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<title>13 Hierarchical clustering of time series – Satellite Image Time Series Analysis on Earth Observation Data Cubes</title>
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<li><a href="#introduction" id="toc-introduction" class="nav-link" data-scroll-target="#introduction"><span class="header-section-number">13.1</span> Introduction</a></li>
<li><a href="#dataset-used-in-this-chapter" id="toc-dataset-used-in-this-chapter" class="nav-link" data-scroll-target="#dataset-used-in-this-chapter"><span class="header-section-number">13.2</span> Dataset used in this chapter</a></li>
<li><a href="#hierarchical-clustering-for-sample-quality-control" id="toc-hierarchical-clustering-for-sample-quality-control" class="nav-link" data-scroll-target="#hierarchical-clustering-for-sample-quality-control"><span class="header-section-number">13.3</span> Hierarchical clustering for sample quality control</a></li>
<li><a href="#summary" id="toc-summary" class="nav-link" data-scroll-target="#summary"><span class="header-section-number">13.4</span> Summary</a></li>
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<span class="chapter-number">13</span> <span class="chapter-title">Hierarchical clustering of time series</span>
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<section id="configurations-to-run-this-chapter" class="level3 unnumbered"><h3 class="unnumbered anchored" data-anchor-id="configurations-to-run-this-chapter">Configurations to run this chapter</h3>
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<span><span class="kw"><a href="https://rdrr.io/r/base/library.html">library</a></span><span class="op">(</span><span class="va"><a href="https://tibble.tidyverse.org/">tibble</a></span><span class="op">)</span></span>
<span><span class="co"># load packages "sits" and "sitsdata"</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html">library</a></span><span class="op">(</span><span class="va"><a href="https://github.com/e-sensing/sits/">sits</a></span><span class="op">)</span></span>
<span><span class="kw"><a href="https://rdrr.io/r/base/library.html">library</a></span><span class="op">(</span><span class="va"><a href="https://github.com/e-sensing/sitsdata/">sitsdata</a></span><span class="op">)</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># load "pysits" library</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> pysits <span class="im">import</span> <span class="op">*</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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</section><section id="introduction" class="level2" data-number="13.1"><h2 data-number="13.1" class="anchored" data-anchor-id="introduction">
<span class="header-section-number">13.1</span> Introduction</h2>
<p>Given a set of training samples, experts should first cross-validate the training set to assess their inherent prediction error. The results show whether the data is internally consistent. Since cross-validation does not predict actual model performance, this chapter provides additional tools for improving the quality of training sets. More detailed information is available in Topic <a href="https://e-sensing.github.io/sitsbook/validation.html">Validation and accuracy measurements</a>.</p>
</section><section id="dataset-used-in-this-chapter" class="level2" data-number="13.2"><h2 data-number="13.2" class="anchored" data-anchor-id="dataset-used-in-this-chapter">
<span class="header-section-number">13.2</span> Dataset used in this chapter</h2>
<p>The examples of this chapter use the <code>cerrado_2classes</code> data, a set of time series for the Cerrado region of Brazil, the second largest biome in South America with an area of more than 2 million <span class="math inline">\(km^2\)</span>. The data contains 746 samples divided into 2 classes (<code>Cerrado</code> and <code>Pasture</code>). Each time series covers 12 months (23 data points) from MOD13Q1 product, and has 2 bands (EVI, and NDVI).</p>
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<span><span class="fu"><a href="https://rdrr.io/r/base/summary.html">summary</a></span><span class="op">(</span><span class="va">cerrado_2classes</span><span class="op">)</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code># A tibble: 2 × 3
label count prop
<chr> <int> <dbl>
1 Cerrado 400 0.536
2 Pasture 346 0.464</code></pre>
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<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>summary(cerrado_2classes)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code> label count prop
0 Cerrado 400 0.536193
1 Pasture 346 0.463807</code></pre>
