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  <title>Chapter 35 Spatially Continuous Data III | An Introduction to Spatial Data Analysis and Statistics: A Course in R</title>
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  <meta property="og:description" content="This book was created as a resource for teaching applied spatial statistics at McMaster University by Antonio Paez, with support from Anastassios Dardas, Rajveer Ubhi, Megan Coad, and Alexis Polidoro. The book is published with support of an Open Educational Resources grant from MacPherson Institute, McMaster University." />
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<li><a href="./">Spatial Analysis in R</a></li>

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<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#spatial-analysis-and-spatial-statistics"><i class="fa fa-check"></i>Spatial Analysis and Spatial Statistics</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#why-this-text"><i class="fa fa-check"></i>Why this Text?</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#plan"><i class="fa fa-check"></i>Plan</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#audience"><i class="fa fa-check"></i>Audience</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#requisites"><i class="fa fa-check"></i>Requisites</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#words-of-appreciation"><i class="fa fa-check"></i>Words of Appreciation</a></li>
<li class="chapter" data-level="" data-path="preface.html"><a href="preface.html#versioning"><i class="fa fa-check"></i>Versioning</a></li>
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<li class="part"><span><b>I Part I: Getting to Know the Technology</b></span></li>
<li class="chapter" data-level="1" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html"><i class="fa fa-check"></i><b>1</b> Preliminaries: Installing <code>R</code> and RStudio</a>
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<li class="chapter" data-level="1.1" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#introduction"><i class="fa fa-check"></i><b>1.1</b> Introduction</a></li>
<li class="chapter" data-level="1.2" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#learning-objectives"><i class="fa fa-check"></i><b>1.2</b> Learning Objectives</a></li>
<li class="chapter" data-level="1.3" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#r-the-open-statistical-computing-project"><i class="fa fa-check"></i><b>1.3</b> <code>R</code>: The Open Statistical Computing Project</a>
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<li class="chapter" data-level="1.3.1" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#what-is-r"><i class="fa fa-check"></i><b>1.3.1</b> What is <code>R</code>?</a></li>
<li class="chapter" data-level="1.3.2" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#the-rstudio-ide"><i class="fa fa-check"></i><b>1.3.2</b> The RStudio IDE</a></li>
</ul></li>
<li class="chapter" data-level="1.4" data-path="preliminaries-installing-r-and-rstudio.html"><a href="preliminaries-installing-r-and-rstudio.html#packages-in-r"><i class="fa fa-check"></i><b>1.4</b> Packages in R</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html"><i class="fa fa-check"></i><b>2</b> Basic Operations and Data Structures in <code>R</code></a>
<ul>
<li class="chapter" data-level="2.1" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#learning-objectives-1"><i class="fa fa-check"></i><b>2.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="2.2" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#rstudio-ide"><i class="fa fa-check"></i><b>2.2</b> RStudio IDE</a></li>
<li class="chapter" data-level="2.3" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#some-basic-operations"><i class="fa fa-check"></i><b>2.3</b> Some Basic Operations</a></li>
<li class="chapter" data-level="2.4" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#data-classes-in-r"><i class="fa fa-check"></i><b>2.4</b> Data Classes in R</a></li>
<li class="chapter" data-level="2.5" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#data-types-in-r"><i class="fa fa-check"></i><b>2.5</b> Data Types in R</a></li>
<li class="chapter" data-level="2.6" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#indexing-and-data-transformations"><i class="fa fa-check"></i><b>2.6</b> Indexing and Data Transformations</a></li>
<li class="chapter" data-level="2.7" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#visualization"><i class="fa fa-check"></i><b>2.7</b> Visualization</a></li>
<li class="chapter" data-level="2.8" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#creating-a-simple-map"><i class="fa fa-check"></i><b>2.8</b> Creating a Simple Map</a></li>
<li class="chapter" data-level="2.9" data-path="basic-operations-and-data-structures-in-r.html"><a href="basic-operations-and-data-structures-in-r.html#references"><i class="fa fa-check"></i><b>2.9</b> References</a></li>
</ul></li>
<li class="part"><span><b>II Part II: Statistics and Maps</b></span></li>
<li class="chapter" data-level="3" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html"><i class="fa fa-check"></i><b>3</b> Introduction to Mapping in <code>R</code></a>
<ul>
<li class="chapter" data-level="3.1" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#learning-objectives-2"><i class="fa fa-check"></i><b>3.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="3.2" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#suggested-readings"><i class="fa fa-check"></i><b>3.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="3.3" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#preliminaries"><i class="fa fa-check"></i><b>3.3</b> Preliminaries</a></li>
<li class="chapter" data-level="3.4" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#packages"><i class="fa fa-check"></i><b>3.4</b> Packages</a></li>
<li class="chapter" data-level="3.5" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#exploring-dataframes-and-a-simple-proportional-symbols-map"><i class="fa fa-check"></i><b>3.5</b> Exploring Dataframes and a Simple Proportional Symbols Map</a></li>
<li class="chapter" data-level="3.6" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#improving-on-the-proportional-symbols-map"><i class="fa fa-check"></i><b>3.6</b> Improving on the Proportional Symbols Map</a></li>
<li class="chapter" data-level="3.7" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#some-simple-spatial-analysis"><i class="fa fa-check"></i><b>3.7</b> Some Simple Spatial Analysis</a></li>
<li class="chapter" data-level="3.8" data-path="introduction-to-mapping-in-r.html"><a href="introduction-to-mapping-in-r.html#other-resources"><i class="fa fa-check"></i><b>3.8</b> Other Resources</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html"><i class="fa fa-check"></i><b>4</b> Activity 1: Statistical Maps I</a>
<ul>
<li class="chapter" data-level="4.1" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html#housekeeping-questions"><i class="fa fa-check"></i><b>4.1</b> Housekeeping Questions</a></li>
<li class="chapter" data-level="4.2" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html#learning-objectives-3"><i class="fa fa-check"></i><b>4.2</b> Learning Objectives</a></li>
<li class="chapter" data-level="4.3" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html#preliminaries-1"><i class="fa fa-check"></i><b>4.3</b> Preliminaries</a></li>
<li class="chapter" data-level="4.4" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html#creating-a-simple-thematic-map"><i class="fa fa-check"></i><b>4.4</b> Creating a simple thematic map</a></li>
<li class="chapter" data-level="4.5" data-path="activity-1-statistical-maps-i.html"><a href="activity-1-statistical-maps-i.html#activity"><i class="fa fa-check"></i><b>4.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html"><i class="fa fa-check"></i><b>5</b> Mapping in R: Continued</a>
<ul>
<li class="chapter" data-level="5.1" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#learning-objectives-4"><i class="fa fa-check"></i><b>5.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="5.2" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#suggested-readings-1"><i class="fa fa-check"></i><b>5.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="5.3" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#preliminaries-2"><i class="fa fa-check"></i><b>5.3</b> Preliminaries</a></li>
<li class="chapter" data-level="5.4" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#summarizing-a-dataframe"><i class="fa fa-check"></i><b>5.4</b> Summarizing a Dataframe</a></li>
<li class="chapter" data-level="5.5" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#factors"><i class="fa fa-check"></i><b>5.5</b> Factors</a></li>
<li class="chapter" data-level="5.6" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#subsetting-data"><i class="fa fa-check"></i><b>5.6</b> Subsetting Data</a></li>
<li class="chapter" data-level="5.7" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#pipe-operator"><i class="fa fa-check"></i><b>5.7</b> Pipe Operator</a></li>
<li class="chapter" data-level="5.8" data-path="mapping-in-r-continued.html"><a href="mapping-in-r-continued.html#more-on-visualization"><i class="fa fa-check"></i><b>5.8</b> More on Visualization</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html"><i class="fa fa-check"></i><b>6</b> Activity 2: Statistical Maps II</a>
<ul>
<li class="chapter" data-level="6.1" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html#housekeeping-questions-1"><i class="fa fa-check"></i><b>6.1</b> Housekeeping Questions</a></li>
<li class="chapter" data-level="6.2" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html#learning-objectives-5"><i class="fa fa-check"></i><b>6.2</b> Learning objectives</a></li>
<li class="chapter" data-level="6.3" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html#suggested-reading"><i class="fa fa-check"></i><b>6.3</b> Suggested reading</a></li>
<li class="chapter" data-level="6.4" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html#preliminaries-3"><i class="fa fa-check"></i><b>6.4</b> Preliminaries</a></li>
<li class="chapter" data-level="6.5" data-path="activity-2-statistical-maps-ii.html"><a href="activity-2-statistical-maps-ii.html#activity-1"><i class="fa fa-check"></i><b>6.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><i class="fa fa-check"></i><b>7</b> Maps as Processes: Null Landscapes, Spatial Processes, and Statistical Maps</a>
<ul>
<li class="chapter" data-level="7.1" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#learning-objectives-6"><i class="fa fa-check"></i><b>7.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="7.2" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#suggested-readings-2"><i class="fa fa-check"></i><b>7.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="7.3" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#preliminaries-4"><i class="fa fa-check"></i><b>7.3</b> Preliminaries</a></li>
<li class="chapter" data-level="7.4" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#random-numbers"><i class="fa fa-check"></i><b>7.4</b> Random Numbers</a></li>
<li class="chapter" data-level="7.5" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#null-landscapes"><i class="fa fa-check"></i><b>7.5</b> Null Landscapes</a></li>
<li class="chapter" data-level="7.6" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#stochastic-processes"><i class="fa fa-check"></i><b>7.6</b> Stochastic Processes</a></li>
<li class="chapter" data-level="7.7" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#simulating-spatial-processes"><i class="fa fa-check"></i><b>7.7</b> Simulating Spatial Processes</a></li>
<li class="chapter" data-level="7.8" data-path="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html"><a href="maps-as-processes-null-landscapes-spatial-processes-and-statistical-maps.html#processes-and-patterns"><i class="fa fa-check"></i><b>7.8</b> Processes and Patterns</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html"><i class="fa fa-check"></i><b>8</b> Activity 3: Maps as Processes</a>
<ul>
<li class="chapter" data-level="8.1" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html#practice-questions"><i class="fa fa-check"></i><b>8.1</b> Practice Questions</a></li>
<li class="chapter" data-level="8.2" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html#learning-objectives-7"><i class="fa fa-check"></i><b>8.2</b> Learning Objectives</a></li>
<li class="chapter" data-level="8.3" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html#suggested-reading-1"><i class="fa fa-check"></i><b>8.3</b> Suggested Reading</a></li>
<li class="chapter" data-level="8.4" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html#preliminaries-5"><i class="fa fa-check"></i><b>8.4</b> Preliminaries</a></li>
<li class="chapter" data-level="8.5" data-path="activity-3-maps-as-processes.html"><a href="activity-3-maps-as-processes.html#activity-2"><i class="fa fa-check"></i><b>8.5</b> Activity</a></li>
</ul></li>
<li class="part"><span><b>III Part III: Analysis of Point Patterns</b></span></li>
<li class="chapter" data-level="9" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html"><i class="fa fa-check"></i><b>9</b> Point Pattern Analysis I</a>
<ul>
<li class="chapter" data-level="9.1" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#learning-objectives-8"><i class="fa fa-check"></i><b>9.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="9.2" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#suggested-readings-3"><i class="fa fa-check"></i><b>9.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="9.3" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#preliminaries-6"><i class="fa fa-check"></i><b>9.3</b> Preliminaries</a></li>
<li class="chapter" data-level="9.4" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#point-patterns"><i class="fa fa-check"></i><b>9.4</b> Point Patterns</a></li>
