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  <title>Chapter 11 Point Pattern Analysis II | 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="point-pattern-analysis-ii" class="section level1" number="11">
<h1><span class="header-section-number">Chapter 11</span> Point Pattern Analysis II</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 last practice/session your learning objectives included:</p>
<ol style="list-style-type: decimal">
<li>A formal definition of point pattern.</li>
<li>Processes and point patterns.</li>
<li>The concepts of intensity and density.</li>
<li>The concept of quadrats and how to create density maps.</li>
<li>More ways to control the look of your plots, in particular faceting and adding lines.</li>
</ol>
<p>Please review the previous practices if you need a refresher on these concepts.</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-10" class="section level2" number="11.1">
<h2><span class="header-section-number">11.1</span> Learning Objectives</h2>
<p>In this practice, you will learn:</p>
<ol style="list-style-type: decimal">
<li>The intuition behind the quadrat-based test of independence.</li>
<li>About the limitations of quadrat-based analysis.</li>
<li>The concept of kernel density.</li>
<li>More ways to manipulate objects to do point pattern analysis using <code>spatstat</code>.</li>
</ol>
</div>
<div id="suggested-readings-4" class="section level2" number="11.2">
<h2><span class="header-section-number">11.2</span> Suggested Readings</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, Chapter 3. Longman: Essex.</li>
<li>Baddeley A, Rubak E, Turner R <span class="citation">(<a href="#ref-Baddeley2015" role="doc-biblioref">2016</a>)</span> Spatial Point Pattern: Methodology and Applications with R, Chapter 6. CRC: Boca Raton.</li>
<li>Bivand RS, Pebesma E, Gomez-Rubio V <span class="citation">(<a href="#ref-Bivand2008" role="doc-biblioref">2008</a>)</span> Applied Spatial Data Analysis with R, Chapter 7. 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, 6.1 - 6.6. Sage: Los Angeles.</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, Chapter 5. John Wiley &amp; Sons: New Jersey.</li>
</ul>
</div>
<div id="preliminaries-8" class="section level2" number="11.3">
<h2><span class="header-section-number">11.3</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="cb224"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb224-1"><a href="point-pattern-analysis-ii.html#cb224-1" 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="cb225"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb225-1"><a href="point-pattern-analysis-ii.html#cb225-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(geog4ga3)</span>
<span id="cb225-2"><a href="point-pattern-analysis-ii.html#cb225-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(spatstat)</span>
<span id="cb225-3"><a href="point-pattern-analysis-ii.html#cb225-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code></pre></div>
<p>Load the datasets that you will use for this practice:</p>
<div class="sourceCode" id="cb226"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb226-1"><a href="point-pattern-analysis-ii.html#cb226-1" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(<span class="st">&quot;PointPatterns&quot;</span>)</span>
<span id="cb226-2"><a href="point-pattern-analysis-ii.html#cb226-2" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(<span class="st">&quot;pp0_df&quot;</span>)</span></code></pre></div>
<p><code>PointPatterns</code> is a data frame with four sets of spatial events, labeled as “Pattern 1”, “Pattern 2”, “Pattern 3”, and “Pattern 4”. Each set has <span class="math inline">\(n=60\)</span> events. You can check the class of this object by means of the function class <code>class()</code>.</p>
<div class="sourceCode" id="cb227"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb227-1"><a href="point-pattern-analysis-ii.html#cb227-1" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(PointPatterns)</span></code></pre></div>
<pre><code>## [1] &quot;data.frame&quot;</code></pre>
<p>The second data frame (i.e., <code>pp0_df</code>) includes the coordinates <code>x</code> and <code>y</code> of two sets of spatial events, labeled as “Pattern 1” and “Pattern 2”.</p>
<p>The summary for <code>PointPatterns</code> shows that these point patterns are located in a square-unit window (check the max and min values of x and y):</p>
