgerblue;">systematic data exploration approach</span>:</p>
+<ul>
+<li>For predictors under consideration, explore the <span style="color: red;">cross-correlation</span> – that is the (normalized) correlation between your response and multiple lags of your potential predictor.</li>
+</ul>
+<p>Then you can choose to use lags that are highly related to your response as possible model predictors.</p>
+</section>
+<section class="slide level2">
+
+<p><span style="color: ForestGreen;">Example</span>: maybe we think that the amount of precipitation on previous days might be a good predictor for temperature in Roanoke. We can check this using the <code>ccf</code> function in R:</p>
+<div class="columns">
+<div class="column" style="width:70%;">
+<div class="cell" data-layout-align="center">
+<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a></a><span class="fu">ccf</span>(weather<span class="sc">$</span>PRCP, weather<span class="sc">$</span>temp)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output-display">
+<div class="quarto-figure quarto-figure-center">
+<figure>
+<p><img data-src="VB_TimeDepData_files/figure-revealjs/unnamed-chunk-30-1.png" class="quarto-figure quarto-figure-center" width="672"></p>
+</figure>
+</div>
+</div>
+</div>
+</div><div class="column" style="width:30%;">
+<p><br></p>
+<p><span style="color: dodgerblue;">Not much going on here</span> –</p>
+<p>maybe a slight relationship with precipitation about 10 days back.</p>
+</div>
+</div>
+</section>
+<section id="coming-up" class="slide level2">
+<h2>Coming up!</h2>
+<p>The tools here are good, but not the best:</p>
+<ul>
+<li><p>In many situations you want to allow for <span class="math inline">\(\beta\)</span> or <span class="math inline">\(\sigma\)</span> parameters that can change in time.</p></li>
+<li><p>This can leave us with some left-over autocorrelation.</p></li>
+<li><p>We’ll talk more about more sophisticated models over the next couple of days.</p></li>
+</ul>
+</section>
+<section id="practice" class="slide level2">
+<h2>Practice</h2>
+<p>Now we’ll practice combining our regression tools with these additional techniques for time-dependent data.</p>
+<p><br></p>
+<p><strong><span style="color: red;">Remember:</span></strong></p>
+<ul>
+<li>Also ways check your residual plots to ensure that your assumptions have been met</li>
+<li>Transformations are your friend!</li>
+<li>Think carefully about how to line up your lagged predictors</li>
+</ul>
+
+<div class="quarto-auto-generated-content">
+<div class="footer footer-default">
+
+</div>
+</div>
+</section>
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diff --git a/docs/search.json b/docs/search.json
index b15bfa3..83df908 100644
--- a/docs/search.json
+++ b/docs/search.json
@@ -140,11 +140,249 @@
     "text": "Footnotes\n\n\nWhat is the difference between VectorBiTE and VectorByte? We are glad you asked! VectorBiTE was an RCN or a research coordination network funded by a 5 year grant from the BBSRC. VectorByte is hosting this training which is a newly funded NSF grant to establish a global open access data platform to study disease vectors. All the databases have transitioned to VectorByte but the legacy options will still be available on the VectorBiTE website.↩︎"
   },
   {
-    "objectID": "index.html",
-    "href": "index.html",
-    "title": "Welcome to the 2024 VectorByte Training Workshop",
+    "objectID": "VB_TimeDepData.html#learning-objectives",
+    "href": "VB_TimeDepData.html#learning-objectives",
+    "title": "VectorByte Methods Training",
+    "section": "Learning Objectives",
+    "text": "Learning Objectives"
+  },
+  {
+    "objectID": "VB_TimeDepData.html#time-dependent-data",
+    "href": "VB_TimeDepData.html#time-dependent-data",
+    "title": "VectorByte Methods Training",
+    "section": "Time Dependent Data",
+    "text": "Time Dependent Data\nMuch of the data we collect in VBD applications depends on time such as\n\nobserved cases in a city or country\nnumber of mosquitoes over time\n\nAdditionally, we often assume that the reasons these change over time may be because covariates (e.g., temperature, precipitation, insecticide spraying) change over time.\nHow do we incorporate these time varying factors into our regression models?"
