diff --git a/bayesian-process-models.html b/bayesian-process-models.html
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+++ b/bayesian-process-models.html
@@ -560,7 +560,7 @@ <h2 data-number="19.1" class="anchored" data-anchor-id="running-mcmc-on-the-fore
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 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>accept <span class="sc">/</span> (num_iter <span class="sc">*</span> num_pars)</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] 0.3535556</code></pre>
+<pre><code>[1] 0.3555556</code></pre>
 </div>
 </div>
 <p>Process MCMC chain by removing the first 500 iterations and pivoting to a long format</p>
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diff --git a/data/terrestrial_daily-2025-02-21-bookcast_forest.csv b/data/terrestrial_daily-2025-02-21-bookcast_forest.csv
new file mode 100644
index 00000000..6ab27c7d
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diff --git a/evaluating-forecasts.html b/evaluating-forecasts.html
index ff50c457..0e177ee5 100644
--- a/evaluating-forecasts.html
+++ b/evaluating-forecasts.html
@@ -483,7 +483,7 @@ <h3 data-number="11.3.2" class="anchored" data-anchor-id="example-of-a-crps-calc
 <div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(scoringRules)</span>
 <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="fu">crps_sample</span>(<span class="at">y =</span> y, <span class="at">dat =</span> x)</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] 0.2384334</code></pre>
+<pre><code>[1] 0.237732</code></pre>
 </div>
 </div>
 </section>
diff --git a/evaluating-forecasts_files/figure-html/unnamed-chunk-9-1.png b/evaluating-forecasts_files/figure-html/unnamed-chunk-9-1.png
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index 303b675e..74934368 100644
--- a/index.html
+++ b/index.html
@@ -7,7 +7,7 @@
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
 <meta name="author" content="R. Quinn Thomas">
-<meta name="dcterms.date" content="2025-02-19">
+<meta name="dcterms.date" content="2025-02-21">
 
 <title>A Practical Guide to Ecosystem Forecasting</title>
 <style>
@@ -405,7 +405,7 @@ <h1 class="title">A Practical Guide to Ecosystem Forecasting</h1>
     <div>
     <div class="quarto-title-meta-heading">Published</div>
     <div class="quarto-title-meta-contents">
-      <p class="date">February 19, 2025</p>
+      <p class="date">February 21, 2025</p>
     </div>
   </div>
   
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index 18a94ba0..7b718697 100644
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+++ b/process-model-forecast-evaluation.html
@@ -436,7 +436,7 @@ <h2 data-number="22.2" class="anchored" data-anchor-id="aggregated-scores"><span
 <pre><code># A tibble: 1 × 1
   mean_crps
       &lt;dbl&gt;
-1     0.872</code></pre>
+1     0.870</code></pre>
 </div>
 </div>
 </section>
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index a4e3eefc..47029ebc 100644
--- a/process-model-forecast.html
+++ b/process-model-forecast.html
@@ -576,7 +576,7 @@ <h2 data-number="21.4" class="anchored" data-anchor-id="step-4-save-forecast-and
 <pre><code>validating that file matches required standard</code></pre>
 </div>
 <div class="cell-output cell-output-stderr">
-<pre><code>data/terrestrial_daily-2025-02-19-bookcast_forest.csv</code></pre>
+<pre><code>data/terrestrial_daily-2025-02-21-bookcast_forest.csv</code></pre>
 </div>
 <div class="cell-output cell-output-stderr">
 <pre><code>✔ file has model_id column
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diff --git a/project1.html b/project1.html
index 13486871..ede80841 100644
--- a/project1.html
+++ b/project1.html
@@ -641,7 +641,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>split</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>&lt;Training/Testing/Total&gt;
-&lt;4159/1040/5199&gt;</code></pre>
+&lt;4189/1048/5237&gt;</code></pre>
 </div>
 </div>
 <p>To get the training and testing data we need to apply the <code>training()</code> and <code>testing()</code> functions to the split.</p>
@@ -653,7 +653,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <div class="cell">
 <div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>train_data</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># A tibble: 4,159 × 5
+<pre><code># A tibble: 4,189 × 5
    datetime   site_id temperature air_temperature   doy
    &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt;
  1 2017-08-27 BARC           31.5              NA   239
@@ -666,7 +666,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
  8 2017-09-03 BARC           31.1              NA   246
  9 2017-09-06 BARC           30.8              NA   249
 10 2017-09-07 BARC           30.5              NA   250
-# ℹ 4,149 more rows</code></pre>
+# ℹ 4,179 more rows</code></pre>
 </div>
 </div>
 <p><em>Feature engineering using a recipe</em></p>
@@ -796,7 +796,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 
 Coefficients:
     (Intercept)      site_idSUGG  air_temperature              doy  
-       5.142973        -0.785496         0.880690         0.002305  </code></pre>
+       5.108053        -0.725307         0.875235         0.002883  </code></pre>
 </div>
 </div>
 <p><em>Step 5: Predict Test Data</em></p>
@@ -808,7 +808,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <div class="cell">
 <div class="sourceCode cell-code" id="cb41"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a>predictions</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># A tibble: 1,040 × 1
+<pre><code># A tibble: 1,048 × 1
    .pred
    &lt;dbl&gt;
  1    NA
@@ -821,7 +821,7 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
  8    NA
  9    NA
 10    NA
-# ℹ 1,030 more rows</code></pre>
+# ℹ 1,038 more rows</code></pre>
 </div>
 </div>
 <p>We need to combine the <code>.pred</code> column with the testing data using the <code>bind_cols</code> function</p>
@@ -832,20 +832,20 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <div class="cell">
 <div class="sourceCode cell-code" id="cb44"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a>pred_test</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># A tibble: 1,040 × 6
+<pre><code># A tibble: 1,048 × 6
    datetime   site_id temperature air_temperature   doy .pred
    &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;
  1 2017-09-04 BARC           31.4              NA   247    NA
  2 2017-09-05 BARC           31.0              NA   248    NA
  3 2017-09-14 BARC           28.5              NA   257    NA
- 4 2017-09-19 BARC           28.8              NA   262    NA
- 5 2017-09-22 BARC           29.0              NA   265    NA
- 6 2017-09-29 BARC           30.1              NA   272    NA
- 7 2017-10-04 BARC           27.6              NA   277    NA
- 8 2017-10-06 BARC           27.6              NA   279    NA
- 9 2017-10-11 BARC           29.6              NA   284    NA
-10 2017-10-14 BARC           28.5              NA   287    NA
-# ℹ 1,030 more rows</code></pre>
+ 4 2017-09-15 BARC           29.3              NA   258    NA
+ 5 2017-09-19 BARC           28.8              NA   262    NA
+ 6 2017-09-22 BARC           29.0              NA   265    NA
+ 7 2017-09-29 BARC           30.1              NA   272    NA
+ 8 2017-10-04 BARC           27.6              NA   277    NA
+ 9 2017-10-06 BARC           27.6              NA   279    NA
+10 2017-10-11 BARC           29.6              NA   284    NA
+# ℹ 1,038 more rows</code></pre>
 </div>
 </div>
 <p><em>Step 6: Evaluate model</em></p>
@@ -863,8 +863,8 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <pre><code># A tibble: 2 × 3
   .metric .estimator .estimate
   &lt;chr&gt;   &lt;chr&gt;          &lt;dbl&gt;
-1 rmse    standard       2.37 
-2 rsq     standard       0.829</code></pre>
+1 rmse    standard       2.42 
+2 rsq     standard       0.819</code></pre>
 </div>
 </div>
 <p><em>Step 7: Deploy model</em></p>
@@ -884,16 +884,16 @@ <h4 data-number="12.2.2.2" class="anchored" data-anchor-id="step-2-pre-process-d
 <pre><code># A tibble: 68 × 5
    datetime   site_id air_temperature temperature   doy
    &lt;date&gt;     &lt;chr&gt;             &lt;dbl&gt; &lt;lgl&gt;       &lt;dbl&gt;
- 1 2025-02-19 BARC               17.1 NA             50
- 2 2025-02-20 BARC               16.5 NA             51
- 3 2025-02-21 BARC               13.4 NA             52
- 4 2025-02-22 BARC               12.9 NA             53
- 5 2025-02-23 BARC               15.0 NA             54
- 6 2025-02-24 BARC               19.0 NA             55
- 7 2025-02-25 BARC               14.1 NA             56
- 8 2025-02-26 BARC               11.9 NA             57
- 9 2025-02-27 BARC               12.3 NA             58
-10 2025-02-28 BARC               14.7 NA             59
+ 1 2025-02-21 BARC               7.26 NA             52
+ 2 2025-02-22 BARC               9.64 NA             53
+ 3 2025-02-23 BARC              11.7  NA             54
+ 4 2025-02-24 BARC              10.3  NA             55
+ 5 2025-02-25 BARC              12.0  NA             56
+ 6 2025-02-26 BARC              15.1  NA             57
+ 7 2025-02-27 BARC              16.0  NA             58
+ 8 2025-02-28 BARC              14.8  NA             59
+ 9 2025-03-01 BARC              11.3  NA             60
+10 2025-03-02 BARC              15.1  NA             61
 # ℹ 58 more rows</code></pre>
 </div>
 </div>
@@ -998,7 +998,7 @@ <h3 data-number="12.3.1" class="anchored" data-anchor-id="step-1-set-up-targets"
 <pre><code># A tibble: 1 × 2
   start_date     h
   &lt;date&gt;     &lt;dbl&gt;
-1 2025-01-30    55</code></pre>
+1 2025-02-18    38</code></pre>
 </div>
 </div>
 <p>The targets have to be converted into a time-series tibble (called a <code>tsibble</code>). To make a <code>tsibble</code> you need to define the grouping variables (<code>key</code>) and the common that defines the time-series (<code>index</code>). Many modeling approaches require the datetime to be continuous (no gaps) but allow na values for values in the gaps. The <code>fill_gaps()</code> function fills the gaps with <code>NA</code>s.</p>
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diff --git a/search.json b/search.json
index a5b8b9de..d9bc683a 100644
--- a/search.json
+++ b/search.json
@@ -396,7 +396,7 @@
     "href": "evaluating-forecasts.html#accuracy-precision",
     "title": "11  Evaluating forecasts",
     "section": "11.3 Accuracy + Precision",