<div class="sourceCode" id="cb229"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb229-1"><a href="point-pattern-analysis-ii.html#cb229-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(PointPatterns)</span></code></pre></div>
<pre><code>##        x                y                 Pattern
##  Min.   :0.0169   Min.   :0.005306   Pattern 1:60
##  1st Qu.:0.2731   1st Qu.:0.289020   Pattern 2:60
##  Median :0.4854   Median :0.550000   Pattern 3:60
##  Mean   :0.5074   Mean   :0.538733   Pattern 4:60
##  3rd Qu.:0.7616   3rd Qu.:0.797850
##  Max.   :0.9990   Max.   :0.999808</code></pre>
<p>The same is true for <code>pp0_df</code>:</p>
<div class="sourceCode" id="cb231"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb231-1"><a href="point-pattern-analysis-ii.html#cb231-1" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(pp0_df)</span></code></pre></div>
<pre><code>##        x                y                 marks
##  Min.   :0.0456   Min.   :0.03409   Pattern 1:36
##  1st Qu.:0.2251   1st Qu.:0.22963   Pattern 2:36
##  Median :0.4282   Median :0.43363
##  Mean   :0.4916   Mean   :0.47952
##  3rd Qu.:0.7812   3rd Qu.:0.77562
##  Max.   :0.9564   Max.   :0.94492</code></pre>
<p>As seen in the previous practice and activity, the package <code>spatstat</code> employs a type of object called <code>ppp</code> (for <em>p</em>lanar <em>p</em>oint <em>p</em>attern). Fortunately, it is relatively simple to convert a data frame into a <code>ppp</code> object by means of <code>as.ppp()</code>. This function requires that you define a window for the point pattern, something we can do by means of the <code>owin</code> function:</p>
<div class="sourceCode" id="cb233"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb233-1"><a href="point-pattern-analysis-ii.html#cb233-1" aria-hidden="true" tabindex="-1"></a><span class="co"># &quot;W&quot; will appear in your environment as a defined window with boundaries of (1,1)</span></span>
<span id="cb233-2"><a href="point-pattern-analysis-ii.html#cb233-2" aria-hidden="true" tabindex="-1"></a>W <span class="ot">&lt;-</span> <span class="fu">owin</span>(<span class="at">xrange =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>), <span class="at">yrange =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>))</span></code></pre></div>
<p>Then the data frames are converted using the <code>as.ppp</code> function:</p>
<div class="sourceCode" id="cb234"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb234-1"><a href="point-pattern-analysis-ii.html#cb234-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Converts the data frame to planar point pattern using the defined window &quot;W&quot;</span></span>
<span id="cb234-2"><a href="point-pattern-analysis-ii.html#cb234-2" aria-hidden="true" tabindex="-1"></a>pp0.ppp <span class="ot">&lt;-</span> <span class="fu">as.ppp</span>(pp0_df, <span class="at">W =</span> W)</span>
<span id="cb234-3"><a href="point-pattern-analysis-ii.html#cb234-3" aria-hidden="true" tabindex="-1"></a>PointPatterns.ppp <span class="ot">&lt;-</span> <span class="fu">as.ppp</span>(PointPatterns, <span class="at">W =</span> W)</span></code></pre></div>
<p>You can verify that the new objects are indeed of <code>ppp</code>-class:</p>
<div class="sourceCode" id="cb235"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb235-1"><a href="point-pattern-analysis-ii.html#cb235-1" aria-hidden="true" tabindex="-1"></a><span class="co">#&quot;class&quot; is an excellent tool to use when verifying the execution of a previous line of code </span></span>
<span id="cb235-2"><a href="point-pattern-analysis-ii.html#cb235-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb235-3"><a href="point-pattern-analysis-ii.html#cb235-3" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(pp0.ppp)</span></code></pre></div>
<pre><code>## [1] &quot;ppp&quot;</code></pre>
<div class="sourceCode" id="cb237"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb237-1"><a href="point-pattern-analysis-ii.html#cb237-1" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(PointPatterns.ppp)</span></code></pre></div>
<pre><code>## [1] &quot;ppp&quot;</code></pre>
</div>
<div id="a-quadrat-based-test-for-spatial-independence" class="section level2" number="11.4">
<h2><span class="header-section-number">11.4</span> A Quadrat-based Test for Spatial Independence</h2>
<p>In the preceding activity, you used a quadrat-based spatial independence test to help you decide whether a pattern was random (the function was <code>quadrat.test</code>). We will now review the intuition of the test.</p>