<li class="chapter" data-level="9.5" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#processes-and-point-patterns"><i class="fa fa-check"></i><b>9.5</b> Processes and Point Patterns</a></li>
<li class="chapter" data-level="9.6" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#intensity-and-density"><i class="fa fa-check"></i><b>9.6</b> Intensity and Density</a></li>
<li class="chapter" data-level="9.7" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#quadrats-and-density-maps"><i class="fa fa-check"></i><b>9.7</b> Quadrats and Density Maps</a></li>
<li class="chapter" data-level="9.8" data-path="point-pattern-analysis-i.html"><a href="point-pattern-analysis-i.html#defining-the-region-for-analysis"><i class="fa fa-check"></i><b>9.8</b> Defining the Region for Analysis</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html"><i class="fa fa-check"></i><b>10</b> Activity 4: Point Pattern Analysis I</a>
<ul>
<li class="chapter" data-level="10.1" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html#practice-questions-1"><i class="fa fa-check"></i><b>10.1</b> Practice questions</a></li>
<li class="chapter" data-level="10.2" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html#learning-objectives-9"><i class="fa fa-check"></i><b>10.2</b> Learning objectives</a></li>
<li class="chapter" data-level="10.3" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html#suggested-reading-2"><i class="fa fa-check"></i><b>10.3</b> Suggested reading</a></li>
<li class="chapter" data-level="10.4" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html#preliminaries-7"><i class="fa fa-check"></i><b>10.4</b> Preliminaries</a></li>
<li class="chapter" data-level="10.5" data-path="activity-4-point-pattern-analysis-i.html"><a href="activity-4-point-pattern-analysis-i.html#activity-3"><i class="fa fa-check"></i><b>10.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html"><i class="fa fa-check"></i><b>11</b> Point Pattern Analysis II</a>
<ul>
<li class="chapter" data-level="11.1" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#learning-objectives-10"><i class="fa fa-check"></i><b>11.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="11.2" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#suggested-readings-4"><i class="fa fa-check"></i><b>11.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="11.3" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#preliminaries-8"><i class="fa fa-check"></i><b>11.3</b> Preliminaries</a></li>
<li class="chapter" data-level="11.4" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#a-quadrat-based-test-for-spatial-independence"><i class="fa fa-check"></i><b>11.4</b> A Quadrat-based Test for Spatial Independence</a></li>
<li class="chapter" data-level="11.5" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#limitations-of-quadrat-analysis-size-and-number-of-quadrats"><i class="fa fa-check"></i><b>11.5</b> Limitations of Quadrat Analysis: Size and Number of Quadrats</a></li>
<li class="chapter" data-level="11.6" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#limitations-of-quadrat-analysis-relative-position-of-events"><i class="fa fa-check"></i><b>11.6</b> Limitations of Quadrat Analysis: Relative Position of Events</a></li>
<li class="chapter" data-level="11.7" data-path="point-pattern-analysis-ii.html"><a href="point-pattern-analysis-ii.html#kernel-density"><i class="fa fa-check"></i><b>11.7</b> Kernel Density</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html"><i class="fa fa-check"></i><b>12</b> Activity 5: Point Pattern Analysis II</a>
<ul>
<li class="chapter" data-level="12.1" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html#practice-questions-2"><i class="fa fa-check"></i><b>12.1</b> Practice questions</a></li>
<li class="chapter" data-level="12.2" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html#learning-objectives-11"><i class="fa fa-check"></i><b>12.2</b> Learning objectives</a></li>
<li class="chapter" data-level="12.3" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html#suggested-reading-3"><i class="fa fa-check"></i><b>12.3</b> Suggested reading</a></li>
<li class="chapter" data-level="12.4" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html#preliminaries-9"><i class="fa fa-check"></i><b>12.4</b> Preliminaries</a></li>
<li class="chapter" data-level="12.5" data-path="activity-5-point-pattern-analysis-ii.html"><a href="activity-5-point-pattern-analysis-ii.html#activity-4"><i class="fa fa-check"></i><b>12.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html"><i class="fa fa-check"></i><b>13</b> Point Pattern Analysis III</a>
<ul>
<li class="chapter" data-level="13.1" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#learning-objectives-12"><i class="fa fa-check"></i><b>13.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="13.2" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#suggested-readings-5"><i class="fa fa-check"></i><b>13.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="13.3" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#preliminaries-10"><i class="fa fa-check"></i><b>13.3</b> Preliminaries</a></li>
<li class="chapter" data-level="13.4" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#motivation"><i class="fa fa-check"></i><b>13.4</b> Motivation</a></li>
<li class="chapter" data-level="13.5" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#nearest-neighbors"><i class="fa fa-check"></i><b>13.5</b> Nearest Neighbors</a></li>
<li class="chapter" data-level="13.6" data-path="point-pattern-analysis-iii.html"><a href="point-pattern-analysis-iii.html#g-function"><i class="fa fa-check"></i><b>13.6</b> <span class="math inline">\(G\)</span>-function</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html"><i class="fa fa-check"></i><b>14</b> Activity 6: Point Pattern Analysis III</a>
<ul>
<li class="chapter" data-level="14.1" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html#practice-questions-3"><i class="fa fa-check"></i><b>14.1</b> Practice questions</a></li>
<li class="chapter" data-level="14.2" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html#learning-objectives-13"><i class="fa fa-check"></i><b>14.2</b> Learning objectives</a></li>
<li class="chapter" data-level="14.3" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html#suggested-reading-4"><i class="fa fa-check"></i><b>14.3</b> Suggested reading</a></li>
<li class="chapter" data-level="14.4" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html#preliminaries-11"><i class="fa fa-check"></i><b>14.4</b> Preliminaries</a></li>
<li class="chapter" data-level="14.5" data-path="activity-6-point-pattern-analysis-iii.html"><a href="activity-6-point-pattern-analysis-iii.html#activity-5"><i class="fa fa-check"></i><b>14.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html"><i class="fa fa-check"></i><b>15</b> Point Pattern Analysis IV</a>
<ul>
<li class="chapter" data-level="15.1" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#learning-objectives-14"><i class="fa fa-check"></i><b>15.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="15.2" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#suggested-readings-6"><i class="fa fa-check"></i><b>15.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="15.3" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#preliminaries-12"><i class="fa fa-check"></i><b>15.3</b> Preliminaries</a></li>
<li class="chapter" data-level="15.4" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#motivation-1"><i class="fa fa-check"></i><b>15.4</b> Motivation</a></li>
<li class="chapter" data-level="15.5" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#f-function"><i class="fa fa-check"></i><b>15.5</b> F-function</a></li>
<li class="chapter" data-level="15.6" data-path="point-pattern-analysis-iv.html"><a href="point-pattern-analysis-iv.html#hatk-function"><i class="fa fa-check"></i><b>15.6</b> <span class="math inline">\(\hat{K}\)</span>-function</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html"><i class="fa fa-check"></i><b>16</b> Activity 7: Point Pattern Analysis IV</a>
<ul>
<li class="chapter" data-level="16.1" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html#practice-questions-4"><i class="fa fa-check"></i><b>16.1</b> Practice questions</a></li>
<li class="chapter" data-level="16.2" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html#learning-objectives-15"><i class="fa fa-check"></i><b>16.2</b> Learning objectives</a></li>
<li class="chapter" data-level="16.3" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html#suggested-reading-5"><i class="fa fa-check"></i><b>16.3</b> Suggested reading</a></li>
<li class="chapter" data-level="16.4" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html#preliminaries-13"><i class="fa fa-check"></i><b>16.4</b> Preliminaries</a></li>
<li class="chapter" data-level="16.5" data-path="activity-7-point-pattern-analysis-iv.html"><a href="activity-7-point-pattern-analysis-iv.html#activity-6"><i class="fa fa-check"></i><b>16.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html"><i class="fa fa-check"></i><b>17</b> Point Pattern Analysis V</a>
<ul>
<li class="chapter" data-level="17.1" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#learning-objectives-16"><i class="fa fa-check"></i><b>17.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="17.2" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#suggested-readings-7"><i class="fa fa-check"></i><b>17.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="17.3" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#preliminaries-14"><i class="fa fa-check"></i><b>17.3</b> Preliminaries</a></li>
<li class="chapter" data-level="17.4" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#motivation-hypothesis-testing"><i class="fa fa-check"></i><b>17.4</b> Motivation: Hypothesis Testing</a></li>
<li class="chapter" data-level="17.5" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#null-landscapes-revisited"><i class="fa fa-check"></i><b>17.5</b> Null Landscapes Revisited</a></li>
<li class="chapter" data-level="17.6" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#simulation-envelopes"><i class="fa fa-check"></i><b>17.6</b> Simulation Envelopes</a></li>
<li class="chapter" data-level="17.7" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#things-to-keep-in-mind"><i class="fa fa-check"></i><b>17.7</b> Things to Keep in Mind!</a>
<ul>
<li class="chapter" data-level="17.7.1" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#definition-of-a-region"><i class="fa fa-check"></i><b>17.7.1</b> Definition of a Region</a></li>
<li class="chapter" data-level="17.7.2" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#edge-effects"><i class="fa fa-check"></i><b>17.7.2</b> Edge Effects</a></li>
<li class="chapter" data-level="17.7.3" data-path="point-pattern-analysis-v.html"><a href="point-pattern-analysis-v.html#sampled-point-patterns"><i class="fa fa-check"></i><b>17.7.3</b> Sampled Point Patterns</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="18" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html"><i class="fa fa-check"></i><b>18</b> Activity 8: Point Pattern Analysis V</a>
<ul>
<li class="chapter" data-level="18.1" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html#practice-questions-5"><i class="fa fa-check"></i><b>18.1</b> Practice questions</a></li>
<li class="chapter" data-level="18.2" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html#learning-objectives-17"><i class="fa fa-check"></i><b>18.2</b> Learning objectives</a></li>
<li class="chapter" data-level="18.3" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html#suggested-reading-6"><i class="fa fa-check"></i><b>18.3</b> Suggested reading</a></li>
<li class="chapter" data-level="18.4" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html#preliminaries-15"><i class="fa fa-check"></i><b>18.4</b> Preliminaries</a></li>
<li class="chapter" data-level="18.5" data-path="activity-8-point-pattern-analysis-v.html"><a href="activity-8-point-pattern-analysis-v.html#activity-7"><i class="fa fa-check"></i><b>18.5</b> Activity</a></li>
</ul></li>
<li class="part"><span><b>IV Part IV: Data in Areal Units</b></span></li>
<li class="chapter" data-level="19" data-path="area-data-i.html"><a href="area-data-i.html"><i class="fa fa-check"></i><b>19</b> Area Data I</a>
<ul>
<li class="chapter" data-level="19.1" data-path="area-data-i.html"><a href="area-data-i.html#learning-objectives-18"><i class="fa fa-check"></i><b>19.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="19.2" data-path="area-data-i.html"><a href="area-data-i.html#suggested-readings-8"><i class="fa fa-check"></i><b>19.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="19.3" data-path="area-data-i.html"><a href="area-data-i.html#preliminaries-16"><i class="fa fa-check"></i><b>19.3</b> Preliminaries</a></li>
<li class="chapter" data-level="19.4" data-path="area-data-i.html"><a href="area-data-i.html#area-data"><i class="fa fa-check"></i><b>19.4</b> Area Data</a></li>
<li class="chapter" data-level="19.5" data-path="area-data-i.html"><a href="area-data-i.html#processes-and-area-data"><i class="fa fa-check"></i><b>19.5</b> Processes and Area Data</a></li>
<li class="chapter" data-level="19.6" data-path="area-data-i.html"><a href="area-data-i.html#visualizing-area-data-choropleth-maps"><i class="fa fa-check"></i><b>19.6</b> Visualizing Area Data: Choropleth Maps</a></li>
<li class="chapter" data-level="19.7" data-path="area-data-i.html"><a href="area-data-i.html#visualizing-area-data-cartograms"><i class="fa fa-check"></i><b>19.7</b> Visualizing Area Data: Cartograms</a></li>
</ul></li>
<li class="chapter" data-level="20" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html"><i class="fa fa-check"></i><b>20</b> Activity 9: Area Data I</a>
<ul>