-    "text": "11.3 Accuracy + Precision\nIt is important to consider uncertainty because intuitively a forecast that has the correct mean but a large uncertainty (so it puts very little confidence on the observed value) should have a lower score than a forecast with a mean that is slightly off but with lower uncertainty (so it puts more confidence on the observed value). The Continuous Ranked Probability Score (CRPS) is one metric that evaluates the forecast distribution. Another is the ignorance score (also called the Good or the log score). The CRPS is described below.\nAdd the concept of proper and local?\n\n11.3.1 Continuous Ranked Probability Score\nForecasts can scored using the continuous ranked probability score (CRPS), a proper scoring rule for evaluating forecasts presented as distributions or ensembles (Gneiting and Raftery (2007)). The CRPS compares the forecast probability distribution to that of the validation observation and assigns a score based on both the accuracy and precision of the forecast. We will use the ‘crps_sample’ function from the scoringRules package in R to calculate the CRPS for each forecast.\nCRPS is a calculation for each forecast-observation pair (e.g., a single datetime from a reference_datetime). CRPS can then be aggregated across all sites and forecast horizons to understand the aggregate performance.\nImportantly, we use the convention for CRPS where zero is the lowest and best possible score, therefore forecasts want to achieve the lowest score. CPRS can also be expressed as a negative number with zero as the highest and best possible score (Gneiting and Raftery (2007)). The scoringRules package we use follows the 0 or greater convention.\n\nlibrary(scoringRules)\nlibrary(tidyverse)\n\n\n\n11.3.2 Example of a CRPS calculation from an ensemble forecast\nThis section aims to provide an intuition for the CRPS metric.\nFirst, create a random sample from a probability distribution. This is the “forecast” for a particular point in time. For simplicity, we will use a normal distribution with a mean of 8 and a standard deviation of 1\n\nx &lt;- rnorm(1000, mean = 8, sd = 1.0)\n\nSecond, we have our data point (i.e., the target) that we set to 8 as well.\n\ny &lt;- 8\n\nNow calculate using the crps_sample() function in the scoringRules package\n\nlibrary(scoringRules)\ncrps_sample(y = y, dat = x)\n\n[1] 0.2384334\n\n\n\n\n11.3.3 Exploring the scoring surface\nNow lets see how the CRPS changes as the mean and standard deviation of the forecasted distribution change\nFirst, set vectors for the different mean and SD values we want to explore\n\nforecast_mean &lt;- seq(4, 12, 0.05)\nforecast_sd &lt;- seq(0.1, 10, 0.05)\n\nSecond, set our observed value to 8 for simplicity\n\ny &lt;- 8\n\nNow calculate the CRPS at each combination of forest mean and SD. We used crps_norm because we know the forecast is normally distributed\n\ncombined &lt;- array(NA, dim = c(length(forecast_mean), length(forecast_sd)))\nfor(i in 1:length(forecast_mean)){\n  for(j in 1:length(forecast_sd)){\n    combined[i, j] &lt;- crps_norm(y = y, mean = forecast_mean[i], forecast_sd[j])\n  }\n}\n\nFinally, visualize the scoring surface with the observed value represented by the red line\n\ncontour(x = forecast_mean, y = forecast_sd, z = as.matrix(combined),nlevels = 20, xlab = \"Forecast mean\", ylab = \"forecast SD\", main = \"CPRS score at different forecasted mean and sd\\nThe observation is at 8\")\nabline(v = y, col = \"red\")\nabline(v = 6, col = \"darkgreen\")\n\n\n\n\n\n\n\n\nThe contour surface highlights the trade-off between the mean and standard deviation. Each isocline is the same CRPS score. Holding the mean constant at the observed value (vertical red line at 8) but increases the uncertainty (increasing SD) results in a higher (worse) score. However, a mean of 6 (green line) with an SD of 2 has a similar score to a mean of 8 and a standard deviation of 1. The following figure capture this idea. The two lines are the distribution of the mean = 8, sd = 4 and mean = 6, sd = 2 forecasts. The red line is the observation of 8. The forecast distribution with the mean of 6 actually has more density at x = 8 (the observation) and scores slightly better.\n\nplot(density(rnorm(10000, mean = 8, sd = 4)), ylim = c(0, 0.2), main = \"\", xlab = \"forecasted value\")\nlines(density(rnorm(10000, mean = 6, sd = 2)), col = \"darkgreen\")\nabline(v = 8, col = \"red\")",
+    "text": "11.3 Accuracy + Precision\nIt is important to consider uncertainty because intuitively a forecast that has the correct mean but a large uncertainty (so it puts very little confidence on the observed value) should have a lower score than a forecast with a mean that is slightly off but with lower uncertainty (so it puts more confidence on the observed value). The Continuous Ranked Probability Score (CRPS) is one metric that evaluates the forecast distribution. Another is the ignorance score (also called the Good or the log score). The CRPS is described below.\nAdd the concept of proper and local?\n\n11.3.1 Continuous Ranked Probability Score\nForecasts can scored using the continuous ranked probability score (CRPS), a proper scoring rule for evaluating forecasts presented as distributions or ensembles (Gneiting and Raftery (2007)). The CRPS compares the forecast probability distribution to that of the validation observation and assigns a score based on both the accuracy and precision of the forecast. We will use the ‘crps_sample’ function from the scoringRules package in R to calculate the CRPS for each forecast.\nCRPS is a calculation for each forecast-observation pair (e.g., a single datetime from a reference_datetime). CRPS can then be aggregated across all sites and forecast horizons to understand the aggregate performance.\nImportantly, we use the convention for CRPS where zero is the lowest and best possible score, therefore forecasts want to achieve the lowest score. CPRS can also be expressed as a negative number with zero as the highest and best possible score (Gneiting and Raftery (2007)). The scoringRules package we use follows the 0 or greater convention.\n\nlibrary(scoringRules)\nlibrary(tidyverse)\n\n\n\n11.3.2 Example of a CRPS calculation from an ensemble forecast\nThis section aims to provide an intuition for the CRPS metric.\nFirst, create a random sample from a probability distribution. This is the “forecast” for a particular point in time. For simplicity, we will use a normal distribution with a mean of 8 and a standard deviation of 1\n\nx &lt;- rnorm(1000, mean = 8, sd = 1.0)\n\nSecond, we have our data point (i.e., the target) that we set to 8 as well.\n\ny &lt;- 8\n\nNow calculate using the crps_sample() function in the scoringRules package\n\nlibrary(scoringRules)\ncrps_sample(y = y, dat = x)\n\n[1] 0.237732\n\n\n\n\n11.3.3 Exploring the scoring surface\nNow lets see how the CRPS changes as the mean and standard deviation of the forecasted distribution change\nFirst, set vectors for the different mean and SD values we want to explore\n\nforecast_mean &lt;- seq(4, 12, 0.05)\nforecast_sd &lt;- seq(0.1, 10, 0.05)\n\nSecond, set our observed value to 8 for simplicity\n\ny &lt;- 8\n\nNow calculate the CRPS at each combination of forest mean and SD. We used crps_norm because we know the forecast is normally distributed\n\ncombined &lt;- array(NA, dim = c(length(forecast_mean), length(forecast_sd)))\nfor(i in 1:length(forecast_mean)){\n  for(j in 1:length(forecast_sd)){\n    combined[i, j] &lt;- crps_norm(y = y, mean = forecast_mean[i], forecast_sd[j])\n  }\n}\n\nFinally, visualize the scoring surface with the observed value represented by the red line\n\ncontour(x = forecast_mean, y = forecast_sd, z = as.matrix(combined),nlevels = 20, xlab = \"Forecast mean\", ylab = \"forecast SD\", main = \"CPRS score at different forecasted mean and sd\\nThe observation is at 8\")\nabline(v = y, col = \"red\")\nabline(v = 6, col = \"darkgreen\")\n\n\n\n\n\n\n\n\nThe contour surface highlights the trade-off between the mean and standard deviation. Each isocline is the same CRPS score. Holding the mean constant at the observed value (vertical red line at 8) but increases the uncertainty (increasing SD) results in a higher (worse) score. However, a mean of 6 (green line) with an SD of 2 has a similar score to a mean of 8 and a standard deviation of 1. The following figure capture this idea. The two lines are the distribution of the mean = 8, sd = 4 and mean = 6, sd = 2 forecasts. The red line is the observation of 8. The forecast distribution with the mean of 6 actually has more density at x = 8 (the observation) and scores slightly better.\n\nplot(density(rnorm(10000, mean = 8, sd = 4)), ylim = c(0, 0.2), main = \"\", xlab = \"forecasted value\")\nlines(density(rnorm(10000, mean = 6, sd = 2)), col = \"darkgreen\")\nabline(v = 8, col = \"red\")",
     "crumbs": [
       "Iterative forecasting using empirical models",
       "<span class='chapter-number'>11</span>  <span class='chapter-title'>Evaluating forecasts</span>"
@@ -484,7 +484,7 @@
     "href": "project1.html#intro-to-tidymodels",
     "title": "12  Build your own forecast",
     "section": "12.2 Intro to Tidymodels",
-    "text": "12.2 Intro to Tidymodels\nTidmodels is a meta-package that provides a standardized interface with many machine-learning algorithms. This brief tutorial that provides an example application with tidymodels builds on the machine learning module in the FREC 3044: Environmental Data Science course at Virginia Tech. https://github.com/frec-3044/machine-learning-template.\nTidymodels does not estimate uncertainty so forecast uncertainty needs to be generated output the tidymodels framework. For example, driver uncertainty can be generated by creating tidymodel forecasts for each meteorological ensemble member.\n\n12.2.1 Overview of tidymodels steps\nStep 1: Obtain data\nRead in data to memory. In the course, we have accessed CSV, excel, and database forms located in the GitHub repo, on websites, and through an API.\nStep 2: Pre-process data\nSplit data: divide data into the training and test sets.\n\nYou will use the training set to fit your model and the testing set to evaluate the model performance.\nWe only want to evaluate our model using data that the model has not used yet. Otherwise, we can only say that the model is good at fitting the data that was used to train it. This does not help us understand how it would perform as a tool to generate new predictions.\nYou want most of your data in the training set because machine learning models can be quite “hungry for data”.\nA typical split is 80% training and 20% testing but there isn’t a required split.\nData are randomly assigned to the two splits.\nYour data may have obvious groupings that you want to be equally represented in both the training and testing sets. For example, if you have cats and dogs as a variable you are using to predict tail length, you may want to be sure that your training set is not randomly full of dogs because that means it may not predict the cats well in the testing set. You can prevent this by defining the strata used when assigning the training and testing sets.