<p>Let’s begin by plotting the patterns. You can use <code>split</code> to do plots for each pattern separately, instead of putting all of them in a single plot (this approach is not as refined as <code>ggplot2</code>, where we have greater control of the aspect of the plots; on the other hand, it is quick):</p>
<div class="sourceCode" id="cb239"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb239-1"><a href="point-pattern-analysis-ii.html#cb239-1" aria-hidden="true" tabindex="-1"></a><span class="co">#The split functions separates without defining a window. </span></span>
<span id="cb239-2"><a href="point-pattern-analysis-ii.html#cb239-2" aria-hidden="true" tabindex="-1"></a><span class="co"># This is a quicker option to get relative results</span></span>
<span id="cb239-3"><a href="point-pattern-analysis-ii.html#cb239-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb239-4"><a href="point-pattern-analysis-ii.html#cb239-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">split</span>(PointPatterns.ppp))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-162-1.png" width="672" /></p>
<p>Recall that you can also plot individual patterns by using <code>$</code> followed by the factor that identifies the desired pattern (this is a way of indexing different patterns in <code>ppp</code>-class objects):</p>
<div class="sourceCode" id="cb240"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb240-1"><a href="point-pattern-analysis-ii.html#cb240-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Using &quot;$&quot; acts as a call sign to retrieve information from a data frame. </span></span>
<span id="cb240-2"><a href="point-pattern-analysis-ii.html#cb240-2" aria-hidden="true" tabindex="-1"></a><span class="co"># In this case, you are calling &quot;Pattern 4&quot; from &quot;PointPatterns.ppp&quot;</span></span>
<span id="cb240-3"><a href="point-pattern-analysis-ii.html#cb240-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 4&quot;</span>)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-163-1.png" width="672" /></p>
<p>Now calculate the quadrat-based test of independence:</p>
<div class="sourceCode" id="cb241"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb241-1"><a href="point-pattern-analysis-ii.html#cb241-1" aria-hidden="true" tabindex="-1"></a><span class="co"># `quadrat.test()` generates a quadrat-based test of independence, in this case, </span></span>
<span id="cb241-2"><a href="point-pattern-analysis-ii.html#cb241-2" aria-hidden="true" tabindex="-1"></a><span class="co"># for &quot;Pattern 2&quot; called from &quot;PointPatterns.ppp&quot;, using 3 quadrats in the direction </span></span>
<span id="cb241-3"><a href="point-pattern-analysis-ii.html#cb241-3" aria-hidden="true" tabindex="-1"></a><span class="co"># of the x-axis and 3 quadrats in the direction of the y-axis </span></span>
<span id="cb241-4"><a href="point-pattern-analysis-ii.html#cb241-4" aria-hidden="true" tabindex="-1"></a>q_test <span class="ot">&lt;-</span> <span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 2&quot;</span>, <span class="at">nx =</span> <span class="dv">3</span>, <span class="at">ny =</span> <span class="dv">3</span>)</span>
<span id="cb241-5"><a href="point-pattern-analysis-ii.html#cb241-5" aria-hidden="true" tabindex="-1"></a>q_test</span></code></pre></div>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 2&quot;
## X2 = 48, df = 8, p-value = 1.976e-07
## alternative hypothesis: two.sided
##
## Quadrats: 3 by 3 grid of tiles</code></pre>
<p>Plot the results of the quadrat test:</p>
<div class="sourceCode" id="cb243"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb243-1"><a href="point-pattern-analysis-ii.html#cb243-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(q_test)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-165-1.png" width="672" /></p>
<p>As seen in the preceding chapter, the expected distribution of events on quadrats under the null landscape tends to be quite even. This is because each quadrat has equal probability of having the same number of events (depending on size, when the quadrats are not all the same size the number will be proportional to the size of the quadrat).</p>
<p>If you check the plot of the quadrat test above, you will notice that the first number (top left corner) is the number of events in the quadrat. The second number (top right corner) is the <em>expected number of events</em> for a null landscape. The third number is a <em>residual</em>, based on the difference between the observed and expected number of events. More specifically, the residual is a <em>Pearson residual</em>, defined as follows:
<span class="math display">\[
r_i=\frac{O_i - E_i}{\sqrt{E_i}},
\]</span>
where <span class="math inline">\(O_i\)</span> is the number of observed events in quadrat <span class="math inline">\(i\)</span> and <span class="math inline">\(E_i\)</span> is the number of expected events in quadrat <span class="math inline">\(i\)</span>. When the number of observed events is similar to the number of expected events, <span class="math inline">\(r_i\)</span> will tend to be a small value. As their difference grows, the residual will also grow.</p>
<p>The independence test is calculated from the residuals as:
<span class="math display">\[
X^2=\sum_{i=1}^{Q}r_i^2,
\]</span>
where <span class="math inline">\(Q\)</span> is the number of quadrats. In other words, the test is based on the squared sum of the Pearson residuals. The smaller this number is, the more likely that the observed pattern of events is not different from a null landscape (i.e., a random process), and the larger it is, the more likely that it is different from a null landscape. This is reflected by the <span class="math inline">\(p\)</span>-value of the test (technically, the <span class="math inline">\(p\)</span>-value is obtained by comparing the test to the <span class="math inline">\(\chi^2\)</span> distribution, pronounced “kay-square”).</p>
<p>Consider for instance the first pattern in the examples:</p>
<div class="sourceCode" id="cb244"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb244-1"><a href="point-pattern-analysis-ii.html#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 1&quot;</span>, <span class="at">nx =</span> <span class="dv">3</span>, <span class="at">ny =</span> <span class="dv">3</span>))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-166-1.png" width="672" /></p>
<p>You can see that the Pearson residual of the top left quadrat is indeed -0.6567673, the next to its right is -0.2704336, and so on. The value of the test statistic should be then:</p>
<div class="sourceCode" id="cb245"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb245-1"><a href="point-pattern-analysis-ii.html#cb245-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The &quot;Paste&quot; function joins together several arguments as characters. </span></span>
<span id="cb245-2"><a href="point-pattern-analysis-ii.html#cb245-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Here, this is a string of values for &quot;X2&quot;, where X2&quot; is the squared </span></span>
<span id="cb245-3"><a href="point-pattern-analysis-ii.html#cb245-3" aria-hidden="true" tabindex="-1"></a><span class="co"># sum of the residuals</span></span>
<span id="cb245-4"><a href="point-pattern-analysis-ii.html#cb245-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb245-5"><a href="point-pattern-analysis-ii.html#cb245-5" aria-hidden="true" tabindex="-1"></a><span class="fu">paste</span>(<span class="st">&quot;X2 = &quot;</span>, (<span class="sc">-</span><span class="fl">0.65</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="sc">-</span><span class="fl">0.26</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="fl">0.52</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="sc">-</span><span class="fl">0.26</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="fl">0.9</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="fl">0.52</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="sc">-</span><span class="dv">1</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="fl">0.13</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> (<span class="fl">0.13</span>)<span class="sc">^</span><span class="dv">2</span>)</span></code></pre></div>
<pre><code>## [1] &quot;X2 =  2.9423&quot;</code></pre>
<p>Which you can confirm by examining the results of the test (the small difference is due to rounding errors):</p>
<div class="sourceCode" id="cb247"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb247-1"><a href="point-pattern-analysis-ii.html#cb247-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 1&quot;</span>, <span class="at">nx =</span> <span class="dv">3</span>, <span class="at">ny =</span> <span class="dv">3</span>)</span></code></pre></div>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 1&quot;
## X2 = 3, df = 8, p-value = 0.1313
## alternative hypothesis: two.sided
##
## Quadrats: 3 by 3 grid of tiles</code></pre>
<p>Explore the remaining patterns. You will notice that the residuals and test statistic tend to grow as more events are concentrated in space. In this way, the test is a test of density of the quadrats: is their density similar to what would be expected from a null landscape?</p>
