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<li><a href="#observation-window" id="toc-observation-window" class="nav-link active" data-scroll-target="#observation-window"><span class="header-section-number">11.1</span> Observation window</a></li>
<li><a href="#coordinate-reference-systems" id="toc-coordinate-reference-systems" class="nav-link" data-scroll-target="#coordinate-reference-systems"><span class="header-section-number">11.2</span> Coordinate reference systems</a></li>
<li><a href="#marked-point-patterns-points-on-linear-networks" id="toc-marked-point-patterns-points-on-linear-networks" class="nav-link" data-scroll-target="#marked-point-patterns-points-on-linear-networks"><span class="header-section-number">11.3</span> Marked point patterns, points on linear networks</a></li>
<li><a href="#spatial-sampling-and-simulating-a-point-process" id="toc-spatial-sampling-and-simulating-a-point-process" class="nav-link" data-scroll-target="#spatial-sampling-and-simulating-a-point-process"><span class="header-section-number">11.4</span> Spatial sampling and simulating a point process</a></li>
<li><a href="#sec-simsphere" id="toc-sec-simsphere" class="nav-link" data-scroll-target="#sec-simsphere"><span class="header-section-number">11.5</span> Simulating points on the sphere</a></li>
<li><a href="#exercises" id="toc-exercises" class="nav-link" data-scroll-target="#exercises"><span class="header-section-number">11.6</span> Exercises</a></li>
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<div class="quarto-title-block"><div><h1 class="title"><span id="sec-pointpatterns" class="quarto-section-identifier"><span class="chapter-number">11</span> <span class="chapter-title">Point Pattern Analysis</span></span></h1><button type="button" class="btn code-tools-button dropdown-toggle" id="quarto-code-tools-menu" data-bs-toggle="dropdown" aria-expanded="false"><i class="bi"></i> Code</button><ul class="dropdown-menu dropdown-menu-end" aria-labelelledby="quarto-code-tools-menu"><li><a id="quarto-show-all-code" class="dropdown-item" href="javascript:void(0)" role="button">Show All Code</a></li><li><a id="quarto-hide-all-code" class="dropdown-item" href="javascript:void(0)" role="button">Hide All Code</a></li><li><hr class="dropdown-divider"></li><li><a id="quarto-view-source" class="dropdown-item" href="javascript:void(0)" role="button">View Source</a></li></ul></div></div>
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<p> </p>
<p>Point pattern analysis is concerned with describing patterns of points over space and making inference about the process that could have generated an observed pattern. The main focus here lies on the information carried in the locations of the points, and typically these locations are not controlled by sampling but a result of a process we are interested in, such as animal sightings, accidents, disease cases, or tree locations. This is opposed to geostatistical processes (<a href="12-Interpolation.html"><span>Chapter 12</span></a>) where we have values of some phenomenon everywhere but observations limited to a set of locations <em>that we can control</em>, at least in principle. Hence, in geostatistical problems the prime interest is not in the observation locations but in estimating the value of the observed phenomenon at unobserved locations. Point pattern analysis typically assumes that for an observed area, all points are available, meaning that locations without a point are not unobserved as in a geostatistical process, but are observed and contain no point. In terms of random processes, in point processes locations are random variables, where, in geostatistical processes the measured variable is a random field with locations fixed.</p>
<p>This chapter is confined to describing the very basics of point pattern analysis, using package <strong>spatstat</strong> <span class="citation" data-cites="R-spatstat">(<a href="references.html#ref-R-spatstat" role="doc-biblioref">Baddeley, Turner, and Rubak 2022</a>)</span>, and related packages by the same authors. The <strong>spatstat</strong> book of <span class="citation" data-cites="baddeley2015spatial">Baddeley, Rubak, and Turner (<a href="references.html#ref-baddeley2015spatial" role="doc-biblioref">2015</a>)</span> gives a comprehensive introduction to point pattern theory and the use of the <strong>spatstat</strong> package family, which we will not try to copy. Inclusion of particular topics in this chapter should not be seen as an expression that these are more relevant than others. In particular, this chapter tries to illustrate interfaces existing between <strong>spatstat</strong> and the more spatial data science oriented packages <strong>sf</strong> and <strong>stars</strong>. A further book that introduces point patterns analysis is <span class="citation" data-cites="STOYAN2017125">Stoyan et al. (<a href="references.html#ref-STOYAN2017125" role="doc-biblioref">2017</a>)</span>. R package <strong>stpp</strong> <span class="citation" data-cites="R-stpp">(<a href="references.html#ref-R-stpp" role="doc-biblioref">Gabriel et al. 2022</a>)</span> for analysing spatiotemporal point processes is discussed in <span class="citation" data-cites="stpp">Gabriel, Rowlingson, and Diggle (<a href="references.html#ref-stpp" role="doc-biblioref">2013</a>)</span>.</p>
<p>Important concepts of point patterns analysis are the distinction between a point <em>pattern</em> and a point <em>process</em>: the latter is the stochastic process that, when sampled, generates a point pattern. A dataset is always a point pattern, and inference involves figuring out the properties of a process that could have generated a pattern like the one we observed. Properties of a spatial point process include</p>
<ul>
<li>first order properties: the intensity function measures the number of points per area unit; this function is spatially varying for a <em>inhomogeneous</em> point process</li>
<li>second order properties: given a constant or varying intensity function, describe whether points are distributed independently <em>from one another</em>, tend to attract each other (clustering), or repulse each other (more regularly distributed than under complete spatial randomness)</li>