\n\nRecipes: Modify predictors/features/covariates/independent variables for analysis\n\nModify the data set so that variables are correctly formatted for a particular model. For example, a linear regression requires groups to be dummy variables. Therefore, a column called “ecosystem” with two values: “forest” and “grass” would be converted to two columns: ecosystem_forest with a value of 0 or 1 and ecosystem_grass with a value of 0 and 1)\nRemoving highly correlated predictors\nRescaling predictor (e.g., converting to 0 mean and 1 standard deviation)\ntransforming predictors (e.g., log)\n\nStep 3: Specify model and workflow\nA workflow combines the model and recipe in a single “object” that you can use for training, testing, and predicting new data.\n\nDefine model type: linear regression, regression tree, neutral net, etc.\nDefine model engine: particular R package (lm, ranger, etc.)\nDefine model mode: regression or classification\nDefine workflow: combine recipe and model definition\n\nStep 4: Train model\n\nTune hyper-parameters: hyper-parameters are configuration settings that govern how a particular ML method is fit to data. They are called “hyper” because “regular” parameters are the parameters within the model that are learned by the ML method. For example, a method called “random forecast” requires a hyper-parameter that controls the minimum size of a regression tree that is allowed. This parameter (called min_n) could be directly provided by you or could be tuned. The tuning process involves repeatedly fitting the model using different values of the hyper-parameter and using the hyper-parameter values that yield the best fit to the data. Importantly: not all ML methods have hyper-parameters (e.g., linear regression using the lm engine does not have hyper-parameters). We don’t tune hyperparameters in this example. See example-with-tuning.Rmd for an extension of this application that includes hyperparameter tuning.\nFit model (using best hyper-parameter if they are tuned). The model is fit to the training data.\n\nStep 5: Predict\n\nPredict testing data using the model that was fit to the training data. This step is critical because we only want to evaluate our model using data that the model has not used yet. Otherwise, we can only say that the model is good at fitting the data that was used to train it. This does not help understand how it would perform as a tool to generate new predictions.\nPredictions are quite easy using the predict() function.\n\nStep 6: Evaluate model\n\nIt is important to use the appropriate metrics to evaluate how the model performs. Some metrics only apply to classification problems and others only apply to regression problems. A list of metric types can be found here\nWe will be focusing on root-mean-squared error (rmse), a metric that subtracts each observation from the predictions, squares it, averages all the squared errors for all the data points, and then takes the square root of the mean squared error.\n\nR-squared (rsq) is another metric that we used in the lake ice module.\n\nStep 7: Deploy model\n\nOne of the main points of using machine learning is to develop a tool that can be used with new predictors to predict data that wasn’t involved in the training and testing. These are data that have all the columns necessary to make predictions but lack data in the column you are trying to predict.\n\nThis step is simple because it involves using the predict() function with the same trained model but with new data.\n\n\n\n12.2.2 Application: Predicting water temperature at NEON sites\n\nlibrary(tidymodels)\ntidymodels_prefer()\nset.seed(100) #for random number generation\n\n\n12.2.2.1 Step 1: Obtain data\n\nlake_sites &lt;- c(\"BARC\", \"SUGG\")\n\n\ntargets &lt;- read_csv('https://data.ecoforecast.org/neon4cast-targets/aquatics/aquatics-targets.csv.gz',\n                    show_col_types = FALSE) |&gt; \n  filter(site_id %in% lake_sites)\n\n\ntargets &lt;- targets |&gt; \n  filter(site_id %in% lake_sites,\n         variable == \"temperature\")\n\n\n# past stacked weather\ndf_past &lt;- neon4cast::noaa_stage3()\n\nvariables &lt;- c(\"air_temperature\")\n\nnoaa_past &lt;- df_past |&gt; \n  dplyr::filter(site_id %in% lake_sites,\n                datetime &gt;= ymd('2017-01-01'),\n                variable %in% variables) |&gt; \n  dplyr::collect()\n\n\n# aggregate the past to mean values\nnoaa_past_mean &lt;- noaa_past |&gt; \n  mutate(datetime = as_date(datetime)) |&gt; \n  group_by(datetime, site_id, variable) |&gt; \n  summarize(prediction = mean(prediction, na.rm = TRUE), .groups = \"drop\") |&gt; \n  pivot_wider(names_from = variable, values_from = prediction) |&gt; \n  # convert air temp to C\n  mutate(air_temperature = air_temperature - 273.15)\n\n\nforecast_date &lt;- Sys.Date() \nnoaa_date &lt;- forecast_date - lubridate::days(2)\n\ndf_future &lt;- neon4cast::noaa_stage2(start_date = noaa_date)\n\nvariables &lt;- c(\"air_temperature\")\n\nnoaa_future &lt;- df_future |&gt; \n  dplyr::filter(reference_datetime == noaa_date,\n                datetime &gt;= forecast_date,\n                site_id %in% lake_sites,\n                variable %in% variables) |&gt; \n  dplyr::collect()\n\n\nnoaa_future_daily &lt;- noaa_future |&gt; \n  mutate(datetime = as_date(datetime)) |&gt; \n  # mean daily forecasts at each site per ensemble\n  summarize(prediction = mean(prediction), .by = c(\"datetime\", \"site_id\", \"parameter\", \"variable\")) |&gt;\n  pivot_wider(names_from = variable, values_from = prediction) |&gt;\n  # convert to Celsius\n  mutate(air_temperature = air_temperature - 273.15) |&gt; \n  select(datetime, site_id, air_temperature, parameter)\n\n\ntargets_df &lt;- targets |&gt; \n  filter(variable == 'temperature') |&gt;\n  pivot_wider(names_from = 'variable', values_from = 'observation') |&gt; \n  left_join(noaa_past_mean, \n            by = c(\"datetime\",\"site_id\")) |&gt; \n  mutate(doy = yday(datetime))\n\n\n\n12.2.2.2 Step 2: Pre-process data\nSplit data into training/testing sets\nWe are going to split the data into training and testing sets using the initial_split function. prop = 0.80 says to use 80% of the data in the training set.\n\nsplit &lt;- initial_split(targets_df, prop = 0.80, strata = site_id)\n\nOur split should reflect the 80/20 that we defined using prop\n\nsplit\n\n&lt;Training/Testing/Total&gt;\n&lt;4159/1040/5199&gt;\n\n\nTo get the training and testing data we need to apply the training() and testing() functions to the split.\n\ntrain_data &lt;- training(split)\ntest_data &lt;- testing(split)\n\nYou can see that train_data is a data frame that we can work with.\n\ntrain_data\n\n# A tibble: 4,159 × 5\n   datetime   site_id temperature air_temperature   doy\n   &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt;\n 1 2017-08-27 BARC           31.5              NA   239\n 2 2017-08-28 BARC           31.1              NA   240\n 3 2017-08-29 BARC           31.1              NA   241\n 4 2017-08-30 BARC           31.4              NA   242\n 5 2017-08-31 BARC           31.7              NA   243\n 6 2017-09-01 BARC           31.6              NA   244\n 7 2017-09-02 BARC           31.1              NA   245\n 8 2017-09-03 BARC           31.1              NA   246\n 9 2017-09-06 BARC           30.8              NA   249\n10 2017-09-07 BARC           30.5              NA   250\n# ℹ 4,149 more rows\n\n\nFeature engineering using a recipe\n\nrequires starting with dataset that is used to provide the columns.\na formula with the dependent variable and the predictors. If . is used as the predictor, that means using all columns other than the dependent variable.\nSteps that modify the data\n\nWe will use the following steps:\n\nstep_rm because we don’t want to use the datetime in the fit.\nstep_naomit because there are na values in the temperature and air_temperature columns. This is used here for illustrative purposes and can also be done by filtering the target data before it is split into training and testing groups\n\nHere are the different recipe steps used above\n\nFilter rows and select columns\n\nHere is our recipe:\n\nour_recipe &lt;- train_data |&gt; \n  recipe(temperature ~ . ) |&gt; \n  step_rm(datetime) |&gt;\n  step_naomit(air_temperature, temperature)\n\nThe recipe should show the steps that will be performed when applying the recipe. Importantly, these steps have not yet been applied, we just have a recipe of what to do.\n\nour_recipe\n\n\n\n\n── Recipe ──────────────────────────────────────────────────────────────────────\n\n\n\n\n\n── Inputs \n\n\nNumber of variables by role\n\n\noutcome:   1\npredictor: 4\n\n\n\n\n\n── Operations \n\n\n• Variables removed: datetime\n\n\n• Removing rows with NA values in: air_temperature temperature\n\n\nStep 3: Specify model, engine, and workflow\nWe need to create the model and engine that we will use. In this example, we are using linear_reg with the mode of regression (as opposed to classification). Setting the mode for a linear regression is not necessary because it only allows regressions but it is included here for completeness.\nThe engine is lm because we are using the standard R function lm. There are a ton of other functions for linear regression modeling that we could use. They would be specified as a different engine.\n\nour_model &lt;- linear_reg(mode = \"regression\") |&gt; \n  set_engine(\"lm\")\n\nYou will see the model, mode, and engine in the model object\n\nour_model \n\nLinear Regression Model Specification (regression)\n\nComputational engine: lm \n\n\nWe now combine the model and the recipe to make a workflow that can be used to fit the training and testing data. workflow() initiates the workflow and add_model and add_recipe add those components to the workflow. Importantly, the workflow has not yet been applied, we just have a description of what to do.\n\nwflow &lt;-\n  workflow() |&gt; \n  add_model(our_model) |&gt; \n  add_recipe(our_recipe)\n\nYou can see that the workflow object has all the components together\n\nwflow\n\n══ Workflow ════════════════════════════════════════════════════════════════════\nPreprocessor: Recipe\nModel: linear_reg()\n\n── Preprocessor ────────────────────────────────────────────────────────────────\n2 Recipe Steps\n\n• step_rm()\n• step_naomit()\n\n── Model ───────────────────────────────────────────────────────────────────────\nLinear Regression Model Specification (regression)\n\nComputational engine: lm \n\n\nStep 4: Train model on Training Data\nWe will use the workflow object to train the model. We need to provide the workflow object and the dataset to the fit function to fit (i.e., train the model)\n\nfit &lt;- wflow |&gt; \n  fit(data = train_data)\n\nYou can see that the fit object is the workflow object + the results of the model fitting\n\nfit\n\n══ Workflow [trained] ══════════════════════════════════════════════════════════\nPreprocessor: Recipe\nModel: linear_reg()\n\n── Preprocessor ────────────────────────────────────────────────────────────────\n2 Recipe Steps\n\n• step_rm()\n• step_naomit()\n\n── Model ───────────────────────────────────────────────────────────────────────\n\nCall:\nstats::lm(formula = ..y ~ ., data = data)\n\nCoefficients:\n    (Intercept)      site_idSUGG  air_temperature              doy  \n       5.142973        -0.785496         0.880690         0.002305  \n\n\nStep 5: Predict Test Data\nNow we will predict the testing data using the model that was fit to the training data.