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<title>Classifier chains</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Classifier chains</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Classifier chains</b> is a <a href="Machine_learning" title="Machine learning">machine learning</a> method for problem transformation in <a href="Multi-label_classification" title="Multi-label classification">multi-label classification</a>. It combines the computational efficiency of the <a href="Binary_relevance" class="mw-redirect" title="Binary relevance">binary relevance</a> method while still being able to take the label dependencies into account for <a href="Classification_in_machine_learning" class="mw-redirect" title="Classification in machine learning">classification</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Problem_transformation">Problem transformation</h2></div>

<p>Several problem transformation methods exist. One of them is the Binary Relevance method (BR). Given a set of labels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {L}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {L}}\,}</annotation>

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</math></span><img src="./46f21c9275ecf6f718f284b5b2afea187e873d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.845ex; height:2.176ex;" alt="{\displaystyle {\mathit {L}}\,}" loading="lazy"></span> and a data set with instances of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {(x,Y)}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {(x,Y)}}\,}</annotation>

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</math></span><img src="./28479f9f6953ed1f88c7e5c9e268e57ea73a33ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.196ex; height:2.843ex;" alt="{\displaystyle {\mathit {(x,Y)}}\,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {x}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {x}}\,}</annotation>

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</math></span><img src="./46830ab2c3def79bbef7b0c1dd8348eb11ed236b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.466ex; height:1.676ex;" alt="{\displaystyle {\mathit {x}}\,}" loading="lazy"></span> is a <a href="Feature_vector" class="mw-redirect" title="Feature vector">feature vector</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subseteq L}">

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<annotation encoding="application/x-tex">{\displaystyle Y\subseteq L}</annotation>

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</math></span><img src="./14c4fcd4584539bcaae667d8a4b34c4cccc260bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.455ex; height:2.343ex;" alt="{\displaystyle Y\subseteq L}" loading="lazy"></span> is a set of labels assigned to the instance. BR transforms the data set into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert L\right\vert }">

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<annotation encoding="application/x-tex">{\displaystyle \left\vert L\right\vert }</annotation>

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</math></span><img src="./559b27704cdf4f7e16e0deb1d69afa4a7a220a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.877ex; height:2.843ex;" alt="{\displaystyle \left\vert L\right\vert }" loading="lazy"></span> data sets and learns <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert L\right\vert }">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \left\vert L\right\vert }</annotation>

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</math></span><img src="./559b27704cdf4f7e16e0deb1d69afa4a7a220a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.877ex; height:2.843ex;" alt="{\displaystyle \left\vert L\right\vert }" loading="lazy"></span> binary classifiers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:X\rightarrow \{l,\neg l\}}">

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<annotation encoding="application/x-tex">{\displaystyle H:X\rightarrow \{l,\neg l\}}</annotation>

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</math></span><img src="./5151eb8850922f3c5956bf79761002a8233e491f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.891ex; height:2.843ex;" alt="{\displaystyle H:X\rightarrow \{l,\neg l\}}" loading="lazy"></span> for each label <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l\in L}">

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<annotation encoding="application/x-tex">{\displaystyle l\in L}</annotation>

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</math></span><img src="./4af31b518b8600dc8ff15041dc353769574d4631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.117ex; height:2.176ex;" alt="{\displaystyle l\in L}" loading="lazy"></span>. During this process the information about dependencies between labels is not preserved. This can lead to a situation where a set of labels is assigned to an instance although these labels never co-occur together in the data set. Thus, information about label co-occurrence can help to assign correct label combinations. Loss of this information can in some cases lead to a decrease in classification performance.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Another approach, which takes into account label correlations, is the Label Powerset method (LP). Each combination of labels in a data set is considered to be a single label. After transformation a single-label classifier <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H:X\rightarrow {\mathcal {P}}(L)}">

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<annotation encoding="application/x-tex">{\displaystyle H:X\rightarrow {\mathcal {P}}(L)}</annotation>