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</section><section id="hierarchical-clustering-for-sample-quality-control" class="level2" data-number="13.3"><h2 data-number="13.3" class="anchored" data-anchor-id="hierarchical-clustering-for-sample-quality-control">
<span class="header-section-number">13.3</span> Hierarchical clustering for sample quality control</h2>
<p>The package provides two clustering methods to assess sample quality: Agglomerative Hierarchical Clustering (AHC) and Self-organizing Maps (SOM). These methods have different computational complexities. AHC has a computational complexity of <span class="math inline">\(\mathcal{O}(n^2)\)</span>, given the number of time series <span class="math inline">\(n\)</span>, whereas SOM complexity is linear. For large data, AHC requires substantial memory and running time; in these cases, SOM is recommended. This section describes how to run AHC in <code>sits</code>. The SOM-based technique is presented in the next section.</p>
<p>AHC computes the dissimilarity between any two elements from a dataset. Depending on the distance functions and linkage criteria, the algorithm decides which two clusters are merged at each iteration. This approach is helpful for exploring samples due to its visualization power and ease of use <span class="citation" data-cites="Keogh2003"><a href="#ref-Keogh2003" role="doc-biblioref">[1]</a></span>. In <code>sits</code>, AHC is implemented using <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_dendro.html">sits_cluster_dendro()</a></code>.</p>
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<div class="code-copy-outer-scaffold"><div class="sourceCode" id="cb7"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="co"># Take a set of patterns for 2 classes</span></span>
<span><span class="co"># Create a dendrogram, plot, and get the optimal cluster based on ARI index</span></span>
<span><span class="va">clusters</span> <span class="op"><-</span> <span class="fu"><a href="https://e-sensing.github.io/sits/reference/sits_cluster_dendro.html">sits_cluster_dendro</a></span><span class="op">(</span></span>
<span> samples <span class="op">=</span> <span class="va">cerrado_2classes</span>, </span>
<span> bands <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span><span class="op">(</span><span class="st">"NDVI"</span>, <span class="st">"EVI"</span><span class="op">)</span>,</span>
<span> dist_method <span class="op">=</span> <span class="st">"dtw_basic"</span>,</span>
<span> linkage <span class="op">=</span> <span class="st">"ward.D2"</span><span class="op">)</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<img src="ts_cluster_files/figure-html/fig-ts-cludendro-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:90.0%">
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-ts-cludendro-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 13.1: Example of hierarchical clustering for a two class set.
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<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb8"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Take a set of patterns for 2 classes</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Create a dendrogram, plot, and get the optimal cluster based on ARI index</span></span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>clusters <span class="op">=</span> sits_cluster_dendro(</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a> samples <span class="op">=</span> cerrado_2classes, </span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a> bands <span class="op">=</span> (<span class="st">"NDVI"</span>, <span class="st">"EVI"</span>),</span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a> dist_method <span class="op">=</span> <span class="st">"dtw_basic"</span>,</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a> linkage <span class="op">=</span> <span class="st">"ward.D2"</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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Figure 13.2: Example of hierarchical clustering for a two class set.
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<p>The <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_dendro.html">sits_cluster_dendro()</a></code> function has one mandatory parameter (<code>samples</code>), with the samples to be evaluated. Optional parameters include <code>bands</code>, <code>dist_method</code>, and <code>linkage</code>. The <code>dist_method</code> parameter specifies how to calculate the distance between two time series. We recommend a metric that uses dynamic time warping (DTW) <span class="citation" data-cites="Petitjean2012"><a href="#ref-Petitjean2012" role="doc-biblioref">[2]</a></span>, as DTW is a reliable method for measuring differences between satellite image time series <span class="citation" data-cites="Maus2016"><a href="#ref-Maus2016" role="doc-biblioref">[3]</a></span>. The options available in <code>sits</code> are based on those provided by package <code>dtwclust</code>, which include <code>dtw_basic</code>, <code>dtw_lb</code>, and <code>dtw2</code>. Please check <code><a href="https://rdrr.io/pkg/dtwclust/man/tsclust.html">?dtwclust::tsclust</a></code> for more information on DTW distances.</p>