\n\npredictions &lt;- predict(fit, new_data = test_data)\n\nThe predictions are a single column called .pred\n\npredictions\n\n# A tibble: 1,040 × 1\n   .pred\n   &lt;dbl&gt;\n 1    NA\n 2    NA\n 3    NA\n 4    NA\n 5    NA\n 6    NA\n 7    NA\n 8    NA\n 9    NA\n10    NA\n# ℹ 1,030 more rows\n\n\nWe need to combine the .pred column with the testing data using the bind_cols function\n\npred_test &lt;- bind_cols(test_data, predictions)\n\nNow we have a data frame with the prediction and all the predictors used to predict it.\n\npred_test\n\n# A tibble: 1,040 × 6\n   datetime   site_id temperature air_temperature   doy .pred\n   &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;\n 1 2017-09-04 BARC           31.4              NA   247    NA\n 2 2017-09-05 BARC           31.0              NA   248    NA\n 3 2017-09-14 BARC           28.5              NA   257    NA\n 4 2017-09-19 BARC           28.8              NA   262    NA\n 5 2017-09-22 BARC           29.0              NA   265    NA\n 6 2017-09-29 BARC           30.1              NA   272    NA\n 7 2017-10-04 BARC           27.6              NA   277    NA\n 8 2017-10-06 BARC           27.6              NA   279    NA\n 9 2017-10-11 BARC           29.6              NA   284    NA\n10 2017-10-14 BARC           28.5              NA   287    NA\n# ℹ 1,030 more rows\n\n\nStep 6: Evaluate model\nWe will evaluate the performance of our predictions of the testing data using two metrics (rmse and rsq). The function metric_set defines the set of metrics we will be using them. It creates a function called multi_metric() that we will use to calculate the metrics. We pipe in the predicted test data (pred_test) and tell the function that our truth (i.e., observed data) is the temperature column and the predictions (i.e., estimate) is the .pred column\n\nmulti_metric &lt;- metric_set(rmse, rsq)\n\nmetric_table &lt;- pred_test |&gt; \n  multi_metric(truth = temperature, estimate = .pred)\n\nThe resulting table has the metrics for evaluation\n\nmetric_table\n\n# A tibble: 2 × 3\n  .metric .estimator .estimate\n  &lt;chr&gt;   &lt;chr&gt;          &lt;dbl&gt;\n1 rmse    standard       2.37 \n2 rsq     standard       0.829\n\n\nStep 7: Deploy model\nThe final step is to apply your model to predict new data.\nNow read in the new data.\n\ntargets_future &lt;- noaa_future_daily |&gt; \n  mutate(temperature = NA,\n         doy = yday(datetime)) |&gt; \n  filter(parameter == 1) |&gt; \n  select(-parameter)\n\nYou will notice that the temperature is all NA because you don’t know what the carbon stock is of the plot.\n\ntargets_future\n\n# A tibble: 68 × 5\n   datetime   site_id air_temperature temperature   doy\n   &lt;date&gt;     &lt;chr&gt;             &lt;dbl&gt; &lt;lgl&gt;       &lt;dbl&gt;\n 1 2025-02-19 BARC               17.1 NA             50\n 2 2025-02-20 BARC               16.5 NA             51\n 3 2025-02-21 BARC               13.4 NA             52\n 4 2025-02-22 BARC               12.9 NA             53\n 5 2025-02-23 BARC               15.0 NA             54\n 6 2025-02-24 BARC               19.0 NA             55\n 7 2025-02-25 BARC               14.1 NA             56\n 8 2025-02-26 BARC               11.9 NA             57\n 9 2025-02-27 BARC               12.3 NA             58\n10 2025-02-28 BARC               14.7 NA             59\n# ℹ 58 more rows\n\n\nAs in “Step 5: Predict Test Data”, use the model fitting on the training data (fit) to predict the new data.\n\nnew_predictions &lt;- predict(fit, new_data = targets_future)\n\n\ntargets_future &lt;- noaa_future_daily |&gt; \n  mutate(temperature = NA,\n         doy = yday(datetime))\n\ntidymodels_forecast &lt;- data.frame()\n\nfor(i in unique(targets_future$parameter)){\n  curr_ens &lt;- targets_future |&gt; \n    filter(parameter == i) |&gt; \n    select(-parameter)\n  \n  new_predictions &lt;- predict(fit, new_data = curr_ens)\n  curr_ens &lt;- bind_cols(curr_ens, new_predictions) |&gt; \n    mutate(parameter = i)\n  tidymodels_forecast &lt;- bind_rows(tidymodels_forecast, curr_ens)\n}\n\n\ntidymodels_forecasts_EFI &lt;- tidymodels_forecast %&gt;%\n  rename(prediction = .pred) %&gt;%\n  mutate(variable = \"temperature\") |&gt; \n  # For the EFI challenge we only want the forecast for future\n  filter(datetime &gt; Sys.Date()) %&gt;%\n  group_by(site_id, variable) %&gt;%\n  mutate(reference_datetime = min(datetime) - lubridate::days(1),\n         family = \"ensemble\",\n         model_id = \"tidymodels_lm\") %&gt;%\n  select(model_id, datetime, reference_datetime, site_id, family, parameter, variable, prediction)\n\n\ntidymodels_forecasts_EFI |&gt;\n  filter(variable == \"temperature\") |&gt;\n  ggplot(aes(x = datetime, y = prediction, group = parameter)) +\n  geom_line() + \n  facet_wrap(~site_id)\n\n\n\n\n\n\n\n\n\n\n\n12.2.3 Tidymodel extras\n\n12.2.3.1 Changing the model approach\nYou can easily change the modeling approach by changing the model and engine. For example, the following code replaces the linear regression model with a random forest model from the ranger package. All other code used to fit and predict with the model stays the same.\n\nour_model &lt;- rand_forest(mode = \"regression\") |&gt; \n  set_engine(\"ranger\")\n\nA list of models that are available for you to use can be found here. Not all models are designed for all applications. For example, forecasting water temperature requires regression models rather than classification models.\n\n\n12.2.3.2 Tuning hyper-parameters\nSome modeling approaches have parameters that govern how the model fitting is done. These are called hyperparameters. Hyperparameters are different from the parameters that are optimized using data in the modeling fitting step. Tidymodels provides tools to also fit hyperparameters. An example can be found here.",
+    "text": "12.2 Intro to Tidymodels\nTidmodels is a meta-package that provides a standardized interface with many machine-learning algorithms. This brief tutorial that provides an example application with tidymodels builds on the machine learning module in the FREC 3044: Environmental Data Science course at Virginia Tech. https://github.com/frec-3044/machine-learning-template.\nTidymodels does not estimate uncertainty so forecast uncertainty needs to be generated output the tidymodels framework. For example, driver uncertainty can be generated by creating tidymodel forecasts for each meteorological ensemble member.\n\n12.2.1 Overview of tidymodels steps\nStep 1: Obtain data\nRead in data to memory. In the course, we have accessed CSV, excel, and database forms located in the GitHub repo, on websites, and through an API.\nStep 2: Pre-process data\nSplit data: divide data into the training and test sets.\n\nYou will use the training set to fit your model and the testing set to evaluate the model performance.\nWe only want to evaluate our model using data that the model has not used yet. Otherwise, we can only say that the model is good at fitting the data that was used to train it. This does not help us understand how it would perform as a tool to generate new predictions.\nYou want most of your data in the training set because machine learning models can be quite “hungry for data”.\nA typical split is 80% training and 20% testing but there isn’t a required split.\nData are randomly assigned to the two splits.\nYour data may have obvious groupings that you want to be equally represented in both the training and testing sets. For example, if you have cats and dogs as a variable you are using to predict tail length, you may want to be sure that your training set is not randomly full of dogs because that means it may not predict the cats well in the testing set. You can prevent this by defining the strata used when assigning the training and testing sets.\n\nRecipes: Modify predictors/features/covariates/independent variables for analysis\n\nModify the data set so that variables are correctly formatted for a particular model. For example, a linear regression requires groups to be dummy variables. Therefore, a column called “ecosystem” with two values: “forest” and “grass” would be converted to two columns: ecosystem_forest with a value of 0 or 1 and ecosystem_grass with a value of 0 and 1)\nRemoving highly correlated predictors\nRescaling predictor (e.g., converting to 0 mean and 1 standard deviation)\ntransforming predictors (e.g., log)\n\nStep 3: Specify model and workflow\nA workflow combines the model and recipe in a single “object” that you can use for training, testing, and predicting new data.\n\nDefine model type: linear regression, regression tree, neutral net, etc.\nDefine model engine: particular R package (lm, ranger, etc.)\nDefine model mode: regression or classification\nDefine workflow: combine recipe and model definition\n\nStep 4: Train model\n\nTune hyper-parameters: hyper-parameters are configuration settings that govern how a particular ML method is fit to data. They are called “hyper” because “regular” parameters are the parameters within the model that are learned by the ML method. For example, a method called “random forecast” requires a hyper-parameter that controls the minimum size of a regression tree that is allowed. This parameter (called min_n) could be directly provided by you or could be tuned. The tuning process involves repeatedly fitting the model using different values of the hyper-parameter and using the hyper-parameter values that yield the best fit to the data. Importantly: not all ML methods have hyper-parameters (e.g., linear regression using the lm engine does not have hyper-parameters). We don’t tune hyperparameters in this example. See example-with-tuning.Rmd for an extension of this application that includes hyperparameter tuning.\nFit model (using best hyper-parameter if they are tuned). The model is fit to the training data.\n\nStep 5: Predict\n\nPredict testing data using the model that was fit to the training data. This step is critical because we only want to evaluate our model using data that the model has not used yet. Otherwise, we can only say that the model is good at fitting the data that was used to train it. This does not help understand how it would perform as a tool to generate new predictions.\nPredictions are quite easy using the predict() function.\n\nStep 6: Evaluate model\n\nIt is important to use the appropriate metrics to evaluate how the model performs. Some metrics only apply to classification problems and others only apply to regression problems. A list of metric types can be found here\nWe will be focusing on root-mean-squared error (rmse), a metric that subtracts each observation from the predictions, squares it, averages all the squared errors for all the data points, and then takes the square root of the mean squared error.