+  },
+  {
+    "objectID": "VB_TimeDepData.html#types-of-time-dependent-data",
+    "href": "VB_TimeDepData.html#types-of-time-dependent-data",
+    "title": "VectorByte Methods Training",
+    "section": "Types of time dependent data",
+    "text": "Types of time dependent data\nThe most common type of time-dependent data that statisticians talk about is time series data. These are data where observations are evenly spaced with no (or very little) missing observations.\n\nAlthough evenly spaced data are ideal (and the most common methods are designed for them), in VBD survey data we often don’t have evenly spaced observations. These data don’t have a specific name, and most time-series methods can’t be directly used with them."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#time-series-data-and-dependence",
+    "href": "VB_TimeDepData.html#time-series-data-and-dependence",
+    "title": "VectorByte Methods Training",
+    "section": "Time series data and dependence",
+    "text": "Time series data and dependence\nTime-series data are simply a collection of observations gathered over time. For example, suppose \\(y_1, \\ldots, y_T\\) are\n\ndaily temperature,\nsolar activity,\nCO\\(_2\\) levels,\nyearly population size.\n\nIn each case, we might expect what happens at time \\(t\\) to be correlated with time \\(t-1\\)."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#checking-for-dependence",
+    "href": "VB_TimeDepData.html#checking-for-dependence",
+    "title": "VectorByte Methods Training",
+    "section": "Checking for dependence",
+    "text": "Checking for dependence\nTo see if \\(Y_{t-1}\\) would be useful for predicting \\(Y_t\\), just plot them together and see if there is a relationship.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nCorrelation between \\(Y_t\\) and \\(Y_{t-1}\\) is called autocorrelation."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#autocorrelation-for-time-series-data",
+    "href": "VB_TimeDepData.html#autocorrelation-for-time-series-data",
+    "title": "VectorByte Methods Training",
+    "section": "Autocorrelation (for time series data)",
+    "text": "Autocorrelation (for time series data)\nTo summarize the time-varying dependence, compute lag-\\(\\ell\\) correlations for \\(\\ell=1,2,3,\\ldots\\)\nIn general, the autocorrelation function (ACF) for \\(Y\\) is \\[\\color{red}{r(\\ell) = \\mathrm{cor}(Y_t, Y_{t-\\ell})}\\]\nFor our Roanoke temperature data:\n\nprint(acf(weather$temp))\n\n     0      1      2      3      4      5      6      7      8    \n 1.000  0.658  0.298  0.263  0.297  0.177  0.111  0.008 -0.099   \n    9     10    11     12     13     14     15     16     17 \n-0.045 0.071 -0.020 -0.157 -0.156 -0.146 -0.278 -0.346 -0.314"
+  },
+  {
+    "objectID": "VB_TimeDepData.html#autoregression",
+    "href": "VB_TimeDepData.html#autoregression",
+    "title": "VectorByte Methods Training",
+    "section": "Autoregression",
+    "text": "Autoregression\nHow do we model data that exhibits autocorrelation?\nSuppose \\(Y_1 = \\varepsilon_1\\), \\(Y_2 = \\varepsilon_{1} + \\varepsilon_{2}\\), \\(Y_3 = \\varepsilon_{1} + \\varepsilon_{2} + \\varepsilon_{3}\\), etc.\n\nThen \\(\\color{dodgerblue}{Y_t =  \\sum_{i=1}^{t}\\varepsilon_i = Y_{t-1} + \\varepsilon_t}\\) and \\(\\color{dodgerblue}{ \\mathbb{E}[Y_t] = Y_{t-1}}\\).\n\nThis is called a random walk model for \\(Y_t\\):\n\nthe expectation of what will happen is always what happened most recently."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#random-walk",
+    "href": "VB_TimeDepData.html#random-walk",
+    "title": "VectorByte Methods Training",
+    "section": "Random walk",
+    "text": "Random walk\nIn a random walk, the series just wanders around.\n\n\\(\\beta_1 = 1\\)"
+  },
+  {
+    "objectID": "VB_TimeDepData.html#exploding-series",
+    "href": "VB_TimeDepData.html#exploding-series",
+    "title": "VectorByte Methods Training",
+    "section": "Exploding series",
+    "text": "Exploding series\nFor AR term \\(&gt;1\\), the \\(Y_t\\)’s move exponentially far from \\(Y_1\\).\n\n\\(\\beta_1 = 1.02\\)\n\n\n\nUseless for modeling and prediction."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#stationary-series",
+    "href": "VB_TimeDepData.html#stationary-series",
+    "title": "VectorByte Methods Training",
+    "section": "Stationary series",
+    "text": "Stationary series\nFor \\(\\beta_1&lt;1\\), \\(Y_t\\) is always pulled back towards the mean.\n\n\\(\\beta_1 = 0.8\\)\n\n\n\nThese are the most common and useful type of AR series."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#mean-reversion",
+    "href": "VB_TimeDepData.html#mean-reversion",
+    "title": "VectorByte Methods Training",
+    "section": "Mean reversion",
+    "text": "Mean reversion\nAn important property of stationary series is mean reversion.\nThink about shifting both \\(Y_t\\) and \\(Y_{t-1}\\) by their mean \\(\\mu\\). \\[\n\\color{dodgerblue}{Y_t - \\mu = \\beta_1 (Y_{t-1} - \\mu) +\\varepsilon_t}\n\\] Since \\(|\\beta_1| &lt; 1\\), \\(Y_t\\) is expected to be closer to \\(\\mu\\) than \\(Y_{t-1}\\).\nMean reversion is all over, and helps predict future behavior:\n\nweekly sales numbers,\ndaily temperature."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#negative-correlation",
