CS 171: Visualization High-Dimensional Data Hanspeter Pfister - PowerPoint PPT Presentation

CS 171: Visualization High-Dimensional Data Hanspeter Pfister pfister@seas.harvard.edu

This Week • PII due Monday, April 8 • Friday lab (10-11:30 am, MD G115): Hands-On D3 with Azalea, Sofia, and Billy (our last lab!)

Th: Alberto Cairo A Functional Art: Storytelling with Data, Graphs, Maps, and Diagrams

High-Dimensional Data

Item

Attribute

Taxonomy • Based on number of attributes • 1: Univariate • 2: Bivariate • 3: Trivariate • >3: Multivariate

Tableau

Linked Views

Multivariate Plots ggplot2

Multivariate Plots R

Heatmap ggplot2

Hierarchical Heatmap A. Lex

3D Scatter Plots R, lattice

3D Continuous Plots R, lattice

Small Multiples Tableau

Small Multiples Protovis

Horizon Graphs

Becker 1996 19

D3

EnRoute A. Lex

Parallel Coordinates

Use more than two axes “Hyperdimensional Data Analysis Using Parallel Coordinates”, Wegman, 1990 Based on slide from Munzner

Parallel Coordinates

Parallel Coordinates

Parallel Coordinates

Correlation “Hyperdimensional Data Analysis Using Parallel Coordinates”, Wegman, 1990 Based on slide from Munzner

Filtering & Brushing D3

Parallel Sets D3

StratomeX A. Lex

Glyphs

Star Plots • Space variables around a circle • Encode values on “spokes” • Data point is now a shape

C. Nussbaumer

Velocity Vorticity (magnitude & (scalar, CW/CCW) direction) Turbulent Charge (vector & scalar) Strain Tensor (second order) M. Kirby, H. Marmanis, and D. Laidlaw

M. Kirby, H. Marmanis, and D. Laidlaw

M. Kirby, H. Marmanis, and D. Laidlaw

M. Kirby, H. Marmanis, and D. Laidlaw

42 G. Kindlmann 2006

G. Kindlmann 2006

G. Kindlmann 2006

Chernoff Faces

Dimensionality Reduction

What about very high- dimensional data? Based on slide from P . Liang

Basic Idea Project the high-dimensional data onto a lower- dimensional subspace using linear or non-linear transformations y 2 < 10 x 2 < 64 × 64 = < 4096 y = Ux Based on slide from P . Liang

Linear Methods • Does the data lie mostly in a hyperplane? • If so, what is its dimensionality? Based on slide from F. Sha

h"p://www.youtube.com/watch?v=4pnQd6jnCWk

PCA Project data to a subspace such as to maximize the variance of the projected data PC vectors are orthogonal Based on slide from J. Leskovec

MusicBox [Anita Lillie]

Variance and Covariance • Variance: • How far are data points spread? • Covariance: • How much do variables change together

Covariance Matrix x1 x1 x2 x1 1 0 x2 0 1 x2

Covariance Matrix x1 x1 x2 x1 1 0.7 x2 0.7 1 x2

Covariance Matrix x1 x1 x2 x1 1 -­‑0.7 x2 -­‑0.7 1 x2

PCA x1 x2

PCA x1 x2

PCA x1 PC ¡1 x2 PC ¡2

How many PC vectors? Enough PC vectors to cover 80-90% of the variance Screeplot Based on slide from J. Leskovec

PCA for Handwritten Digits HasGe ¡et ¡al.,”The ¡Elements ¡of ¡StaGsGcal ¡Learning: ¡Data ¡Mining, ¡Inference, ¡and ¡PredicGon”, ¡Springer ¡(2009)

Eigenfaces Gunnar ¡Grimnes: h"p:// www.flickr.com/ photos/gromgull/ 3329844591/in/ photostream/

PCA ¡VisualizaGon ¡ • Mondrian ¡painGngs • h"p://www.youtube.com/watch?v=xiWpZ5jhvx4 h"p://www.youtube.com/watch?v=7jLXDyQxck

Gene Array Data First two PC First three PC directions directions

Text Documents >45 features, projected onto two PC dimensions

Multidimensional Scaling (MDS)

Multi-Dimensional Scaling • A ¡different ¡goal ¡: – Find ¡a ¡set ¡of ¡points ¡whose ¡pairwise ¡distances ¡match ¡a ¡ given ¡distance ¡matrix p5 p1 p2 p3 p4 p5 1 1 p1 0 1 2 3 1 1 p1 p2 p2 1 0 2 4 1 2 p3 2 2 0 1 3 2 3 p4 3 4 1 0 1 4 p3 p5 1 1 3 1 0 1 p4

European Cities Data • Distances between European cities:

Result of MDS ! ! Based on slide from T. Yang

Color Images N. Bonneel

Facebook Friends – Distance ¡= ¡1 ¡for ¡friends – Distance ¡= ¡2 ¡for ¡friends ¡of ¡friends ¡; ¡etc. N. Bonneel

IN-SPIRE, PNNL

What if data is non-linear? • Classic “Swiss Roll” example PCA x i Based on slide from F. Sha

Non-Linear Methods • Intuition: Distortion in local areas, but faithful in the global structure Based on slide from F. Sha

Dimensionality Reduction • Linear methods: • Principal Component Analysis (PCA) – Hotelling[33] • Multidimensional Scaling (MDS) – Young[38] • Nonnegative Matrix Factorization (NMF) – Lee[99] • Nonlinear methods: • Locally Linear Embeddings (LLE) – Roweis[00] • IsoMap – Tenenbaum[00]