<li class="chapter" data-level="20.1" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html#practice-questions-6"><i class="fa fa-check"></i><b>20.1</b> Practice questions</a></li>
<li class="chapter" data-level="20.2" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html#learning-objectives-19"><i class="fa fa-check"></i><b>20.2</b> Learning objectives</a></li>
<li class="chapter" data-level="20.3" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html#suggested-reading-7"><i class="fa fa-check"></i><b>20.3</b> Suggested reading</a></li>
<li class="chapter" data-level="20.4" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html#preliminaries-17"><i class="fa fa-check"></i><b>20.4</b> Preliminaries</a></li>
<li class="chapter" data-level="20.5" data-path="activity-9-area-data-i.html"><a href="activity-9-area-data-i.html#activity-8"><i class="fa fa-check"></i><b>20.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="21" data-path="area-data-ii.html"><a href="area-data-ii.html"><i class="fa fa-check"></i><b>21</b> Area Data II</a>
<ul>
<li class="chapter" data-level="21.1" data-path="area-data-ii.html"><a href="area-data-ii.html#learning-objectives-20"><i class="fa fa-check"></i><b>21.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="21.2" data-path="area-data-ii.html"><a href="area-data-ii.html#suggested-readings-9"><i class="fa fa-check"></i><b>21.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="21.3" data-path="area-data-ii.html"><a href="area-data-ii.html#preliminaries-18"><i class="fa fa-check"></i><b>21.3</b> Preliminaries</a></li>
<li class="chapter" data-level="21.4" data-path="area-data-ii.html"><a href="area-data-ii.html#proximity-in-area-data"><i class="fa fa-check"></i><b>21.4</b> Proximity in Area Data</a></li>
<li class="chapter" data-level="21.5" data-path="area-data-ii.html"><a href="area-data-ii.html#spatial-weights-matrices"><i class="fa fa-check"></i><b>21.5</b> Spatial Weights Matrices</a></li>
<li class="chapter" data-level="21.6" data-path="area-data-ii.html"><a href="area-data-ii.html#creating-spatial-weights-matrices-in-r"><i class="fa fa-check"></i><b>21.6</b> Creating Spatial Weights Matrices in <code>R</code></a></li>
<li class="chapter" data-level="21.7" data-path="area-data-ii.html"><a href="area-data-ii.html#spatial-moving-averages"><i class="fa fa-check"></i><b>21.7</b> Spatial Moving Averages</a></li>
<li class="chapter" data-level="21.8" data-path="area-data-ii.html"><a href="area-data-ii.html#other-criteria-for-coding-proximity"><i class="fa fa-check"></i><b>21.8</b> Other Criteria for Coding Proximity</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html"><i class="fa fa-check"></i><b>22</b> Activity 10: Area Data II</a>
<ul>
<li class="chapter" data-level="22.1" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html#practice-questions-7"><i class="fa fa-check"></i><b>22.1</b> Practice questions</a></li>
<li class="chapter" data-level="22.2" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html#learning-objectives-21"><i class="fa fa-check"></i><b>22.2</b> Learning objectives</a></li>
<li class="chapter" data-level="22.3" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html#suggested-reading-8"><i class="fa fa-check"></i><b>22.3</b> Suggested reading</a></li>
<li class="chapter" data-level="22.4" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html#preliminaries-19"><i class="fa fa-check"></i><b>22.4</b> Preliminaries</a></li>
<li class="chapter" data-level="22.5" data-path="activity-10-area-data-ii.html"><a href="activity-10-area-data-ii.html#activity-9"><i class="fa fa-check"></i><b>22.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="area-data-iii.html"><a href="area-data-iii.html"><i class="fa fa-check"></i><b>23</b> Area Data III</a>
<ul>
<li class="chapter" data-level="23.1" data-path="area-data-iii.html"><a href="area-data-iii.html#learning-objectives-22"><i class="fa fa-check"></i><b>23.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="23.2" data-path="area-data-iii.html"><a href="area-data-iii.html#suggested-readings-10"><i class="fa fa-check"></i><b>23.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="23.3" data-path="area-data-iii.html"><a href="area-data-iii.html#preliminaries-20"><i class="fa fa-check"></i><b>23.3</b> Preliminaries</a></li>
<li class="chapter" data-level="23.4" data-path="area-data-iii.html"><a href="area-data-iii.html#spatial-moving-averages-and-simulation"><i class="fa fa-check"></i><b>23.4</b> Spatial Moving Averages and Simulation</a></li>
<li class="chapter" data-level="23.5" data-path="area-data-iii.html"><a href="area-data-iii.html#the-spatial-moving-average-as-a-smoother"><i class="fa fa-check"></i><b>23.5</b> The Spatial Moving Average as a Smoother</a></li>
<li class="chapter" data-level="23.6" data-path="area-data-iii.html"><a href="area-data-iii.html#spatial-moving-average-scatterplots"><i class="fa fa-check"></i><b>23.6</b> Spatial Moving Average Scatterplots</a></li>
<li class="chapter" data-level="23.7" data-path="area-data-iii.html"><a href="area-data-iii.html#spatial-autocorrelation-and-morans-i-coefficient"><i class="fa fa-check"></i><b>23.7</b> Spatial Autocorrelation and Moran’s <span class="math inline">\(I\)</span> coefficient</a></li>
<li class="chapter" data-level="23.8" data-path="area-data-iii.html"><a href="area-data-iii.html#morans-i-and-morans-scatterplot"><i class="fa fa-check"></i><b>23.8</b> Moran’s <span class="math inline">\(I\)</span> and Moran’s Scatterplot</a></li>
<li class="chapter" data-level="23.9" data-path="area-data-iii.html"><a href="area-data-iii.html#hypothesis-testing-for-spatial-autocorrelation"><i class="fa fa-check"></i><b>23.9</b> Hypothesis Testing for Spatial Autocorrelation</a></li>
</ul></li>
<li class="chapter" data-level="24" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html"><i class="fa fa-check"></i><b>24</b> Activity 11: Area Data III</a>
<ul>
<li class="chapter" data-level="24.1" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html#practice-questions-8"><i class="fa fa-check"></i><b>24.1</b> Practice questions</a></li>
<li class="chapter" data-level="24.2" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html#learning-objectives-23"><i class="fa fa-check"></i><b>24.2</b> Learning objectives</a></li>
<li class="chapter" data-level="24.3" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html#suggested-reading-9"><i class="fa fa-check"></i><b>24.3</b> Suggested reading</a></li>
<li class="chapter" data-level="24.4" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html#preliminaries-21"><i class="fa fa-check"></i><b>24.4</b> Preliminaries</a></li>
<li class="chapter" data-level="24.5" data-path="activity-11-area-data-iii.html"><a href="activity-11-area-data-iii.html#activity-10"><i class="fa fa-check"></i><b>24.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="area-data-iv.html"><a href="area-data-iv.html"><i class="fa fa-check"></i><b>25</b> Area Data IV</a>
<ul>
<li class="chapter" data-level="25.1" data-path="area-data-iv.html"><a href="area-data-iv.html#learning-objectives-24"><i class="fa fa-check"></i><b>25.1</b> Learning objectives</a></li>
<li class="chapter" data-level="25.2" data-path="area-data-iv.html"><a href="area-data-iv.html#suggested-readings-11"><i class="fa fa-check"></i><b>25.2</b> Suggested readings</a></li>
<li class="chapter" data-level="25.3" data-path="area-data-iv.html"><a href="area-data-iv.html#preliminaries-22"><i class="fa fa-check"></i><b>25.3</b> Preliminaries</a></li>
<li class="chapter" data-level="25.4" data-path="area-data-iv.html"><a href="area-data-iv.html#decomposing-morans-i"><i class="fa fa-check"></i><b>25.4</b> Decomposing Moran’s <span class="math inline">\(I\)</span></a></li>
<li class="chapter" data-level="25.5" data-path="area-data-iv.html"><a href="area-data-iv.html#local-morans-i-and-mapping"><i class="fa fa-check"></i><b>25.5</b> Local Moran’s <span class="math inline">\(I\)</span> and Mapping</a></li>
<li class="chapter" data-level="25.6" data-path="area-data-iv.html"><a href="area-data-iv.html#a-quick-note-on-functions"><i class="fa fa-check"></i><b>25.6</b> A Quick Note on Functions</a></li>
<li class="chapter" data-level="25.7" data-path="area-data-iv.html"><a href="area-data-iv.html#a-concentration-approach-for-local-analysis-of-spatial-association"><i class="fa fa-check"></i><b>25.7</b> A Concentration approach for Local Analysis of Spatial Association</a></li>
<li class="chapter" data-level="25.8" data-path="area-data-iv.html"><a href="area-data-iv.html#a-short-note-on-hypothesis-testing"><i class="fa fa-check"></i><b>25.8</b> A Short Note on Hypothesis Testing</a></li>
<li class="chapter" data-level="25.9" data-path="area-data-iv.html"><a href="area-data-iv.html#detection-of-hot-and-cold-spots"><i class="fa fa-check"></i><b>25.9</b> Detection of Hot and Cold Spots</a></li>
<li class="chapter" data-level="25.10" data-path="area-data-iv.html"><a href="area-data-iv.html#other-resources-1"><i class="fa fa-check"></i><b>25.10</b> Other Resources</a></li>
</ul></li>
<li class="chapter" data-level="26" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html"><i class="fa fa-check"></i><b>26</b> Activity 12: Area Data IV</a>
<ul>
<li class="chapter" data-level="26.1" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html#practice-questions-9"><i class="fa fa-check"></i><b>26.1</b> Practice questions</a></li>
<li class="chapter" data-level="26.2" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html#learning-objectives-25"><i class="fa fa-check"></i><b>26.2</b> Learning objectives</a></li>
<li class="chapter" data-level="26.3" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html#suggested-reading-10"><i class="fa fa-check"></i><b>26.3</b> Suggested reading</a></li>
<li class="chapter" data-level="26.4" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html#preliminaries-23"><i class="fa fa-check"></i><b>26.4</b> Preliminaries</a></li>
<li class="chapter" data-level="26.5" data-path="activity-12-area-data-iv.html"><a href="activity-12-area-data-iv.html#activity-11"><i class="fa fa-check"></i><b>26.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="27" data-path="area-data-v.html"><a href="area-data-v.html"><i class="fa fa-check"></i><b>27</b> Area Data V</a>
<ul>
<li class="chapter" data-level="27.1" data-path="area-data-v.html"><a href="area-data-v.html#learning-objectives-26"><i class="fa fa-check"></i><b>27.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="27.2" data-path="area-data-v.html"><a href="area-data-v.html#suggested-readings-12"><i class="fa fa-check"></i><b>27.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="27.3" data-path="area-data-v.html"><a href="area-data-v.html#preliminaries-24"><i class="fa fa-check"></i><b>27.3</b> Preliminaries</a></li>
<li class="chapter" data-level="27.4" data-path="area-data-v.html"><a href="area-data-v.html#regression-analysis-in-r"><i class="fa fa-check"></i><b>27.4</b> Regression Analysis in <code>R</code></a></li>
<li class="chapter" data-level="27.5" data-path="area-data-v.html"><a href="area-data-v.html#autocorrelation-as-a-model-diagnostic"><i class="fa fa-check"></i><b>27.5</b> Autocorrelation as a Model Diagnostic</a></li>
<li class="chapter" data-level="27.6" data-path="area-data-v.html"><a href="area-data-v.html#variable-transformations"><i class="fa fa-check"></i><b>27.6</b> Variable Transformations</a></li>
<li class="chapter" data-level="27.7" data-path="area-data-v.html"><a href="area-data-v.html#a-note-about-spatial-autocorrelation-in-regression-analysis"><i class="fa fa-check"></i><b>27.7</b> A Note about Spatial Autocorrelation in Regression Analysis</a></li>
</ul></li>
<li class="chapter" data-level="28" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html"><i class="fa fa-check"></i><b>28</b> Activity 13: Area Data V</a>
<ul>
<li class="chapter" data-level="28.1" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html#practice-questions-10"><i class="fa fa-check"></i><b>28.1</b> Practice questions</a></li>
<li class="chapter" data-level="28.2" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html#learning-objectives-27"><i class="fa fa-check"></i><b>28.2</b> Learning objectives</a></li>
<li class="chapter" data-level="28.3" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html#suggested-reading-11"><i class="fa fa-check"></i><b>28.3</b> Suggested reading</a></li>
<li class="chapter" data-level="28.4" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html#preliminaries-25"><i class="fa fa-check"></i><b>28.4</b> Preliminaries</a></li>
<li class="chapter" data-level="28.5" data-path="activity-13-area-data-v.html"><a href="activity-13-area-data-v.html#activity-12"><i class="fa fa-check"></i><b>28.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="29" data-path="area-data-vi.html"><a href="area-data-vi.html"><i class="fa fa-check"></i><b>29</b> Area Data VI</a>
<ul>
<li class="chapter" data-level="29.1" data-path="area-data-vi.html"><a href="area-data-vi.html#learning-objectives-28"><i class="fa fa-check"></i><b>29.1</b> Learning Objectives</a></li>
<li class="chapter" data-level="29.2" data-path="area-data-vi.html"><a href="area-data-vi.html#suggested-readings-13"><i class="fa fa-check"></i><b>29.2</b> Suggested Readings</a></li>
<li class="chapter" data-level="29.3" data-path="area-data-vi.html"><a href="area-data-vi.html#preliminaries-26"><i class="fa fa-check"></i><b>29.3</b> Preliminaries</a></li>
<li class="chapter" data-level="29.4" data-path="area-data-vi.html"><a href="area-data-vi.html#residual-spatial-autocorrelation-revisited"><i class="fa fa-check"></i><b>29.4</b> Residual spatial autocorrelation revisited</a>
<ul>
<li class="chapter" data-level="29.4.1" data-path="area-data-vi.html"><a href="area-data-vi.html#incorrect-functional-form"><i class="fa fa-check"></i><b>29.4.1</b> Incorrect Functional Form</a></li>
