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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 <- c(0.1, 0.3, 0.5)
b <- 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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<li>ilr</li><li>invilr</li>
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<ul>
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</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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