Graphics: Effect Ordering Packages: seriation, gclus, corrgram - - PowerPoint PPT Presentation

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E ULERIAN TOUR ALGORITHMS FOR DATA VISUALIZATION AND THE P AIR V IZ PACKAGE Catherine Hurley R.W. Oldford NUI Maynooth U. Waterloo July 8 2009 UseR! Monday 13 July 2009 Graphics: Effect Ordering Packages: seriation, gclus, corrgram

SLIDE 1

EULERIAN TOUR ALGORITHMS

FOR DATA VISUALIZATION AND THE PAIRVIZ PACKAGE

Catherine Hurley NUI Maynooth R.W. Oldford

U. Waterloo

July 8 2009 UseR!

Monday 13 July 2009

SLIDE 2

Graphics: Effect Ordering

Packages: seriation, gclus, corrgram
Example: PCP Flea data

Standard order

Tars1 Tars2 Aede1 Aede2 Head Aede3

Correlation order

Tars2 Aede1 Aede2 Aede3 Tars1 Head

0.2

Monday 13 July 2009

SLIDE 3

Pairviz: relationship ordering

Statistical graphics are about comparisons

between variables, cases, groups, models

Aede3 Aede2 Aede1 Tars2 Aede2 Aede1 Aede3 Tars2 Tars1 Head Tars2 Tars1 Aede1 Head Aede2 Tars1 Aede3 Head

0.0 0.6

Flea data: correlation order

Monday 13 July 2009

SLIDE 4

A graph model

Build a graph where nodes are statistical objects
Edges are relationships
Example:

Node Vis Edge Vis Group Boxplot Two groups CI for mean diff Var Hist Two vars Scatterplot 2 vars Scat 4-d space Dynamic scat Model Resid 2 Models PCP

A B C D E F

Monday 13 July 2009

SLIDE 5

Example: planned comparisons

Mice in 5 diet groups, response is lifetime Nodes are treatments, edges are planned comparisons Weights are p-values

0.0083 0.0147 0.3111 N/N85 N/R40 N/R50 NP R/R50 lopro

N/R50 N/N85 NP lopro N/R50 N/R40 R/R50 N/R50 10 20 30 40 50

Planned comparisons of diets

Lifetime

5 10

Differences

Reducing calories and protein increases lifetime

Monday 13 July 2009

SLIDE 6

Graph Traversal

Traverse all nodes: hamiltonian path
Traverse all edges: eulerian path
Use gclus, seriation: hamiltonian paths on complete graphs
PairViz: eulerian paths

A B C D E F G H A B C D E F G H

Open hamiltonian path Closed hamiltonian path Closed eulerian path on K7

A B C D E F G

Monday 13 July 2009

SLIDE 7

Graph Structures

Complete graph: all

comparisons are interesting

Edge-weighted graphs: low

weight edges are more interesting

Bipartite graph

eg only treatment-control comparisons are of interest

Aede3 Aede2 Aede1 Tars2 Aede2 Aede1 Aede3 Tars2 Tars1 Head Tars2 Tars1 Aede1 Head Aede2 Tars1 Aede3 Head

0.0 0.6

Weight edges by 1-corr, eulerian follows low weight edges

X1 X2 X3 Y1 Y2

Monday 13 July 2009

SLIDE 8

Hypercube graph
r model selection:

Each node in G is a predictor subset edge: add/drop predictor

Graph Structures- cont’d

Line graph

transform G to L(G)

eg Each node in G is a var, each node in L(G) is var pair, edge is 3-d transition

Cube for factorial experiment

000 001 010 011 100 101 110 111

A B C D

AB AC AD BC BD CD

Monday 13 July 2009

SLIDE 9

Algorithms- Complete graph

Closed eulerian path exists when each node has odd number of vertices: ie for K2n+1
Hamiltonian decomposition of graph
into hamiltonian cycles: eulerian for K2n+1
into hamiltonian paths: approx eulerian for K2n
classical algorithm: hpaths
WHam: weighted_hpaths: pick best for H1, best orientaton and order for others.

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Monday 13 July 2009

SLIDE 10

Algorithms-Complete graph cont’d

Recursive algorithm: eseq:
Start with eulerian on Kn, append edges to get eulerian on Kn+2

1 2 3 4 5 6 7 Monday 13 July 2009

SLIDE 11

Algorithms- general

Eulerian graph: connected, all nodes have even number of edges
Otherwise, add edges, pairing up odd nodes
Classical algorithm (Hierholzer, Fleury)
Our version GrEul, (etour) follows weight increasing edges

Chinese postman does this in optimal way

0.0083 0.0147 0.3111 N/N85 N/R40 N/R50 NP R/R50 lopro

Monday 13 July 2009

SLIDE 12

Algorithms comparison

Complete-no weights

5 10 15 20 25 30 35 2 4 6 8

Etour 9

5 10 15 20 25 30 35 2 4 6 8

Eseq 9

5 10 15 20 25 30 35 2 4 6 8

hpaths 9

prefers low vertices prefers low edges 4 hamiltonians

Monday 13 July 2009

SLIDE 13

Algorithms: complete, weighted

50 100 150 200 1000 2000 3000 4000

Algorithm eseq: Eurodist edge weights

50 100 150 200 1000 2000 3000 4000

Weighted etour on Eurodist

50 100 150 200 1000 2000 3000 4000 Weighted hamiltonians on Eurodist 1 2 3 4 5 6 7 8 9 10

ignores weights Starts in Geneva

hamiltonian decomp, with increasing path lengths

Eurodist: 21 European cities

Monday 13 July 2009

SLIDE 14

Example: model selection

Mammal sleep data Y= log brain wt. Predictors A= non dreaming sleep, B=dreaming sleep, C=log body wt, D=life span

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

Hypercube graph represents possible moves in a

stepwise regression algorithm

Graph Qn is hamiltonian, and eulerian for even n
Edge weights: change in SSE
Eulerian starting with full model
All models with C are good
Bar chart: change in SSE

Sleep data: Model residuals.

ABCD BCD CD ACD ABCD ABC BC C AC ABC AB A AD ABD BD D AD ACD AC A D CD C B BD BCD BC B AB ABD ABCD

Monday 13 July 2009

SLIDE 15

More variables

Sleep data: 10 vars (nodes) 45 edges Eulerian has length 50

Eulerian on scagnostics: Outlying

GP Bd L Br Bd SW PS TS SE PS TS D L P L PS Br P TS Bd TS PS P D D Br P D 0.0 0.3 0.6

Using outlying index from scagnostics package for eulerian traversal zoom on first half of display

Monday 13 July 2009

SLIDE 16

More variables-cont’d

Reduce the graph NN graph: eliminate edges with outlier index < .2 Reduces graph from 10 to 5 nodes, and 45 to 5 edges Other nodes have no edges

NN Eulerian on scagnostics: Outlying

GP L Bd SW L Br GP 0.0 0.3 0.6

SW Bd Br L GP Monday 13 July 2009

SLIDE 17

IN CONCLUSION..

Pairviz package: relationship ordering for data visualisation
Current version: algorithms presented here
Thanks to graph, igraph
Work in progress: ordering dynamic visualisations via ggobi.

with Adrian Waddell, UW

Monday 13 July 2009