<li class="chapter" data-level="29.4.2" data-path="area-data-vi.html"><a href="area-data-vi.html#omitted-variables"><i class="fa fa-check"></i><b>29.4.2</b> Omitted Variables</a></li>
</ul></li>
<li class="chapter" data-level="29.5" data-path="area-data-vi.html"><a href="area-data-vi.html#remedial-action"><i class="fa fa-check"></i><b>29.5</b> Remedial Action</a></li>
<li class="chapter" data-level="29.6" data-path="area-data-vi.html"><a href="area-data-vi.html#flexible-functional-forms-and-models-with-spatially-varying-coefficients"><i class="fa fa-check"></i><b>29.6</b> Flexible Functional Forms and Models with Spatially-varying Coefficients</a>
<ul>
<li class="chapter" data-level="29.6.1" data-path="area-data-vi.html"><a href="area-data-vi.html#trend-surface-analysis"><i class="fa fa-check"></i><b>29.6.1</b> Trend Surface Analysis</a></li>
<li class="chapter" data-level="29.6.2" data-path="area-data-vi.html"><a href="area-data-vi.html#models-with-spatially-varying-coefficients"><i class="fa fa-check"></i><b>29.6.2</b> Models with Spatially-varying Coefficients</a></li>
</ul></li>
<li class="chapter" data-level="29.7" data-path="area-data-vi.html"><a href="area-data-vi.html#spatial-error-model-sem"><i class="fa fa-check"></i><b>29.7</b> Spatial Error Model (SEM)</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html"><i class="fa fa-check"></i><b>30</b> Activity 14: Area Data VI</a>
<ul>
<li class="chapter" data-level="30.1" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#practice-questions-11"><i class="fa fa-check"></i><b>30.1</b> Practice questions</a></li>
<li class="chapter" data-level="30.2" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#learning-objectives-29"><i class="fa fa-check"></i><b>30.2</b> Learning objectives</a></li>
<li class="chapter" data-level="30.3" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#suggested-reading-12"><i class="fa fa-check"></i><b>30.3</b> Suggested reading</a></li>
<li class="chapter" data-level="30.4" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#preliminaries-27"><i class="fa fa-check"></i><b>30.4</b> Preliminaries</a>
<ul>
<li class="chapter" data-level="30.4.1" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#new-york-leukemia-data"><i class="fa fa-check"></i><b>30.4.1</b> New York leukemia data</a></li>
<li class="chapter" data-level="30.4.2" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#pennsylvania-lung-cancer"><i class="fa fa-check"></i><b>30.4.2</b> Pennsylvania lung cancer</a></li>
</ul></li>
<li class="chapter" data-level="30.5" data-path="activity-14-area-data-vi.html"><a href="activity-14-area-data-vi.html#activity-13"><i class="fa fa-check"></i><b>30.5</b> Activity</a></li>
</ul></li>
<li class="part"><span><b>V Part V: Analysis and Prediction of Fields</b></span></li>
<li class="chapter" data-level="31" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html"><i class="fa fa-check"></i><b>31</b> Spatially Continuous Data I</a>
<ul>
<li class="chapter" data-level="31.1" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#learning-objectives-30"><i class="fa fa-check"></i><b>31.1</b> Learning objectives</a></li>
<li class="chapter" data-level="31.2" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#suggested-readings-14"><i class="fa fa-check"></i><b>31.2</b> Suggested readings</a></li>
<li class="chapter" data-level="31.3" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#preliminaries-28"><i class="fa fa-check"></i><b>31.3</b> Preliminaries</a></li>
<li class="chapter" data-level="31.4" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#spatially-continuous-field-data"><i class="fa fa-check"></i><b>31.4</b> Spatially continuous (field) data</a></li>
<li class="chapter" data-level="31.5" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#exploratory-visualization"><i class="fa fa-check"></i><b>31.5</b> Exploratory visualization</a></li>
<li class="chapter" data-level="31.6" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#tile-based-methods"><i class="fa fa-check"></i><b>31.6</b> Tile-based methods</a></li>
<li class="chapter" data-level="31.7" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#inverse-distance-weighting-idw"><i class="fa fa-check"></i><b>31.7</b> Inverse distance weighting (IDW)</a></li>
<li class="chapter" data-level="31.8" data-path="spatially-continuous-data-i.html"><a href="spatially-continuous-data-i.html#k-point-means"><i class="fa fa-check"></i><b>31.8</b> <span class="math inline">\(k\)</span>-point means</a></li>
</ul></li>
<li class="chapter" data-level="32" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html"><i class="fa fa-check"></i><b>32</b> Activity 15: Spatially Continuous Data I</a>
<ul>
<li class="chapter" data-level="32.1" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html#practice-questions-12"><i class="fa fa-check"></i><b>32.1</b> Practice questions</a></li>
<li class="chapter" data-level="32.2" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html#learning-objectives-31"><i class="fa fa-check"></i><b>32.2</b> Learning objectives</a></li>
<li class="chapter" data-level="32.3" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html#suggested-reading-13"><i class="fa fa-check"></i><b>32.3</b> Suggested reading</a></li>
<li class="chapter" data-level="32.4" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html#preliminaries-29"><i class="fa fa-check"></i><b>32.4</b> Preliminaries</a></li>
<li class="chapter" data-level="32.5" data-path="activity-15-spatially-continuous-data-i.html"><a href="activity-15-spatially-continuous-data-i.html#activity-14"><i class="fa fa-check"></i><b>32.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html"><i class="fa fa-check"></i><b>33</b> Spatially Continuous Data II</a>
<ul>
<li class="chapter" data-level="33.1" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#learning-objectives-32"><i class="fa fa-check"></i><b>33.1</b> Learning objectives</a></li>
<li class="chapter" data-level="33.2" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#suggested-readings-15"><i class="fa fa-check"></i><b>33.2</b> Suggested readings</a></li>
<li class="chapter" data-level="33.3" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#preliminaries-30"><i class="fa fa-check"></i><b>33.3</b> Preliminaries</a></li>
<li class="chapter" data-level="33.4" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#uncertainty-in-the-predictions"><i class="fa fa-check"></i><b>33.4</b> Uncertainty in the predictions</a></li>
<li class="chapter" data-level="33.5" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#trend-surface-analysis-1"><i class="fa fa-check"></i><b>33.5</b> Trend surface analysis</a></li>
<li class="chapter" data-level="33.6" data-path="spatially-continuous-data-ii.html"><a href="spatially-continuous-data-ii.html#accuracy-and-precision"><i class="fa fa-check"></i><b>33.6</b> Accuracy and precision</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html"><i class="fa fa-check"></i><b>34</b> Activity 16: Spatially Continuous Data II</a>
<ul>
<li class="chapter" data-level="34.1" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html#practice-questions-13"><i class="fa fa-check"></i><b>34.1</b> Practice questions</a></li>
<li class="chapter" data-level="34.2" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html#learning-objectives-33"><i class="fa fa-check"></i><b>34.2</b> Learning objectives</a></li>
<li class="chapter" data-level="34.3" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html#suggested-reading-14"><i class="fa fa-check"></i><b>34.3</b> Suggested reading</a></li>
<li class="chapter" data-level="34.4" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html#preliminaries-31"><i class="fa fa-check"></i><b>34.4</b> Preliminaries</a></li>
<li class="chapter" data-level="34.5" data-path="activity-16-spatially-continuous-data-ii.html"><a href="activity-16-spatially-continuous-data-ii.html#activity-15"><i class="fa fa-check"></i><b>34.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="35" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html"><i class="fa fa-check"></i><b>35</b> Spatially Continuous Data III</a>
<ul>
<li class="chapter" data-level="35.1" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#learning-objectives-34"><i class="fa fa-check"></i><b>35.1</b> Learning objectives</a></li>
<li class="chapter" data-level="35.2" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#suggested-reading-15"><i class="fa fa-check"></i><b>35.2</b> Suggested reading</a></li>
<li class="chapter" data-level="35.3" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#preliminaries-32"><i class="fa fa-check"></i><b>35.3</b> Preliminaries</a></li>
<li class="chapter" data-level="35.4" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#residual-spatial-pattern"><i class="fa fa-check"></i><b>35.4</b> Residual spatial pattern</a></li>
<li class="chapter" data-level="35.5" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#measuring-spatial-dependence-in-spatially-continuous-data"><i class="fa fa-check"></i><b>35.5</b> Measuring spatial dependence in spatially continuous data</a></li>
<li class="chapter" data-level="35.6" data-path="spatially-continuous-data-iii.html"><a href="spatially-continuous-data-iii.html#variographic-analyisis"><i class="fa fa-check"></i><b>35.6</b> Variographic analyisis</a></li>
</ul></li>
<li class="chapter" data-level="36" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html"><i class="fa fa-check"></i><b>36</b> Activity 17: Spatially Continuous Data III</a>
<ul>
<li class="chapter" data-level="36.1" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html#practice-questions-14"><i class="fa fa-check"></i><b>36.1</b> Practice questions</a></li>
<li class="chapter" data-level="36.2" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html#learning-objectives-35"><i class="fa fa-check"></i><b>36.2</b> Learning objectives</a></li>
<li class="chapter" data-level="36.3" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html#suggested-reading-16"><i class="fa fa-check"></i><b>36.3</b> Suggested reading</a></li>
<li class="chapter" data-level="36.4" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html#preliminaries-33"><i class="fa fa-check"></i><b>36.4</b> Preliminaries</a></li>
<li class="chapter" data-level="36.5" data-path="activity-17-spatially-continuous-data-iii.html"><a href="activity-17-spatially-continuous-data-iii.html#activity-16"><i class="fa fa-check"></i><b>36.5</b> Activity</a></li>
</ul></li>
<li class="chapter" data-level="37" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html"><i class="fa fa-check"></i><b>37</b> Spatially Continuous Data IV</a>
<ul>
<li class="chapter" data-level="37.1" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html#learning-objectives-36"><i class="fa fa-check"></i><b>37.1</b> Learning objectives</a></li>
<li class="chapter" data-level="37.2" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html#suggested-reading-17"><i class="fa fa-check"></i><b>37.2</b> Suggested reading</a></li>
<li class="chapter" data-level="37.3" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html#preliminaries-34"><i class="fa fa-check"></i><b>37.3</b> Preliminaries</a></li>
<li class="chapter" data-level="37.4" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html#using-residual-spatial-pattern-to-estimate-prediction-errors"><i class="fa fa-check"></i><b>37.4</b> Using residual spatial pattern to estimate prediction errors</a></li>
<li class="chapter" data-level="37.5" data-path="spatially-continuous-data-iv.html"><a href="spatially-continuous-data-iv.html#kriging-a-method-for-optimal-prediction."><i class="fa fa-check"></i><b>37.5</b> Kriging: a method for optimal prediction.</a></li>
</ul></li>
<li class="chapter" data-level="38" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html"><i class="fa fa-check"></i><b>38</b> Activity 18: Spatially Continuous Data IV</a>
<ul>
<li class="chapter" data-level="38.1" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html#practice-questions-15"><i class="fa fa-check"></i><b>38.1</b> Practice questions</a></li>
<li class="chapter" data-level="38.2" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html#learning-objectives-37"><i class="fa fa-check"></i><b>38.2</b> Learning objectives</a></li>
<li class="chapter" data-level="38.3" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html#suggested-reading-18"><i class="fa fa-check"></i><b>38.3</b> Suggested reading</a></li>
<li class="chapter" data-level="38.4" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html#preliminaries-35"><i class="fa fa-check"></i><b>38.4</b> Preliminaries</a></li>
<li class="chapter" data-level="38.5" data-path="activity-18-spatially-continuous-data-iv.html"><a href="activity-18-spatially-continuous-data-iv.html#activity-17"><i class="fa fa-check"></i><b>38.5</b> Activity</a></li>
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<div id="spatially-continuous-data-iii" class="section level1" number="35">
<h1><span class="header-section-number">Chapter 35</span> Spatially Continuous Data III</h1>
<p><em>NOTE</em>: You can download the source files for this book from <a href="https://github.com/paezha/Spatial-Statistics-Course">here</a>. The source files are in the format of R Notebooks. Notebooks are pretty neat, because the allow you execute code within the notebook, so that you can work interactively with the notes.</p>
<p>In the previous practice you were introduced to the concept of fields/spatially continuous data.</p>
<p>If you wish to work interactively with this chapter you will need the following:</p>
<ul>
<li><p>An R markdown notebook version of this document (the source file).</p></li>
<li><p>A package called <code>geog4ga3</code>.</p></li>
</ul>
<div id="learning-objectives-34" class="section level2" number="35.1">
<h2><span class="header-section-number">35.1</span> Learning objectives</h2>
