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

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

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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

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CS 171: Visualization

High-Dimensional Data

Hanspeter Pfister pfister@seas.harvard.edu

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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!)

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Th: Alberto Cairo

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

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High-Dimensional Data

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Item

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Attribute

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Taxonomy

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

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Linked Views

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Multivariate Plots

ggplot2

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Multivariate Plots

R

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Heatmap

ggplot2

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  • A. Lex

Hierarchical Heatmap

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3D Scatter Plots

R, lattice

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R, lattice

3D Continuous Plots

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Small Multiples

Tableau

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Small Multiples

Protovis

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Horizon Graphs

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19

Becker 1996

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D3

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EnRoute

  • A. Lex
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Parallel Coordinates

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Use more than two axes

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

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Parallel Coordinates

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Parallel Coordinates

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Parallel Coordinates

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Correlation

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

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Filtering & Brushing

D3

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Parallel Sets

D3

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StratomeX

  • A. Lex
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Glyphs

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Star Plots

  • Space variables around a circle
  • Encode values on “spokes”
  • Data point is now a shape
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  • C. Nussbaumer
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  • M. Kirby, H. Marmanis, and D. Laidlaw

Velocity (magnitude & direction) Vorticity (scalar, CW/CCW) Strain Tensor (second order) Turbulent Charge (vector & scalar)

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  • M. Kirby, H. Marmanis, and D. Laidlaw
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  • M. Kirby, H. Marmanis, and D. Laidlaw
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  • M. Kirby, H. Marmanis, and D. Laidlaw
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42

  • G. Kindlmann 2006
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  • G. Kindlmann 2006
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  • G. Kindlmann 2006
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Chernoff Faces

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Dimensionality Reduction

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What about very high- dimensional data?

Based on slide from P . Liang

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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

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Linear Methods

  • Does the data lie mostly in a hyperplane?
  • If so, what is its dimensionality?

Based on slide from F. Sha

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

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PCA

Project data to a subspace such as to maximize the variance of the projected data

Based on slide from J. Leskovec

PC vectors are orthogonal

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MusicBox [Anita Lillie]

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Variance and Covariance

  • Variance:
  • How far are data points spread?
  • Covariance:
  • How much do variables change together
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x1 x2 x1 1 x2 1

x2 x1

Covariance Matrix

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x1 x2 x1 1 0.7 x2 0.7 1

x2 x1

Covariance Matrix

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x1 x2 x1 1

  • ­‑0.7

x2

  • ­‑0.7

1

x2 x1

Covariance Matrix

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x2 x1

PCA

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x2 x1

PCA

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x2 x1

PCA

PC ¡1 PC ¡2

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How many PC vectors?

Enough PC vectors to cover 80-90% of the variance

Based on slide from J. Leskovec

Screeplot

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

PCA for Handwritten Digits

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

Eigenfaces

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PCA ¡VisualizaGon ¡

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

h"p://www.youtube.com/watch?v=7jLXDyQxck

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Gene Array Data

First two PC directions First three PC directions

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Text Documents

>45 features, projected onto two PC dimensions

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Multidimensional Scaling (MDS)

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  • A ¡different ¡goal ¡:

–Find ¡a ¡set ¡of ¡points ¡whose ¡pairwise ¡distances ¡match ¡a ¡ given ¡distance ¡matrix

p1 p2 p3 p4 p5 p1 1 2 3 1 p2 1 2 4 1 p3 2 2 1 3 p4 3 4 1 1 p5 1 1 3 1

p1 p4 p2 p3 p5

1 2 2 3 4 1 1 1

Multi-Dimensional Scaling

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European Cities Data

  • Distances between European cities:
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Result of MDS

Based on slide from T. Yang

! !

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Color Images

  • N. Bonneel
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Facebook Friends

– Distance ¡= ¡1 ¡for ¡friends – Distance ¡= ¡2 ¡for ¡friends ¡of ¡friends ¡; ¡etc.

  • N. Bonneel
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IN-SPIRE, PNNL

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What if data is non-linear?

  • Classic “Swiss Roll” example

PCA

xi

Based on slide from F. Sha

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Non-Linear Methods

  • Intuition: Distortion in local areas, but faithful in

the global structure

Based on slide from F. Sha

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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]