\n\nR-squared (rsq) is another metric that we used in the lake ice module.\n\nStep 7: Deploy model\n\nOne of the main points of using machine learning is to develop a tool that can be used with new predictors to predict data that wasn’t involved in the training and testing. These are data that have all the columns necessary to make predictions but lack data in the column you are trying to predict.\n\nThis step is simple because it involves using the predict() function with the same trained model but with new data.\n\n\n\n12.2.2 Application: Predicting water temperature at NEON sites\n\nlibrary(tidymodels)\ntidymodels_prefer()\nset.seed(100) #for random number generation\n\n\n12.2.2.1 Step 1: Obtain data\n\nlake_sites &lt;- c(\"BARC\", \"SUGG\")\n\n\ntargets &lt;- read_csv('https://data.ecoforecast.org/neon4cast-targets/aquatics/aquatics-targets.csv.gz',\n                    show_col_types = FALSE) |&gt; \n  filter(site_id %in% lake_sites)\n\n\ntargets &lt;- targets |&gt; \n  filter(site_id %in% lake_sites,\n         variable == \"temperature\")\n\n\n# past stacked weather\ndf_past &lt;- neon4cast::noaa_stage3()\n\nvariables &lt;- c(\"air_temperature\")\n\nnoaa_past &lt;- df_past |&gt; \n  dplyr::filter(site_id %in% lake_sites,\n                datetime &gt;= ymd('2017-01-01'),\n                variable %in% variables) |&gt; \n  dplyr::collect()\n\n\n# aggregate the past to mean values\nnoaa_past_mean &lt;- noaa_past |&gt; \n  mutate(datetime = as_date(datetime)) |&gt; \n  group_by(datetime, site_id, variable) |&gt; \n  summarize(prediction = mean(prediction, na.rm = TRUE), .groups = \"drop\") |&gt; \n  pivot_wider(names_from = variable, values_from = prediction) |&gt; \n  # convert air temp to C\n  mutate(air_temperature = air_temperature - 273.15)\n\n\nforecast_date &lt;- Sys.Date() \nnoaa_date &lt;- forecast_date - lubridate::days(2)\n\ndf_future &lt;- neon4cast::noaa_stage2(start_date = noaa_date)\n\nvariables &lt;- c(\"air_temperature\")\n\nnoaa_future &lt;- df_future |&gt; \n  dplyr::filter(reference_datetime == noaa_date,\n                datetime &gt;= forecast_date,\n                site_id %in% lake_sites,\n                variable %in% variables) |&gt; \n  dplyr::collect()\n\n\nnoaa_future_daily &lt;- noaa_future |&gt; \n  mutate(datetime = as_date(datetime)) |&gt; \n  # mean daily forecasts at each site per ensemble\n  summarize(prediction = mean(prediction), .by = c(\"datetime\", \"site_id\", \"parameter\", \"variable\")) |&gt;\n  pivot_wider(names_from = variable, values_from = prediction) |&gt;\n  # convert to Celsius\n  mutate(air_temperature = air_temperature - 273.15) |&gt; \n  select(datetime, site_id, air_temperature, parameter)\n\n\ntargets_df &lt;- targets |&gt; \n  filter(variable == 'temperature') |&gt;\n  pivot_wider(names_from = 'variable', values_from = 'observation') |&gt; \n  left_join(noaa_past_mean, \n            by = c(\"datetime\",\"site_id\")) |&gt; \n  mutate(doy = yday(datetime))\n\n\n\n12.2.2.2 Step 2: Pre-process data\nSplit data into training/testing sets\nWe are going to split the data into training and testing sets using the initial_split function. prop = 0.80 says to use 80% of the data in the training set.\n\nsplit &lt;- initial_split(targets_df, prop = 0.80, strata = site_id)\n\nOur split should reflect the 80/20 that we defined using prop\n\nsplit\n\n&lt;Training/Testing/Total&gt;\n&lt;4189/1048/5237&gt;\n\n\nTo get the training and testing data we need to apply the training() and testing() functions to the split.\n\ntrain_data &lt;- training(split)\ntest_data &lt;- testing(split)\n\nYou can see that train_data is a data frame that we can work with.\n\ntrain_data\n\n# A tibble: 4,189 × 5\n   datetime   site_id temperature air_temperature   doy\n   &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt;\n 1 2017-08-27 BARC           31.5              NA   239\n 2 2017-08-28 BARC           31.1              NA   240\n 3 2017-08-29 BARC           31.1              NA   241\n 4 2017-08-30 BARC           31.4              NA   242\n 5 2017-08-31 BARC           31.7              NA   243\n 6 2017-09-01 BARC           31.6              NA   244\n 7 2017-09-02 BARC           31.1              NA   245\n 8 2017-09-03 BARC           31.1              NA   246\n 9 2017-09-06 BARC           30.8              NA   249\n10 2017-09-07 BARC           30.5              NA   250\n# ℹ 4,179 more rows\n\n\nFeature engineering using a recipe\n\nrequires starting with dataset that is used to provide the columns.\na formula with the dependent variable and the predictors. If . is used as the predictor, that means using all columns other than the dependent variable.\nSteps that modify the data\n\nWe will use the following steps:\n\nstep_rm because we don’t want to use the datetime in the fit.\nstep_naomit because there are na values in the temperature and air_temperature columns. This is used here for illustrative purposes and can also be done by filtering the target data before it is split into training and testing groups\n\nHere are the different recipe steps used above\n\nFilter rows and select columns\n\nHere is our recipe:\n\nour_recipe &lt;- train_data |&gt; \n  recipe(temperature ~ . ) |&gt; \n  step_rm(datetime) |&gt;\n  step_naomit(air_temperature, temperature)\n\nThe recipe should show the steps that will be performed when applying the recipe. Importantly, these steps have not yet been applied, we just have a recipe of what to do.\n\nour_recipe\n\n\n\n\n── Recipe ──────────────────────────────────────────────────────────────────────\n\n\n\n\n\n── Inputs \n\n\nNumber of variables by role\n\n\noutcome:   1\npredictor: 4\n\n\n\n\n\n── Operations \n\n\n• Variables removed: datetime\n\n\n• Removing rows with NA values in: air_temperature temperature\n\n\nStep 3: Specify model, engine, and workflow\nWe need to create the model and engine that we will use. In this example, we are using linear_reg with the mode of regression (as opposed to classification). Setting the mode for a linear regression is not necessary because it only allows regressions but it is included here for completeness.\nThe engine is lm because we are using the standard R function lm. There are a ton of other functions for linear regression modeling that we could use. They would be specified as a different engine.\n\nour_model &lt;- linear_reg(mode = \"regression\") |&gt; \n  set_engine(\"lm\")\n\nYou will see the model, mode, and engine in the model object\n\nour_model \n\nLinear Regression Model Specification (regression)\n\nComputational engine: lm \n\n\nWe now combine the model and the recipe to make a workflow that can be used to fit the training and testing data. workflow() initiates the workflow and add_model and add_recipe add those components to the workflow. Importantly, the workflow has not yet been applied, we just have a description of what to do.\n\nwflow &lt;-\n  workflow() |&gt; \n  add_model(our_model) |&gt; \n  add_recipe(our_recipe)\n\nYou can see that the workflow object has all the components together\n\nwflow\n\n══ Workflow ════════════════════════════════════════════════════════════════════\nPreprocessor: Recipe\nModel: linear_reg()\n\n── Preprocessor ────────────────────────────────────────────────────────────────\n2 Recipe Steps\n\n• step_rm()\n• step_naomit()\n\n── Model ───────────────────────────────────────────────────────────────────────\nLinear Regression Model Specification (regression)\n\nComputational engine: lm \n\n\nStep 4: Train model on Training Data\nWe will use the workflow object to train the model. We need to provide the workflow object and the dataset to the fit function to fit (i.e., train the model)\n\nfit &lt;- wflow |&gt; \n  fit(data = train_data)\n\nYou can see that the fit object is the workflow object + the results of the model fitting\n\nfit\n\n══ Workflow [trained] ══════════════════════════════════════════════════════════\nPreprocessor: Recipe\nModel: linear_reg()\n\n── Preprocessor ────────────────────────────────────────────────────────────────\n2 Recipe Steps\n\n• step_rm()\n• step_naomit()\n\n── Model ───────────────────────────────────────────────────────────────────────\n\nCall:\nstats::lm(formula = ..y ~ ., data = data)\n\nCoefficients:\n    (Intercept)      site_idSUGG  air_temperature              doy  \n       5.108053        -0.725307         0.875235         0.002883  \n\n\nStep 5: Predict Test Data\nNow we will predict the testing data using the model that was fit to the training data.\n\npredictions &lt;- predict(fit, new_data = test_data)\n\nThe predictions are a single column called .pred\n\npredictions\n\n# A tibble: 1,048 × 1\n   .pred\n   &lt;dbl&gt;\n 1    NA\n 2    NA\n 3    NA\n 4    NA\n 5    NA\n 6    NA\n 7    NA\n 8    NA\n 9    NA\n10    NA\n# ℹ 1,038 more rows\n\n\nWe need to combine the .pred column with the testing data using the bind_cols function\n\npred_test &lt;- bind_cols(test_data, predictions)\n\nNow we have a data frame with the prediction and all the predictors used to predict it.\n\npred_test\n\n# A tibble: 1,048 × 6\n   datetime   site_id temperature air_temperature   doy .pred\n   &lt;date&gt;     &lt;chr&gt;         &lt;dbl&gt;           &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;\n 1 2017-09-04 BARC           31.4              NA   247    NA\n 2 2017-09-05 BARC           31.0              NA   248    NA\n 3 2017-09-14 BARC           28.5              NA   257    NA\n 4 2017-09-15 BARC           29.3              NA   258    NA\n 5 2017-09-19 BARC           28.8              NA   262    NA\n 6 2017-09-22 BARC           29.0              NA   265    NA\n 7 2017-09-29 BARC           30.1              NA   272    NA\n 8 2017-10-04 BARC           27.6              NA   277    NA\n 9 2017-10-06 BARC           27.6              NA   279    NA\n10 2017-10-11 BARC           29.6              NA   284    NA\n# ℹ 1,038 more rows\n\n\nStep 6: Evaluate model\nWe will evaluate the performance of our predictions of the testing data using two metrics (rmse and rsq). The function metric_set defines the set of metrics we will be using them. It creates a function called multi_metric() that we will use to calculate the metrics. We pipe in the predicted test data (pred_test) and tell the function that our truth (i.e., observed data) is the temperature column and the predictions (i.e., estimate) is the .pred column\n\nmulti_metric &lt;- metric_set(rmse, rsq)\n\nmetric_table &lt;- pred_test |&gt; \n  multi_metric(truth = temperature, estimate = .pred)\n\nThe resulting table has the metrics for evaluation\n\nmetric_table\n\n# A tibble: 2 × 3\n  .metric .estimator .estimate\n  &lt;chr&gt;   &lt;chr&gt;          &lt;dbl&gt;\n1 rmse    standard       2.42 \n2 rsq     standard       0.819\n\n\nStep 7: Deploy model\nThe final step is to apply your model to predict new data.