</div>
<div id="limitations-of-quadrat-analysis-size-and-number-of-quadrats" class="section level2" number="11.5">
<h2><span class="header-section-number">11.5</span> Limitations of Quadrat Analysis: Size and Number of Quadrats</h2>
<p>As hinted by the previous activity, one issue with quadrat analysis is the selection of the size for the quadrats. Changing the size of the quadrats has an impact on the counts, and in turn on the aspect of density plots and even the results of the test of independence.</p>
<p>For example, the results of the test for “Pattern 2” in the dataset change when the number of quadrats is modified. For instance, with a small number of quadrats:</p>
<div class="sourceCode" id="cb249"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb249-1"><a href="point-pattern-analysis-ii.html#cb249-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 2&quot;</span>, <span class="at">nx =</span> <span class="dv">2</span>, <span class="at">ny =</span> <span class="dv">1</span>)</span></code></pre></div>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 2&quot;
## X2 = 1.6667, df = 1, p-value = 0.3934
## alternative hypothesis: two.sided
##
## Quadrats: 2 by 1 grid of tiles</code></pre>
<p>Compare to four quadrats:</p>
<div class="sourceCode" id="cb251"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb251-1"><a href="point-pattern-analysis-ii.html#cb251-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 2&quot;</span>, <span class="at">nx =</span> <span class="dv">2</span>, <span class="at">ny =</span> <span class="dv">2</span>)</span></code></pre></div>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 2&quot;
## X2 = 6, df = 3, p-value = 0.2232
## alternative hypothesis: two.sided
##
## Quadrats: 2 by 2 grid of tiles</code></pre>
<p>And:</p>
<div class="sourceCode" id="cb253"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb253-1"><a href="point-pattern-analysis-ii.html#cb253-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 2&quot;</span>, <span class="at">nx =</span> <span class="dv">3</span>, <span class="at">ny =</span> <span class="dv">2</span>)</span></code></pre></div>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 2&quot;
## X2 = 23.2, df = 5, p-value = 0.0006182
## alternative hypothesis: two.sided
##
## Quadrats: 3 by 2 grid of tiles</code></pre>
<p>Why is the statistic generally smaller when there are fewer quadrats?</p>
<p>A different issue emerges when the number of quadrats is large:</p>
<div class="sourceCode" id="cb255"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb255-1"><a href="point-pattern-analysis-ii.html#cb255-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(<span class="fu">split</span>(PointPatterns.ppp)<span class="sc">$</span><span class="st">&quot;Pattern 2&quot;</span>, <span class="at">nx =</span> <span class="dv">4</span>, <span class="at">ny =</span> <span class="dv">4</span>)</span></code></pre></div>
<pre><code>## Warning: Some expected counts are small; chi^2 approximation may be inaccurate</code></pre>
<pre><code>##
##  Chi-squared test of CSR using quadrat counts
##
## data:  split(PointPatterns.ppp)$&quot;Pattern 2&quot;
## X2 = 47.2, df = 15, p-value = 6.84e-05
## alternative hypothesis: two.sided
##
## Quadrats: 4 by 4 grid of tiles</code></pre>
<p>A warning now tells you that some expected counts are small: space has been divided so minutely, that the expected number of events per quadrat has become too thin; as a consequence, the approximation to the probability distribution may be inaccurate.</p>
<p>While there are no hard rules to select the size/number of quadrats, the following rules of thumb are sometimes suggested:</p>
<ol style="list-style-type: decimal">
<li>Each quadrat should have a minimum of two events.</li>
<li>The number of quadrats is selected based on the area (A) of the region, and the number of events (n):
<span class="math display">\[
Q=\frac{2A}{N}
\]</span>
Caution should be exercised when interpreting the results of the analysis based on quadrats, due to the issue of size/number of quadrats.</li>
</ol>
</div>
<div id="limitations-of-quadrat-analysis-relative-position-of-events" class="section level2" number="11.6">
<h2><span class="header-section-number">11.6</span> Limitations of Quadrat Analysis: Relative Position of Events</h2>