</ul>
<section id="observation-window" class="level2" data-number="11.1">
<h2 data-number="11.1" class="anchored" data-anchor-id="observation-window"><span class="header-section-number">11.1</span> Observation window</h2>
<p></p>
<p>Point patterns have an observation window. Consider the points generated randomly by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(sf)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Linking to GEOS 3.11.1, GDAL 3.6.2, PROJ 9.1.1; sf_use_s2() is TRUE</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>n <span class="ot"><-</span> <span class="dv">30</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">13531</span>) <span class="co"># remove this to create another random sequence</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>xy <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="at">x =</span> <span class="fu">runif</span>(n), <span class="at">y =</span> <span class="fu">runif</span>(n)) <span class="sc">|></span> </span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_as_sf</span>(<span class="at">coords =</span> <span class="fu">c</span>(<span class="st">"x"</span>, <span class="st">"y"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>then these points are (by construction) uniformly distributed, or completely spatially random, over the domain <span class="math inline">\([0,1] \times [0,1]\)</span>. For a larger domain, they are not uniform, for the two square windows <code>w1</code> and <code>w2</code> created by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>w1 <span class="ot"><-</span> <span class="fu">st_bbox</span>(<span class="fu">c</span>(<span class="at">xmin =</span> <span class="dv">0</span>, <span class="at">ymin =</span> <span class="dv">0</span>, <span class="at">xmax =</span> <span class="dv">1</span>, <span class="at">ymax =</span> <span class="dv">1</span>)) <span class="sc">|></span> </span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_as_sfc</span>() </span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>w2 <span class="ot"><-</span> <span class="fu">st_sfc</span>(<span class="fu">st_point</span>(<span class="fu">c</span>(<span class="dv">1</span>, <span class="fl">0.5</span>))) <span class="sc">|></span> <span class="fu">st_buffer</span>(<span class="fl">1.2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>this is shown in <a href="#fig-pp1">Figure <span>11.1</span></a>.</p>
<div class="cell">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>), <span class="at">mar =</span> <span class="fu">c</span>(<span class="fl">2.1</span>, <span class="fl">2.1</span>, <span class="fl">0.1</span>, <span class="fl">0.5</span>), <span class="at">xaxs =</span> <span class="st">"i"</span>, <span class="at">yaxs =</span> <span class="st">"i"</span>)</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(w1, <span class="at">axes =</span> <span class="cn">TRUE</span>, <span class="at">col =</span> <span class="st">'grey'</span>)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xy, <span class="at">add =</span> <span class="cn">TRUE</span>)</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(w2, <span class="at">axes =</span> <span class="cn">TRUE</span>, <span class="at">col =</span> <span class="st">'grey'</span>)</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xy, <span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">cex =</span> .<span class="dv">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-pp1" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-pp1-1.png" class="img-fluid figure-img" style="width:100.0%"></p>
<figcaption class="figure-caption">Figure 11.1: Depending on the observation window (grey), the same point pattern can appear completely spatially random (left), or clustered (right)</figcaption>
</figure>
</div>
</div>
</div>
<p> </p>
<p>Point patterns in <strong>spatstat</strong> are objects of class <code>ppp</code> that contain points and an observation window (an object of class <code>owin</code>). We can create a <code>ppp</code> from points by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(spatstat) <span class="sc">|></span> <span class="fu">suppressPackageStartupMessages</span>()</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="fu">as.ppp</span>(xy)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Planar point pattern: 30 points</span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="co"># window: rectangle = [0.009, 0.999] x [0.103, 0.996] units</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div style="page-break-after: always;"></div>
<p>where we see that the bounding box of the points is used as observation window when no window is specified. If we add a polygonal geometry as the first feature of the dataset, then this is used as observation window:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>(pp1 <span class="ot"><-</span> <span class="fu">c</span>(w1, <span class="fu">st_geometry</span>(xy)) <span class="sc">|></span> <span class="fu">as.ppp</span>())</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Planar point pattern: 30 points</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a><span class="co"># window: polygonal boundary</span></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># enclosing rectangle: [0, 1] x [0, 1] units</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>c1 <span class="ot"><-</span> <span class="fu">st_buffer</span>(<span class="fu">st_centroid</span>(w2), <span class="fl">1.2</span>)</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>(pp2 <span class="ot"><-</span> <span class="fu">c</span>(c1, <span class="fu">st_geometry</span>(xy)) <span class="sc">|></span> <span class="fu">as.ppp</span>())</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Planar point pattern: 30 points</span></span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="co"># window: polygonal boundary</span></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="co"># enclosing rectangle: [-0.2, 2.2] x [-0.7, 1.7] units</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p></p>