<p>The <code>linkage</code> parameter defines the distance metric between clusters. The recommended linkage criteria are: <code>complete</code> or <code>ward.D2</code>. Complete linkage prioritizes the within-cluster dissimilarities, producing clusters with shorter distance samples, but results are sensitive to outliers. As an alternative, Ward proposes to use the sum-of-squares error to minimize data variance <span class="citation" data-cites="Ward1963"><a href="#ref-Ward1963" role="doc-biblioref">[4]</a></span>; his method is available as <code>ward.D2</code> option to the <code>linkage</code> parameter. To cut the dendrogram, the <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_dendro.html">sits_cluster_dendro()</a></code> function computes the adjusted rand index (ARI) <span class="citation" data-cites="Rand1971"><a href="#ref-Rand1971" role="doc-biblioref">[5]</a></span>, returning the height where the cut of the dendrogram maximizes the index. In the example, the ARI index indicates that there are six clusters. The result of <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_dendro.html">sits_cluster_dendro()</a></code> is a time series tibble with one additional column called “cluster”. The function <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_frequency.html">sits_cluster_frequency()</a></code> provides information on the composition of each cluster.</p>
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<div class="code-copy-outer-scaffold"><div class="sourceCode" id="cb9"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="co"># Show clusters samples frequency</span></span>
<span><span class="fu"><a href="https://e-sensing.github.io/sits/reference/sits_cluster_frequency.html">sits_cluster_frequency</a></span><span class="op">(</span><span class="va">clusters</span><span class="op">)</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code>
1 2 3 4 5 6 Total
Cerrado 203 13 23 80 1 80 400
Pasture 2 176 28 0 140 0 346
Total 205 189 51 80 141 80 746</code></pre>
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<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Show clusters samples frequency</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>sits_cluster_frequency(clusters)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code> 1 2 3 4 5 6 Total
Cerrado 203.0 13.0 23.0 80.0 1.0 80.0 400.0
Pasture 2.0 176.0 28.0 0.0 140.0 0.0 346.0
Total 205.0 189.0 51.0 80.0 141.0 80.0 746.0</code></pre>
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<p>The cluster frequency table shows that each cluster has a predominance of either <code>Cerrado</code> or <code>Pasture</code> labels, except for cluster 3, which has a mix of samples from both labels. Such confusion may have resulted from incorrect labeling, inadequacy of selected bands and spatial resolution, or even a natural confusion due to the variability of the land classes. To remove cluster 3, use <code><a href="https://dplyr.tidyverse.org/reference/filter.html">dplyr::filter()</a></code>. The resulting clusters still contain mixed labels, possibly resulting from outliers. In this case, <code><a href="https://e-sensing.github.io/sits/reference/sits_cluster_clean.html">sits_cluster_clean()</a></code> removes the outliers, leaving only the most frequent label. After cleaning the samples, the resulting set of samples is likely to improve the classification results.</p>
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<div class="code-copy-outer-scaffold"><div class="sourceCode" id="cb13"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="co"># Remove cluster 3 from the samples</span></span>
<span><span class="va">clusters_new</span> <span class="op"><-</span> <span class="fu">dplyr</span><span class="fu">::</span><span class="fu"><a href="https://dplyr.tidyverse.org/reference/filter.html">filter</a></span><span class="op">(</span><span class="va">clusters</span>, <span class="va">cluster</span> <span class="op">!=</span> <span class="fl">3</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Clear clusters, leaving only the majority label</span></span>
<span><span class="va">clean</span> <span class="op"><-</span> <span class="fu"><a href="https://e-sensing.github.io/sits/reference/sits_cluster_clean.html">sits_cluster_clean</a></span><span class="op">(</span><span class="va">clusters_new</span><span class="op">)</span></span>