<p>Another issue with quadrat analysis is that it is not sensitive to the relative position of the events within the quadrats.</p>
<p>Consider for instance the following two patterns in <code>pp0</code>:</p>
<div class="sourceCode" id="cb258"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb258-1"><a href="point-pattern-analysis-ii.html#cb258-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">split</span>(pp0.ppp))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-173-1.png" width="672" /></p>
<p>These two patterns look quite different. And yet, when we count the events by quadrats:</p>
<div class="sourceCode" id="cb259"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb259-1"><a href="point-pattern-analysis-ii.html#cb259-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">quadratcount</span>(<span class="fu">split</span>(pp0.ppp), <span class="at">nx =</span> <span class="dv">3</span>, <span class="at">ny =</span> <span class="dv">3</span>))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-174-1.png" width="672" /></p>
<p>This example highlights how quadrats are relatively coarse measures of density, and fail to distinguish between fairly different event distributions, in particular because quadrat analysis does not take into account the relative position of the events with respect to each other.</p>
</div>
<div id="kernel-density" class="section level2" number="11.7">
<h2><span class="header-section-number">11.7</span> Kernel Density</h2>
<p>In order to better take into account the relative position of the events with respect to each other, a different technique can be devised.</p>
<p>Imagine that a quadrat is a kind of “window”. We use it to observe the landscape. When we count the number of events in a quadrat, we simply peek through that particular window: all events inside the “window” are simply counted, and all events outside the “window” are ignored. Then we visit another quadrat and do the same, until we have visited all quadrats.</p>
<p>Imagine now that we define a window that, unlike the quadrats which are fixed, can move and visit different points in space. This window also has the property that, instead of counting the events that are in the window, it gives greater weight to events that are close to the center of the window, and less weight to events that are more distant from the center of the window.</p>
<p>We can define such a window by selecting a function that declines with increasing distance. We will call this function a <em>kernel</em>. An example of a function that can work as a moving window is the following.</p>
<div class="sourceCode" id="cb260"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb260-1"><a href="point-pattern-analysis-ii.html#cb260-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Here we create a data.frame to use for plotting; it includes a single column </span></span>
<span id="cb260-2"><a href="point-pattern-analysis-ii.html#cb260-2" aria-hidden="true" tabindex="-1"></a><span class="co"># with a variable called `dist` for distance, that varies between -3 and 3; </span></span>
<span id="cb260-3"><a href="point-pattern-analysis-ii.html#cb260-3" aria-hidden="true" tabindex="-1"></a><span class="co"># the function `stat_function()` is used in `ggplot2` to transform an input </span></span>
<span id="cb260-4"><a href="point-pattern-analysis-ii.html#cb260-4" aria-hidden="true" tabindex="-1"></a><span class="co"># by means of a function, which in this case is `dnorm` the normal distribution!</span></span>
<span id="cb260-5"><a href="point-pattern-analysis-ii.html#cb260-5" aria-hidden="true" tabindex="-1"></a><span class="co"># `ylim()` sets the limits of the plot in the y-axis </span></span>
<span id="cb260-6"><a href="point-pattern-analysis-ii.html#cb260-6" aria-hidden="true" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> <span class="fu">data.frame</span>(<span class="at">dist =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">3</span>, <span class="dv">3</span>)), <span class="fu">aes</span>(dist)) <span class="sc">+</span></span>
<span id="cb260-7"><a href="point-pattern-analysis-ii.html#cb260-7" aria-hidden="true" tabindex="-1"></a>  <span class="fu">stat_function</span>(<span class="at">fun =</span> dnorm, <span class="at">n =</span> <span class="dv">101</span>, <span class="at">args =</span> <span class="fu">list</span>(<span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">1</span>)) <span class="sc">+</span></span>
