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Principal Component Analysis

Principal Component Analysis. Zelin Jia Shengbin Lin 10/20/2015. What is PCA?. An orthogonal transformation Convert correlated variables to an artificial variable(Principle Component) The resulting vectors are an orthogonal basis set A tool in  exploratory data analysis.

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Principal Component Analysis

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  1. Principal Component Analysis Zelin Jia Shengbin Lin 10/20/2015

  2. What is PCA? • An orthogonal transformation • Convert correlated variables to an artificial variable(Principle Component) • The resulting vectors are an orthogonal basis set • A tool in exploratory data analysis https://en.wikipedia.org/wiki/Principal_component_analysis

  3. Why use PCA? • Reduce the dimensionality of the data • Compress the data • Prepare the data for further analysis using other techniques • Understand your data better by interpreting the loadings, and by graphing the derived variables http://psych.colorado.edu/wiki/lib/exe/fetch.php?media=labs:learnr:emily_-_principal_components_analysis_in_r:pca_how_to.pdf Dr. Peter Westfall

  4. How PCA works • PCA begin with covariance matrix: Cov(X)=XTX • For the covariance matrix, calculate its eigenvectors and eigenvalues. • Get sets of eigenvectors zi and eigenvaluesλi (Constraint: ziT zi=1) • arrange the eigenvectors in decreasing order of the eigenvalues • Pick eigenvectors, multiple by original data matrix(X), we will get PC matrix. https://www.riskprep.com/all-tutorials/36-exam-22/132-understanding-principal-component-analysis-pca

  5. Example of how PCA works (by R) • A financial sample data with 8 variables and 25obs • Perform PCA on this data and reduce the number of variables from 8 to something more manageable https://www.riskprep.com/all-tutorials/36-exam-22/132-understanding-principal-component-analysis-pca

  6. Simulate PC on uncorrelated data and highly correlated data (by R) • PCA is better for more highly correlated data in that greater reduction is achievable.  Provided by Dr. Peter Westfall

  7. PCA standardization Why: The variable with the smaller numbers – even though this may be the more important number – will be overwhelmed by the other larger numbers in what it contributes to the covariance https://www.riskprep.com/all-tutorials/36-exam-22/132-understanding-principal-component-analysis-pca

  8. properties of PC • The number of principal components is less than or equal to the number of original variables. • The first principal component has the largest possible variance. • Each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. https://en.wikipedia.org/wiki/Principal_component_analysis

  9. What is SVD? Applied_Regression_Analysis_A_Research_Tool.pdf

  10. Relationship between SVD and PCA • From SVD we have X = UL1/2ZT-> W = XZ = UL1/2 • If X is an n × p matrix of observations on p variables, each column of W is a new variable defined as a linear transformation of the original variables. Applied_Regression_Analysis_A_Research_Tool.pdf

  11. EFA vs PCA • EFA: EFA provides a model to explain why the data looks like it does. • PCA: PC is not a model  that explains how the data looks.  There is no model at all. Provided by Dr. Peter Westfall

  12. EFA vs PCA http://www.gac-usp.com.br/resources/use_of_exploratory_factor_analysis_park_dailey.pdf

  13. EFA vs PCA EFA: in EFA one postulates that there is a smaller set of unobserved (latent) variables or constructs underlying the variables actually observed or measured (this is commonly done to assess validity) PCA: in PCA one is simply trying to mathematically derive a relatively small number of variables to use to convey as much of the information in the observed/measured variables as possible http://www.gac-usp.com.br/resources/use_of_exploratory_factor_analysis_park_dailey.pdf

  14. Application of PCA • Data visualization • Image compression

  15. Data visualization • If a multivariate dataset is visualized as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture. https://en.wikipedia.org/wiki/Principal_component_analysis

  16. PCA using on compressing image • The PCA formulation may be used as a digital image compression algorithm with a low level of loss. http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1679-45082012000200004

  17. princomp vs prcomp • For prcomp: • The calculation is done by a singular value decomposition of the (centered and possibly scaled) data matrix, not by using eigen on the covariance matrix. This is generally the preferred method for numerical accuracy. • For princomp: • The calculation is done using eigen on the correlation or covariance matrix, as determined by cor. This is done for compatibility with the S-PLUS result. A preferred method of calculation is to use svd on x, as is done in prcomp." http://stats.stackexchange.com/questions/20101/what-is-the-difference-between-r-functions-prcomp-and-princomp

  18. Thanks!

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