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Multivariate Data Plots
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?
Cross-sectional/correlation analysis misses the big picture!
The original 2D sample
• 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 nonconventional methods to present multidimensional
information for human visual analysis
Chernoff faces
Idea: We are good at reading facial emotions
Chernoff faces
Chernoff, H. (1973): The use of faces to represent statistical association, JASA, 68, pp. 361–368.
Chernoff faces
Example: Economy is going to recession
Kruskal’s
Multidimensional Scaling
Idea: Redraw a p-dimensional point set in
dimension q < p preserving as much pair-wise
distances as possible.
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,…,xN in q-dimensional
space that
• Kruskal stress is a measure of discrepancy between the true
and estimated similarities (distances)
Kruskal MDS results
for distance between
European cities (“eurodist”)
Actual map of Europe
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. USA 101, 7499-7504 (2004).
http://www-personal.umich.edu/~mejn/cart/
http://www-personal.umich.edu/~mejn/cartograms/
Good old physical map
World Population
Gross Domestic Product
Child Mortality
US Elections 2012
Mitt Romney
Barak Obama
US Elections 2012
by population
Mitt Romney
Barak Obama
US Elections 2012
Mitt Romney
Barak Obama
US Elections 2012
by population
Mitt Romney
Barak Obama
A Brief Summary
of Multivariate Techniques
Car marketing analysis
Comfort
Price
Car marketing analysis
Comfort
Very good (and probably unrealistic) situation
Price
Car marketing analysis
Comfort
Very bad (and very realistic) situation
Price
Car marketing analysis
Comfort
Very bad (and very realistic) situation:
A historic car
Price
Car marketing analysis
Comfort
Realistic (common) situations
Price
Car marketing analysis
Comfort
Price
Car marketing analysis
Comfort
Price
Car marketing analysis
Comfort
Good value!
Bad value!
Price