<span id="cb260-8"><a href="point-pattern-analysis-ii.html#cb260-8" aria-hidden="true" tabindex="-1"></a>  <span class="fu">ylim</span>(<span class="fu">c</span>(<span class="dv">0</span>, <span class="fl">0.45</span>))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-175-1.png" width="672" /></p>
<p>As you can see, the value of the function declines with increasing distance from the center of the window (when dist == 0; note that the value never becomes zero!). Since we used the normal distribution, this is a <em>Gaussian kernel</em>. The shape of the Gaussian kernel depends on the standard deviation, which controls how “big” the window is, or alternatively, how quickly the function decays. We will call the standard deviation the <em>kernel bandwidth</em> of the function.</p>
<p>Since the bandwidth controls how rapidly the weight assigned to distant events decays, if the argument changes, so will the shape of the kernel function. As an experiment, change the value of the argument <code>sd</code> in the chunk above. You will see that as it becomes smaller, the slope of the kernel becomes steeper (and distant observations are downweighted more rapidly). On the contrary, as it becomes larger, the slope becomes less steep (and distant events are weighted almost as highly as close events).</p>
<p>Kernel density estimates are usually obtained by creating a fine grid that is superimposed on the region. The kernel function then visits each point on the grid and obtains an estimate of the density by summing the weights of all events as per the kernel function.</p>
<p>Kernel density is implemented in <code>spatstat</code> and can be used as follows.</p>
<p>The input is a <code>ppp</code> object, and optionally a <code>sigma</code> argument that corresponds to the bandwidth of the kernel:</p>
<div class="sourceCode" id="cb261"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb261-1"><a href="point-pattern-analysis-ii.html#cb261-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The &quot;density&quot; function computes estimates of kernel density. Here we are creating </span></span>
<span id="cb261-2"><a href="point-pattern-analysis-ii.html#cb261-2" aria-hidden="true" tabindex="-1"></a><span class="co"># a Kernel Density estimate using &quot;pp0.ppp&quot; from our data frame by means of a </span></span>
<span id="cb261-3"><a href="point-pattern-analysis-ii.html#cb261-3" aria-hidden="true" tabindex="-1"></a><span class="co"># bandwidth defined by &quot;sigma&quot;</span></span>
<span id="cb261-4"><a href="point-pattern-analysis-ii.html#cb261-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb261-5"><a href="point-pattern-analysis-ii.html#cb261-5" aria-hidden="true" tabindex="-1"></a>kernel_density <span class="ot">&lt;-</span> <span class="fu">density</span>(<span class="fu">split</span>(pp0.ppp), <span class="at">sigma =</span> <span class="fl">0.1</span>)</span>
<span id="cb261-6"><a href="point-pattern-analysis-ii.html#cb261-6" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(kernel_density)</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-176-1.png" width="672" /></p>
<p>Compare to the distribution of events:</p>
<div class="sourceCode" id="cb262"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb262-1"><a href="point-pattern-analysis-ii.html#cb262-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(<span class="fu">split</span>(pp0.ppp))</span></code></pre></div>
<p><img src="spatial-analysis-R_files/figure-html/unnamed-chunk-177-1.png" width="672" /></p>
<p>It is important to note that the gradation of colors is different in the two kernel density plots. Whereas the smallest value in the plot on the left is less than 20 and the largest is greater than 100, on the other plot the range is only between 45 to approximately 50. Thus, the intensity of the process is much higher at places in Pattern 1 that in Pattern 2.</p>
<p>The plots above illustrate how the map of the kernel density is better able to capture the variations in density across the region. In fact, kernel density is a smooth estimate of the underlying intensity of the process, and the degree of smoothing is controlled by the bandwidth.</p>

</div>
</div>
<h3>References</h3>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-Baddeley2015" class="csl-entry">
Baddeley, Adrian, Ege Rubak, and Rolf Turner. 2016. <em>Spatial Point Patterns: Methodology and Applications with r</em>. Book. Chapman; Hall/CRC.
</div>
<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-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>
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