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Pseudo Inverse Heisenberg Uncertainty for Data Mining Explicit Principal Components Implicit Principal Components NIPALS Algorithm for Eigenvalues and Eigenvectors Scripts - PCA transformation of data - Pharma-plots - PCA training and testing - Bootstrap PCA - NIPALS and other PCA algorithms Examples Feature selection Principal Component Regression Analysis

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Pseudo inverse Penrose inverse Least-Squares Optimization Classical Regression Analysis 1 T mnnm T mn XXX

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The Machine Learning Paradox 1 XX T If data are can learned from, they must have redundancy If there is redundancy, (X T X) -1 is ill-conditioned - similar data patterns - closely correlated descriptive features

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Beyond Regression Paul Werbos motivated beyond regression in 1972 In addition, there are related statistical duals (PCA, PLS, SVM) Principal component analysis: T mhnmnh hmnhnm BXT BTX h = # Principal components Trick: eliminate poor conditioning by using h PCs (largest ) Now matrix to invert is small and well-conditioned Generally include ~ 2 - 4 - 6 PCAs A Better PCA Regression is PLS (Please Listen to Savanti Wold) A Better PLS is nonlinear PNLS

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Explicit PCA Regression We had Assume we derive PCA features for A according to We now have T mhnmnh hmnhnm BXT BTX h = # Principal components

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Explicit PCA Regression on training/test set We have for training set: And for the test set:

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Implicit PCA Regression T mhnmnh hmnhnm BXT BTX h = # Principal components How to apply? Calculate T and B with NIPALS algorithm Determine b, and apply to data matrix

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T mhnmnh hmnhnm BXT BTX h = # Principal components The B matrix is a matrix of eigenvectors of the correlation matrix C If the features are zero centered we have: We only consider the h eigenvectors corresponding to largest eigenvalues The eigenvalues are the variances Eigenvectors are normalized to 1 and solutions of: Use NIPALS algorithm to build up B and T Algorithm

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NIPALS Algorithm: Part 2 T mhnmnh hmnhnm BXT BTX h = # Principal components

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PRACTICAL TIPS FOR PCA NIPALS algorithm assumes the features are zero centered It is standard practice to do a Mahalanobis scaling of the data PCA regression does not consider the response data The ts are called the scores Use 3-10 PCAs I usually use 4 PCAs It is common practice to drop 4 sigma outlier features (if there are many features)

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PCA with Analyze Several options: option #17 for training and #18 for testing (the weight vectors after training is in file bbmatrixx.txt) The file num_eg.txt contains a number equal to # PCAs Option –17 is the NIPALS algorithm and generally faster than 17 SAnalyze has options for calculating Ts, Bs and s - option #36 transforms a data matrix to its PCAs - option #36 also saves eigenvalues and eigenvectors of X T X Analyze has also option for bootstrap PCA (-33)

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StripMiner Scripts last lecture: iris_pca.bat (make PCAs and visualize) iris.bat (split up data in training and validation set and predict) iris_boot.bat (bootstrap prediction)

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Bootstrap Prediction (iris_boo.bat) Make different models for training set Predict Test set on average model

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Neural Network Interpretation of PCA

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PCA in DATA SPACE Σ Σ Σ x1x1 xMxM xixi Weights correspond to H eigenvectors corresponding to largest eigenvalues of X T X Σ Σ Σ Σ Σ Σ Σ Σ... Weights correspond to the scores or PCAs for the entire training set Weights correspond to the dependent variable for the entire training data Means that the similarity score with each data point will be weighed (i.e.., effectively incorporating Mahalanobis scaling in data space) This layer gives a similarity score with each datapoint Kind of a nearest neighbor weighted prediction score

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