<p>To test for homogeneity, one could carry out a quadrat count, using an appropriate quadrat layout (a 3 <span class="math inline">\(\times\)</span> 3 layout is shown in <a href="#fig-quadrat">Figure <span>11.2</span></a>)</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>), <span class="at">mar =</span> <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">4</span>))</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>q1 <span class="ot"><-</span> <span class="fu">quadratcount</span>(pp1, <span class="at">nx=</span><span class="dv">3</span>, <span class="at">ny=</span><span class="dv">3</span>)</span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>q2 <span class="ot"><-</span> <span class="fu">quadratcount</span>(pp2, <span class="at">nx=</span><span class="dv">3</span>, <span class="at">ny=</span><span class="dv">3</span>)</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(q1, <span class="at">main =</span> <span class="st">""</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xy, <span class="at">add =</span> <span class="cn">TRUE</span>)</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(q2, <span class="at">main =</span> <span class="st">""</span>)</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xy, <span class="at">add =</span> <span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div id="fig-quadrat" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-quadrat-1.png" class="img-fluid figure-img" style="width:100.0%"></p>
<figcaption class="figure-caption">Figure 11.2: 3 <span class="math inline">\(\times\)</span> 3 quadrat counts for the two point patterns</figcaption>
</figure>
</div>
</div>
</div>
<p> and carry out a <span class="math inline">\(\chi^2\)</span> test on these counts:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(pp1, <span class="at">nx=</span><span class="dv">3</span>, <span class="at">ny=</span><span class="dv">3</span>)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Warning: Some expected counts are small; chi^2 approximation may be</span></span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="co"># inaccurate</span></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Chi-squared test of CSR using quadrat counts</span></span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a><span class="co"># data: pp1</span></span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="co"># X2 = 8, df = 8, p-value = 0.9</span></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a><span class="co"># alternative hypothesis: two.sided</span></span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Quadrats: 9 tiles (irregular windows)</span></span>
<span id="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></a><span class="fu">quadrat.test</span>(pp2, <span class="at">nx=</span><span class="dv">3</span>, <span class="at">ny=</span><span class="dv">3</span>)</span>
<span id="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Warning: Some expected counts are small; chi^2 approximation may be</span></span>
<span id="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></a><span class="co"># inaccurate</span></span>
<span id="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-16"><a href="#cb7-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Chi-squared test of CSR using quadrat counts</span></span>
<span id="cb7-17"><a href="#cb7-17" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-18"><a href="#cb7-18" aria-hidden="true" tabindex="-1"></a><span class="co"># data: pp2</span></span>
<span id="cb7-19"><a href="#cb7-19" aria-hidden="true" tabindex="-1"></a><span class="co"># X2 = 43, df = 8, p-value = 2e-06</span></span>
<span id="cb7-20"><a href="#cb7-20" aria-hidden="true" tabindex="-1"></a><span class="co"># alternative hypothesis: two.sided</span></span>
<span id="cb7-21"><a href="#cb7-21" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb7-22"><a href="#cb7-22" aria-hidden="true" tabindex="-1"></a><span class="co"># Quadrats: 9 tiles (irregular windows)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>which indicates that for the second case we have an indication that this is not a CSR (completely spatially random) pattern. As indicated by the warning, we should take the p-values with a large grain of salt because we have too small expected counts.</p>
<p> Kernel densities can be computed using <code>density</code>, where kernel shape and bandwidth can be controlled. Here, cross-validation is used by function <code>bw.diggle</code> to specify the bandwidth parameter <code>sigma</code>; plots are shown in <a href="#fig-bwdiggle">Figure <span>11.3</span></a>.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>den1 <span class="ot"><-</span> <span class="fu">density</span>(pp1, <span class="at">sigma =</span> bw.diggle)</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>den2 <span class="ot"><-</span> <span class="fu">density</span>(pp2, <span class="at">sigma =</span> bw.diggle)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>), <span class="at">mar =</span> <span class="fu">c</span>(<span class="dv">0</span>,<span class="dv">0</span>,<span class="fl">1.1</span>,<span class="dv">2</span>))</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(den1)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(pp1, <span class="at">add=</span><span class="cn">TRUE</span>)</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(den2)</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(pp1, <span class="at">add=</span><span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-bwdiggle" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-bwdiggle-1.png" class="img-fluid figure-img" style="width:100.0%"></p>
<figcaption class="figure-caption">Figure 11.3: Kernel densities for both point patterns</figcaption>
</figure>
</div>
</div>
</div>
<p>The density maps created this way are obviously raster images, and we can convert them into stars object by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(stars)</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Loading required package: abind</span></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>s1 <span class="ot"><-</span> <span class="fu">st_as_stars</span>(den1)</span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>(s2 <span class="ot"><-</span> <span class="fu">st_as_stars</span>(den2))</span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a><span class="co"># stars object with 2 dimensions and 1 attribute</span></span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a><span class="co"># attribute(s):</span></span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Min. 1st Qu. Median Mean 3rd Qu. Max. NA's</span></span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a><span class="co"># v 6.28e-15 0.000153 0.304 6.77 13.1 42.7 3492</span></span>
