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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>
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    <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

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