+    "href": "VB_TimeDepData.html#negative-correlation",
+    "title": "VectorByte Methods Training",
+    "section": "Negative correlation",
+    "text": "Negative correlation\nIt is also possible to have negatively correlated AR(1) series.\n\n\\(\\beta_1 = -0.8\\)\n\n\n\nBut you see these far less often in practice."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#summary-of-ar1-behavior",
+    "href": "VB_TimeDepData.html#summary-of-ar1-behavior",
+    "title": "VectorByte Methods Training",
+    "section": "Summary of AR(1) behavior",
+    "text": "Summary of AR(1) behavior\n\n\\(\\color{dodgerblue}{|\\beta_1|&lt;1|}\\): The series has a mean level to which it reverts over time (stationary). For \\(+\\beta_1\\), the series tends to wander above or below the mean level for a while. For \\(-\\beta_1\\), the series tends to flip back and forth around the mean.\n\\(\\color{dodgerblue}{|\\beta_1|=1|}\\): A random walk series. The series has no mean level and, thus, is called nonstationary. The drift parameter \\(\\beta_0\\) is the direction in which the series wanders.\n\\(\\color{dodgerblue}{|\\beta_1|&gt;1|}\\): The series explodes, is nonstationary, and pretty much useless for prediction."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#arp-models",
+    "href": "VB_TimeDepData.html#arp-models",
+    "title": "VectorByte Methods Training",
+    "section": "AR(\\(p\\)) models",
+    "text": "AR(\\(p\\)) models\nIt is possible to expand the AR idea to higher lags \\[\n\\color{red}{AR(p): Y_t = \\beta_0 + \\beta_1Y_{t-1} + \\cdots + \\beta_pY_{t-p} + \\varepsilon}.\n\\]\nHowever, it is seldom necessary to fit AR lags for \\(p&gt;1\\).\n\nLike having polynomial terms higher than 2, this just isn’t usually required in practice.\nYou lose all of the stationary/nonstationary intuition.\nOften, the need for higher lags is symptomatic of (missing) a more persistent trend or periodicity in the data, or needing predictors ..."
+  },
+  {
+    "objectID": "VB_TimeDepData.html#trending-series",
+    "href": "VB_TimeDepData.html#trending-series",
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+    "section": "Trending series",
+    "text": "Trending series\nOften, you’ll have a linear trend in your time series.\n\\(\\Rightarrow\\) AR structure, sloping up or down in time."
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+    "text": "Periodic models\nIt is very common to see seasonality or periodicity in series.\n\nTemperature goes up in Summer and down in Winter.\nGas consumption in Blacksburg would do the opposite.\n\nRecall the monthly lung infection data:\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAppears to oscillate on a 12-month cycle."
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+    "section": "Alternative Periodicity",
+    "text": "Alternative Periodicity\nAn alternative way to add periodicity would be to simply add a dummy variable for each month (feb, mar, apr, ...).\n\nThis achieves basically the same fit as above, without requiring you to add sine or cosine.\nHowever, this takes 11 periodic parameters while we use only 2."
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+    "section": "Non-time series data",
+    "text": "Non-time series data\n\nWhat happens if data aren’t evenly sampled?\n\n\nAll of the models/tools we explored that incorporate auto-correlation are not valid if data are not evenly spaced.\n\nYou can’t calculate an auto-correlation if the gap between data points and the earlier points aren’t all the same because we don’t expect all lags to have the same correlation.\n\n\nSo what can we do?"
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+    "section": "Time Dependent Predictors",
+    "text": "Time Dependent Predictors\nOften we have additional measurements of possible covariates that might impact the time-dependent responses that we want to model. E.g. in VBD systems:\n\nweather variables: temperature, rainfall, humidity\nhabitat/climate variables: greenness, ENSO, land use, container densities\nsocio-economic variables: bed net coverage, insecticide spraying\n\nThese may all depend on time, and can be incorporated into a model for all time dependent data (including time series!)."
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+    "href": "VB_TimeDepData.html#time-lagged-predictors",
+    "title": "VectorByte Methods Training",
+    "section": "Time-Lagged Predictors",
+    "text": "Time-Lagged Predictors\nAdditionally, sometimes there may be a lag between an observed covariate and the response.\n\nExample: The number of people being hospitalized for dengue on a particular day reflect the number of people infected days before, and potentially mosquitoes infected days before that!\nThus, proxies of mosquito abundance, like temperature or humidity, weeks earlier may be appropriate predictors.\n\nHow can we determine an appropriate lag for a predictor?"
