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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/master/R/ilr.R'><code>R/ilr.R</code></a></small>
<div class="hidden name"><code>ilr.Rd</code></div>
</div>
<div class="ref-description">
<p>This implementation is a special case of the class of isometric log-ratio
transformations.</p>
</div>
<pre class="usage"><span class='fu'>ilr</span><span class='op'>(</span><span class='va'>x</span><span class='op'>)</span>
<span class='fu'>invilr</span><span class='op'>(</span><span class='va'>x</span><span class='op'>)</span></pre>
<h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
<table class="ref-arguments">
<colgroup><col class="name" /><col class="desc" /></colgroup>
<tr>
<th>x</th>
<td><p>A numeric vector. Naturally, the forward transformation is only
sensible for vectors with all elements being greater than zero.</p></td>
</tr>
</table>
<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>
<div class='dont-index'><p>Another implementation can be found in R package
<code>robCompositions</code>.</p></div>
<h2 class="hasAnchor" id="author"><a class="anchor" href="#author"></a>Author</h2>
<p>René Lehmann and Johannes Ranke</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='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>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>
</div><div class='output co'>#> [1] -1.628174 -2.820079</div><div class='input'><span class='fu'>ilr</span><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='fl'>10</span>, <span class='fl'>1</span>, <span class='fl'>0.1</span><span class='op'>)</span><span class='op'>)</span>
</div><div class='output co'>#> [1] 1.628174 2.820079</div><div class='input'><span class='co'># Equal entries give ilr transformations with zeros as elements</span>
<span class='fu'>ilr</span><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='fl'>3</span>, <span class='fl'>3</span>, <span class='fl'>3</span><span class='op'>)</span><span class='op'>)</span>
</div><div class='output co'>#> [1] 0 0</div><div class='input'><span class='co'># Almost equal entries give small numbers</span>
<span class='fu'>ilr</span><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='fl'>0.3</span>, <span class='fl'>0.4</span>, <span class='fl'>0.3</span><span class='op'>)</span><span class='op'>)</span>
</div><div class='output co'>#> [1] -0.2034219 0.1174457</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='op'>(</span><span class='fu'>ilr</span><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='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>
</div><div class='output co'>#> [1] 0.70 0.29 0.01</div><div class='input'><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'>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>
</div><div class='output co'>#> [1] 0.70 0.29 0.01</div><div class='input'><span class='co'># Inverse transformation of larger numbers gives unequal elements</span>
<span class='fu'>invilr</span><span class='op'>(</span><span class='op'>-</span><span class='fl'>10</span><span class='op'>)</span>
</div><div class='output co'>#> [1] 7.213536e-07 9.999993e-01</div><div class='input'><span class='fu'>invilr</span><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='op'>-</span><span class='fl'>10</span>, <span class='fl'>0</span><span class='op'>)</span><span class='op'>)</span>
</div><div class='output co'>#> [1] 7.207415e-07 9.991507e-01 8.486044e-04</div><div class='input'><span class='co'># The sum of the elements of the inverse ilr is 1</span>
<span class='fu'><a href='https://rdrr.io/r/base/sum.html'>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'>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>
</div><div class='output co'>#> [1] 1</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='va'>a</span> <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='fl'>0.1</span>, <span class='fl'>0.3</span>, <span class='fl'>0.5</span><span class='op'>)</span>
<span class='va'>b</span> <span class='op'><-</span> <span class='fu'>invilr</span><span class='op'>(</span><span class='va'>a</span><span class='op'>)</span>
<span class='fu'><a href='https://rdrr.io/r/base/length.html'>length</a></span><span class='op'>(</span><span class='va'>b</span><span class='op'>)</span> <span class='co'># Four elements</span>
</div><div class='output co'>#> [1] 4</div><div class='input'><span class='fu'>ilr</span><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='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'>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>
</div><div class='output co'>#> [1] 0.1 0.3 0.5</div><div class='input'>
</div></pre>
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