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</math></span><img src="./48c30638dfd0759fd7c78fb517a7124a1140543d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.691ex; height:2.843ex;" alt="{\displaystyle H:X\rightarrow {\mathcal {P}}(L)}" loading="lazy"></span> is trained where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(L)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(L)}</annotation>

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</math></span><img src="./369bb773ef614224e895841a855c99e52ba04c41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.096ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(L)}" loading="lazy"></span> is the power set of all labels in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {L}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {L}}}</annotation>

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</math></span><img src="./3264e77f188f45fbffeb9c58323c61f659721c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.002ex; width:1.46ex; height:2.176ex;" alt="{\displaystyle {\mathit {L}}}" loading="lazy"></span>. The main drawback of this approach is that the number of label combinations grows exponentially with the number of labels. For example, a multi-label data set with 10 labels can have up to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{10}=1024}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{10}=1024}</annotation>

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</math></span><img src="./13588ba2bcf107e75098e0aac63663fa4f147e45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.787ex; height:2.676ex;" alt="{\displaystyle 2^{10}=1024}" loading="lazy"></span> label combinations. This increases the run-time of classification.

</p><p>The Classifier Chains method is based on the BR method and it is efficient even on a big number of labels. Furthermore, it considers dependencies between labels.

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<div class="mw-heading mw-heading2"><h2 id="Method_description">Method description</h2></div>

<p>For a given set of labels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {L}}\,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathit {L}}\,}</annotation>

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</math></span><img src="./46f21c9275ecf6f718f284b5b2afea187e873d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.845ex; height:2.176ex;" alt="{\displaystyle {\mathit {L}}\,}" loading="lazy"></span> the Classifier Chain model (CC) learns <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert L\right\vert }">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert L\right\vert }</annotation>

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</math></span><img src="./559b27704cdf4f7e16e0deb1d69afa4a7a220a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.877ex; height:2.843ex;" alt="{\displaystyle \left\vert L\right\vert }" loading="lazy"></span> classifiers as in the Binary Relevance method. All classifiers are linked in a chain through feature space.

</p><p>Given a data set where the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>-th instance has the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {(x_{i},Y_{i})}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {(x_{i},Y_{i})}}\,}</annotation>

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</math></span><img src="./27b58126a75c53164fa017efd4611fe539391816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.671ex; height:2.843ex;" alt="{\displaystyle {\mathit {(x_{i},Y_{i})}}\,}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {Y_{i}}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {Y_{i}}}\,}</annotation>

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</math></span><img src="./54e9c781eeb59e091ca63cd1cc594bec7c791152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.851ex; height:2.509ex;" alt="{\displaystyle {\mathit {Y_{i}}}\,}" loading="lazy"></span> is a subset of labels, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {x_{i}}}\,}">

<msub>

</mrow>

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</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {x_{i}}}\,}</annotation>

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</math></span><img src="./60f4c8780a72ceb8f96c48488ffb160cba793ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle {\mathit {x_{i}}}\,}" loading="lazy"></span> is a set of features. The data set is transformed in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert L\right\vert }">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert L\right\vert }</annotation>

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</math></span><img src="./559b27704cdf4f7e16e0deb1d69afa4a7a220a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.877ex; height:2.843ex;" alt="{\displaystyle \left\vert L\right\vert }" loading="lazy"></span> data sets where instances of the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>-th data set has the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x_{i},l_{1},...,l_{j-1}),l_{j}),l_{j}\in \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle ((x_{i},l_{1},...,l_{j-1}),l_{j}),l_{j}\in \{0,1\}}</annotation>

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</math></span><img src="./d00828c8cfbc400d7869b68daaba2cbfdbecff1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.201ex; height:3.009ex;" alt="{\displaystyle ((x_{i},l_{1},...,l_{j-1}),l_{j}),l_{j}\in \{0,1\}}" loading="lazy"></span>. If the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>-th label was assigned to the instance then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {l_{j}}}\,}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {l_{j}}}\,}</annotation>

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</math></span><img src="./fae3feaaa938200e461c39f3bee428421b2cb0ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.72ex; height:2.843ex;" alt="{\displaystyle {\mathit {l_{j}}}\,}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span>, otherwise it is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span>. Thus, classifiers build a chain where each of them learns binary classification of a single label. The features given to each classifier are extended with binary values that indicate which of previous labels were assigned to the instance.