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+    "text": "Two Strategies\nThe first is what we might call a scientific approach:\n\nUsing our system knowledge, we can define what might be feasible time lags to include in a model, given evenly sampled predictor data. We decide and include just those a priori lags, and maybe do model/feature selection to narrow down.\n\nThis approach may miss a best lag for time series data, but is often the main way we can try to find appropriate lags for unevenly sampled data.\n(Note, we almost always assume a lag of at least 1.)"
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+    "section": "Coming up!",
+    "text": "Coming up!\nThe tools here are good, but not the best:\n\nIn many situations you want to allow for \\(\\beta\\) or \\(\\sigma\\) parameters that can change in time.\nThis can leave us with some left-over autocorrelation.\nWe’ll talk more about more sophisticated models over the next couple of days."
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+    "title": "VectorByte Methods Training",
+    "section": "Practice",
+    "text": "Practice\nNow we’ll practice combining our regression tools with these additional techniques for time-dependent data.\n\nRemember:\n\nAlso ways check your residual plots to ensure that your assumptions have been met\nTransformations are your friend!\nThink carefully about how to line up your lagged predictors"
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+    "title": "VectorByte Methods Training",
+    "section": "1. Fit the linear regression model. Plot the data and fitted line.",
+    "text": "1. Fit the linear regression model. Plot the data and fitted line.\n\n## fit models\nlm1 &lt;- lm(Y1 ~ X1)\n\n## plot points and fitted lines\nplot(X1, Y1, col=1, main=\"I\"); abline(lm1, col=2)"
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+    "title": "VectorByte Methods Training",
+    "section": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.",
+    "text": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.\n\npar(mfrow=c(1,3), mar=c(4,4,2,0.5))   \n\n## studentized residuals vs fitted\nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", \n     ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\n\n## qq plot of studentized residuals\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2, col=2)\n\n## histogram of studentized residuals\nhist(rstudent(lm1), col=1, \n     xlab=\"Studentized Residuals\", \n     main=\"\", border=8)"
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+    "title": "VectorByte Methods Training",
+    "section": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.",
+    "text": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.\nXs are clumpy AND the variance seems non-constant. It looks a lot like the GDP data from class. Since both Xs and Ys are strictly positive, we can try a log-log transform."
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+    "href": "VB_RegDiagTrans_practical.html#determine-the-data-transformation-to-correct-the-problems-in-3-fit-the-corresponding-regression-model-and-plot-the-transformed-data-with-new-fitted-line.",
+    "title": "VectorByte Methods Training",
+    "section": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.",
+    "text": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.\n\n### the fix is as follows:\nlogX1&lt;- log(X1)\nlogY1 &lt;- log(Y1)\n\n### re-run the regressions and residual plots to show this worked\nlm1 &lt;- lm(logY1 ~ logX1)\n\n## plot points and lines\nplot(logX1, logY1, col=1, main=\"I\"); abline(lm1, col=2)"
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+    "title": "VectorByte Methods Training",
+    "section": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.",
+    "text": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.\n\n## studentized residuals vs fitted\n\npar(mfrow=c(1,3), mar=c(4,4,2,0.5))  \nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", \n     ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\n\n## Q-Q plots\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2, col=2)\n\n## histograms of studentized residuals\nhist(rstudent(lm1), col=1, \n     xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\n\n\n\n\n\n\n\n\nThis is much better! The histogram still maybe looks a little funny, but given that the qq-plot looks pretty good, I think we’ve made a good transformation."
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+    "section": "Your Turn!",
+    "text": "Your Turn!\nRepeat this process with the other 3 datasets, and see if you can figure out a appropriate transformations for each dataset."
+  },
+  {
+    "objectID": "schedule2024.html",
+    "href": "schedule2024.html",
+    "title": "2024 Training Schedule",
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-    "text": "Check out our about and schedule: coming soon! pages to continue.\nInformation about pre-workshop preparation – including software installation, expectations for what you should already be familiar with, and review materials – is available in the pre-work portion of the materials page."
+    "text": "Main materials\n\nPre-workshop\nInformation about pre-workshop preparation – including software installation, expectations for what you should already be familiar with, and review materials – is available in the pre-work portion of the materials page.\n  \n\n\n22 July 2024\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nArrival\n\n\n\n\n  \n\n\n23 July 2024 (08:30 - 17:00)\n\n\n\n\n\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n12:00\nLunch\n\n\n\n\n  \n\n\n24 July 2024 (08:30 - 17:00)\n\n\n\n\n\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n\n\n\n\n\n12:00\nLunch\n\n\n\n\n\n\n\n\n\n\n\n25 July 2024 (08:30 - 17:00)\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n\n\n\n26 July 2024\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nTravel\n\n\n\n\n\n\nPost-workshop\nEnjoy using these new techniques and databases!"