\nNow read in the new data.\n\ntargets_future &lt;- noaa_future_daily |&gt; \n  mutate(temperature = NA,\n         doy = yday(datetime)) |&gt; \n  filter(parameter == 1) |&gt; \n  select(-parameter)\n\nYou will notice that the temperature is all NA because you don’t know what the carbon stock is of the plot.\n\ntargets_future\n\n# A tibble: 68 × 5\n   datetime   site_id air_temperature temperature   doy\n   &lt;date&gt;     &lt;chr&gt;             &lt;dbl&gt; &lt;lgl&gt;       &lt;dbl&gt;\n 1 2025-02-21 BARC               7.26 NA             52\n 2 2025-02-22 BARC               9.64 NA             53\n 3 2025-02-23 BARC              11.7  NA             54\n 4 2025-02-24 BARC              10.3  NA             55\n 5 2025-02-25 BARC              12.0  NA             56\n 6 2025-02-26 BARC              15.1  NA             57\n 7 2025-02-27 BARC              16.0  NA             58\n 8 2025-02-28 BARC              14.8  NA             59\n 9 2025-03-01 BARC              11.3  NA             60\n10 2025-03-02 BARC              15.1  NA             61\n# ℹ 58 more rows\n\n\nAs in “Step 5: Predict Test Data”, use the model fitting on the training data (fit) to predict the new data.\n\nnew_predictions &lt;- predict(fit, new_data = targets_future)\n\n\ntargets_future &lt;- noaa_future_daily |&gt; \n  mutate(temperature = NA,\n         doy = yday(datetime))\n\ntidymodels_forecast &lt;- data.frame()\n\nfor(i in unique(targets_future$parameter)){\n  curr_ens &lt;- targets_future |&gt; \n    filter(parameter == i) |&gt; \n    select(-parameter)\n  \n  new_predictions &lt;- predict(fit, new_data = curr_ens)\n  curr_ens &lt;- bind_cols(curr_ens, new_predictions) |&gt; \n    mutate(parameter = i)\n  tidymodels_forecast &lt;- bind_rows(tidymodels_forecast, curr_ens)\n}\n\n\ntidymodels_forecasts_EFI &lt;- tidymodels_forecast %&gt;%\n  rename(prediction = .pred) %&gt;%\n  mutate(variable = \"temperature\") |&gt; \n  # For the EFI challenge we only want the forecast for future\n  filter(datetime &gt; Sys.Date()) %&gt;%\n  group_by(site_id, variable) %&gt;%\n  mutate(reference_datetime = min(datetime) - lubridate::days(1),\n         family = \"ensemble\",\n         model_id = \"tidymodels_lm\") %&gt;%\n  select(model_id, datetime, reference_datetime, site_id, family, parameter, variable, prediction)\n\n\ntidymodels_forecasts_EFI |&gt;\n  filter(variable == \"temperature\") |&gt;\n  ggplot(aes(x = datetime, y = prediction, group = parameter)) +\n  geom_line() + \n  facet_wrap(~site_id)\n\n\n\n\n\n\n\n\n\n\n\n12.2.3 Tidymodel extras\n\n12.2.3.1 Changing the model approach\nYou can easily change the modeling approach by changing the model and engine. For example, the following code replaces the linear regression model with a random forest model from the ranger package. All other code used to fit and predict with the model stays the same.\n\nour_model &lt;- rand_forest(mode = \"regression\") |&gt; \n  set_engine(\"ranger\")\n\nA list of models that are available for you to use can be found here. Not all models are designed for all applications. For example, forecasting water temperature requires regression models rather than classification models.\n\n\n12.2.3.2 Tuning hyper-parameters\nSome modeling approaches have parameters that govern how the model fitting is done. These are called hyperparameters. Hyperparameters are different from the parameters that are optimized using data in the modeling fitting step. Tidymodels provides tools to also fit hyperparameters. An example can be found here.",
     "crumbs": [
       "Iterative forecasting using empirical models",
       "<span class='chapter-number'>12</span>  <span class='chapter-title'>Build your own forecast</span>"
@@ -495,7 +495,7 @@
     "href": "project1.html#intro-to-fable-models",
     "title": "12  Build your own forecast",
     "section": "12.3 Intro to Fable Models",
-    "text": "12.3 Intro to Fable Models\nThe Fable modeling framework provides an interface for many common models used in forecasting. The useful thing about Fable is that it is specifically focused on forecasting so uncertainty estimation is a core feature. Fable also has an associated ebook called “Forecasting: Principles and Practice” that is an excellent resource for learning empirical methods in time-series forecasting. Similar to Tidymodels, once you have learned the general structure of how the package works, it opens up many different modeling approaches.\nThe example below forecasts using a simple random walk. Since the random walk starts at the last observation, it shows how to generate a forecast for variables that have a gap between the last observation and the current day.\n\nlibrary(tsibble)\nlibrary(fable)\n\n\n12.3.1 Step 1: Set up targets\n\ntargets &lt;- read_csv('https://data.ecoforecast.org/neon4cast-targets/aquatics/aquatics-targets.csv.gz',\n                    show_col_types = FALSE)\n\n\nmax_horizon &lt;- 35\nvar &lt;- \"temperature\"\nsite &lt;- \"BARC\"\n\nWe want to collect when to start our forecast. If we have observations from today then our forecast starts today. But if there is a gap between our last observation and today, our forecast needs to start on the date of the last observation and forecast until today. Then the forecast needs to extend into the future for our desired horizon. As a result, the horizon that we provide the fable model may be longer than the horizon that we want to forecast into the future. The code below calculates the start date of the forecast and the full horizon (the desired horizon in the future + the gap between the last observation and today)\n\nforecast_starts &lt;- targets |&gt; \n  dplyr::filter(!is.na(observation) & site_id == site & variable == var) |&gt; \n  # Start the day after the most recent non-NA value\n  dplyr::summarise(start_date = max(datetime) + lubridate::days(1)) |&gt;  # Date\n  dplyr::mutate(h = (Sys.Date() - start_date) + max_horizon,\n                h = as.numeric(h)) |&gt;  # Horizon value\n  dplyr::ungroup()\n\nforecast_starts\n\n# A tibble: 1 × 2\n  start_date     h\n  &lt;date&gt;     &lt;dbl&gt;\n1 2025-01-30    55\n\n\nThe targets have to be converted into a time-series tibble (called a tsibble). To make a tsibble you need to define the grouping variables (key) and the common that defines the time-series (index). Many modeling approaches require the datetime to be continuous (no gaps) but allow na values for values in the gaps. The fill_gaps() function fills the gaps with NAs.\n\ntargets_use &lt;- targets |&gt; \n  dplyr::filter(site_id == site,\n                variable == var) %&gt;%\n  tsibble::as_tsibble(key = c('variable', 'site_id'), index = 'datetime') |&gt; \n  tsibble::fill_gaps()   # add NA values up to today (index)\n\nOnce you have a tsibble, you combine it with a modeling approach to generate a model that is ready to be used. In this case, the model is a random walk (RW) with the target of observation.\n\nRW_model &lt;- targets_use |&gt; \n  fabletools::model(RW = fable::RW(observation))\n\nThe model is then used to generate a forecast for h number of time-steps in the future. The uncertainty is calculated using a bootstrap method (bootstrap= T) with 200 ensemble members, e.g. samples (times = 200).\n\nforecast &lt;- RW_model |&gt;  \n  fabletools::generate(h = forecast_starts$h, bootstrap = T, times = 200)\n\nThe forecast is converted into a standard format\n\nRW_forecasts_EFI &lt;- forecast %&gt;%\n  rename(parameter = .rep,\n         prediction = .sim) %&gt;%\n  # For the EFI challenge we only want the forecast for future\n  #filter(datetime &gt; Sys.Date()) %&gt;%\n  group_by(site_id, variable) %&gt;%\n  mutate(reference_datetime = Sys.Date(),\n         family = \"ensemble\",\n         model_id = \"persistenceRW\") %&gt;%\n  select(model_id, datetime, reference_datetime, site_id, family, parameter, variable, prediction)\n\nand plotted. The reference_datetime is shown in blue.\n\nRW_forecasts_EFI |&gt;\n  filter(variable == \"temperature\") |&gt;\n  ggplot(aes(x = datetime, y = prediction, group = parameter)) +\n  geom_line() + \n  geom_vline(aes(xintercept = reference_datetime), color = \"blue\") +\n  facet_wrap(~site_id)\n\n\n\n\n\n\n\n\n\n\n12.3.2 Fable extras\n\n12.3.2.1 Other models\nOther models can be added at the step that the targets are combined with the modeling approach. This example is a time-series linear model with a trend and seasonal term.\n\nRW_model &lt;- targets_use |&gt; \n  fabletools::model(lm = TSLM(observation ~ trend() + season()))\n\nA list of the other models are are: https://fable.tidyverts.org/reference/index.html",
+    "text": "12.3 Intro to Fable Models\nThe Fable modeling framework provides an interface for many common models used in forecasting. The useful thing about Fable is that it is specifically focused on forecasting so uncertainty estimation is a core feature. Fable also has an associated ebook called “Forecasting: Principles and Practice” that is an excellent resource for learning empirical methods in time-series forecasting. Similar to Tidymodels, once you have learned the general structure of how the package works, it opens up many different modeling approaches.\nThe example below forecasts using a simple random walk. Since the random walk starts at the last observation, it shows how to generate a forecast for variables that have a gap between the last observation and the current day.\n\nlibrary(tsibble)\nlibrary(fable)\n\n\n12.3.1 Step 1: Set up targets\n\ntargets &lt;- read_csv('https://data.ecoforecast.org/neon4cast-targets/aquatics/aquatics-targets.csv.gz',\n                    show_col_types = FALSE)\n\n\nmax_horizon &lt;- 35\nvar &lt;- \"temperature\"\nsite &lt;- \"BARC\"\n\nWe want to collect when to start our forecast. If we have observations from today then our forecast starts today. But if there is a gap between our last observation and today, our forecast needs to start on the date of the last observation and forecast until today. Then the forecast needs to extend into the future for our desired horizon. As a result, the horizon that we provide the fable model may be longer than the horizon that we want to forecast into the future. The code below calculates the start date of the forecast and the full horizon (the desired horizon in the future + the gap between the last observation and today)\n\nforecast_starts &lt;- targets |&gt; \n  dplyr::filter(!is.na(observation) & site_id == site & variable == var) |&gt; \n  # Start the day after the most recent non-NA value\n  dplyr::summarise(start_date = max(datetime) + lubridate::days(1)) |&gt;  # Date\n  dplyr::mutate(h = (Sys.Date() - start_date) + max_horizon,\n                h = as.numeric(h)) |&gt;  # Horizon value\n  dplyr::ungroup()\n\nforecast_starts\n\n# A tibble: 1 × 2\n  start_date     h\n  &lt;date&gt;     &lt;dbl&gt;\n1 2025-02-18    38\n\n\nThe targets have to be converted into a time-series tibble (called a tsibble). To make a tsibble you need to define the grouping variables (key) and the common that defines the time-series (index). Many modeling approaches require the datetime to be continuous (no gaps) but allow na values for values in the gaps. The fill_gaps() function fills the gaps with NAs.