<p>Previously, in Chapter <a href="spatially-continuous-data-ii.html#spatially-continuous-data-ii">33</a>, we discussed some limitations of tile-based approaches, inverse distance weighting, and <span class="math inline">\(k\)</span>-points mean. Particularly, these methods do not provide estimates of the uncertainty of point estimates when doing spatial interpolation. Trend surface analysis was introduced as a method for spatial interpolation that also provides estimates of the standard error. However, we saw that it is possible for the residuals of a trend surface model to be autocorrelated: this is an indication that there is still systematic variation in the residuals that was not fully captured by the model. To more fully exploit that residual pattern we need some additional tools. In this practice, you will learn some of said tools, as follows:</p>
<ol style="list-style-type: decimal">
<li>About the implications of residual spatial pattern for predictions.</li>
<li>The measurement of spatial dependence in fields.</li>
<li>Variographic analysis.</li>
</ol>
</div>
<div id="suggested-reading-15" class="section level2" number="35.2">
<h2><span class="header-section-number">35.2</span> Suggested reading</h2>
<ul>
<li>Bailey TC and Gatrell AC <span class="citation">(<a href="#ref-Bailey1995" role="doc-biblioref">1995</a>)</span> Interactive Spatial Data Analysis, Chapters 5 and 6. Longman: Essex.</li>
<li>Bivand RS, Pebesma E, and Gomez-Rubio V <span class="citation">(<a href="#ref-Bivand2008" role="doc-biblioref">2008</a>)</span> Applied Spatial Data Analysis with R, Chapter 8. Springer: New York.</li>
<li>Brunsdon C and Comber L <span class="citation">(<a href="#ref-Brunsdon2015R" role="doc-biblioref">2015</a>)</span> An Introduction to R for Spatial Analysis and Mapping, Chapter 6, Sections 6.7 and 6.8. Sage: Los Angeles.</li>
<li>Isaaks EH and Srivastava RM <span class="citation">(<a href="#ref-Isaaks1989applied" role="doc-biblioref">1989</a>)</span> An Introduction to Applied Geostatistics, <strong>CHAPTERS</strong>. Oxford University Press: Oxford.</li>
<li>O’Sullivan D and Unwin D <span class="citation">(<a href="#ref-Osullivan2010" role="doc-biblioref">2010</a>)</span> Geographic Information Analysis, 2nd Edition, Chapters 9 and 10. John Wiley &amp; Sons: New Jersey.</li>
</ul>
</div>
<div id="preliminaries-32" class="section level2" number="35.3">
<h2><span class="header-section-number">35.3</span> Preliminaries</h2>
<p>As usual, it is good practice to clear the working space to make sure that you do not have extraneous items there when you begin your work. The command in <code>R</code> to clear the workspace is <code>rm</code> (for “remove”), followed by a list of items to be removed. To clear the workspace from <em>all</em> objects, do the following:</p>
<div class="sourceCode" id="cb763"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb763-1"><a href="spatially-continuous-data-iii.html#cb763-1" aria-hidden="true" tabindex="-1"></a><span class="fu">rm</span>(<span class="at">list =</span> <span class="fu">ls</span>())</span></code></pre></div>
<p>Note that <code>ls()</code> lists all objects currently on the workspace.</p>
<p>Load the libraries you will use in this activity:</p>
<div class="sourceCode" id="cb764"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb764-1"><a href="spatially-continuous-data-iii.html#cb764-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(geog4ga3)</span>
<span id="cb764-2"><a href="spatially-continuous-data-iii.html#cb764-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(gstat)</span>
<span id="cb764-3"><a href="spatially-continuous-data-iii.html#cb764-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(plotly)</span>
<span id="cb764-4"><a href="spatially-continuous-data-iii.html#cb764-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(spdep)</span>
<span id="cb764-5"><a href="spatially-continuous-data-iii.html#cb764-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code></pre></div>
<p>Begin by loading the data file:</p>
<div class="sourceCode" id="cb765"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb765-1"><a href="spatially-continuous-data-iii.html#cb765-1" aria-hidden="true" tabindex="-1"></a><span class="co"># We have been working with the Walker Lake dataset for the last few chapters.</span></span>
<span id="cb765-2"><a href="spatially-continuous-data-iii.html#cb765-2" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(<span class="st">&quot;Walker_Lake&quot;</span>)  </span></code></pre></div>
<p>You can verify the contents of the dataframe:</p>
<div class="sourceCode" id="cb766"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb766-1"><a href="spatially-continuous-data-iii.html#cb766-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(Walker_Lake)</span></code></pre></div>
<pre><code>##       ID                  X               Y               V
##  Length:470         Min.   :  8.0   Min.   :  8.0   Min.   :   0.0
##  Class :character   1st Qu.: 51.0   1st Qu.: 80.0   1st Qu.: 182.0
##  Mode  :character   Median : 89.0   Median :139.5   Median : 425.2
##                     Mean   :111.1   Mean   :141.3   Mean   : 435.4
##                     3rd Qu.:170.0   3rd Qu.:208.0   3rd Qu.: 644.4
##                     Max.   :251.0   Max.   :291.0   Max.   :1528.1
##
##        U           T
##  Min.   :   0.00   1: 45
##  1st Qu.:  83.95   2:425
##  Median : 335.00
##  Mean   : 613.27
##  3rd Qu.: 883.20
##  Max.   :5190.10
##  NA&#39;s   :195</code></pre>
</div>
<div id="residual-spatial-pattern" class="section level2" number="35.4">
<h2><span class="header-section-number">35.4</span> Residual spatial pattern</h2>
<p>In Chapter @ref{spatially-continuous-data-i} we used trend surface analysis for spatial interpolation. Trend surface analysis improves on methods such as Voronoi polygons, IDW, and <span class="math inline">\(k\)</span>-point means by providing a built-in mechanism for estimating the uncertainty in the predictions. Let us quickly revisit this idea.</p>
<p>The objective of interpolation is to provide the following estimates:
<span class="math display">\[
\hat{z}_p + \hat{\epsilon}_p
\]</span></p>
<p>Trend surface analysis provides interpolated values by generating a trend surface as follows:
<span class="math display">\[
\hat{z} = f(x, y)
\]</span>
from which estimates of <span class="math inline">\(\hat{z}_p\)</span> can be obtained by using suitable prediction coordinates <span class="math inline">\((x_p, y_p)\)</span>.</p>
<p>Next, although trend surface analysis does not provide an estimate of the prediction error <span class="math inline">\(\hat{\epsilon}_p\)</span> (since we do <em>not</em> know the true value of the field at <span class="math inline">\(p\)</span>), it provides confidence intervals for the prediction. In this way we can at the very least bound the prediction error as follows:
<span class="math display">\[
CI_{z_p} = [z_{p_{lwr}}, z_{p_{upr}}].
\]</span></p>
<p>As previously seen, however, use of trend surface analysis does not guarantee that the residuals of the model will be independent.</p>
<p>Let us revisit the model for Walker Lake.</p>
<p>As before, we first calculate the polynomial terms of the coordinates:</p>
<div class="sourceCode" id="cb768"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb768-1"><a href="spatially-continuous-data-iii.html#cb768-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Here we use `mutate()` to calculate the polynomial terms of the coordinates.</span></span>
<span id="cb768-2"><a href="spatially-continuous-data-iii.html#cb768-2" aria-hidden="true" tabindex="-1"></a>Walker_Lake <span class="ot">&lt;-</span> <span class="fu">mutate</span>(Walker_Lake,</span>
<span id="cb768-3"><a href="spatially-continuous-data-iii.html#cb768-3" aria-hidden="true" tabindex="-1"></a>                        <span class="at">X3 =</span> X<span class="sc">^</span><span class="dv">3</span>, <span class="at">X2Y =</span> X<span class="sc">^</span><span class="dv">2</span> <span class="sc">*</span> Y, <span class="at">X2 =</span> X<span class="sc">^</span><span class="dv">2</span>, </span>
<span id="cb768-4"><a href="spatially-continuous-data-iii.html#cb768-4" aria-hidden="true" tabindex="-1"></a>                        <span class="at">XY =</span> X <span class="sc">*</span> Y,</span>
<span id="cb768-5"><a href="spatially-continuous-data-iii.html#cb768-5" aria-hidden="true" tabindex="-1"></a>                        <span class="at">Y2 =</span> Y<span class="sc">^</span><span class="dv">2</span>, <span class="at">XY2 =</span> X <span class="sc">*</span> Y<span class="sc">^</span><span class="dv">2</span>, <span class="at">Y3 =</span> Y<span class="sc">^</span><span class="dv">3</span>)</span></code></pre></div>
<p>And proceed to estimate the following cubic trend surface model, which provided the best fit to the data:</p>
<div class="sourceCode" id="cb769"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb769-1"><a href="spatially-continuous-data-iii.html#cb769-1" aria-hidden="true" tabindex="-1"></a>WL.trend3 <span class="ot">&lt;-</span> <span class="fu">lm</span>(<span class="at">formula =</span> V <span class="sc">~</span> X3 <span class="sc">+</span> X2Y <span class="sc">+</span> X2 <span class="sc">+</span> X <span class="sc">+</span> XY <span class="sc">+</span> Y <span class="sc">+</span> Y2 <span class="sc">+</span> XY2 <span class="sc">+</span> Y3, </span>
<span id="cb769-2"><a href="spatially-continuous-data-iii.html#cb769-2" aria-hidden="true" tabindex="-1"></a>                <span class="at">data =</span> Walker_Lake) <span class="co">#Recall use of the linear model for walker lake</span></span>
<span id="cb769-3"><a href="spatially-continuous-data-iii.html#cb769-3" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(WL.trend3)</span></code></pre></div>
<pre><code>##
## Call:
## lm(formula = V ~ X3 + X2Y + X2 + X + XY + Y + Y2 + XY2 + Y3,
##     data = Walker_Lake)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -564.19 -197.41    7.91  194.25  929.72
##
## Coefficients:
##               Estimate Std. Error t value Pr(&gt;|t|)
## (Intercept) -8.620e+00  1.227e+02  -0.070 0.944035
## X3           1.533e-04  4.806e-05   3.190 0.001522 **
## X2Y          6.139e-05  3.909e-05   1.570 0.117000
## X2          -6.651e-02  1.838e-02  -3.618 0.000330 ***
## X            9.172e+00  2.386e+00   3.844 0.000138 ***
## XY          -4.420e-02  1.430e-02  -3.092 0.002110 **
## Y            4.794e+00  2.040e+00   2.350 0.019220 *
## Y2          -1.806e-03  1.327e-02  -0.136 0.891822
## XY2          7.679e-05  2.956e-05   2.598 0.009669 **
## Y3          -4.170e-05  2.819e-05  -1.479 0.139759
## ---
## Signif. codes:  0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1
##
## Residual standard error: 276.7 on 460 degrees of freedom
## Multiple R-squared:  0.1719, Adjusted R-squared:  0.1557
## F-statistic: 10.61 on 9 and 460 DF,  p-value: 5.381e-15</code></pre>
<p>To examine the residuals, first we label them as “positive” or “negative”:</p>
<div class="sourceCode" id="cb771"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb771-1"><a href="spatially-continuous-data-iii.html#cb771-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The function `ifelse()` is used to label the residuals as &quot;Positive&quot; if they are greater than zero, or &quot;Negative&quot; if they are zero or less.</span></span>
<span id="cb771-2"><a href="spatially-continuous-data-iii.html#cb771-2" aria-hidden="true" tabindex="-1"></a>Walker_Lake <span class="ot">&lt;-</span> Walker_Lake <span class="sc">%&gt;%</span></span>
<span id="cb771-3"><a href="spatially-continuous-data-iii.html#cb771-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">residual3 =</span> <span class="fu">ifelse</span>(WL.trend3<span class="sc">$</span>residuals <span class="sc">&gt;</span> <span class="dv">0</span>, </span>
<span id="cb771-4"><a href="spatially-continuous-data-iii.html#cb771-4" aria-hidden="true" tabindex="-1"></a>                            <span class="st">&quot;Positive&quot;</span>, </span>
<span id="cb771-5"><a href="spatially-continuous-data-iii.html#cb771-5" aria-hidden="true" tabindex="-1"></a>                            <span class="st">&quot;Negative&quot;</span>))</span></code></pre></div>
<p>Once the residuals have been labeled we can be plotted as follows:</p>
<div class="sourceCode" id="cb772"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb772-1"><a href="spatially-continuous-data-iii.html#cb772-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> Walker_Lake, </span>
<span id="cb772-2"><a href="spatially-continuous-data-iii.html#cb772-2" aria-hidden="true" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> X, <span class="at">y =</span> Y, <span class="at">color =</span> residual3)) <span class="sc">+</span> <span class="co"># Note color is only applied to results of positive or negative residuals</span></span>
<span id="cb772-3"><a href="spatially-continuous-data-iii.html#cb772-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
<span id="cb772-4"><a href="spatially-continuous-data-iii.html#cb772-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">coord_equal</span>() <span class="co"># Ensures equal scales for both axes</span></span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-552-1.png" width="672" /></p>
<p>As seen before, there is considerable spatial autocorrelation as confirmed by Moran’s <span class="math inline">\(I\)</span> coefficient:</p>
<div class="sourceCode" id="cb773"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb773-1"><a href="spatially-continuous-data-iii.html#cb773-1" aria-hidden="true" tabindex="-1"></a>WL.listw <span class="ot">&lt;-</span> <span class="fu">as.matrix</span>(Walker_Lake[,<span class="dv">2</span><span class="sc">:</span><span class="dv">3</span>]) <span class="sc">%&gt;%</span> <span class="co"># Take the coordinates of Walker Lake and convert to matrix.</span></span>