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+    "href": "VB_RegDiagTrans_practical_soln.html#fit-the-linear-regression-model.-plot-the-data-and-fitted-line.",
+    "title": "VectorByte Methods Training",
+    "section": "1. Fit the linear regression model. Plot the data and fitted line.",
+    "text": "1. Fit the linear regression model. Plot the data and fitted line.\n\n## fit models\nattach(D &lt;- read.csv(\"data/transforms.csv\"))\nlm1 &lt;- lm(Y1 ~ X1)\nlm2 &lt;- lm(Y2 ~ X2)\nlm3 &lt;- lm(Y3 ~ X3)\nlm4 &lt;- lm(Y4 ~ X4)\n\n## plot points and lines\npar(mfrow=c(2,2), mar=c(3,2,2,1))\nplot(X1, Y1, col=1, main=\"I\"); abline(lm1, col=1)\nplot(X2, Y2, col=2, main=\"II\"); abline(lm2, col=2)\nplot(X3, Y3, col=3, main=\"III\"); abline(lm3, col=3)\nplot(X4, Y4, col=4, main=\"IV\"); abline(lm4, col=4)"
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+    "title": "VectorByte Methods Training",
+    "section": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.",
+    "text": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.\n\npar(mfrow=c(3,4), mar=c(4,4,2,0.5))   # you might have to make \n                                      # the plot window big to \n                                      # fit everything\nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\nplot(lm2$fitted, rstudent(lm2), col=2,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"II\")\nplot(lm3$fitted, rstudent(lm3), col=3,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"III\")\nplot(lm4$fitted, rstudent(lm4), col=4,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"IV\")\n\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm2), pch=20, col=2, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm3), pch=20, col=3, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm4), pch=20, col=4, main=\"\" )\nabline(a=0,b=1,lty=2)\n\nhist(rstudent(lm1), col=1, xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\nhist(rstudent(lm2), col=2, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm3), col=3, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm4), col=4, xlab=\"Studentized Residuals\", main=\"\")"
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+    "title": "VectorByte Methods Training",
+    "section": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.",
+    "text": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.\nSet 1: Xs are clumpy AND the variance seems non-constant. It looks a lot like the GDP data from class. Since both Xs and Ys are strictly positive, we can try a log-log transform.\nSet 2: Data have non-constant variance – should probably log transform the Ys\nSet 3: Data have an underlying non-linear pattern. Add in an x^2 and x^3 term in this case.\nSet 4: X values are very clumpy and all positive. Try log transform of the Xs"
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+    "href": "VB_RegDiagTrans_practical_soln.html#determine-the-data-transformation-to-correct-the-problems-in-3-fit-the-corresponding-regression-model-and-plot-the-transformed-data-with-new-fitted-line.",
+    "title": "VectorByte Methods Training",
+    "section": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.",
+    "text": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.\n\n### the fixes are as follows:\nlogX1&lt;- log(X1)\nlogY1 &lt;- log(Y1)\nlogY2 &lt;- log(Y2)\nX3sq &lt;- X3^2\nX3cube&lt;-X3^3\nlogX4 &lt;- log(X4)\n\n\n### re-run the regressions and residual plots to show this worked\nlm1 &lt;- lm(logY1 ~ logX1)\nlm2 &lt;- lm(logY2 ~ X2)\nlm3 &lt;- lm(Y3 ~ X3+ X3sq + X3cube)\nlm4 &lt;- lm(Y4 ~ logX4)\n\n## plot points and lines\npar(mfrow=c(2,2), mar=c(3,2,2,1))\nplot(logX1, logY1, col=1, main=\"I\"); abline(lm1, col=1)\nplot(X2, logY2, col=2, main=\"II\"); abline(lm2, col=2)\nplot(X3, Y3, col=3, main=\"III\")\nxx3 &lt;- seq(min(X3), max(X3), length=1000)\nlines(xx3, lm3$coef[1] + lm3$coef[2]*xx3 + \n        lm3$coef[3]*xx3^2+lm3$coef[3]*xx3^3, col=3)\nplot(logX4, Y4, col=4, main=\"IV\"); abline(lm4, col=4)"
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+    "title": "VectorByte Methods Training",
+    "section": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.",
