PCA (Principal Component Analysis)
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This document provides an introduction to principal component analysis (PCA), outlining its purpose for data reduction and structural detection. It defines PCA as a linear combination of weighted observed variables. The procedure section discusses assumptions like normality, homoscedasticity, and linearity that are evaluated prior to PCA. Requirements for performing PCA include the variables being at the metric or nominal level, sufficient sample size and variable ratios, and adequate correlations between variables.
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PCA transforms correlated variables into uncorrelated variables called principal components. It finds the directions of maximum variance in high-dimensional data by computing the eigenvectors of the covariance matrix. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. Dimensionality reduction is achieved by ignoring components with small eigenvalues, retaining only the most significant components.
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PCA transforms correlated variables into uncorrelated variables called principal components. It finds the directions of maximum variance in high-dimensional data by computing the eigenvectors of the covariance matrix. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. Dimensionality reduction is achieved by ignoring components with small eigenvalues, retaining only the most significant components.
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This document describes the 5 steps of principal component analysis (PCA):
1) Subtract the mean from each dimension of the data to center it around zero.
2) Calculate the covariance matrix of the data.
3) Calculate the eigenvalues and eigenvectors of the covariance matrix.
4) Form a feature vector by selecting eigenvectors corresponding to largest eigenvalues. Project the data onto this to reduce dimensions.
5) To reconstruct the data, take the transpose of the feature vector and multiply it with the projected data, then add the mean back.
Missing data handling
Missing data handling is typically done in an ad-hoc way. Without understanding the repurcussions of a missing data handling technique, approaches that only let you get to the "next step" in your analytics pipeline leads to terrible outputs, conclusions that aren't robust and biased estimates. Handling missing data in data sets requires a structured approach. In this workshop, we will cover the key tenets of handling missing data in a structured way
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Principal Component Analysis (PCA) is a technique used to reduce the dimensionality of data by transforming it to a new coordinate system. It works by finding the principal components - linear combinations of variables with the highest variance - and using those to project the data to a lower dimensional space. PCA is useful for visualizing high-dimensional data, reducing dimensions without much loss of information, and finding patterns. It involves calculating the covariance matrix and solving the eigenvalue problem to determine the principal components.
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PCA (Principal Component Analysis)
- 1. Seoul National University2/25/2017 1 PRINCIPAL COMPONENT ANALYSIS (PCA) 주성분분석 박정호 박사과정* 서울대학교 기계항공공학부 시스템 건전성 및 리스크 관리 연구실 *hihijung@snu.ac.kr
- 2. Seoul National University Principles of PCA 2/25/2017 2 1) Maximum variance 2) Minimum error • To maximize the variance of the projected data on the certain dimension. Var1 PC1 PC2 PC1 PC2Var2 PC1 PC2 SSE1 SSE2 • To minimize the mean squared distance between the data and their projections. SSE : Sum or squared errors
- 3. Seoul National University Maximum variance – (1) 2/25/2017 3 x 1 xSample set mean : 1 u x u xVariance of the projected data : maximize where is the data covariance matrix defined by x x x x 전개하면 똑같음 Projected data의 variance 를 maximize 하는 것은 결국 를 maximize 하는 것과 동일하게 됨. * 은 data 가 projection 되는 vector 를 말한다. (unit vector 임. 즉, 1)
- 4. Seoul National University Maximum variance – (2) 2/25/2017 4 Formulation of maximization using Lagrange multiplier x, 1 원래식 Constraint (derivative w.r.t. ) • 은 의 eigenvector, 은 eigenvalue 이다. • 1 라는 사실을 이용하면, = 이 된다. 즉, variance 의 maximization 문제가 eigenvalue 의 maximum을 구하는 문제와 같아진다. Summary 1 u x u x → → Variance Different expression of the variance using covariance matrix Eigenvalue of the covariance matrix
- 5. Seoul National University Minimum error – (1) 2/25/2017 5 x u Representation of each data point by a linear combination of the basis vectors : Where δ , i.e. D‐dimensional basis vectors {u } x x u u x u u u → x u Approximation of each data point by a restricted number M < D : x u u
- 6. Seoul National University Minimum error – (2) 2/25/2017 6 Distortion measure : 1 x x Need to be minimized x u u x u x u Derivative w.r.t. & orthonormality Derivative w.r.t. & orthonormality x x x x u u 1 x u x u • 최소의 distortion measure, J를 구하기 위해서는 1~M 까지의 eigenvalue 들의 최대값을 가져야함 ↔ Variance의 maximization 문제와 같은 결론