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  René Lehmann and Johannes Ranke
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      <h1>
  Function to perform isometric log-ratio transformation
</h1>

<div class="row">
  <div class="span8">
    <h2>Usage</h2>
    <pre><div>ilr(x) invilr(x)</div></pre>
    
    <h2>Arguments</h2>
    <dl>
      <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>
    
    <div class="Description">
      <h2>Description</h2>

      <p>This implementation is a special case of the class of isometric log-ratio transformations.</p>
  
    </div>

    <div class="Value">
      <h2>Value</h2>

      <p><dl>
  The result of the forward or backward transformation. The returned components always 
  sum to 1 for the case of the inverse log-ratio transformation.
</dl></p>
  
    </div>

    <div class="References">
      <h2>References</h2>

      <p>Peter Filzmoser, Karel Hron (2008) Outlier Detection for Compositional Data Using Robust Methods. Math Geosci 40 233-248</p>
  
    </div>
    
    <h2 id="examples">Examples</h2>
    <pre class="examples"><div class='input'># Order matters
ilr(c(0.1, 1, 10))
</div>
<div class='output'>[1] -1.628174 -2.820079
</div>
<div class='input'>ilr(c(10, 1, 0.1))
</div>
<div class='output'>[1] 1.628174 2.820079
</div>
<div class='input'># Equal entries give ilr transformations with zeros as elements
ilr(c(3, 3, 3))
</div>
<div class='output'>[1] 0 0
</div>
<div class='input'># Almost equal entries give small numbers
ilr(c(0.3, 0.4, 0.3))
</div>
<div class='output'>[1] -0.2034219  0.1174457
</div>
<div class='input'># Only the ratio between the numbers counts, not their sum
invilr(ilr(c(0.7, 0.29, 0.01)))
</div>
<div class='output'>[1] 0.70 0.29 0.01
</div>
<div class='input'>invilr(ilr(2.1 * c(0.7, 0.29, 0.01)))
</div>
<div class='output'>[1] 0.70 0.29 0.01
</div>
<div class='input'># Inverse transformation of larger numbers gives unequal elements
invilr(-10)
</div>
<div class='output'>[1] 7.213536e-07 9.999993e-01
</div>
<div class='input'>invilr(c(-10, 0))
</div>
<div class='output'>[1] 7.207415e-07 9.991507e-01 8.486044e-04
</div>
<div class='input'># The sum of the elements of the inverse ilr is 1
sum(invilr(c(-10, 0)))
</div>
<div class='output'>[1] 1
</div>
<div class='input'># This is why we do not need all elements of the inverse transformation to go back:
a &lt;- c(0.1, 0.3, 0.5)
b &lt;- invilr(a)
length(b) # Four elements
</div>
<div class='output'>[1] 4
</div>
<div class='input'>ilr(c(b[1:3], 1 - sum(b[1:3]))) # Gives c(0.1, 0.3, 0.5)
</div>
<div class='output'>[1] 0.1 0.3 0.5
</div></pre>
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    <!-- <ul>
      <li>ilr</li><li>invilr</li>
    </ul>
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      <li> manip </li>
    </ul> -->
      
    <h2>See also</h2>
    
  Another implementation can be found in R package <code>robCompositions</code>.

    
    <h2>Author</h2>
    
  René Lehmann and Johannes Ranke

    
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