12th presentation @ Papers We Love - Bucharest Chapter on 16th December, 2016

1. Principal Component Analysis
– the original paper
ADRIAN FLOREA
PAPERS WE LOVE – BUCHAREST CHAPTER – MEETUP #12
DECEMBER 16TH, 2016

2. Karl Pearson, 1901
K. Pearson, "On lines and planes of closest fit to systems of points in space",
The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science,
Sixth Series, 2, pp. 559-572 (1901)
(1857-1936)

3. Dimensionality reduction (Pearson’s words)
(1901)

4. Dimensionality reduction (115 years later)
(2016)
71291 points
200 features
2 principal components
https://projector.tensorflow.org

5. Dimensionality reduction
n
d
n
r
• n observations
• d possibly correlated features
• r << d
• n observations
• r uncorrelated features
• retain most of the data variability
transform
Original data set
New data set

6. Introduction
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7. Goal: find an “optimal” basis
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9. First principal component
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The largest eigenvalue is the projected variance on the associated eigenvector that is the direction of most variance,
called the first principal component.

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12. An equivalent objective function
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13. r=2
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14. PCA algorithm
1. Normalize to zero-mean, unit-variance
2. Calculate the covariance matrix of the normalized data set
3. Find all eigenvalues of and arrange them in descending order
4. Choose r for the top r eigenvalues (and corresponding eigenvectors/principal components)
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15. How to choose r?
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16. PCA Pros & Cons
PROS
• dimensionality reduction can help learning
• removes noise
• can deal with large data sets
CONS
• hard to interpret
• sample dependent
• linear (

17. Bibliography
• A.A. Farag, S. Elhabian, "A Tutorial on Principal Component Analysis" (2009) -
http://dai.fmph.uniba.sk/courses/ml/sl/PCA.pdf
• N. de Freitas, "CPSC 340 - Machine Learning and Data Mining Lectures", University of British Columbia (2012) -
http://www.cs.ubc.ca/~nando/340-2012/lectures.php
• I. Georgescu, “Inteligență computațională“, Editura ASE (2015)
• I.T. Jolliffe, "Principal Component Analysis", 2nd ed., Springer (2002)
• J.N. Kutz, "Data-Driven Modeling & Scientific Computation. Methods for Complex Systems & Big Data", Oxford
University Press (2013)
•J. N. Kutz, "Singular Value Decomposition" - http://courses.washington.edu/am301/page1/page15/video.html
• K. Pearson, "On lines and planes of closest to systems of points in space", Philos. Mag, Sixth Series, 2, pp. 559-572
(1901) - http://stat.smmu.edu.cn/history/pearson1901.pdf
• M.J. Zaki, W. Meira Jr., “Data Mining and Analysis. Fundamental Concepts and Algorithms”, Cambridge University
Press (2014)