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57 lines
2.2 KiB
Markdown
57 lines
2.2 KiB
Markdown
---
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title: Principal Component Analysis Pt. 1
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showthedate: false
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math: true
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---
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## What is PCA?
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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.
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Number of distinct principle components equals $min(\# Variables, \# Observations - 1)$
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The transformation is defined in such a way that the first principle component has the largest possible variance explained in the data.
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Each succeeding component has the highest possible variance under the constraint of having to be orthogonal to the preceding components.
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PCA is sensitive to the relative scaling of the original variables.
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### Results of a PCA
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Results are discussed in terms of *component scores* which is the transformed variables and *loadings* which is the weight by which each original variable should be multiplied to get the component score.
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## Assumptions of PCA
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1. Linearity
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2. Large variances are important and small variances denote noise
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3. Principal components are orthogonal
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## Why perform PCA?
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- Distance measures perform poorly in high-dimensional space (https://stats.stackexchange.com/questions/256172/why-always-doing-dimensionality-reduction-before-clustering)
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- Helps eliminates noise from the dataset (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)
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- One initial cost to help reduce further computations
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## Computing PCA
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1. Subtract off the mean of each measurement type
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2. Compute the covariance matrix
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3. Take the eigenvalues/vectors of the covariance matrix
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## R Code
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```R
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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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}
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```
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