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123 lines
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HTML
123 lines
5 KiB
HTML
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<h1>Principal Component Analysis Pt. 1</h1>
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<h2>What is PCA?</h2>
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<p>Principal component analysis is a statistical procedure that performs an orthogonal transformation to convert a set of variables into a set of linearly uncorrelated variables called principle components.</p>
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<p>Number of distinct principle components equals $min(# Variables, # Observations - 1)$</p>
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<p>The transformation is defined in such a way that the first principle component has the largest possible variance explained in the data.</p>
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<p>Each succeeding component has the highest possible variance under the constraint of having to be orthogonal to the preceding components.</p>
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<p>PCA is sensitive to the relative scaling of the original variables.</p>
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<h3>Results of a PCA</h3>
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<p>Results are discussed in terms of <em>component scores</em> which is the transformed variables and <em>loadings</em> which is the weight by which each original variable should be multiplied to get the component score.</p>
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<h2>Assumptions of PCA</h2>
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<ol>
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<li>Linearity</li>
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<li>Large variances are important and small variances denote noise</li>
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<li>Principal components are orthogonal</li>
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</ol>
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<h2>Why perform PCA?</h2>
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<ul>
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<li>Distance measures perform poorly in high-dimensional space (<a href="https://stats.stackexchange.com/questions/256172/why-always-doing-dimensionality-reduction-before-clustering">https://stats.stackexchange.com/questions/256172/why-always-doing-dimensionality-reduction-before-clustering</a>)</li>
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<li>Helps eliminates noise from the dataset (<a href="https://www.quora.com/Does-it-make-sense-to-perform-principal-components-analysis-before-clustering-if-the-original-data-has-too-many-dimensions-Is-it-theoretically-unsound-to-try-to-cluster-data-with-no-correlation">https://www.quora.com/Does-it-make-sense-to-perform-principal-components-analysis-before-clustering-if-the-original-data-has-too-many-dimensions-Is-it-theoretically-unsound-to-try-to-cluster-data-with-no-correlation</a>)</li>
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<li>One initial cost to help reduce further computations</li>
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</ul>
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<h2>Computing PCA</h2>
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<ol>
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<li>Subtract off the mean of each measurement type</li>
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<li>Compute the covariance matrix</li>
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<li>Take the eigenvalues/vectors of the covariance matrix</li>
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</ol>
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<h2>R Code</h2>
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<pre><code class="language-R">pcal = function(data) {
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centered_data = scale(data)
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covariance = cov(centered_data)
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eigen_stuff = eigen(covariance)
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sorted_indices = sort(eigen_stuff$values,
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index.return = T,
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decreasing = T)$ix
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loadings = eigen_stuff$values[sorted_indices]
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components = eigen_stuff$vectors[sorted_indices,]
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combined_list = list(loadings, components)
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names(combined_list) = c("Loadings", "Components")
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return(combined_list)
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}</code></pre>
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