<span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a><span class="co"># dimension(s):</span></span>
<span id="cb10-10"><a href="#cb10-10" aria-hidden="true" tabindex="-1"></a><span class="co"># from to offset delta x/y</span></span>
<span id="cb10-11"><a href="#cb10-11" aria-hidden="true" tabindex="-1"></a><span class="co"># x 1 128 -0.2 0.01875 [x]</span></span>
<span id="cb10-12"><a href="#cb10-12" aria-hidden="true" tabindex="-1"></a><span class="co"># y 1 128 -0.7 0.01875 [y]</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>and we can verify that the area under the density surface is similar to the sample size (30), by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>s1<span class="sc">$</span>a <span class="ot"><-</span> <span class="fu">st_area</span>(s1) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>s2<span class="sc">$</span>a <span class="ot"><-</span> <span class="fu">st_area</span>(s2) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a><span class="fu">with</span>(s1, <span class="fu">sum</span>(v <span class="sc">*</span> a, <span class="at">na.rm =</span> <span class="cn">TRUE</span>))</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a><span class="co"># [1] 29</span></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="fu">with</span>(s2, <span class="fu">sum</span>(v <span class="sc">*</span> a, <span class="at">na.rm =</span> <span class="cn">TRUE</span>))</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a><span class="co"># [1] 30.7</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p> </p>
<p>More exciting applications involve modelling the density surface as a function of external variables. Suppose we want to model the density of <code>pp2</code> as a Poisson point process (meaning that points do not interact with each other), where the intensity is a function of distance to the centre of the “cluster”, and these distance are available in a <code>stars</code> object:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>pt <span class="ot"><-</span> <span class="fu">st_sfc</span>(<span class="fu">st_point</span>(<span class="fu">c</span>(<span class="fl">0.5</span>, <span class="fl">0.5</span>)))</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="fu">st_as_sf</span>(s2, <span class="at">as_points =</span> <span class="cn">TRUE</span>, <span class="at">na.rm =</span> <span class="cn">FALSE</span>) <span class="sc">|></span></span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_distance</span>(pt) <span class="ot">-></span> s2<span class="sc">$</span>dist</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>we can then model the densities using <code>ppm</code>, where the <em>name</em> of the point pattern object is used on the left-hand side of the <code>formula</code>:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>(m <span class="ot"><-</span> <span class="fu">ppm</span>(pp2 <span class="sc">~</span> dist, <span class="at">data =</span> <span class="fu">list</span>(<span class="at">dist =</span> <span class="fu">as.im</span>(s2[<span class="st">"dist"</span>]))))</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Nonstationary Poisson process</span></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Fitted to point pattern dataset 'pp2'</span></span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Log intensity: ~dist</span></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Fitted trend coefficients:</span></span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a><span class="co"># (Intercept) dist </span></span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a><span class="co"># 4.54 -4.25 </span></span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Estimate S.E. CI95.lo CI95.hi Ztest Zval</span></span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a><span class="co"># (Intercept) 4.54 0.341 3.87 5.21 *** 13.32</span></span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a><span class="co"># dist -4.25 0.701 -5.62 -2.88 *** -6.06</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>The returned object is of class <code>ppm</code>, and can be plotted: <a href="#fig-ppm">Figure <span>11.4</span></a> shows the predicted surface. The prediction standard error can also be plotted.</p>
<div class="cell">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(m, <span class="at">se =</span> <span class="cn">FALSE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-ppm" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-ppm-1.png" class="img-fluid figure-img" style="width:50.0%"></p>
<figcaption class="figure-caption">Figure 11.4: Predicted densities of a ppm model</figcaption>
</figure>
</div>
</div>
</div>
<p>The model also has a <code>predict</code> method, which returns an <code>im</code> object that can be converted into a <code>stars</code> object by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">predict</span>(m, <span class="at">covariates =</span> <span class="fu">list</span>(<span class="at">dist =</span> <span class="fu">as.im</span>(s2[<span class="st">"dist"</span>]))) <span class="sc">|></span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_as_stars</span>()</span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a><span class="co"># stars object with 2 dimensions and 1 attribute</span></span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a><span class="co"># attribute(s):</span></span>
<span id="cb15-5"><a href="#cb15-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Min. 1st Qu. Median Mean 3rd Qu. Max. NA's</span></span>
<span id="cb15-6"><a href="#cb15-6" aria-hidden="true" tabindex="-1"></a><span class="co"># v 0.0694 0.527 2.12 6.62 7.3 89.9 3492</span></span>