\n\ntargets_use &lt;- targets |&gt; \n  dplyr::filter(site_id == site,\n                variable == var) %&gt;%\n  tsibble::as_tsibble(key = c('variable', 'site_id'), index = 'datetime') |&gt; \n  tsibble::fill_gaps()   # add NA values up to today (index)\n\nOnce you have a tsibble, you combine it with a modeling approach to generate a model that is ready to be used. In this case, the model is a random walk (RW) with the target of observation.\n\nRW_model &lt;- targets_use |&gt; \n  fabletools::model(RW = fable::RW(observation))\n\nThe model is then used to generate a forecast for h number of time-steps in the future. The uncertainty is calculated using a bootstrap method (bootstrap= T) with 200 ensemble members, e.g. samples (times = 200).\n\nforecast &lt;- RW_model |&gt;  \n  fabletools::generate(h = forecast_starts$h, bootstrap = T, times = 200)\n\nThe forecast is converted into a standard format\n\nRW_forecasts_EFI &lt;- forecast %&gt;%\n  rename(parameter = .rep,\n         prediction = .sim) %&gt;%\n  # For the EFI challenge we only want the forecast for future\n  #filter(datetime &gt; Sys.Date()) %&gt;%\n  group_by(site_id, variable) %&gt;%\n  mutate(reference_datetime = Sys.Date(),\n         family = \"ensemble\",\n         model_id = \"persistenceRW\") %&gt;%\n  select(model_id, datetime, reference_datetime, site_id, family, parameter, variable, prediction)\n\nand plotted. The reference_datetime is shown in blue.\n\nRW_forecasts_EFI |&gt;\n  filter(variable == \"temperature\") |&gt;\n  ggplot(aes(x = datetime, y = prediction, group = parameter)) +\n  geom_line() + \n  geom_vline(aes(xintercept = reference_datetime), color = \"blue\") +\n  facet_wrap(~site_id)\n\n\n\n\n\n\n\n\n\n\n12.3.2 Fable extras\n\n12.3.2.1 Other models\nOther models can be added at the step that the targets are combined with the modeling approach. This example is a time-series linear model with a trend and seasonal term.\n\nRW_model &lt;- targets_use |&gt; \n  fabletools::model(lm = TSLM(observation ~ trend() + season()))\n\nA list of the other models are are: https://fable.tidyverts.org/reference/index.html",
     "crumbs": [
       "Iterative forecasting using empirical models",
       "<span class='chapter-number'>12</span>  <span class='chapter-title'>Build your own forecast</span>"
@@ -1056,7 +1056,7 @@
     "href": "bayesian-process-models.html",
     "title": "19  Applying Bayesian calibration methods to a process model",
     "section": "",
-    "text": "19.1 Running MCMC on the forest process model\nlibrary(tidyverse)\nlibrary(patchwork)\nsource(\"R/helpers.R\")\nsource(\"R/forest_model.R\")\nset.seed(100)\nsite &lt;- \"OSBS\"\nRead in observations\nobs &lt;- read_csv(\"data/site_carbon_data.csv\", show_col_types = FALSE)\nSet up dates of simulation, parameters, initial conditions, and meteorology inputs\nsim_dates &lt;- seq(as_date(\"2022-01-01\"), Sys.Date() - 2, by = \"1 day\")\n\nens_members &lt;- 1\nparams &lt;- list()\nparams$alpha &lt;- rep(0.02, ens_members)\nparams$SLA &lt;- rep(4.74, ens_members)\nparams$leaf_frac &lt;- rep(0.315, ens_members)\nparams$Ra_frac &lt;- rep(0.5, ens_members)\nparams$Rbasal &lt;- rep(0.002, ens_members)\nparams$Q10 &lt;- rep(2.1, ens_members)\nparams$litterfall_rate &lt;- rep(1/(2.0*365), ens_members) #Two year leaf lifespan\nparams$litterfall_start &lt;- rep(250, ens_members)\nparams$litterfall_length&lt;- rep(60, ens_members)\nparams$mortality &lt;- rep(0.00015, ens_members) #Wood lives about 18 years on average (all trees, branches, roots, course roots)\nparams$sigma.leaf &lt;- rep(0.0, ens_members) #0.01 \nparams$sigma.wood &lt;- rep(0.0, ens_members) #0.01 ## wood biomass\nparams$sigma.soil &lt;- rep(0.0, ens_members)# 0.01\nparams &lt;- as.data.frame(params)\n\n\nstate_init &lt;- rep(NA, 3)\n\nstate_init[1] &lt;- obs |&gt; \n  filter(datetime %in% sim_dates,\n         variable == \"lai\") |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  mutate(observation = observation / (mean(params$SLA) * 0.1)) |&gt; \n  pull(observation)\n\nstate_init[2] &lt;- obs |&gt; \n  filter(variable == \"wood\",\n         datetime %in% sim_dates) |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  pull(observation)\n\nstate_init[3] &lt;- obs |&gt; \n  filter(variable == \"som\") |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  pull(observation)\n\ninputs &lt;- get_historical_met(site = site, sim_dates, use_mean = TRUE)\ninputs_ensemble &lt;- assign_met_ensembles(inputs, ens_members)\nSet up MCMC configuration\n#Set MCMC Configuration\nnum_iter &lt;- 1500\nlog_likelihood_prior_current &lt;- -10000000000\naccept &lt;- 0\n\n\n#Initialize chain\nnum_pars &lt;- 3\njump_params &lt;- c(0.001, 0.0002, 1)\nfit_params &lt;- array(NA, dim = c(num_pars, num_iter))\nfit_params[1, 1] &lt;- params$alpha\nfit_params[2, 1] &lt;- params$Rbasal\nfit_params[3, 1] &lt;- params$litterfall_start\nprior_mean &lt;- c(0.029, 0.002, 200)\nprior_sd &lt;- c(0.005, 0.0005, 10)\nRun MCMC\nfor(iter in 2:num_iter){\n  \n  #Loop through parameter value\n  \n  for(j in 1:num_pars){\n    \n    proposed_pars &lt;- fit_params[, iter - 1]\n    proposed_pars[j] &lt;- rnorm(1, mean = fit_params[j, iter - 1], sd = jump_params[j])\n    \n    log_prior &lt;- dnorm(proposed_pars[1], mean = prior_mean[1], sd = prior_sd[1], log = TRUE) + \n      dnorm(proposed_pars[2], mean = prior_mean[2], sd = prior_sd[2], log = TRUE) +\n      dnorm(proposed_pars[3], mean = prior_mean[3], sd = prior_sd[3], log = TRUE)\n    \n    params$alpha  &lt;- proposed_pars[1]\n    params$Rbasal  &lt;- proposed_pars[2]\n    params$litterfall_start &lt;- proposed_pars[3]\n    \n    #Set initial conditions\n    output &lt;- array(NA, dim = c(length(sim_dates), ens_members, 12)) #12 is the number of outputs\n    output[1, , 1] &lt;- state_init[1]\n    output[1, , 2] &lt;- state_init[2]\n    output[1, , 3] &lt;- state_init[3]\n    \n    for(t in 2:length(sim_dates)){\n      output[t, , ]  &lt;- forest_model(t, \n                                     states = matrix(output[t-1 , , 1:3], nrow = ens_members) , \n                                     parms = params, \n                                     inputs = matrix(inputs_ensemble[t ,, ], nrow = ens_members))\n    }\n    \n    output_df &lt;- output_to_df(output, sim_dates, sim_name = \"fitting\")\n    \n    combined_output_obs &lt;- combine_model_obs(output_df, \n                                             obs,\n                                             variables = c(\"lai\", \"wood\", \"som\", \"nee\"), \n                                             sds = c(0.1, 1, 20, 0.01))\n    \n    log_likelihood &lt;- sum(dnorm(x =  combined_output_obs$observation, \n                                mean = combined_output_obs$prediction, \n                                sd = combined_output_obs$sds, log = TRUE))\n    \n    log_likelihood_prior_proposed &lt;- log_prior + log_likelihood\n    \n    z &lt;- exp(log_likelihood_prior_proposed - log_likelihood_prior_current)\n    \n    r &lt;- runif(1, min = 0, max = 1)\n    \n    if(z &gt;  r){\n      fit_params[j, iter] &lt;- proposed_pars[j]\n      log_likelihood_prior_current &lt;- log_likelihood_prior_proposed\n      accept &lt;- accept + 1\n    }else{\n      fit_params[j, iter] &lt;- fit_params[j, iter - 1]\n      log_likelihood_prior_current &lt;- log_likelihood_prior_current #this calculation isn't necessary but is here to show you the logic\n    }\n  }\n}\nExamine acceptance rate (goal is 23-45%)\naccept / (num_iter * num_pars)\n\n[1] 0.3535556\nProcess MCMC chain by removing the first 500 iterations and pivoting to a long format\nnburn &lt;- 500\nparameter_MCMC &lt;- tibble(iter = nburn:num_iter,\n            alpha = fit_params[1, nburn:num_iter],\n            Rbasal = fit_params[2, nburn:num_iter],\n            litterfall_start = fit_params[3, nburn:num_iter]) %&gt;%\n  pivot_longer(-iter, values_to = \"value\", names_to = \"parameter\")\nFigure 19.1\np1 &lt;- ggplot(parameter_MCMC, aes(x = iter, y = value)) +\n  geom_line() +\n  facet_wrap(~parameter, scales = \"free\") +\n  theme_bw()\n\np2 &lt;- ggplot(parameter_MCMC, aes(x = value)) +\n  geom_histogram() +\n  facet_wrap(~parameter, scales = \"free\") +\n  theme_bw()\n\np1 / p2\n\n\n\n\n\n\n\nFigure 19.1: The MCMC chains and posterior distributions for the calibrated parameters",
+    "text": "19.1 Running MCMC on the forest process model\nlibrary(tidyverse)\nlibrary(patchwork)\nsource(\"R/helpers.R\")\nsource(\"R/forest_model.R\")\nset.seed(100)\nsite &lt;- \"OSBS\"\nRead in observations\nobs &lt;- read_csv(\"data/site_carbon_data.csv\", show_col_types = FALSE)\nSet up dates of simulation, parameters, initial conditions, and meteorology inputs\nsim_dates &lt;- seq(as_date(\"2022-01-01\"), Sys.Date() - 2, by = \"1 day\")\n\nens_members &lt;- 1\nparams &lt;- list()\nparams$alpha &lt;- rep(0.02, ens_members)\nparams$SLA &lt;- rep(4.74, ens_members)\nparams$leaf_frac &lt;- rep(0.315, ens_members)\nparams$Ra_frac &lt;- rep(0.5, ens_members)\nparams$Rbasal &lt;- rep(0.002, ens_members)\nparams$Q10 &lt;- rep(2.1, ens_members)\nparams$litterfall_rate &lt;- rep(1/(2.0*365), ens_members) #Two year leaf lifespan\nparams$litterfall_start &lt;- rep(250, ens_members)\nparams$litterfall_length&lt;- rep(60, ens_members)\nparams$mortality &lt;- rep(0.00015, ens_members) #Wood lives about 18 years on average (all trees, branches, roots, course roots)\nparams$sigma.leaf &lt;- rep(0.0, ens_members) #0.01 \nparams$sigma.wood &lt;- rep(0.0, ens_members) #0.01 ## wood biomass\nparams$sigma.soil &lt;- rep(0.0, ens_members)# 0.01\nparams &lt;- as.data.frame(params)\n\n\nstate_init &lt;- rep(NA, 3)\n\nstate_init[1] &lt;- obs |&gt; \n  filter(datetime %in% sim_dates,\n         variable == \"lai\") |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  mutate(observation = observation / (mean(params$SLA) * 0.1)) |&gt; \n  pull(observation)\n\nstate_init[2] &lt;- obs |&gt; \n  filter(variable == \"wood\",\n         datetime %in% sim_dates) |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  pull(observation)\n\nstate_init[3] &lt;- obs |&gt; \n  filter(variable == \"som\") |&gt; \n  na.omit() |&gt; \n  slice(1) |&gt; \n  pull(observation)\n\ninputs &lt;- get_historical_met(site = site, sim_dates, use_mean = TRUE)\ninputs_ensemble &lt;- assign_met_ensembles(inputs, ens_members)\nSet up MCMC configuration\n#Set MCMC Configuration\nnum_iter &lt;- 1500\nlog_likelihood_prior_current &lt;- -10000000000\naccept &lt;- 0\n\n\n#Initialize chain\nnum_pars &lt;- 3\njump_params &lt;- c(0.001, 0.0002, 1)\nfit_params &lt;- array(NA, dim = c(num_pars, num_iter))\nfit_params[1, 1] &lt;- params$alpha\nfit_params[2, 1] &lt;- params$Rbasal\nfit_params[3, 