<span></span>
<span><span class="co"># Show clusters samples frequency</span></span>
<span><span class="fu"><a href="https://e-sensing.github.io/sits/reference/sits_cluster_frequency.html">sits_cluster_frequency</a></span><span class="op">(</span><span class="va">clean</span><span class="op">)</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code>
1 2 4 5 6 Total
Cerrado 203 0 80 0 80 363
Pasture 0 176 0 140 0 316
Total 203 176 80 140 80 679</code></pre>
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<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a>clusters_new <span class="op">=</span> clusters.query(<span class="st">"cluster != 3"</span>)</span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Clear clusters, leaving only the majority label</span></span>
<span id="cb15-5"><a href="#cb15-5" aria-hidden="true" tabindex="-1"></a>clean <span class="op">=</span> sits_cluster_clean(clusters_new)</span>
<span id="cb15-6"><a href="#cb15-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-7"><a href="#cb15-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Show clusters samples frequency</span></span>
<span id="cb15-8"><a href="#cb15-8" aria-hidden="true" tabindex="-1"></a>sits_cluster_frequency(clean)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
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<pre><code> 1 2 4 5 6 Total
Cerrado 203.0 0.0 80.0 0.0 80.0 363.0
Pasture 0.0 176.0 0.0 140.0 0.0 316.0
Total 203.0 176.0 80.0 140.0 80.0 679.0</code></pre>
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</section><section id="summary" class="level2" data-number="13.4"><h2 data-number="13.4" class="anchored" data-anchor-id="summary">
<span class="header-section-number">13.4</span> Summary</h2>
<p>In this chapter, we present hierarchical clustering to improve the quality of training data. This method works well for up to four classes. Because of its quadratical computational complexity, it is not practical to use it for data sets with many classes. In this case, we suggest the use of self-organized maps (SOM) as shown in the next chapter.</p>
</section><section id="references" class="level2 unnumbered"><h2 class="unnumbered anchored" data-anchor-id="references">References</h2>
<div id="refs" class="references csl-bib-body" data-entry-spacing="0" role="list">
<div id="ref-Keogh2003" class="csl-entry" role="listitem">
<div class="csl-left-margin">[1] </div>
<div class="csl-right-inline">E. Keogh, J. Lin, and W. Truppel, <span>“Clustering of time series subsequences is meaningless: <span>Implications</span> for previous and future research,”</span> in <em>Data <span>Mining</span>, 2003. <span>ICDM</span> 2003. <span>Third IEEE International Conference</span> on</em>, 2003, pp. 115–122.</div>
</div>
<div id="ref-Petitjean2012" class="csl-entry" role="listitem">
<div class="csl-left-margin">[2] </div>
<div class="csl-right-inline">F. Petitjean, J. Inglada, and P. Gancarski, <span>“Satellite <span>Image Time Series Analysis Under Time Warping</span>,”</span> <em>IEEE Transactions on Geoscience and Remote Sensing</em>, vol. 50, no. 8, pp. 3081–3095, 2012, doi: <a href="https://doi.org/10.1109/TGRS.2011.2179050">10.1109/TGRS.2011.2179050</a>.</div>
</div>
<div id="ref-Maus2016" class="csl-entry" role="listitem">
<div class="csl-left-margin">[3] </div>
<div class="csl-right-inline">V. Maus, G. Camara, R. Cartaxo, A. Sanchez, F. M. Ramos, and G. R. Queiroz, <span>“A <span>Time-Weighted Dynamic Time Warping Method</span> for <span>Land-Use</span> and <span>Land-Cover Mapping</span>,”</span> <em>IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing</em>, vol. 9, no. 8, pp. 3729–3739, 2016, doi: <a href="https://doi.org/10.1109/JSTARS.2016.2517118">10.1109/JSTARS.2016.2517118</a>.</div>
</div>
<div id="ref-Ward1963" class="csl-entry" role="listitem">
<div class="csl-left-margin">[4] </div>
<div class="csl-right-inline">J. H. Ward, <span>“Hierarchical grouping to optimize an objective function,”</span> <em>Journal of the American statistical association</em>, vol. 58, no. 301, pp. 236–244, 1963.</div>
</div>
<div id="ref-Rand1971" class="csl-entry" role="listitem">
<div class="csl-left-margin">[5] </div>
<div class="csl-right-inline">W. M. Rand, <span>“Objective <span>Criteria</span> for the <span>Evaluation</span> of <span>Clustering Methods</span>,”</span> <em>Journal of the American Statistical Association</em>, vol. 66, no. 336, pp. 846–850, 1971, doi: <a href="https://doi.org/10.1080/01621459.1971.10482356">10.1080/01621459.1971.10482356</a>.</div>
</div>
</div>
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