</p><p>By classifying new instances the labels are again predicted by building a chain of classifiers. The classification begins with the first classifier <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {C_{1}}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {C_{1}}}\,}</annotation>

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</math></span><img src="./5d0de7d48326a47fbdd9d5bc9c5aeacb4ace8560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.124ex; height:2.509ex;" alt="{\displaystyle {\mathit {C_{1}}}\,}" loading="lazy"></span> and proceeds to the last one <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathit {C_{|L|}}}\,}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathit {C_{|L|}}}\,}</annotation>

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</math></span><img src="./eb47018dd48c9a8660952d783fdf6dbb61949b38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.229ex; height:3.009ex;" alt="{\displaystyle {\mathit {C_{|L|}}}\,}" loading="lazy"></span> by passing label information between classifiers through the feature space. Hence, the inter-label dependency is preserved. However, the result can vary for different order of chains. For example, if a label often co-occur with some other label, then only instances of the label which comes later in the chain will have information about the other one in its feature vector. In order to solve this problem and increase accuracy it is possible to use <a href="Ensemble_learning" title="Ensemble learning">ensemble</a> of classifiers.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In Ensemble of Classifier Chains (ECC) several CC classifiers can be trained with random order of chains (i.e. random order of labels) on a random subset of data set. Labels of a new instance are predicted by each classifier separately. After that, the total number of predictions or "votes" is counted for each label. The label is accepted if it was predicted by a percentage of classifiers that is bigger than some threshold value.

</p>

<div class="mw-heading mw-heading2"><h2 id="Adaptations">Adaptations</h2></div>

<p>There is also regressor chains, which themselves can resemble <a href="Vector_autoregression" title="Vector autoregression">vector autoregression</a> models if the order of the chain makes sure temporal order is respected.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFReadBernhard_PfahringerGeoff_HolmesEibe_Frank2009" class="citation journal cs1">Read, Jesse; Bernhard Pfahringer; Geoff Holmes; Eibe Frank (2009). <a rel="nofollow" class="external text" href="http://www.cs.waikato.ac.nz/~ml/publications/2009/chains.pdf">"Classifier Chains for Multi-label Classification"</a> <span class="cs1-format">(PDF)</span>. <i>Proc 13th European Conference on Principles and Practice of Knowledge Discovery in Databases and 20th European Conference on Machine Learning</i>. <b>2009</b>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFDembczynskiWillem_WaegemanWeiwei_ChengEyke_Hüllermeier2010" class="citation journal cs1">Dembczynski, Krzysztof; Willem Waegeman; Weiwei Cheng; Eyke Hüllermeier (2010). <a rel="nofollow" class="external text" href="http://www.mathematik.uni-marburg.de/~eyke/publications/mld10.pdf">"On label dependence in multi-label classification"</a> <span class="cs1-format">(PDF)</span>. <i>Workshop Proceedings of Learning from Multi-Label Data</i>. <b>2010</b>: <span class="nowrap">5–</span>12.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRokach2010" class="citation journal cs1">Rokach, Lior (2010). <a rel="nofollow" class="external text" href="http://www.ise.bgu.ac.il/faculty/liorr/AI.pdf">"Ensemble-based classifiers"</a> <span class="cs1-format">(PDF)</span>. <i>Artif. Intell. Rev</i>. <b>33</b> (<span class="nowrap">1–</span>2). Norwell, MA, USA: ACM: <span class="nowrap">1–</span>39. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10462-009-9124-7">10.1007/s10462-009-9124-7</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://users.ics.aalto.fi/jesse/talks/UC3M-Charla2.pdf">Better Classifier Chains for Multi-label Classification</a> Presentation on Classifier Chains by Jesse Read and Fernando Pérez Cruz</li></ul></div><div><div class="zim-footer">

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