<span id="cb15-7"><a href="#cb15-7" aria-hidden="true" tabindex="-1"></a><span class="co"># dimension(s):</span></span>
<span id="cb15-8"><a href="#cb15-8" aria-hidden="true" tabindex="-1"></a><span class="co"># from to offset delta x/y</span></span>
<span id="cb15-9"><a href="#cb15-9" aria-hidden="true" tabindex="-1"></a><span class="co"># x 1 128 -0.2 0.01875 [x]</span></span>
<span id="cb15-10"><a href="#cb15-10" aria-hidden="true" tabindex="-1"></a><span class="co"># y 1 128 -0.7 0.01875 [y]</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="coordinate-reference-systems" class="level2" data-number="11.2">
<h2 data-number="11.2" class="anchored" data-anchor-id="coordinate-reference-systems"><span class="header-section-number">11.2</span> Coordinate reference systems</h2>
<p>All routines in <strong>spatstat</strong> are layed out for two-dimensional data with Cartesian coordinates. If we try to convert an object with ellipsoidal coordinates, we get an error:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="fu">system.file</span>(<span class="st">"gpkg/nc.gpkg"</span>, <span class="at">package =</span> <span class="st">"sf"</span>) <span class="sc">|></span> </span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">read_sf</span>() <span class="sc">|></span></span>
<span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_geometry</span>() <span class="sc">|></span></span>
<span id="cb16-4"><a href="#cb16-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_centroid</span>() <span class="sc">|></span></span>
<span id="cb16-5"><a href="#cb16-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">as.ppp</span>()</span>
<span id="cb16-6"><a href="#cb16-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Error: Only projected coordinates may be converted to spatstat</span></span>
<span id="cb16-7"><a href="#cb16-7" aria-hidden="true" tabindex="-1"></a><span class="co"># class objects</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>When converting to a <strong>spatstat</strong> data structure we loose the coordinate reference system we started with. It can be set back to <code>sf</code> or <code>stars</code> objects by using <code>st_set_crs</code>.</p>
</section>
<section id="marked-point-patterns-points-on-linear-networks" class="level2" data-number="11.3">
<h2 data-number="11.3" class="anchored" data-anchor-id="marked-point-patterns-points-on-linear-networks"><span class="header-section-number">11.3</span> Marked point patterns, points on linear networks</h2>
<p></p>
<p>A few more data types can be converted to and from <strong>spatstat</strong>. Marked point patterns are point patterns that have a “mark”, which is either a categorical label or a numeric label for each point. A dataset available in <strong>spatstat</strong> with marks is the <code>longleaf</code> pines dataset, containing diameter at breast height as a numeric mark:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>longleaf</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Marked planar point pattern: 584 points</span></span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a><span class="co"># marks are numeric, of storage type 'double'</span></span>
<span id="cb17-4"><a href="#cb17-4" aria-hidden="true" tabindex="-1"></a><span class="co"># window: rectangle = [0, 200] x [0, 200] metres</span></span>
<span id="cb17-5"><a href="#cb17-5" aria-hidden="true" tabindex="-1"></a>ll <span class="ot"><-</span> <span class="fu">st_as_sf</span>(longleaf)</span>
<span id="cb17-6"><a href="#cb17-6" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(ll, <span class="at">n =</span> <span class="dv">3</span>)</span>
<span id="cb17-7"><a href="#cb17-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Simple feature collection with 585 features and 2 fields</span></span>
<span id="cb17-8"><a href="#cb17-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Geometry type: GEOMETRY</span></span>
<span id="cb17-9"><a href="#cb17-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Dimension: XY</span></span>
<span id="cb17-10"><a href="#cb17-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Bounding box: xmin: 0 ymin: 0 xmax: 200 ymax: 200</span></span>
<span id="cb17-11"><a href="#cb17-11" aria-hidden="true" tabindex="-1"></a><span class="co"># CRS: NA</span></span>
<span id="cb17-12"><a href="#cb17-12" aria-hidden="true" tabindex="-1"></a><span class="co"># First 3 features:</span></span>
<span id="cb17-13"><a href="#cb17-13" aria-hidden="true" tabindex="-1"></a><span class="co"># spatstat.geom..marks.x. label geom</span></span>
<span id="cb17-14"><a href="#cb17-14" aria-hidden="true" tabindex="-1"></a><span class="co"># NA NA window POLYGON ((0 0, 200 0, 200 2...</span></span>
<span id="cb17-15"><a href="#cb17-15" aria-hidden="true" tabindex="-1"></a><span class="co"># 1 32.9 point POINT (200 8.8)</span></span>
<span id="cb17-16"><a href="#cb17-16" aria-hidden="true" tabindex="-1"></a><span class="co"># 2 53.5 point POINT (199 10)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Values can be converted back to <code>ppp</code> with</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="fu">as.ppp</span>(ll)</span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Warning in as.ppp.sf(ll): only first attribute column is used for</span></span>
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a><span class="co"># marks</span></span>
<span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Marked planar point pattern: 584 points</span></span>
<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a><span class="co"># marks are numeric, of storage type 'double'</span></span>
<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a><span class="co"># window: polygonal boundary</span></span>
<span id="cb18-7"><a href="#cb18-7" aria-hidden="true" tabindex="-1"></a><span class="co"># enclosing rectangle: [0, 200] x [0, 200] units</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p> Line segments, in <strong>spatstat</strong> objects of class <code>psp</code> can be converted back and forth to simple feature with <code>LINESTRING</code> geometries following a <code>POLYGON</code> feature with the observation window, as in</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(<span class="fu">st_as_sf</span>(copper<span class="sc">$</span>SouthLines), <span class="at">n =</span> <span class="dv">5</span>)</span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Simple feature collection with 91 features and 1 field</span></span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Geometry type: GEOMETRY</span></span>