<span id="cb773-2"><a href="spatially-continuous-data-iii.html#cb773-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">knearneigh</span>(<span class="at">k =</span> <span class="dv">5</span>) <span class="sc">%&gt;%</span> <span class="co"># Find the 5 nearest neighbors of each observations.</span></span>
<span id="cb773-3"><a href="spatially-continuous-data-iii.html#cb773-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">knn2nb</span>() <span class="sc">%&gt;%</span> <span class="co"># Convert the nearest neighbors to `nb` object.</span></span>
<span id="cb773-4"><a href="spatially-continuous-data-iii.html#cb773-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">nb2listw</span>() <span class="co"># Convert the `nb` object into spatial weights. </span></span>
<span id="cb773-5"><a href="spatially-continuous-data-iii.html#cb773-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb773-6"><a href="spatially-continuous-data-iii.html#cb773-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Use Moran&#39;s test on the residuals of the trend surface model </span></span>
<span id="cb773-7"><a href="spatially-continuous-data-iii.html#cb773-7" aria-hidden="true" tabindex="-1"></a><span class="fu">moran.test</span>(<span class="at">x =</span> WL.trend3<span class="sc">$</span>residuals, <span class="at">listw =</span> WL.listw)</span></code></pre></div>
<pre><code>##
##  Moran I test under randomisation
##
## data:  WL.trend3$residuals
## weights: WL.listw
##
## Moran I statistic standard deviate = 17.199, p-value &lt; 2.2e-16
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance
##      0.4633803457     -0.0021321962      0.0007325452</code></pre>
<p>The fact that the residuals are not independent has important implications for prediction. Consider the following thought experiment.</p>
<p>Imagine that you were asked to guess whether the residual was positive or negative at the locations indicated with triangles in the figure. These are locations where an observation was not made, and we only have the interpolated value of the variable according to the trend surface model:</p>
<div class="sourceCode" id="cb775"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb775-1"><a href="spatially-continuous-data-iii.html#cb775-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> Walker_Lake, </span>
<span id="cb775-2"><a href="spatially-continuous-data-iii.html#cb775-2" aria-hidden="true" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> X, <span class="at">y =</span> Y)) <span class="sc">+</span></span>
<span id="cb775-3"><a href="spatially-continuous-data-iii.html#cb775-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="fu">aes</span>(<span class="at">color =</span> residual3)) <span class="sc">+</span></span>
<span id="cb775-4"><a href="spatially-continuous-data-iii.html#cb775-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">55</span>, <span class="dv">25</span>, <span class="dv">210</span>, <span class="dv">227</span>), <span class="at">y =</span> <span class="fu">c</span>(<span class="dv">200</span>, <span class="dv">90</span>, <span class="dv">90</span>, <span class="dv">230</span>)), <span class="co">#We are adding coordinates for the triangles in the figure</span></span>
<span id="cb775-5"><a href="spatially-continuous-data-iii.html#cb775-5" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> x, <span class="at">y =</span> y), <span class="at">shape =</span> <span class="dv">17</span>, <span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span></span>
<span id="cb775-6"><a href="spatially-continuous-data-iii.html#cb775-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">coord_equal</span>()</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-554-1.png" width="672" /></p>
<p>What would your guess be, and why? Would you say that your guess has a better than 50% chance of being right?</p>
<p>Now imagine that you were asked to guess whether the residual was positive or negative at the locations indicated with squares in the figure:</p>
<div class="sourceCode" id="cb776"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb776-1"><a href="spatially-continuous-data-iii.html#cb776-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> Walker_Lake, </span>
<span id="cb776-2"><a href="spatially-continuous-data-iii.html#cb776-2" aria-hidden="true" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> X, <span class="at">y =</span> Y)) <span class="sc">+</span></span>
<span id="cb776-3"><a href="spatially-continuous-data-iii.html#cb776-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="fu">aes</span>(<span class="at">color =</span> residual3)) <span class="sc">+</span></span>
<span id="cb776-4"><a href="spatially-continuous-data-iii.html#cb776-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">160</span>, <span class="dv">240</span>, <span class="dv">12</span>, <span class="dv">120</span>), <span class="at">y =</span> <span class="fu">c</span>(<span class="dv">38</span>, <span class="dv">280</span>, <span class="dv">240</span>, <span class="dv">180</span>)), <span class="co">#We are adding coordinates for the squares in the figure</span></span>
<span id="cb776-5"><a href="spatially-continuous-data-iii.html#cb776-5" aria-hidden="true" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">x =</span> x, <span class="at">y =</span> y), <span class="at">shape =</span> <span class="dv">15</span>, <span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span></span>
<span id="cb776-6"><a href="spatially-continuous-data-iii.html#cb776-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">coord_equal</span>()</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-555-1.png" width="672" /></p>
<p>Again, what would your guess be, and why? Would you be able to guess this way if the residuals were random?</p>
<p>The fact that you can guess and be fairly sure about the quality of your guess is a consequence of the strong residual pattern. If the residuals were random, there would be no information left to use: the odds of a residual being positive or negative would essentially be 50%. However, when there is residual pattern, this information can be used to enhance the quality of your guesses about the residuals, or in other words, of the <span class="math inline">\(\hat{\epsilon}_p\)</span> terms. At the very least you can guess whether they are positive or negative (therefore reducing their confidence intervals), but possibly you can learn even more from them, as will be seen later.</p>
<p>Before learning how to do this, however, we need to think more about the way in which we measure spatial pattern in spatially continuous data.</p>
</div>
<div id="measuring-spatial-dependence-in-spatially-continuous-data" class="section level2" number="35.5">
<h2><span class="header-section-number">35.5</span> Measuring spatial dependence in spatially continuous data</h2>
<p>In the preceding sections we used Moran’s <span class="math inline">\(I\)</span> coefficient to measure spatial pattern. Moran’s <span class="math inline">\(I\)</span> is, by design, a single-scale statistic, not unlike the case of nearest neighbor analysis in point patterns. The reason for this is that Moran’s <span class="math inline">\(I\)</span> is limited to detecting pattern at the scale at which the spatial weights are defined: for instance, at the level of adjacency, contiguity, or <span class="math inline">\(k\)</span>-nearest neighbors.</p>
<p>While this makes sense (mostly) in the case of area data, since the areas inherently introduce spatial discontinuities, it makes less sense in the case of fields, where the underlying process is typically smooth. In fact, more often we are interested in exploring the properties of the pattern over the field, not just the nearest neighbors.</p>
<p>One way of extending Moran’s <span class="math inline">\(I\)</span> analysis to multiple scales is by means of the <em>correlogram</em>. The correlogram is simply a sequence of Moran’s <span class="math inline">\(I\)</span> coefficients computed at different scales.</p>
<p>Consider for example the following sequence of coefficients, computed for <span class="math inline">\(k\)</span>=10 neighbors to <span class="math inline">\(k\)</span>=30 neighbors. Notice how the <code>for</code> loop calculates spatial weights using the designated number of neighbors, before calculating Moran’s <span class="math inline">\(I\)</span>.</p>
<div class="sourceCode" id="cb777"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb777-1"><a href="spatially-continuous-data-iii.html#cb777-1" aria-hidden="true" tabindex="-1"></a>k <span class="ot">&lt;-</span> <span class="fu">c</span>(<span class="dv">10</span><span class="sc">:</span><span class="dv">30</span>) <span class="co"># Initialize the values of k.</span></span>
<span id="cb777-2"><a href="spatially-continuous-data-iii.html#cb777-2" aria-hidden="true" tabindex="-1"></a>moranI <span class="ot">&lt;-</span> <span class="fu">numeric</span>(<span class="at">length =</span> <span class="fu">length</span>(k)) <span class="co"># Initialize an empty vector to store the results of calculating Moran&#39;s I </span></span>
<span id="cb777-3"><a href="spatially-continuous-data-iii.html#cb777-3" aria-hidden="true" tabindex="-1"></a>correlogram <span class="ot">&lt;-</span> <span class="fu">data.frame</span>(k, moranI) <span class="co"># Initialize an empty dataframe to store the values of k and moranI</span></span>
<span id="cb777-4"><a href="spatially-continuous-data-iii.html#cb777-4" aria-hidden="true" tabindex="-1"></a><span class="co"># A `for` loop is a way of repeating instructions a defined number of times, here from 1 to the length of vector `k`.</span></span>
<span id="cb777-5"><a href="spatially-continuous-data-iii.html#cb777-5" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(k)){</span>
<span id="cb777-6"><a href="spatially-continuous-data-iii.html#cb777-6" aria-hidden="true" tabindex="-1"></a>  listwk <span class="ot">&lt;-</span> Walker_Lake[,<span class="dv">2</span><span class="sc">:</span><span class="dv">3</span>] <span class="sc">%&gt;%</span></span>
<span id="cb777-7"><a href="spatially-continuous-data-iii.html#cb777-7" aria-hidden="true" tabindex="-1"></a>    <span class="fu">as.matrix</span>() <span class="sc">%&gt;%</span></span>
<span id="cb777-8"><a href="spatially-continuous-data-iii.html#cb777-8" aria-hidden="true" tabindex="-1"></a>    <span class="fu">knearneigh</span>(<span class="at">k =</span> k[i]) <span class="sc">%&gt;%</span> <span class="co"># Use the ith element of vector `k` to find the nearest neighbors</span></span>
<span id="cb777-9"><a href="spatially-continuous-data-iii.html#cb777-9" aria-hidden="true" tabindex="-1"></a>    <span class="fu">knn2nb</span>() <span class="sc">%&gt;%</span></span>
<span id="cb777-10"><a href="spatially-continuous-data-iii.html#cb777-10" aria-hidden="true" tabindex="-1"></a>    <span class="fu">nb2listw</span>()</span>
<span id="cb777-11"><a href="spatially-continuous-data-iii.html#cb777-11" aria-hidden="true" tabindex="-1"></a>  m <span class="ot">&lt;-</span> <span class="fu">moran.test</span>(<span class="at">x =</span> WL.trend3<span class="sc">$</span>residuals, <span class="at">listw =</span> listwk) <span class="co">#Moran test for residuals</span></span>
<span id="cb777-12"><a href="spatially-continuous-data-iii.html#cb777-12" aria-hidden="true" tabindex="-1"></a>  correlogram<span class="sc">$</span>moranI[i] <span class="ot">&lt;-</span> m<span class="sc">$</span>estimate[<span class="dv">1</span>] <span class="co">#calling moran&#39;s I values from correlogram dataframe</span></span>
<span id="cb777-13"><a href="spatially-continuous-data-iii.html#cb777-13" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>Given the values of Moran’s <span class="math inline">\(I\)</span> at different scales (i.e., values of <span class="math inline">\(k\)</span>), the correlogram can be plotted as:</p>
<div class="sourceCode" id="cb778"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb778-1"><a href="spatially-continuous-data-iii.html#cb778-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> correlogram, <span class="fu">aes</span>(<span class="at">x =</span> k, <span class="at">y =</span> moranI)) <span class="sc">+</span> </span>
<span id="cb778-2"><a href="spatially-continuous-data-iii.html#cb778-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>()</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-557-1.png" width="672" /></p>
<p>As can be seen in the plot, spatial autocorrelation tends to decline as the number of nearest neighbors used in the test grows - in other words, as the scale of the test increases. This is a common occurrence: when autocorrelation is present, observations tend to be more similar to their closest neighbors than to their more distant neighbors.</p>
<p>The use of <span class="math inline">\(k\)</span>-nearest neighbors points to a problem, however. The scale of the process does not depend on distance, which would be a more natural metric for a continuous process. In this case, <span class="math inline">\(k\)</span>-nearest neighbors were used to ensure that each sum in the coefficient had the same number of observations. However, this means that “neighborhoods” will be geographically smaller where the observations are more dense, and larger where they are sparse.</p>
<p>While this issue is not insurmountable (for instance, instead of <span class="math inline">\(k\)</span>-nearst neighbors we could have used the neighbors found at a certain distance), it points out to the fact that Moran’s <span class="math inline">\(I\)</span> is not by design well suited for the analysis of spatially continuous data.</p>
<p>A different approach, known as variographic analysis, is introduced next.</p>