1] &lt;- params$litterfall_start\nprior_mean &lt;- c(0.029, 0.002, 200)\nprior_sd &lt;- c(0.005, 0.0005, 10)\nRun MCMC\nfor(iter in 2:num_iter){\n  \n  #Loop through parameter value\n  \n  for(j in 1:num_pars){\n    \n    proposed_pars &lt;- fit_params[, iter - 1]\n    proposed_pars[j] &lt;- rnorm(1, mean = fit_params[j, iter - 1], sd = jump_params[j])\n    \n    log_prior &lt;- dnorm(proposed_pars[1], mean = prior_mean[1], sd = prior_sd[1], log = TRUE) + \n      dnorm(proposed_pars[2], mean = prior_mean[2], sd = prior_sd[2], log = TRUE) +\n      dnorm(proposed_pars[3], mean = prior_mean[3], sd = prior_sd[3], log = TRUE)\n    \n    params$alpha  &lt;- proposed_pars[1]\n    params$Rbasal  &lt;- proposed_pars[2]\n    params$litterfall_start &lt;- proposed_pars[3]\n    \n    #Set initial conditions\n    output &lt;- array(NA, dim = c(length(sim_dates), ens_members, 12)) #12 is the number of outputs\n    output[1, , 1] &lt;- state_init[1]\n    output[1, , 2] &lt;- state_init[2]\n    output[1, , 3] &lt;- state_init[3]\n    \n    for(t in 2:length(sim_dates)){\n      output[t, , ]  &lt;- forest_model(t, \n                                     states = matrix(output[t-1 , , 1:3], nrow = ens_members) , \n                                     parms = params, \n                                     inputs = matrix(inputs_ensemble[t ,, ], nrow = ens_members))\n    }\n    \n    output_df &lt;- output_to_df(output, sim_dates, sim_name = \"fitting\")\n    \n    combined_output_obs &lt;- combine_model_obs(output_df, \n                                             obs,\n                                             variables = c(\"lai\", \"wood\", \"som\", \"nee\"), \n                                             sds = c(0.1, 1, 20, 0.01))\n    \n    log_likelihood &lt;- sum(dnorm(x =  combined_output_obs$observation, \n                                mean = combined_output_obs$prediction, \n                                sd = combined_output_obs$sds, log = TRUE))\n    \n    log_likelihood_prior_proposed &lt;- log_prior + log_likelihood\n    \n    z &lt;- exp(log_likelihood_prior_proposed - log_likelihood_prior_current)\n    \n    r &lt;- runif(1, min = 0, max = 1)\n    \n    if(z &gt;  r){\n      fit_params[j, iter] &lt;- proposed_pars[j]\n      log_likelihood_prior_current &lt;- log_likelihood_prior_proposed\n      accept &lt;- accept + 1\n    }else{\n      fit_params[j, iter] &lt;- fit_params[j, iter - 1]\n      log_likelihood_prior_current &lt;- log_likelihood_prior_current #this calculation isn't necessary but is here to show you the logic\n    }\n  }\n}\nExamine acceptance rate (goal is 23-45%)\naccept / (num_iter * num_pars)\n\n[1] 0.3555556\nProcess MCMC chain by removing the first 500 iterations and pivoting to a long format\nnburn &lt;- 500\nparameter_MCMC &lt;- tibble(iter = nburn:num_iter,\n            alpha = fit_params[1, nburn:num_iter],\n            Rbasal = fit_params[2, nburn:num_iter],\n            litterfall_start = fit_params[3, nburn:num_iter]) %&gt;%\n  pivot_longer(-iter, values_to = \"value\", names_to = \"parameter\")\nFigure 19.1\np1 &lt;- ggplot(parameter_MCMC, aes(x = iter, y = value)) +\n  geom_line() +\n  facet_wrap(~parameter, scales = \"free\") +\n  theme_bw()\n\np2 &lt;- ggplot(parameter_MCMC, aes(x = value)) +\n  geom_histogram() +\n  facet_wrap(~parameter, scales = \"free\") +\n  theme_bw()\n\np1 / p2\n\n\n\n\n\n\n\nFigure 19.1: The MCMC chains and posterior distributions for the calibrated parameters",
     "crumbs": [
       "Iterative forecasting using process-models",
       "<span class='chapter-number'>19</span>  <span class='chapter-title'>Applying Bayesian calibration methods to a process model</span>"
@@ -1133,7 +1133,7 @@
     "href": "process-model-forecast.html#step-4-save-forecast-and-data-assimilation-output",
     "title": "21  Forecast - analysis cycle",
     "section": "21.4 Step 4: Save forecast and data assimilation output",
-    "text": "21.4 Step 4: Save forecast and data assimilation output\nConvert forecast to a dataframe.\n\noutput_df &lt;- output_to_df(forecast, sim_dates, sim_name = \"simple_forest_model\")\n\n\nforecast_weighted &lt;- array(NA, dim = c(length(sim_dates), ens_members, 12))\nparams_weighted &lt;- array(NA, dim = c(length(sim_dates) ,ens_members , num_pars))\nfor(t in 1:length(sim_dates)){\n  wt_norm &lt;-  wt[t, ]/sum(wt[t, ])\n  resample_index &lt;- sample(1:ens_members, ens_members, replace = TRUE, prob = wt_norm ) \n  forecast_weighted[t, , ] &lt;- forecast[t, resample_index, 1:12] \n  params_weighted[t, , ] &lt;- fit_params[t,resample_index, ] \n}\noutput_df &lt;- output_to_df(forecast_weighted, sim_dates, sim_name = \"simple_forest_model\")\nparameter_df &lt;- parameters_to_df(params_weighted, sim_dates, sim_name = \"simple_forest_model\", param_names = c(\"alpha\",\"Rbasal\"))\n\nFigure 21.1 is a visualization of the forecast. The vertical line is where the look-back period transitions to the future.\n\n\n\n\n\n\n\n\nFigure 21.1: Forecast median and 90% CI. The look-back and forecast periods are included with the transition denoted with a vertical line.\n\n\n\n\n\nSave the states and weights for use as initial conditions in the next forecast.\n\nanalysis$params &lt;- params\nsave(analysis, file = \"data/PF_analysis_1.Rdata\")\n\nConvert to the format and units required by the NEON Ecological Forecasting Challenge\n\nefi_output &lt;- output_df |&gt; \n  mutate(datetime = as_datetime(datetime),\n         model_id = \"bookcast_forest\",\n         duration = \"P1D\",\n         project_id = \"neon4cast\",\n         site_id = site,\n         family = \"ensemble\",\n         reference_datetime = as_datetime(reference_datetime)) |&gt; \n  rename(parameter = ensemble) |&gt; \n  filter(datetime &gt;= reference_datetime) |&gt; \n  filter(variable == \"nee\") |&gt; \n  mutate(prediction = prediction  / 0.01) #Convert from MgC/ha/Day to gC/m2/day\n\nWrite the forecast to a CSV file\n\nfile_name &lt;- paste0(\"data/terrestrial_daily-\", reference_datetime, \"-bookcast_forest.csv\")\nwrite_csv(efi_output, file_name)\n\nand submit to the Challenge. The submission uses the submit function in the neon4cast package(remotes::install_github(\"eco4cast/neon4cast\"))\n\nneon4cast::submit(file_name, ask = FALSE)\n\nvalidating that file matches required standard\n\n\ndata/terrestrial_daily-2025-02-19-bookcast_forest.csv\n\n\n✔ file has model_id column\n✔ forecasted variables found correct variable + prediction column\n✔ nee is a valid variable name\n✔ file has correct family and parameter columns\n✔ file has site_id column\n✔ file has datetime column\n✔ file has correct datetime column\n✔ file has correct duration column\n✔ file has project_id column\n✔ file has reference_datetime column\nForecast format is valid\n\nThank you for submitting!",
+    "text": "21.4 Step 4: Save forecast and data assimilation output\nConvert forecast to a dataframe.\n\noutput_df &lt;- output_to_df(forecast, sim_dates, sim_name = \"simple_forest_model\")\n\n\nforecast_weighted &lt;- array(NA, dim = c(length(sim_dates), ens_members, 12))\nparams_weighted &lt;- array(NA, dim = c(length(sim_dates) ,ens_members , num_pars))\nfor(t in 1:length(sim_dates)){\n  wt_norm &lt;-  wt[t, ]/sum(wt[t, ])\n  resample_index &lt;- sample(1:ens_members, ens_members, replace = TRUE, prob = wt_norm ) \n  forecast_weighted[t, , ] &lt;- forecast[t, resample_index, 1:12] \n  params_weighted[t, , ] &lt;- fit_params[t,resample_index, ] \n}\noutput_df &lt;- output_to_df(forecast_weighted, sim_dates, sim_name = \"simple_forest_model\")\nparameter_df &lt;- parameters_to_df(params_weighted, sim_dates, sim_name = \"simple_forest_model\", param_names = c(\"alpha\",\"Rbasal\"))\n\nFigure 21.1 is a visualization of the forecast. The vertical line is where the look-back period transitions to the future.\n\n\n\n\n\n\n\n\nFigure 21.1: Forecast median and 90% CI. The look-back and forecast periods are included with the transition denoted with a vertical line.\n\n\n\n\n\nSave the states and weights for use as initial conditions in the next forecast.\n\nanalysis$params &lt;- params\nsave(analysis, file = \"data/PF_analysis_1.Rdata\")\n\nConvert to the format and units required by the NEON Ecological Forecasting Challenge\n\nefi_output &lt;- output_df |&gt; \n  mutate(datetime = as_datetime(datetime),\n         model_id = \"bookcast_forest\",\n         duration = \"P1D\",\n         project_id = \"neon4cast\",\n         site_id = site,\n         family = \"ensemble\",\n         reference_datetime = as_datetime(reference_datetime)) |&gt; \n  rename(parameter = ensemble) |&gt; \n  filter(datetime &gt;= reference_datetime) |&gt; \n  filter(variable == \"nee\") |&gt; \n  mutate(prediction = prediction  / 0.01) #Convert from MgC/ha/Day to gC/m2/day\n\nWrite the forecast to a CSV file\n\nfile_name &lt;- paste0(\"data/terrestrial_daily-\", reference_datetime, \"-bookcast_forest.csv\")\nwrite_csv(efi_output, file_name)\n\nand submit to the Challenge. The submission uses the submit function in the neon4cast package(remotes::install_github(\"eco4cast/neon4cast\"))\n\nneon4cast::submit(file_name, ask = FALSE)\n\nvalidating that file matches required standard\n\n\ndata/terrestrial_daily-2025-02-21-bookcast_forest.csv\n\n\n✔ file has model_id column\n✔ forecasted variables found correct variable + prediction column\n✔ nee is a valid variable name\n✔ file has correct family and parameter columns\n✔ file has site_id column\n✔ file has datetime column\n✔ file has correct datetime column\n✔ file has correct duration column\n✔ file has project_id column\n✔ file has reference_datetime column\nForecast format is valid\n\nThank you for submitting!",
     "crumbs": [
       "Iterative forecasting using process-models",
       "<span class='chapter-number'>21</span>  <span class='chapter-title'>Forecast - analysis cycle</span>"
@@ -1177,7 +1177,7 @@
     "href": "process-model-forecast-evaluation.html#aggregated-scores",
     "title": "22  Analyzing forecasts",
     "section": "22.2 Aggregated scores",
-    "text": "22.2 Aggregated scores\nWe can look at the mean score for the process model but this provides very little context for the quality of forecast. It is more informative to compare the score to the score from other models.\n\ndf |&gt; \n  summarise(mean_crps = mean(crps, na.rm = TRUE))\n\n# A tibble: 1 × 1\n  mean_crps\n      &lt;dbl&gt;\n1     0.872",
+    "text": "22.2 Aggregated scores\nWe can look at the mean score for the process model but this provides very little context for the quality of forecast. It is more informative to compare the score to the score from other models.\n\ndf |&gt; \n  summarise(mean_crps = mean(crps, na.rm = TRUE))\n\n# A tibble: 1 × 1\n  mean_crps\n      &lt;dbl&gt;\n1     0.870",
     "crumbs": [
       "Iterative forecasting using process-models",
       "<span class='chapter-number'>22</span>  <span class='chapter-title'>Analyzing forecasts</span>"