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Dimension: XY</span></span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Bounding box: xmin: -0.335 ymin: 0.19 xmax: 35 ymax: 158</span></span>
<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a><span class="co"># CRS: NA</span></span>
<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a><span class="co"># First 5 features:</span></span>
<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a><span class="co"># label geom</span></span>
<span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a><span class="co"># 1 window POLYGON ((-0.335 0.19, 35 0...</span></span>
<span id="cb19-10"><a href="#cb19-10" aria-hidden="true" tabindex="-1"></a><span class="co"># 2 segment LINESTRING (3.36 0.19, 10.4...</span></span>
<span id="cb19-11"><a href="#cb19-11" aria-hidden="true" tabindex="-1"></a><span class="co"># 3 segment LINESTRING (12.5 0.263, 11....</span></span>
<span id="cb19-12"><a href="#cb19-12" aria-hidden="true" tabindex="-1"></a><span class="co"># 4 segment LINESTRING (11.2 0.197, -0....</span></span>
<span id="cb19-13"><a href="#cb19-13" aria-hidden="true" tabindex="-1"></a><span class="co"># 5 segment LINESTRING (6.35 12.8, 16.5...</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Finally, point patterns on linear networks, in <strong>spatstat</strong> represented by <code>lpp</code> objects, can be converted to <code>sf</code> by</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(<span class="fu">st_as_sf</span>(chicago), <span class="at">n =</span> <span class="dv">5</span>)</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Simple feature collection with 620 features and 4 fields</span></span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Geometry type: GEOMETRY</span></span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Dimension: XY</span></span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Bounding box: xmin: 0.389 ymin: 153 xmax: 1280 ymax: 1280</span></span>
<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a><span class="co"># CRS: NA</span></span>
<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a><span class="co"># First 5 features:</span></span>
<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a><span class="co"># label seg tp marks geom</span></span>
<span id="cb20-9"><a href="#cb20-9" aria-hidden="true" tabindex="-1"></a><span class="co"># 1 window NA NA <NA> POLYGON ((0.389 153, 1282 1...</span></span>
<span id="cb20-10"><a href="#cb20-10" aria-hidden="true" tabindex="-1"></a><span class="co"># 2 segment NA NA <NA> LINESTRING (0.389 1254, 110...</span></span>
<span id="cb20-11"><a href="#cb20-11" aria-hidden="true" tabindex="-1"></a><span class="co"># 3 segment NA NA <NA> LINESTRING (110 1252, 111 1...</span></span>
<span id="cb20-12"><a href="#cb20-12" aria-hidden="true" tabindex="-1"></a><span class="co"># 4 segment NA NA <NA> LINESTRING (110 1252, 198 1...</span></span>
<span id="cb20-13"><a href="#cb20-13" aria-hidden="true" tabindex="-1"></a><span class="co"># 5 segment NA NA <NA> LINESTRING (198 1277, 198 1...</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>where we only see the first five features; the points are also in this object, as variable <code>label</code> indicates</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="fu">table</span>(<span class="fu">st_as_sf</span>(chicago)<span class="sc">$</span>label)</span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="co"># </span></span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a><span class="co"># point segment window </span></span>
<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a><span class="co"># 116 503 1</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Potential information about network <em>structure</em>, how <code>LINESTRING</code> geometries are connected, is not present in the <code>sf</code> object. Package <strong>sfnetworks</strong> <span class="citation" data-cites="R-sfnetworks">(<a href="references.html#ref-R-sfnetworks" role="doc-biblioref">van der Meer et al. 2022</a>)</span> would be a candidate package to hold such information in R or to pass on network data imported from OpenStreetMaps to <strong>spatstat</strong>.</p>
</section>
<section id="spatial-sampling-and-simulating-a-point-process" class="level2" data-number="11.4">
<h2 data-number="11.4" class="anchored" data-anchor-id="spatial-sampling-and-simulating-a-point-process"><span class="header-section-number">11.4</span> Spatial sampling and simulating a point process</h2>
<p> Package <strong>sf</strong> contains an <code>st_sample</code> method that samples points from <code>MULTIPOINT</code>, linear or polygonal geometries, using different spatial sampling strategies. It natively supports strategies “random”, “hexagonal”, “Fibonacci” (<a href="#sec-simsphere"><span>Section 11.5</span></a>), and “regular”, where “regular” refers to sampling on a square regular grid and “hexagonal” essentially gives a triangular grid. For type “random”, it can return exactly the number of requested points, for other types this is approximate.</p>
<p><code>st_sample</code> also interfaces point process simulation functions of <strong>spatstat</strong>, when other values for sampling type are chosen. For instance the <strong>spatstat</strong> function <code>rThomas</code> is invoked when setting <code>type = Thomas</code> (<a href="#fig-rThomas">Figure <span>11.5</span></a>):</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>kappa <span class="ot"><-</span> <span class="dv">30</span> <span class="sc">/</span> <span class="fu">st_area</span>(w2) <span class="co"># intensity</span></span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>th <span class="ot"><-</span> <span class="fu">st_sample</span>(w2, <span class="at">kappa =</span> kappa, <span class="at">mu =</span> <span class="dv">3</span>, <span class="at">scale =</span> <span class="fl">0.05</span>, </span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"Thomas"</span>)</span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a><span class="fu">nrow</span>(th)</span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a><span class="co"># [1] 90</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<div class="cell">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mar =</span> <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">4</span>))</span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(w2)</span>
