1. Multivariate Data Plots

2. Example of conventional analysis of multivariate data Example: A 2D sample of 100 observations is illustrated here using the two 1D cross-sectional histograms. Corr(x,y) = 0.04. Question: Can you guess the shape of the original sample?

3. Cross-sectional/correlation analysis misses the big picture! The original 2D sample

4. Analysis of marginal distributions has limited power • Nothing is better that a human eye • One wishes to “see” the whole picture, but we live in 3D world while the realistic data sets are coming from multi-dimensional spaces • Special visual methods have been developed to visualize multidimensional sets • We proceed with a brief review of different non-conventional methods to present multidimensional information for human visual analysis

5. Chernoff faces Idea: We are good at reading facial emotions

6. Chernoff faces Chernoff, H. (1973): The use of faces to represent statistical association, JASA, 68, pp. 361–368.

7. Chernoff faces Example: Economy is going to recession

8. Kruskal’s Multidimensional Scaling Idea: Redraw a p-dimensional point set in dimension q < p preserving as much pair-wise distances as possible.

9. Kruskal’s Multidimensional Scaling • Consider a set of N objects (points ) in p-dimensional space • Matrix of similarities (or distances) is given by • The goal of MDS is to find such points x1,…,xNin q-dimensional space that • Kruskal stress is a measure of discrepancy between the true and estimated similarities (distances)

10. Kruskal MDS results for distance between European cities (“eurodist”)

11. Actual map of Europe

12. Cartograms Idea: Draw a map with areas corresponding to the quantity of interest, NOT physical area. Michael T. Gastner and M. E. J. Newman, Proc. Natl. Acad. Sci. USA101, 7499-7504 (2004). http://www-personal.umich.edu/~mejn/cart/ http://www-personal.umich.edu/~mejn/cartograms/

13. Good old physical map

14. World Population

15. Gross Domestic Product

16. Child Mortality

17. US Elections 2012 Mitt Romney Barak Obama

18. US Elections 2012 by population Mitt Romney Barak Obama

19. US Elections 2012 Mitt Romney Barak Obama

20. US Elections 2012 by population Mitt Romney Barak Obama

21. A Brief Summary of Multivariate Techniques

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