+    "text": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.\n\npar(mfrow=c(3,4), mar=c(4,4,2,0.5))  \nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\nplot(lm2$fitted, rstudent(lm2), col=2,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"II\")\nplot(lm3$fitted, rstudent(lm3), col=3,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"III\")\nplot(lm4$fitted, rstudent(lm4), col=4,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"IV\")\n\n## Q-Q plots\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm2), pch=20, col=2, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm3), pch=20, col=3, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm4), pch=20, col=4, main=\"\" )\nabline(a=0,b=1,lty=2)\n\n## histograms of studentized residuals\nhist(rstudent(lm1), col=1, xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\nhist(rstudent(lm2), col=2, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm3), col=3, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm4), col=4, xlab=\"Studentized Residuals\", main=\"\")"
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     "section": "Example model violations",
-    "text": "Example model violations\nAnscombe’s quartet comprises four datasets that have similar statistical properties ...\n\n\n\n\nXmean\nYmean\nXsd\nYsd\nXYcor\n\n\n\n\n1\n9.000\n7.501\n3.317\n2.032\n0.816\n\n\n2\n9.000\n7.501\n3.317\n2.032\n0.816\n\n\n3\n9.000\n7.500\n3.317\n2.030\n0.816\n\n\n4\n9.000\n7.501\n3.317\n2.031\n0.817"
+    "text": "Example model violations\nAnscombe’s quartet comprises four datasets that have similar statistical properties …\n\n\n\n\nXmean\nYmean\nXsd\nYsd\nXYcor\n\n\n\n\n1\n9.000\n7.501\n3.317\n2.032\n0.816\n\n\n2\n9.000\n7.501\n3.317\n2.032\n0.816\n\n\n3\n9.000\n7.500\n3.317\n2.030\n0.816\n\n\n4\n9.000\n7.501\n3.317\n2.031\n0.817"
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     "href": "VB_RegDiagTrans.html#outliers-leverage-and-residuals",
     "title": "VectorByte Methods Training",
     "section": "Outliers, leverage, and residuals",
-    "text": "Outliers, leverage, and residuals\nWarning: Unfortunately, outliers with high leverage are hard to catch through \\(\\color{dodgerblue}{r_i}\\) (since the line is pulled towards them).\nMeans get distracted by outliers..."
+    "text": "Outliers, leverage, and residuals\nWarning: Unfortunately, outliers with high leverage are hard to catch through \\(\\color{dodgerblue}{r_i}\\) (since the line is pulled towards them).\nMeans get distracted by outliers…"
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     "href": "VB_RegDiagTrans.html#testing-for-nonlinearity",
     "title": "VectorByte Methods Training",
     "section": "Testing for nonlinearity",
-    "text": "Testing for nonlinearity\nTo see if you need more nonlinearity, try the regression which includes the next polynomial term, and see if it is significant.\nFor example, to see if you need a quadratic term,\n\nfit the model then run the regression \\(\\mathbb{E}[Y] = \\beta_0 + \\beta_1 X +  \\beta_2 X^2\\).\nIf your test implies \\(\\color{dodgerblue}{\\beta_2 \\neq 0}\\), you need \\(\\color{dodgerblue}{X^2}\\) in your model.\n\nNote: \\(p\\)-values are calculated “given the other \\(\\beta\\)’s are nonzero”; i.e., conditional on \\(X\\) being in the model."
+    "text": "Testing for nonlinearity\nTo see if you need more nonlinearity, try the regression which includes the next polynomial term, and see if it is significant.\nFor example, to see if you need a quadratic term,\n\nfit the model then run the regression \\(\\mathbb{E}[Y] = \\beta_0 + \\beta_1 X + \\beta_2 X^2\\).\nIf your test implies \\(\\color{dodgerblue}{\\beta_2 \\neq 0}\\), you need \\(\\color{dodgerblue}{X^2}\\) in your model.\n\nNote: \\(p\\)-values are calculated “given the other \\(\\beta\\)’s are nonzero”; i.e., conditional on \\(X\\) being in the model."
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     "href": "VB_RegDiagTrans.html#the-log-log-model",
     "title": "VectorByte Methods Training",
     "section": "The log-log model",
-    "text": "The log-log model\nThe other common covariate transform is \\(\\log(X)\\).\n\nWhen \\(X\\)-values are bunched up, \\(\\log(X)\\) helps spread them out and reduces the leverage of extreme values.\nRecall that both reduce \\(s_{b_1}\\).\n\nIn practice, this is often used in conjunction with a \\(\\log(Y)\\) response transformation. The log-log model is \\[\n    \\color{red}{\\log(Y) = \\beta_0 + \\beta_1 \\log(X) + \\varepsilon}.\n    \\]\nIt is super useful, and has some special properties ..."