<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(th, <span class="at">add =</span> <span class="cn">TRUE</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-rThomas" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-rThomas-1.png" class="img-fluid figure-img" style="width:60.0%"></p>
<figcaption class="figure-caption">Figure 11.5: Thomas process with mu = 3 and scale = 0.05</figcaption>
</figure>
</div>
</div>
</div>
<p>The help function obtained by <code>?rThomas</code> details the meaning of the parameters <code>kappa</code>, <code>mu</code>, and <code>scale</code>. Simulating point processes means that the intensity is given, not the sample size. The sample size within the observation window obtained this way is a random variable.</p>
</section>
<section id="sec-simsphere" class="level2" data-number="11.5">
<h2 data-number="11.5" class="anchored" data-anchor-id="sec-simsphere"><span class="header-section-number">11.5</span> Simulating points on the sphere</h2>
<p> Another spatial random sampling type supported by <code>sf</code> natively (in <code>st_sample</code>) is simulation of random points on the sphere. An example of this is shown in <a href="#fig-srsglobe">Figure <span>11.6</span></a>, where points were constrained to those in oceans. Points approximately regularly distributed over a sphere are obtained by <code>st_sample</code> with <code>type = "Fibonacci"</code> <span class="citation" data-cites="fibo">(<a href="references.html#ref-fibo" role="doc-biblioref">González 2010</a>)</span>.</p>
<div class="cell" data-messages="false">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">par</span>(<span class="at">mar =</span> <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">4</span>), <span class="at">mfrow =</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>))</span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(s2)</span>
<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a>g <span class="ot"><-</span> <span class="fu">as_s2_geography</span>(<span class="cn">TRUE</span>) <span class="co"># Earth</span></span>
<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a>co <span class="ot"><-</span> <span class="fu">s2_data_countries</span>()</span>
<span id="cb24-5"><a href="#cb24-5" aria-hidden="true" tabindex="-1"></a>oc <span class="ot"><-</span> <span class="fu">s2_difference</span>(g, <span class="fu">s2_union_agg</span>(co)) <span class="co"># oceans</span></span>
<span id="cb24-6"><a href="#cb24-6" aria-hidden="true" tabindex="-1"></a>b <span class="ot"><-</span> <span class="fu">s2_buffer_cells</span>(<span class="fu">as_s2_geography</span>(<span class="st">"POINT(-30 -10)"</span>), <span class="dv">9800000</span>) <span class="co"># visible half</span></span>
<span id="cb24-7"><a href="#cb24-7" aria-hidden="true" tabindex="-1"></a>i <span class="ot"><-</span> <span class="fu">s2_intersection</span>(b, oc) <span class="co"># visible ocean</span></span>
<span id="cb24-8"><a href="#cb24-8" aria-hidden="true" tabindex="-1"></a>co <span class="ot"><-</span> <span class="fu">s2_intersection</span>(b, co)</span>
<span id="cb24-9"><a href="#cb24-9" aria-hidden="true" tabindex="-1"></a>ortho <span class="ot">=</span> <span class="fu">st_crs</span>(<span class="st">"+proj=ortho +lat_0=-10 +lon_0=-30"</span>)</span>
<span id="cb24-10"><a href="#cb24-10" aria-hidden="true" tabindex="-1"></a><span class="co"># background:</span></span>
<span id="cb24-11"><a href="#cb24-11" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(<span class="fu">st_as_sfc</span>(i), ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="st">'lightblue'</span>)</span>
<span id="cb24-12"><a href="#cb24-12" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(<span class="fu">st_as_sfc</span>(co), ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="cn">NA</span>, <span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">border =</span> <span class="st">'grey'</span>)</span>
<span id="cb24-13"><a href="#cb24-13" aria-hidden="true" tabindex="-1"></a><span class="co"># sampling randomly from globe:</span></span>
<span id="cb24-14"><a href="#cb24-14" aria-hidden="true" tabindex="-1"></a><span class="fu">sf_use_s2</span>(<span class="cn">FALSE</span>) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb24-15"><a href="#cb24-15" aria-hidden="true" tabindex="-1"></a><span class="fu">st_as_stars</span>() <span class="sc">|></span> <span class="fu">st_bbox</span>() <span class="sc">|></span> <span class="fu">st_as_sfc</span>() <span class="sc">|></span> <span class="fu">st_sample</span>(<span class="dv">1000</span>, <span class="at">exact =</span> <span class="cn">FALSE</span>) <span class="sc">|></span></span>
<span id="cb24-16"><a href="#cb24-16" aria-hidden="true" tabindex="-1"></a> <span class="fu">suppressMessages</span>() <span class="ot">-></span> pts</span>
<span id="cb24-17"><a href="#cb24-17" aria-hidden="true" tabindex="-1"></a><span class="fu">sf_use_s2</span>(<span class="cn">TRUE</span>) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb24-18"><a href="#cb24-18" aria-hidden="true" tabindex="-1"></a>pts <span class="sc">|></span> <span class="fu">s2_intersection</span>(i) <span class="sc">|></span> <span class="fu">st_as_sfc</span>() <span class="ot">-></span> pts</span>
<span id="cb24-19"><a href="#cb24-19" aria-hidden="true" tabindex="-1"></a><span class="co"># add:</span></span>
<span id="cb24-20"><a href="#cb24-20" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(pts, ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">pch =</span> <span class="dv">3</span>, <span class="at">cex =</span> .<span class="dv">5</span>)</span>
