PCA properties PCA Principal Components Analysis



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PCA properties PCA Principal Components Analysis

California College of Arts and Crafts, US has reference to this Academic Journal, Principal Components Analysis (PCA) 273A Intro Machine Learning Principal Components Analysis We search in consideration of those directions in space that have the highest variance. We then project the data onto the subspace of highest variance. This structure is encoded in the sample co-variance of the data: Note that PCA is a unsupervised learning method (why?) PCA We want so that find the eigenvectors in addition to eigenvalues of this covariance: eigenvalue = variance in direction eigenvector ( in matlab [U,L]=eig(C) ) Orthogonal, unit-length eigenvectors.

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PCA properties (U eigevectors) (u orthonormal ? U rotation) (rank-k approximation) (projection) PCA properties is the optimal rank-k approximation of C in Frobenius norm. I.e. it minimizes the cost-function: Note that there are infinite solutions that minimize this norm. If A is a solution, then is also a solution. The solution provided by PCA is unique because U is orthogonal in addition to ordered by largest eigenvalue. Solution is also nested: if I solve in consideration of a rank-k+1 approximation, I will find that the first k eigenvectors are those found by an rank-k approximation (etc.)

The Early Detection of Disease ?Statistical Challenges Outline Background in addition to Rationale Some Statistical Challenges Early Detection Randomized Clinical Trials Early Detection Clinical Trials Public Health Programs Example : Breast Cancer Screening Using Mammography Over Diagnosis Need in consideration of Models Models Issues in the interpretation of data Length biased sampling Lead Time Bias :Usual care Notes on Modeling Applications so that Breast in addition to Prostate Cancer Use of Model: Evaluating Benefit in consideration of Women Aged 40-49 Equal Intervals Between Exams Numerical Calculation : Prostate Cancer Conclusions My Collaborators Thank you in consideration of coming Why would screening result in benefit ? Natural History of Disease

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