+    "text": "The log-log model\nThe other common covariate transform is \\(\\log(X)\\).\n\nWhen \\(X\\)-values are bunched up, \\(\\log(X)\\) helps spread them out and reduces the leverage of extreme values.\nRecall that both reduce \\(s_{b_1}\\).\n\nIn practice, this is often used in conjunction with a \\(\\log(Y)\\) response transformation. The log-log model is \\[\n    \\color{red}{\\log(Y) = \\beta_0 + \\beta_1 \\log(X) + \\varepsilon}.\n    \\]\nIt is super useful, and has some special properties …"
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     "text": "Practical\nNext we’ll do a short practical to practice:\n\nFitting linear models in R\nChecking diagnostics\nChoosing transformations\nPlotting predictions"
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-    "title": "VectorByte Methods Training",
-    "section": "1. Fit the linear regression model. Plot the data and fitted line.",
-    "text": "1. Fit the linear regression model. Plot the data and fitted line.\n\n## fit models\nattach(D &lt;- read.csv(\"data/transforms.csv\"))\nlm1 &lt;- lm(Y1 ~ X1)\nlm2 &lt;- lm(Y2 ~ X2)\nlm3 &lt;- lm(Y3 ~ X3)\nlm4 &lt;- lm(Y4 ~ X4)\n\n## plot points and lines\npar(mfrow=c(2,2), mar=c(3,2,2,1))\nplot(X1, Y1, col=1, main=\"I\"); abline(lm1, col=1)\nplot(X2, Y2, col=2, main=\"II\"); abline(lm2, col=2)\nplot(X3, Y3, col=3, main=\"III\"); abline(lm3, col=3)\nplot(X4, Y4, col=4, main=\"IV\"); abline(lm4, col=4)"
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-    "title": "VectorByte Methods Training",
-    "section": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.",
-    "text": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.\n\npar(mfrow=c(3,4), mar=c(4,4,2,0.5))   # you might have to make \n                                      # the plot window big to \n                                      # fit everything\nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\nplot(lm2$fitted, rstudent(lm2), col=2,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"II\")\nplot(lm3$fitted, rstudent(lm3), col=3,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"III\")\nplot(lm4$fitted, rstudent(lm4), col=4,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"IV\")\n\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm2), pch=20, col=2, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm3), pch=20, col=3, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm4), pch=20, col=4, main=\"\" )\nabline(a=0,b=1,lty=2)\n\nhist(rstudent(lm1), col=1, xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\nhist(rstudent(lm2), col=2, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm3), col=3, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm4), col=4, xlab=\"Studentized Residuals\", main=\"\")"
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-    "title": "VectorByte Methods Training",
-    "section": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.",
-    "text": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.\nSet 1: Xs are clumpy AND the variance seems non-constant. It looks a lot like the GDP data from class. Since both Xs and Ys are strictly positive, we can try a log-log transform.\nSet 2: Data have non-constant variance – should probably log transform the Ys\nSet 3: Data have an underlying non-linear pattern. Add in an x^2 and x^3 term in this case.\nSet 4: X values are very clumpy and all positive. Try log transform of the Xs"
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-    "title": "VectorByte Methods Training",
-    "section": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.",
-    "text": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.\n\n### the fixes are as follows:\nlogX1&lt;- log(X1)\nlogY1 &lt;- log(Y1)\nlogY2 &lt;- log(Y2)\nX3sq &lt;- X3^2\nX3cube&lt;-X3^3\nlogX4 &lt;- log(X4)\n\n\n### re-run the regressions and residual plots to show this worked\nlm1 &lt;- lm(logY1 ~ logX1)\nlm2 &lt;- lm(logY2 ~ X2)\nlm3 &lt;- lm(Y3 ~ X3+ X3sq + X3cube)\nlm4 &lt;- lm(Y4 ~ logX4)\n\n## plot points and lines\npar(mfrow=c(2,2), mar=c(3,2,2,1))\nplot(logX1, logY1, col=1, main=\"I\"); abline(lm1, col=1)\nplot(X2, logY2, col=2, main=\"II\"); abline(lm2, col=2)\nplot(X3, Y3, col=3, main=\"III\")\nxx3 &lt;- seq(min(X3), max(X3), length=1000)\nlines(xx3, lm3$coef[1] + lm3$coef[2]*xx3 + \n        lm3$coef[3]*xx3^2+lm3$coef[3]*xx3^3, col=3)\nplot(logX4, Y4, col=4, main=\"IV\"); abline(lm4, col=4)"
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-    "title": "VectorByte Methods Training",
-    "section": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.",
-    "text": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.\n\npar(mfrow=c(3,4), mar=c(4,4,2,0.5))  \nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\nplot(lm2$fitted, rstudent(lm2), col=2,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"II\")\nplot(lm3$fitted, rstudent(lm3), col=3,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"III\")\nplot(lm4$fitted, rstudent(lm4), col=4,\n     xlab=\"Fitted Values\", ylab=\"Studentized Residuals\", \n     pch=20, main=\"IV\")\n\n## Q-Q plots\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm2), pch=20, col=2, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm3), pch=20, col=3, main=\"\" )\nabline(a=0,b=1,lty=2)\nqqnorm(rstudent(lm4), pch=20, col=4, main=\"\" )\nabline(a=0,b=1,lty=2)\n\n## histograms of studentized residuals\nhist(rstudent(lm1), col=1, xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\nhist(rstudent(lm2), col=2, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm3), col=3, xlab=\"Studentized Residuals\", main=\"\")\nhist(rstudent(lm4), col=4, xlab=\"Studentized Residuals\", main=\"\")"
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+    "title": "Welcome to the 2024 VectorByte Training Workshop",
     "section": "",
-    "text": "Main materials\n\nPre-workshop\nInformation about pre-workshop preparation – including software installation, expectations for what you should already be familiar with, and review materials – is available in the pre-work portion of the materials page.\n  \n\n\n22 July 2024\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nArrival\n\n\n\n\n  \n\n\n23 July 2024 (08:30 - 17:00)\n\n\n\n\n\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n12:00\nLunch\n\n\n\n\n  \n\n\n24 July 2024 (08:30 - 17:00)\n\n\n\n\n\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n\n\n\n\n\n12:00\nLunch\n\n\n\n\n\n\n\n\n\n\n\n25 July 2024 (08:30 - 17:00)\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nComing Soon!\n\n\n\n\n\n\n26 July 2024\n\n\n\nTime\nActivity\nMaterials\n\n\n\n\n\nTravel\n\n\n\n\n\n\nPost-workshop\nEnjoy using these new techniques and databases!"