<span id="cb24-21"><a href="#cb24-21" aria-hidden="true" tabindex="-1"></a><span class="co"># right: background:</span></span>
<span id="cb24-22"><a href="#cb24-22" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(<span class="fu">st_as_sfc</span>(i), ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="st">'lightblue'</span>)</span>
<span id="cb24-23"><a href="#cb24-23" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(<span class="fu">st_as_sfc</span>(co), ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="cn">NA</span>, <span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">border =</span> <span class="st">'grey'</span>)</span>
<span id="cb24-24"><a href="#cb24-24" aria-hidden="true" tabindex="-1"></a><span class="co"># Fibonacci:</span></span>
<span id="cb24-25"><a href="#cb24-25" aria-hidden="true" tabindex="-1"></a><span class="fu">sf_use_s2</span>(<span class="cn">FALSE</span>) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb24-26"><a href="#cb24-26" aria-hidden="true" tabindex="-1"></a><span class="fu">st_as_stars</span>() <span class="sc">|></span> <span class="fu">st_bbox</span>() <span class="sc">|></span> <span class="fu">st_as_sfc</span>() <span class="sc">|></span> </span>
<span id="cb24-27"><a href="#cb24-27" aria-hidden="true" tabindex="-1"></a> <span class="fu">st_sample</span>(<span class="dv">1000</span>, <span class="at">type =</span> <span class="st">"Fibonacci"</span>, <span class="at">exact =</span> <span class="cn">FALSE</span>) <span class="sc">|></span> <span class="fu">suppressMessages</span>() <span class="ot">-></span> pts</span>
<span id="cb24-28"><a href="#cb24-28" aria-hidden="true" tabindex="-1"></a><span class="fu">sf_use_s2</span>(<span class="cn">TRUE</span>) <span class="sc">|></span> <span class="fu">suppressMessages</span>()</span>
<span id="cb24-29"><a href="#cb24-29" aria-hidden="true" tabindex="-1"></a>pts <span class="sc">|></span> <span class="fu">s2_intersection</span>(i) <span class="sc">|></span> <span class="fu">st_as_sfc</span>() <span class="ot">-></span> pts</span>
<span id="cb24-30"><a href="#cb24-30" aria-hidden="true" tabindex="-1"></a><span class="fu">st_transform</span>(pts, ortho) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">add =</span> <span class="cn">TRUE</span>, <span class="at">pch =</span> <span class="dv">3</span>, <span class="at">cex =</span> .<span class="dv">5</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output-display">
<div id="fig-srsglobe" class="quarto-figure quarto-figure-center anchored">
<figure class="figure">
<p><img src="11-PointPattern_files/figure-html/fig-srsglobe-1.png" class="img-fluid figure-img" style="width:100.0%"></p>
<figcaption class="figure-caption">Figure 11.6: Points sampled on the globe over the oceans: randomly (left) and approximately regular (Fibonacci; right), shown in an orthographic projection</figcaption>
</figure>
</div>
</div>
</div>
</section>
<section id="exercises" class="level2" data-number="11.6">
<h2 data-number="11.6" class="anchored" data-anchor-id="exercises"><span class="header-section-number">11.6</span> Exercises</h2>
<ol type="1">
<li>After loading <strong>spatstat</strong>, recreate the plot obtained by <code>plot(longleaf)</code> by using <strong>ggplot2</strong> and <code>geom_sf()</code>, and by <code>sf::plot()</code>.</li>
<li>Convert the sample locations of the NO<span class="math inline">\(_2\)</span> data used in <a href="12-Interpolation.html"><span>Chapter 12</span></a> to a <code>ppp</code> object, with a proper window.</li>
<li>Compute and plot the density of the NO<span class="math inline">\(_2\)</span> dataset, import the density as a <code>stars</code> object, and compute the volume under the surface.</li>
</ol>
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<div id="refs" class="references csl-bib-body hanging-indent" role="list" style="display: none">
<div id="ref-baddeley2015spatial" class="csl-entry" role="listitem">
Baddeley, Adrian, Ege Rubak, and Rolf Turner. 2015. <em>Spatial Point Patterns: Methodology and Applications with <span>R</span></em>. Chapman & Hall/CRC.
</div>
<div id="ref-R-spatstat" class="csl-entry" role="listitem">
Baddeley, Adrian, Rolf Turner, and Ege Rubak. 2022. <em>Spatstat: Spatial Point Pattern Analysis, Model-Fitting, Simulation, Tests</em>. <a href="http://spatstat.org/">http://spatstat.org/</a>.
</div>
<div id="ref-R-stpp" class="csl-entry" role="listitem">
Gabriel, Edith, Peter J Diggle, Barry Rowlingson, and Francisco J Rodriguez-Cortes. 2022. <em>Stpp: Space-Time Point Pattern Simulation, Visualisation and Analysis</em>. <a href="https://CRAN.R-project.org/package=stpp">https://CRAN.R-project.org/package=stpp</a>.
</div>
<div id="ref-stpp" class="csl-entry" role="listitem">
Gabriel, Edith, Barry Rowlingson, and Peter Diggle. 2013. <span>“Stpp: An <span>R</span> Package for Plotting, Simulating and Analyzing Spatio-Temporal Point Patterns.”</span> <em>Journal of Statistical Software, Articles</em> 53 (2): 1–29. <a href="https://doi.org/10.18637/jss.v053.i02">https://doi.org/10.18637/jss.v053.i02</a>.
</div>
<div id="ref-fibo" class="csl-entry" role="listitem">
González, Álvaro. 2010. <span>“Measurement of Areas on a Sphere Using <span>Fibonacci</span> and Latitude–Longitude Lattices.”</span> <em>Mathematical Geosciences</em> 42 (1): 49–64. <a href="https://arxiv.org/pdf/0912.4540.pdf">https://arxiv.org/pdf/0912.4540.pdf</a>.
</div>
<div id="ref-STOYAN2017125" class="csl-entry" role="listitem">
Stoyan, Dietrich, Francisco J. Rodríguez-Cortés, Jorge Mateu, and Wilfried Gille. 2017. <span>“Mark Variograms for Spatio-Temporal Point Processes.”</span> <em>Spatial Statistics</em> 20: 125–47. <a href="https://doi.org/10.1016/j.spasta.2017.02.006">https://doi.org/10.1016/j.spasta.2017.02.006</a>.
</div>
<div id="ref-R-sfnetworks" class="csl-entry" role="listitem">
van der Meer, Lucas, Lorena Abad, Andrea Gilardi, and Robin Lovelace. 2022. <em>Sfnetworks: Tidy Geospatial Networks</em>. <a href="https://CRAN.R-project.org/package=sfnetworks">https://CRAN.R-project.org/package=sfnetworks</a>.
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window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;