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+ <h1>
+</h1>
+ </div>
+
+
+ <p>This implementation is a special case of the class of isometric log-ratio transformations.</p>
+
+
+ <pre><span class='fu'>ilr</span>(<span class='no'>x</span>)
+ <span class='fu'>invilr</span>(<span class='no'>x</span>)</pre>
+
+ <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a> Arguments</h2>
+ <dl class="dl-horizontal">
+ <dt>x</dt>
+ <dd>
+ A numeric vector. Naturally, the forward transformation is only sensible for
+ vectors with all elements being greater than zero.
+ </dd>
+ </dl>
+
+ <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</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>
+
+ <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
+
+ <p>Peter Filzmoser, Karel Hron (2008) Outlier Detection for Compositional Data Using Robust Methods. Math Geosci 40 233-248</p>
+
+ <h2 class="hasAnchor" id="see-also"><a class="anchor" href="#see-also"></a>See also</h2>
+
+ <p>Another implementation can be found in R package <code>robCompositions</code>.</p>
+
+
+ <h2 class="hasAnchor" id="examples">
+ <a class="anchor" href="#examples"></a>
+ Examples
+ </h2>
+ <pre class="examples"><div class='input'><span class='co'># Order matters</span>
+<span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='fl'>0.1</span>, <span class='fl'>1</span>, <span class='fl'>10</span>))</div><div class='output co'>#&gt; [1] -1.628174 -2.820079
+#&gt; </div><div class='input'><span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='fl'>10</span>, <span class='fl'>1</span>, <span class='fl'>0.1</span>))</div><div class='output co'>#&gt; [1] 1.628174 2.820079
+#&gt; </div><div class='input'><span class='co'># Equal entries give ilr transformations with zeros as elements</span>
+<span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='fl'>3</span>, <span class='fl'>3</span>, <span class='fl'>3</span>))</div><div class='output co'>#&gt; [1] 0 0
+#&gt; </div><div class='input'><span class='co'># Almost equal entries give small numbers</span>
+<span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='fl'>0.3</span>, <span class='fl'>0.4</span>, <span class='fl'>0.3</span>))</div><div class='output co'>#&gt; [1] -0.2034219 0.1174457
+#&gt; </div><div class='input'><span class='co'># Only the ratio between the numbers counts, not their sum</span>
+<span class='fu'>invilr</span>(<span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='fl'>0.7</span>, <span class='fl'>0.29</span>, <span class='fl'>0.01</span>)))</div><div class='output co'>#&gt; [1] 0.70 0.29 0.01
+#&gt; </div><div class='input'><span class='fu'>invilr</span>(<span class='fu'>ilr</span>(<span class='fl'>2.1</span> * <span class='fu'>c</span>(<span class='fl'>0.7</span>, <span class='fl'>0.29</span>, <span class='fl'>0.01</span>)))</div><div class='output co'>#&gt; [1] 0.70 0.29 0.01
+#&gt; </div><div class='input'><span class='co'># Inverse transformation of larger numbers gives unequal elements</span>
+<span class='fu'>invilr</span>(-<span class='fl'>10</span>)</div><div class='output co'>#&gt; [1] 7.213536e-07 9.999993e-01
+#&gt; </div><div class='input'><span class='fu'>invilr</span>(<span class='fu'>c</span>(-<span class='fl'>10</span>, <span class='fl'>0</span>))</div><div class='output co'>#&gt; [1] 7.207415e-07 9.991507e-01 8.486044e-04
+#&gt; </div><div class='input'><span class='co'># The sum of the elements of the inverse ilr is 1</span>
+<span class='fu'>sum</span>(<span class='fu'>invilr</span>(<span class='fu'>c</span>(-<span class='fl'>10</span>, <span class='fl'>0</span>)))</div><div class='output co'>#&gt; [1] 1
+#&gt; </div><div class='input'><span class='co'># This is why we do not need all elements of the inverse transformation to go back:</span>
+<span class='no'>a</span> <span class='kw'>&lt;-</span> <span class='fu'>c</span>(<span class='fl'>0.1</span>, <span class='fl'>0.3</span>, <span class='fl'>0.5</span>)
+<span class='no'>b</span> <span class='kw'>&lt;-</span> <span class='fu'>invilr</span>(<span class='no'>a</span>)
+<span class='fu'>length</span>(<span class='no'>b</span>) <span class='co'># Four elements</span></div><div class='output co'>#&gt; [1] 4
+#&gt; </div><div class='input'><span class='fu'>ilr</span>(<span class='fu'>c</span>(<span class='no'>b</span>[<span class='fl'>1</span>:<span class='fl'>3</span>], <span class='fl'>1</span> - <span class='fu'>sum</span>(<span class='no'>b</span>[<span class='fl'>1</span>:<span class='fl'>3</span>]))) <span class='co'># Gives c(0.1, 0.3, 0.5)</span></div><div class='output co'>#&gt; [1] 0.1 0.3 0.5
+#&gt; </div></pre>
+ </div>
+ <div class="col-md-3 hidden-xs hidden-sm" id="sidebar">
+ <h2>Contents</h2>
+ <ul class="nav nav-pills nav-stacked">
+ <li><a href="#arguments">Arguments</a></li>
+
+ <li><a href="#value">Value</a></li>
+
+ <li><a href="#references">References</a></li>
+
+ <li><a href="#see-also">See also</a></li>
+
+ <li><a href="#examples">Examples</a></li>
+ </ul>
+
+ <h2>Author</h2>
+
+ René Lehmann and Johannes Ranke
+
+ </div>
+</div>
+
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