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+      <h1>Function to perform isometric log-ratio transformation</h1>
+      <small class="dont-index">Source: <a href="https://github.com/jranke/mkin/blob/HEAD/R/ilr.R" class="external-link"><code>R/ilr.R</code></a></small>
+      <div class="d-none name"><code>ilr.Rd</code></div>
+    </div>
+
+    <div class="ref-description section level2">
+    <p>This implementation is a special case of the class of isometric log-ratio
+transformations.</p>
+    </div>
+
+    <div class="section level2">
+    <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage"></a></h2>
+    <div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">ilr</span><span class="op">(</span><span class="va">x</span><span class="op">)</span></span>
+<span></span>
+<span><span class="fu">invilr</span><span class="op">(</span><span class="va">x</span><span class="op">)</span></span></code></pre></div>
+    </div>
+
+    <div class="section level2">
+    <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#arguments"></a></h2>
+
+
+<dl><dt id="arg-x">x<a class="anchor" aria-label="anchor" href="#arg-x"></a></dt>
+<dd><p>A numeric vector. Naturally, the forward transformation is only
+sensible for vectors with all elements being greater than zero.</p></dd>
+
+</dl></div>
+    <div class="section level2">
+    <h2 id="value">Value<a class="anchor" aria-label="anchor" href="#value"></a></h2>
+    <p>The result of the forward or backward transformation. The returned
+components always sum to 1 for the case of the inverse log-ratio
+transformation.</p>
+    </div>
+    <div class="section level2">
+    <h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a></h2>
+    <p>Peter Filzmoser, Karel Hron (2008) Outlier Detection for
+Compositional Data Using Robust Methods. Math Geosci 40 233-248</p>
+    </div>
+    <div class="section level2">
+    <h2 id="see-also">See also<a class="anchor" aria-label="anchor" href="#see-also"></a></h2>
+    <div class="dont-index"><p>Another implementation can be found in R package
+<code>robCompositions</code>.</p></div>
+    </div>
+    <div class="section level2">
+    <h2 id="author">Author<a class="anchor" aria-label="anchor" href="#author"></a></h2>
+    <p>René Lehmann and Johannes Ranke</p>
+    </div>
+
+    <div class="section level2">
+    <h2 id="ref-examples">Examples<a class="anchor" aria-label="anchor" href="#ref-examples"></a></h2>
+    <div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span></span></span>
+<span class="r-in"><span><span class="co"># Order matters</span></span></span>
+<span class="r-in"><span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.1</span>, <span class="fl">1</span>, <span class="fl">10</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] -1.628174 -2.820079</span>
+<span class="r-in"><span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">10</span>, <span class="fl">1</span>, <span class="fl">0.1</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 1.628174 2.820079</span>
+<span class="r-in"><span><span class="co"># Equal entries give ilr transformations with zeros as elements</span></span></span>
+<span class="r-in"><span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">3</span>, <span class="fl">3</span>, <span class="fl">3</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 0 0</span>
+<span class="r-in"><span><span class="co"># Almost equal entries give small numbers</span></span></span>
+<span class="r-in"><span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.3</span>, <span class="fl">0.4</span>, <span class="fl">0.3</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] -0.2034219  0.1174457</span>
+<span class="r-in"><span><span class="co"># Only the ratio between the numbers counts, not their sum</span></span></span>
+<span class="r-in"><span><span class="fu">invilr</span><span class="op">(</span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.7</span>, <span class="fl">0.29</span>, <span class="fl">0.01</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 0.70 0.29 0.01</span>
+<span class="r-in"><span><span class="fu">invilr</span><span class="op">(</span><span class="fu">ilr</span><span class="op">(</span><span class="fl">2.1</span> <span class="op">*</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.7</span>, <span class="fl">0.29</span>, <span class="fl">0.01</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 0.70 0.29 0.01</span>
+<span class="r-in"><span><span class="co"># Inverse transformation of larger numbers gives unequal elements</span></span></span>
+<span class="r-in"><span><span class="fu">invilr</span><span class="op">(</span><span class="op">-</span><span class="fl">10</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 7.213536e-07 9.999993e-01</span>
+<span class="r-in"><span><span class="fu">invilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="op">-</span><span class="fl">10</span>, <span class="fl">0</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 7.207415e-07 9.991507e-01 8.486044e-04</span>
+<span class="r-in"><span><span class="co"># The sum of the elements of the inverse ilr is 1</span></span></span>
+<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/sum.html" class="external-link">sum</a></span><span class="op">(</span><span class="fu">invilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="op">-</span><span class="fl">10</span>, <span class="fl">0</span><span class="op">)</span><span class="op">)</span><span class="op">)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 1</span>
+<span class="r-in"><span><span class="co"># This is why we do not need all elements of the inverse transformation to go back:</span></span></span>
+<span class="r-in"><span><span class="va">a</span> <span class="op">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="fl">0.1</span>, <span class="fl">0.3</span>, <span class="fl">0.5</span><span class="op">)</span></span></span>
+<span class="r-in"><span><span class="va">b</span> <span class="op">&lt;-</span> <span class="fu">invilr</span><span class="op">(</span><span class="va">a</span><span class="op">)</span></span></span>
+<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/length.html" class="external-link">length</a></span><span class="op">(</span><span class="va">b</span><span class="op">)</span> <span class="co"># Four elements</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 4</span>
+<span class="r-in"><span><span class="fu">ilr</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">c</a></span><span class="op">(</span><span class="va">b</span><span class="op">[</span><span class="fl">1</span><span class="op">:</span><span class="fl">3</span><span class="op">]</span>, <span class="fl">1</span> <span class="op">-</span> <span class="fu"><a href="https://rdrr.io/r/base/sum.html" class="external-link">sum</a></span><span class="op">(</span><span class="va">b</span><span class="op">[</span><span class="fl">1</span><span class="op">:</span><span class="fl">3</span><span class="op">]</span><span class="op">)</span><span class="op">)</span><span class="op">)</span> <span class="co"># Gives c(0.1, 0.3, 0.5)</span></span></span>
+<span class="r-out co"><span class="r-pr">#&gt;</span> [1] 0.1 0.3 0.5</span>
+<span class="r-in"><span></span></span>
+</code></pre></div>
+    </div>
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside></div>
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