Principal component analysis
- 1. PRINCIPAL COMPONENT ANALYSIS Partha Sarathi Kar IVSM 166777 1
- 2. CONTENTS • WHAT IS PCA • HOW IT WORKS • HISTORY OF PCA • PCA IMPLEMENTATION • USES of PCA • LIMITATION OF PCA 2/28
- 3. WHAT IS PCA PCA takes a dataset with a lots of dimension (i.e. Lots of Cells) and flattens it to 2 or 3 dimensions so we can look on it. 3/28 Principal component analysis (PCA) is a technique used to emphasize variation and bring out strong patterns in a dataset. It's often used to make data easy to explore and visualize.
- 4. HOW IT WORKS 4/28 Eating in the UK (a 17D example) Here's the plot of the data along the first principal component. Already we can see something is different about Northern Ireland. Northern Irish eat way more grams of fresh potatoes and way fewer of fresh fruits, cheese, fish and alcoholic drinks
- 5. HISTORY OF PCA • PCA was invented in 1901 by Karl Pearson • as an analogue of the principal axis theorem in mechanics 5/28 src: https://commons.wikimedia.org/wiki/File:GaussianScatterPCA.svg
- 6. HISTORY OF PCA Depending on the field of application, it is also named: 6/28 • discrete Kosambi-Karhunen–Loève transform (KLT) in signal processing, • the Hotelling transform in multivariate quality control, • proper orthogonal decomposition (POD) in mechanical engineering, • singular value decomposition (SVD) of X (Golub and Van Loan, 1983), • eigenvalue decomposition (EVD) of XTX in linear algebra, • Eckart–Young theorem (Harman, 1960), or Schmidt–Mirsky theorem in psychometrics, • empirical orthogonal functions (EOF) in meteorological science, • empirical eigenfunction decomposition (Sirovich, 1987) etc
- 7. PCA IMPLEMENTATION PCA could have different implementations. But most popular ones are • eigenvalue decomposition (EVD) and • singular value decomposition (SVD). 7/28
- 8. PCA IMPLEMENTATION Eigenvalue decomposition (EVD) 8/28 NxM > NxK (K<=M) X = original data matrix W and D new Matrix from W contains all principal component vectors, while D contains all ranks of those vectors (ordered from the largest variance to the least one XXTW=WD
- 9. PCA IMPLEMENTATION Singular Value decomposition (SVD) 9/28 NxM > NxK (K<=M) X = original data matrix X=UΣVT XV=UΣ
- 14. PCA IMPLEMENTATION 14/28 EXAMPLE Dot are spread out along a diagonal line and maximum variation of data is between the two end points of line
- 15. PCA IMPLEMENTATION 15/28 EXAMPLE Dot are spread out along a diagonal line and maximum variation of data is between the two end points of line Dots are also spread out a little above and below the first line and 2nd largest amount of variation is at the endpoints of the new line
- 17. PCA IMPLEMENTATION 17/28 EXAMPLE These two new axes that describe the variation in the data are “Principal Components”
- 26. USES OF PCA 26/28 PCA is mostly used as a tool for Compression and Simplifying data for easier learning in exploratory data analysis and for making predictive models. 1- Better Perspective and less Complexity 2 - Better visualization 3- Reduce size 4- Different perspective:
- 27. LIMITATION OF PCA 27/28 If the data does not follow a multidimensional normal (Gaussian) distribution, PCA may not give the best principal components
- 28. REFERENCES 28/28 Figure: pixel representation Information and Image Credit : • http://www.mit.edu/~gari/teaching/6.555/LECTURE_NOTES/ch28_bss.pdf • https://www.quora.com/What-are-some-of-the-limitations-of-principal-component-analysis • http://mengnote.blogspot.com.ee/2013/05/an-intuitive-explanation-of-pca.html • https://www.youtube.com/watch?v=_UVHneBUBW0&t=2s • http://setosa.io/ev/principal-component-analysis/
- 29. 29 THANKS