</div>
<div id="variographic-analyisis" class="section level2" number="35.6">
<h2><span class="header-section-number">35.6</span> Variographic analyisis</h2>
<p>To introduce variographic analysis it is worthwhile to recall the definition of the covariance between two variables, say <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span>:
<span class="math display">\[
C(X,Y) = E[{(X_i^2 - \bar{X})(Y_i^2 - \bar{Y})}]
\]</span>
Where <span class="math inline">\(\bar{X}\)</span> and <span class="math inline">\(\bar{Y}\)</span> are the means of <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span> respectively.</p>
<p>The expectation operator <span class="math inline">\(E[]\)</span> turns out to be the mean:
<span class="math display">\[
C(X,Y) = \frac{1}{n}\sum_{i=1}^{n}{(X_i^2 - \bar{X})(Y_j^2 - \bar{Y})}
\]</span></p>
<p>The observations <span class="math inline">\(X_i\)</span> and <span class="math inline">\(Y_i\)</span> in the covariance formula can be seen as a points in a scatterplot, with the axes shifted to the means of <span class="math inline">\(X\)</span> and <span class="math inline">\(Y\)</span>, as shown in Figure @ref{fig:covariance-as-scatterplot}.</p>
<div class="figure"><span style="display:block;" id="fig:covariance-as-a-scatterplot"></span>
<img src="figures/35-Figure-1.jpg" alt="\label{fig:covariance-as-scatterplot}Observations of the covariance as a scatterplot" width="640" />
<p class="caption">
FIGURE 35.1: Observations of the covariance as a scatterplot
</p>
</div>
<p>The autocovariance of variable <span class="math inline">\(z\)</span> can be defined in a similar way, the difference being that instead of two variables, it is the covariance of a variable with <em>itself</em> but at a different location (i.e., between locations <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span>):
<span class="math display">\[
C(z_i,z_j) = E[{(z_i^2 - \bar{z})(z_j^2 - \bar{z})}]
\]</span></p>
<p>To implement the spatial autocovariance we need some criterion to explicitly define the spatial relationship between locations <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span>. A useful criterion in this case is as follows:</p>
<p><span class="math display">\[
w_{ij}(h)=\bigg\{\begin{array}{l l}
1\text{ if } d_{ij} = h\\
0\text{ otherwise}\\
\end{array}
\]</span>
In other words, <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> are considered to be spatially related for the purposes of calculating the autocovariance, if the distance between the two locations is equal to some predefined spatial lag <span class="math inline">\(h\)</span>.</p>
<p>The above criterion makes explicit the assumption that the autocovariance is a function of the separation <span class="math inline">\(h\)</span> between two observations, but not of other factors, such as the angle between observations. This assumption is called <em>isotropy</em>.</p>
<p>Further, if we assume that the variance of <span class="math inline">\(z\)</span> is constant, and the correlation between observations does not depend on location (an assumption called <em>intrinsic stationarity</em>), we can <em>pool</em> observations from across the map to create a scatterplot to form the basis of the autocovariance calculations.</p>
<p>Consider the (regular) arrangement of observations spaced at <span class="math inline">\(h\)</span> in Figure @ref{fig:autocovariance}. Each observation generally has four neighbors, with the exception of those in the edges, which have fewer neighbors at spatial lag <span class="math inline">\(h\)</span>. This means that most observations will contribute four points to the scatterplot (<span class="math inline">\(z_i\)</span> and <span class="math inline">\(z_j\)</span>, <span class="math inline">\(z_k\)</span>, <span class="math inline">\(z_l\)</span>, and <span class="math inline">\(z_m\)</span>), and others will contribute three or at least two (those in the corners).</p>
<div class="figure"><span style="display:block;" id="fig:autocovariance"></span>
<img src="figures/35-Figure-2.jpg" alt="\label{fig:autocovariance}Finding spatial pairs for the calculation of the autocovariance" width="640" />
<p class="caption">
FIGURE 35.2: Finding spatial pairs for the calculation of the autocovariance
</p>
</div>
<p>Given those pairs of observations, the autocovariance at lag <span class="math inline">\(h\)</span> can be calculated as:
<span class="math display">\[
C_{z}(h) = \frac{\sum_{i=1}^{n}{w_{ij}(h)(z_i^2 - \bar{z})(z_j^2 - \bar{z})}}{\sum_{i=1}^n{w_{ij}(h)}}
\]</span></p>
<p>Changing the spatial lag <span class="math inline">\(h\)</span> allows us to calculate the autocovariance at different scales. The plot of the autocovariance at different scales is called a <em>covariogram</em>.</p>
<p>A related quantity that is more commonly used (mainly for historical reasons) is the <em>semivariance</em>.</p>
<p>The semivariance is defined as follows, calculated based on the difference between <span class="math inline">\(z_i\)</span> and <span class="math inline">\(z_j\)</span>:
<span class="math display">\[
\hat{\gamma}_{z}(h) = \frac{\sum_{i=1}^{n}{w_{ij}(h)(z_i - z_j)^2}}{2\sum_{i=1}^n{w_{ij}(h)}}
\]</span>
The plot of the semivariance at different lags <span class="math inline">\(h\)</span> is called a <em>semivariogram</em>.</p>
<p>The covariogram and semivariogram are related by the following formula:
<span class="math display">\[
C_{z}(h) =\sigma^2 - \hat{\gamma}_{z}(h)
\]</span>
where <span class="math inline">\(\sigma^2\)</span> is the sample variance.</p>
<p>The condition that <span class="math inline">\(d_{ij} = h\)</span> is, with the exception of gridded data, too strict, and is often relaxed in the following way:</p>
<p><span class="math display">\[
w_{ij}(\tilde{h})=\bigg\{\begin{array}{l l}
1\text{ if } d_{ij}\simeq h\\
0\text{ otherwise}\\
\end{array}
\]</span></p>
<p>In this way, the distance between observations <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> does not need to be <em>exactly</em>, but can be an approximation. The approximation can be defined explicitly as follows:</p>
<p><span class="math display">\[
w_{ij}(\tilde{h})=\bigg\{\begin{array}{l l}
1\text{ if } h - \Delta h &lt; d_{ij} &lt; h + \Delta h\\
0\text{ otherwise}\\
\end{array}
\]</span>
Instead of forming pairs with observations that are at exactly a distance <span class="math inline">\(h\)</span> (which would lead in many cases to too few pairs), pairs are formed with observations at approximately lag <span class="math inline">\(h\)</span> (or <span class="math inline">\(\tilde{h}\)</span>), with a tolerance given by <span class="math inline">\(\Delta h\)</span>.</p>
<p>Analysis based on the semivariogram (called <em>variographic analysis</em>) is implemented in <code>R</code> in the <code>gstat</code> package.</p>
<p>We will illustrate the use of the semivariogram by means of the Walker Lake data. The package <code>gstat</code> accepts simple features objects of the <code>sf</code> package, so we convert our dataframe into such an object:</p>
<div class="sourceCode" id="cb779"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb779-1"><a href="spatially-continuous-data-iii.html#cb779-1" aria-hidden="true" tabindex="-1"></a>Walker_Lake.sf <span class="ot">&lt;-</span> <span class="fu">st_as_sf</span>(Walker_Lake, <span class="at">coords =</span> <span class="fu">c</span>(<span class="st">&quot;X&quot;</span>, <span class="st">&quot;Y&quot;</span>))</span>
<span id="cb779-2"><a href="spatially-continuous-data-iii.html#cb779-2" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(Walker_Lake.sf)</span></code></pre></div>
<pre><code>## [1] &quot;sf&quot;         &quot;tbl_df&quot;     &quot;tbl&quot;        &quot;data.frame&quot;</code></pre>
<p>The empirical semivariogram is calculated by means of the <code>gstat::variogram</code> function, as follows:</p>
<div class="sourceCode" id="cb781"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb781-1"><a href="spatially-continuous-data-iii.html#cb781-1" aria-hidden="true" tabindex="-1"></a>variogram_z <span class="ot">&lt;-</span> <span class="fu">variogram</span>(V <span class="sc">~</span> <span class="dv">1</span>, <span class="at">data =</span> Walker_Lake.sf) <span class="co"># `variogram()` calculates the sample semivariogram from data, or if a linear model is given, for the residuals; in this case, the formula `V ~ 1` means that we are not using a model</span></span>
<span id="cb781-2"><a href="spatially-continuous-data-iii.html#cb781-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb781-3"><a href="spatially-continuous-data-iii.html#cb781-3" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> variogram_z, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span> <span class="co">#Note we are plotting the data of &#39;variogram_z&#39;</span></span>
<span id="cb781-4"><a href="spatially-continuous-data-iii.html#cb781-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span> </span>
<span id="cb781-5"><a href="spatially-continuous-data-iii.html#cb781-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> np), <span class="at">nudge_y =</span> <span class="sc">-</span><span class="dv">1500</span>) <span class="sc">+</span> <span class="co"># Add labels to indicate the number of pairs of observations used in the calculation of each point in the variogram</span></span>
<span id="cb781-6"><a href="spatially-continuous-data-iii.html#cb781-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">&quot;Distance&quot;</span>) <span class="sc">+</span> <span class="fu">ylab</span>(<span class="st">&quot;Semivariance&quot;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-559-1.png" width="672" /></p>
<p>The numbers indicate the number of pairs of observations used to calculate the semivariance at the corresponding lag.</p>
<p>Since the sample variance is:</p>
<div class="sourceCode" id="cb782"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb782-1"><a href="spatially-continuous-data-iii.html#cb782-1" aria-hidden="true" tabindex="-1"></a>s2 <span class="ot">&lt;-</span> <span class="fu">var</span>(Walker_Lake<span class="sc">$</span>V) <span class="co"># We are calculating the variance of X</span></span>
<span id="cb782-2"><a href="spatially-continuous-data-iii.html#cb782-2" aria-hidden="true" tabindex="-1"></a>s2</span></code></pre></div>
<pre><code>## [1] 90694.59</code></pre>
<p>It follows that the covariogram in this case is:</p>
<div class="sourceCode" id="cb784"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb784-1"><a href="spatially-continuous-data-iii.html#cb784-1" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> variogram_z, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> s2 <span class="sc">-</span> gamma)) <span class="sc">+</span></span>
<span id="cb784-2"><a href="spatially-continuous-data-iii.html#cb784-2" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span> </span>
<span id="cb784-3"><a href="spatially-continuous-data-iii.html#cb784-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> np), <span class="at">nudge_y =</span> <span class="sc">-</span><span class="dv">1500</span>) <span class="sc">+</span></span>
<span id="cb784-4"><a href="spatially-continuous-data-iii.html#cb784-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">&quot;Distance&quot;</span>) <span class="sc">+</span> </span>
<span id="cb784-5"><a href="spatially-continuous-data-iii.html#cb784-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylab</span>(<span class="st">&quot;Autocovariance&quot;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-561-1.png" width="672" /></p>
<p>As expected, the autocovariance (and hence, the autocorrelation) is stronger at short spatial lags, and declines at larger spatial lags.</p>
<p>The above plots are the <em>empirical</em> semivariogram and covariogram. These plots are used to model a theoretical semivariogram, a function that can be used to estimate spatial dependence at any lag within the domain of the - and not just at the distances for which we have points in the empirical variogram.</p>
<p>Since the semivariogram is the expectation of the square, the function selected for modeling the theoretical semivariogram must be non-negative. Several functions satisfy this condition, a list of which are available in <code>gstat</code> as shown below:</p>
<div class="sourceCode" id="cb785"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb785-1"><a href="spatially-continuous-data-iii.html#cb785-1" aria-hidden="true" tabindex="-1"></a><span class="fu">vgm</span>() <span class="co">#this function generates a variogram mode. Here, we are able to view the list of possible models for a semivariogram</span></span></code></pre></div>
<pre><code>##    short                                      long
## 1    Nug                              Nug (nugget)
## 2    Exp                         Exp (exponential)
## 3    Sph                           Sph (spherical)
## 4    Gau                            Gau (gaussian)
## 5    Exc        Exclass (Exponential class/stable)
## 6    Mat                              Mat (Matern)
## 7    Ste Mat (Matern, M. Stein&#39;s parameterization)
## 8    Cir                            Cir (circular)
## 9    Lin                              Lin (linear)
## 10   Bes                              Bes (bessel)
## 11   Pen                      Pen (pentaspherical)
## 12   Per                            Per (periodic)
## 13   Wav                                Wav (wave)
## 14   Hol                                Hol (hole)
## 15   Log                         Log (logarithmic)
## 16   Pow                               Pow (power)
## 17   Spl                              Spl (spline)
## 18   Leg                            Leg (Legendre)
## 19   Err                   Err (Measurement error)
## 20   Int                           Int (Intercept)</code></pre>
<p>The anatomy of a semivariogram includes a range, a sill, and possibly a nugget. These elements are shown in Figure @ref{fig:semivariogram}.</p>