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-    "title": "VectorByte Methods Training",
-    "section": "1. Fit the linear regression model. Plot the data and fitted line.",
-    "text": "1. Fit the linear regression model. Plot the data and fitted line.\n\n## fit models\nlm1 &lt;- lm(Y1 ~ X1)\n\n## plot points and fitted lines\nplot(X1, Y1, col=1, main=\"I\"); abline(lm1, col=2)"
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-    "title": "VectorByte Methods Training",
-    "section": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.",
-    "text": "2. Provide a scatterplot, normal Q-Q plot, and histogram for the studentized regression residuals.\n\npar(mfrow=c(1,3), mar=c(4,4,2,0.5))   \n\n## studentized residuals vs fitted\nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", \n     ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\n\n## qq plot of studentized residuals\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2, col=2)\n\n## histogram of studentized residuals\nhist(rstudent(lm1), col=1, \n     xlab=\"Studentized Residuals\", \n     main=\"\", border=8)"
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-    "title": "VectorByte Methods Training",
-    "section": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.",
-    "text": "3. Using the residual scatterplots, state how the SLR model assumptions are violated.\nXs are clumpy AND the variance seems non-constant. It looks a lot like the GDP data from class. Since both Xs and Ys are strictly positive, we can try a log-log transform."
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-    "href": "VB_RegDiagTrans_practical.html#determine-the-data-transformation-to-correct-the-problems-in-3-fit-the-corresponding-regression-model-and-plot-the-transformed-data-with-new-fitted-line.",
-    "title": "VectorByte Methods Training",
-    "section": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.",
-    "text": "4. Determine the data transformation to correct the problems in 3, fit the corresponding regression model, and plot the transformed data with new fitted line.\n\n### the fix is as follows:\nlogX1&lt;- log(X1)\nlogY1 &lt;- log(Y1)\n\n### re-run the regressions and residual plots to show this worked\nlm1 &lt;- lm(logY1 ~ logX1)\n\n## plot points and lines\nplot(logX1, logY1, col=1, main=\"I\"); abline(lm1, col=2)"
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-    "title": "VectorByte Methods Training",
-    "section": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.",
-    "text": "5. Provide plots to show that your transformations have (mostly) fixed the model violations.\n\n## studentized residuals vs fitted\n\npar(mfrow=c(1,3), mar=c(4,4,2,0.5))  \nplot(lm1$fitted, rstudent(lm1), col=1,\n     xlab=\"Fitted Values\", \n     ylab=\"Studentized Residuals\", \n     pch=20, main=\"I\")\n\n## Q-Q plots\nqqnorm(rstudent(lm1), pch=20, col=1, main=\"\" )\nabline(a=0,b=1,lty=2, col=2)\n\n## histograms of studentized residuals\nhist(rstudent(lm1), col=1, \n     xlab=\"Studentized Residuals\", \n     main=\"\", border=8)\n\n\n\n\n\n\n\n\nThis is much better! The histogram still maybe looks a little funny, but given that the qq-plot looks pretty good, I think we’ve made a good transformation."
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-    "section": "Your Turn!",
-    "text": "Your Turn!\nRepeat this process with the other 3 datasets, and see if you can figure out a appropriate transformations for each dataset."
+    "text": "Check out our about and schedule: coming soon! pages to continue.\nInformation about pre-workshop preparation – including software installation, expectations for what you should already be familiar with, and review materials – is available in the pre-work portion of the materials page."
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