<div class="figure"><span style="display:block;" id="fig:anatomy-semivariogram"></span>
<img src="figures/35-Figure-2.jpg" alt="\label{fig:semivarigoram}Anatomy of a semivariogram" width="640" />
<p class="caption">
FIGURE 35.3: Anatomy of a semivariogram
</p>
</div>
<p>Since the semivariogram is calculated based on the square of the differences <span class="math inline">\(z_i - z_j\)</span>, the smaller the semivariance is, the more similar observations tend to be. In principle, the semivariogram begins at zero, because at distance zero an observation is identical to itself (i.e., <span class="math inline">\(z_i - z_i\)</span>). The range is the distance at which the sill is reached. The sill, on the other hand, is the point at which the semivariance becomes simply the variance, meaning that there is no more or less similarity between observations than would be implied by the variance of the sample.</p>
<p>An additional element is the nugget. While the semivariogram in principle begins at zero, sometime discontinuities near the origin can be observed. The terminology is from mining, and reflects the fact that a nugget could be very different from the material around it, hence the jump in the semivariogram.</p>
<p>Some theoretical functions are shown next.</p>
<p>Exponential semivariogram:</p>
<div class="sourceCode" id="cb787"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb787-1"><a href="spatially-continuous-data-iii.html#cb787-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">variogramLine</span>(<span class="fu">vgm</span>(<span class="dv">1</span>, <span class="st">&quot;Exp&quot;</span>, <span class="dv">1</span>), <span class="dv">10</span>), <span class="at">type =</span> <span class="st">&#39;l&#39;</span>) <span class="co">#We use &quot;exp&quot; to denote the use of an exponential semivariogram. Refer to the list on line 297 and explore the different outcomes of the listed variogram models! </span></span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-563-1.png" width="672" /></p>
<p>Spherical semivariogram:</p>
<div class="sourceCode" id="cb788"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb788-1"><a href="spatially-continuous-data-iii.html#cb788-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">variogramLine</span>(<span class="fu">vgm</span>(<span class="dv">1</span>, <span class="st">&quot;Sph&quot;</span>, <span class="dv">1</span>), <span class="dv">10</span>), <span class="at">type =</span> <span class="st">&#39;l&#39;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-564-1.png" width="672" /></p>
<p>Gaussian semivariogram:</p>
<div class="sourceCode" id="cb789"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb789-1"><a href="spatially-continuous-data-iii.html#cb789-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">variogramLine</span>(<span class="fu">vgm</span>(<span class="dv">1</span>, <span class="st">&quot;Gau&quot;</span>, <span class="dv">1</span>), <span class="dv">10</span>), <span class="at">type =</span> <span class="st">&#39;l&#39;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-565-1.png" width="672" /></p>
<p>These plots illustrate some differences in the behavior of the models. For identical parameters, the Gaussian model provides smoother changes near the origin. The spherical model reaches the sill more rapidly than the other models.</p>
<p>To fit a theoretical semivariogram to the empirical one, the function <code>fit.variogram</code> is used:</p>
<div class="sourceCode" id="cb790"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb790-1"><a href="spatially-continuous-data-iii.html#cb790-1" aria-hidden="true" tabindex="-1"></a>variogram_z.t <span class="ot">&lt;-</span> <span class="fu">fit.variogram</span>(variogram_z, <span class="at">model =</span> <span class="fu">vgm</span>(<span class="st">&quot;Exp&quot;</span>)) <span class="co">#&#39;fit_variogram&#39; selects the type of model that will fit the empirical semivariogram best</span></span></code></pre></div>
<p>The results of which can be plotted after passing the model the the function <code>variogramLine</code>:</p>
<div class="sourceCode" id="cb791"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb791-1"><a href="spatially-continuous-data-iii.html#cb791-1" aria-hidden="true" tabindex="-1"></a>gamma.t <span class="ot">&lt;-</span> <span class="fu">variogramLine</span>(variogram_z.t, <span class="at">maxdist =</span> <span class="dv">130</span>) <span class="co">#Notice how &#39;maxdist&#39; is 130, and the model does not exceed that value. </span></span>
<span id="cb791-2"><a href="spatially-continuous-data-iii.html#cb791-2" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> variogram_z, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span></span>
<span id="cb791-3"><a href="spatially-continuous-data-iii.html#cb791-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span> </span>
<span id="cb791-4"><a href="spatially-continuous-data-iii.html#cb791-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> gamma.t, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span></span>
<span id="cb791-5"><a href="spatially-continuous-data-iii.html#cb791-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">&quot;Distance&quot;</span>) <span class="sc">+</span> <span class="fu">ylab</span>(<span class="st">&quot;Semivariance&quot;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-567-1.png" width="672" /></p>
<p>A set of models can be passed as an argument to <code>fit.variogram</code>, in which case the value (output) of the function is the model that provides the best fit to the empirical semivariogram:</p>
<div class="sourceCode" id="cb792"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb792-1"><a href="spatially-continuous-data-iii.html#cb792-1" aria-hidden="true" tabindex="-1"></a>variogram_z.t <span class="ot">&lt;-</span> <span class="fu">fit.variogram</span>(variogram_z, <span class="at">model =</span> <span class="fu">vgm</span>(<span class="st">&quot;Exp&quot;</span>, <span class="st">&quot;Sph&quot;</span>, <span class="st">&quot;Gau&quot;</span>)) <span class="co">#Choosing the best fit again</span></span>
<span id="cb792-2"><a href="spatially-continuous-data-iii.html#cb792-2" aria-hidden="true" tabindex="-1"></a>variogram_z.t</span></code></pre></div>
<pre><code>##   model     psill    range
## 1   Nug  4045.567  0.00000
## 2   Exp 90703.773 12.52591</code></pre>
<p>In this case, it can be seen that the best fitting model is the exponential, as follows:</p>
<div class="sourceCode" id="cb794"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb794-1"><a href="spatially-continuous-data-iii.html#cb794-1" aria-hidden="true" tabindex="-1"></a>gamma.t <span class="ot">&lt;-</span> <span class="fu">variogramLine</span>(variogram_z.t, <span class="at">maxdist =</span> <span class="dv">130</span>)</span>
<span id="cb794-2"><a href="spatially-continuous-data-iii.html#cb794-2" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> variogram_z, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span></span>
<span id="cb794-3"><a href="spatially-continuous-data-iii.html#cb794-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">size =</span> <span class="dv">3</span>) <span class="sc">+</span> </span>
<span id="cb794-4"><a href="spatially-continuous-data-iii.html#cb794-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> gamma.t, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span></span>
<span id="cb794-5"><a href="spatially-continuous-data-iii.html#cb794-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">&quot;Distance&quot;</span>) <span class="sc">+</span> <span class="fu">ylab</span>(<span class="st">&quot;Semivariance&quot;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-569-1.png" width="672" /></p>
<p>For comparison, we will do the variographic analysis of a simulated random dataset.</p>
<p>Generate coordinates for observations and expand on a grid:</p>
<div class="sourceCode" id="cb795"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb795-1"><a href="spatially-continuous-data-iii.html#cb795-1" aria-hidden="true" tabindex="-1"></a><span class="co">#We are generating a regular sequence of coordinates by means of &#39;seq&#39; </span></span>
<span id="cb795-2"><a href="spatially-continuous-data-iii.html#cb795-2" aria-hidden="true" tabindex="-1"></a>x <span class="ot">&lt;-</span> <span class="fu">seq</span>(<span class="at">from =</span> <span class="dv">0</span>, <span class="at">to =</span> <span class="dv">250</span>, <span class="at">by =</span> <span class="dv">10</span>)</span>
<span id="cb795-3"><a href="spatially-continuous-data-iii.html#cb795-3" aria-hidden="true" tabindex="-1"></a>y <span class="ot">&lt;-</span> <span class="fu">seq</span>(<span class="at">from =</span> <span class="dv">0</span>, <span class="at">to =</span> <span class="dv">290</span>, <span class="at">by =</span> <span class="dv">10</span>)</span>
<span id="cb795-4"><a href="spatially-continuous-data-iii.html#cb795-4" aria-hidden="true" tabindex="-1"></a>df <span class="ot">&lt;-</span> <span class="fu">expand.grid</span>(<span class="at">x =</span> x, <span class="at">y =</span> y) <span class="co">#df is our dataframe </span></span></code></pre></div>
<p>Then, create a random variable for this coordinates:</p>
<div class="sourceCode" id="cb796"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb796-1"><a href="spatially-continuous-data-iii.html#cb796-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">100</span>) <span class="co">#&#39;set.seed&#39; is used for replicability: it uses the seed in the argument for generating random numbers</span></span>
<span id="cb796-2"><a href="spatially-continuous-data-iii.html#cb796-2" aria-hidden="true" tabindex="-1"></a>df<span class="sc">$</span>z <span class="ot">&lt;-</span> <span class="fu">rnorm</span>(<span class="at">n =</span> <span class="dv">780</span>, <span class="at">mean =</span> <span class="dv">500</span>, <span class="at">sd =</span> <span class="dv">300</span>)</span></code></pre></div>
<p>Finally, convert to a simple features object:</p>
<div class="sourceCode" id="cb797"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb797-1"><a href="spatially-continuous-data-iii.html#cb797-1" aria-hidden="true" tabindex="-1"></a>df <span class="ot">&lt;-</span> <span class="fu">st_as_sf</span>(df, <span class="at">coords =</span> <span class="fu">c</span>(<span class="st">&quot;x&quot;</span>, <span class="st">&quot;y&quot;</span>)) </span></code></pre></div>
<p>The empirical variogram is:</p>
<div class="sourceCode" id="cb798"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb798-1"><a href="spatially-continuous-data-iii.html#cb798-1" aria-hidden="true" tabindex="-1"></a>variogram_df <span class="ot">&lt;-</span> <span class="fu">variogram</span>(z <span class="sc">~</span> <span class="dv">1</span>, <span class="at">data =</span> df) <span class="co">#establishing the foundation of the variogram</span></span>
<span id="cb798-2"><a href="spatially-continuous-data-iii.html#cb798-2" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> variogram_df, <span class="fu">aes</span>(<span class="at">x =</span> dist, <span class="at">y =</span> gamma)) <span class="sc">+</span></span>
<span id="cb798-3"><a href="spatially-continuous-data-iii.html#cb798-3" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span> </span>
<span id="cb798-4"><a href="spatially-continuous-data-iii.html#cb798-4" aria-hidden="true" tabindex="-1"></a>  <span class="fu">geom_text</span>(<span class="fu">aes</span>(<span class="at">label =</span> np), <span class="at">nudge_y =</span> <span class="sc">-</span><span class="dv">1500</span>) <span class="sc">+</span> </span>
<span id="cb798-5"><a href="spatially-continuous-data-iii.html#cb798-5" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylim</span>(<span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">98100</span>)) <span class="sc">+</span></span>
<span id="cb798-6"><a href="spatially-continuous-data-iii.html#cb798-6" aria-hidden="true" tabindex="-1"></a>  <span class="fu">xlab</span>(<span class="st">&quot;Distance&quot;</span>) <span class="sc">+</span> <span class="fu">ylab</span>(<span class="st">&quot;Semivariance&quot;</span>)  <span class="co">#add labels to axes</span></span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-573-1.png" width="672" /></p>
<p>The range of the semivariogram appears to be zero, or alternatively, there seems to be a pure nugget effect. This is as expected. Since the data are spatially random, they are not more similar at shorter distances than they would be at longer distances.</p>

</div>
</div>
<h3>References</h3>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-Bailey1995" class="csl-entry">
Bailey, T. C., and A. C. Gatrell. 1995. <em>Interactive Spatial Data Analysis</em>. Book. Essex: Addison Wesley Longman.
</div>
<div id="ref-Bivand2008" class="csl-entry">
Bivand, R. S., E. J. Pebesma, and V. Gómez-Rubio. 2008. <em>Applied Spatial Data Analysis with r</em>. Book. New York: Springer Science+Business Media.
</div>
<div id="ref-Brunsdon2015R" class="csl-entry">
Brunsdon, Chris, and Lex Comber. 2015. <em>An Introduction to r for Spatial Analysis and Mapping</em>. Book. Sage.
</div>
<div id="ref-Isaaks1989applied" class="csl-entry">
Isaaks, E. H., and R. M. Srivastava. 1989. <em>Applied Geostatistics</em>. Book. New York: Oxford University Press.
</div>
<div id="ref-Osullivan2010" class="csl-entry">
O’Sullivan, David, and David Unwin. 2010. <em>Geographic Information Analysis</em>. Book. 2nd. Edition. Hoboken, New Jersey: John Wiley &amp; Sons.
</div>
</div>
            </section>

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