Ordering and Visualisation of Many-objective Populations David - - PowerPoint PPT Presentation

Ordering and Visualisation

• f Many-objective Populations

David Walker, Richard Everson and Jonathan Fieldsend

University of Exeter

19 July 2010

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 1 / 17

Introduction

Ordering Many-objective populations League tables are used to rank performance Often multiple key performance indicators measure a single item Information must be drawn from all of these KPIs to come up with a single measure of performance We construct a league table with multiple KPIs applying a single-objective ranking method to a many-objective data set — the Power Index Visualising Many-objective populations Ordering methods can be used to support visualisation We present

A new dominance/graph theory-based method A method to improve the comprehension of heatmaps

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 2 / 17

League Table Construction

An example — university league tables The Times Good University Guide 2009 8 KPIs Weighted sum approach Choosing weights can be difficult Many-objective Optimisation Concepts In MOEAs, dominance approaches replaced weighted sum methods The job of a university is to maximise their performance over all KPIs simultaneously Let y = (y1, . . . , yk), for k KPIs. 1 – Oxford 2 – Cambridge 3 – Imperial 4 – LSE 5 – St Andrews 6 – Warwick 7 – UCL 8 – Durham 9 – York 10 – King’s 11 – Bristol 12 – L’borough 13 – Exeter 14 – Leicester 15 – Bath

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 3 / 17

Pareto Sorting

Use dominance to locate the Pareto shells Provides a partial ordering

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 4 / 17

A Graphical Visualisation

Each university is represented by a node An edge between two universities means that the

• rigin university dominates

the destination university Universities are shown in their Pareto shells For clarity, edges between universities are only shown from one shell to the next

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 5 / 17

A Graphical Visualisation

Each university is represented by a node An edge between two universities means that the

• rigin university dominates

the destination university Universities are shown in their Pareto shells For clarity, edges between universities are only shown from one shell to the next

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Pareto

• ptimal

Shell 1 Shell 2 Shell 3 Shell 4 Shell 5 Shell 6 Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 5 / 17

Constructing the Graph

Probability of Dominance

p(yi ≻ yj) = α + β α — the fraction of objectives

• n which yi is better than yj.

β — half of the fraction of

• bjectives on which yi is

equal to yj. The probability that yi would beat yj in a tournament on a single randomly selected

• bjective

Adjacency matrix W Wij = p(yi ≻ yj)

20 40 60 80 100 20 40 60 80 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 6 / 17

Outflow (van den Brink and Gilles, 2009)

Consider the population as a graph, and rank the individuals according to the amount of outgoing traffic at each node

Outflow σout

i

σout

i

=

• j

Wij yi is ranked higher than yj if σout

yi

> σout

yj

Wij represents a transition from node i to node j. Equivalent to the Average Rank

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 8 / 17

Power Index

Look for the most influential individual Rank the individuals that dominate the most powerful individuals highest 1 – Oxford 2 – Imperial 3 – Cambridge 4 – UCL 5 – LSE 6 – Warwick 7 – St Andrews 8 – Durham 9 – King’s 10 – Bristol 11 – York 12 – N’ham 13 – Bath 14 – S’ton 15 – Edinburgh

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 9 / 17

Power Index

The power index is the limit of the sequence ut = Wut−1 u0 = 1 u1 = Wu0 (equivalent to the average rank and outflow, u1 = σout

i

) u2 = Wu1 (the average rank, with some proportion of the average rank of dominated universities) . . . u = limt→∞

ut

• i ut

i Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 10 / 17

Power Index

Blue universities have the most power and tend to define the next shell in terms

• f the universities they

dominate Red universities have the least power

Oxford (1) St Andrews (7) Warwick (6) Durham (8) York (11) Bristol (10) King's (9) Loughborough (25) Exeter (16) Leicester (17) Nottingham (12) Southampton (14) Edinburgh (15) Lancaster (18) Glasgow (19) Aberdeen (28) Manchester (21) Strathclyde (34) Cambridge (3) Imperial (2) LSE (5) UCL (4) SOAS (23) Sheffield (22) East Anglia (36) Cardiff (27) Reading (33) Liverpool (32) Kent (40) Sussex (37) Essex (42) Hull (49) Royal Holloway (35) Bradford (43) Bedfordshire (77) Abertay (90) Bath (13) Newcastle (20) Surrey (39) Keele (41) Birmingham (24) Aston (30) Queen's Belfast (26) Queen Mary (29) Dundee (38) Heriot-Watt (45) City (50) Robert Gordon (55) N'ham Trent (56) Bournemouth (58) Brighton (60) Napier (62) UWIC Cardiff (89) Stirling (48) Brunel (46) Ulster (52) B'ham City (59) Glamorgan (64) Hertfordshire (72) Roehampton (76) Leeds (31) Oxford Brookes (53) Staffordshire (70) Coventry (67) Aberystwyth (47) Bangor (54) Swansea (44) Goldsmiths (51) Portsmouth (63) Plymouth (57) Central Lancs (68) West England (71) Winchester (69) Glasgow Cal (73) Lampeter (79) Bath Spa (75) Northumbria (74)

• U. Arts (65)

S'field Hallam (78) De Montfort (82) Canterbury CC (85) Sunderland (86) Salford (84) Chester (91) Huddersfield (92) York St John (94) Manchester Met (99) Leeds Met (95) Anglia Ruskin (105) Bucks New (106) QM Edinburgh (80) Chichester (61) Gloucestershire (66) Derby (93) West Scotland (102) Edge Hill (101) Cumbria (96) Teesside (87) Middlesex (88) East London (98) Worcester (83) Northampton (81) Kingston (100) Soton Solent (110) Wolverhampton (108) London S Bank (112) Liverpool JM (103) Greenwich (107) Thames Valley (113) Westminster (97) Bolton (111) UWCN (109) Lincoln (104)

Pareto

• ptimal

Shell 1 Shell 2 Shell 3 Shell 4 Shell 5 Shell 6 Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 11 / 17

Summary — Power Index

We have demonstrated a method based on graph theory and dominance for visualising many-objectives We cast dominance in a probabilistic framework The power index has been applied to many-objective populations for the first time We have improved the information which can be understood from the dominance graph by colouring by the power index

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 12 / 17

Seriation of Heatmaps

1 2 3 4 5 6 7 8 KPIs Universities 15 30 45 60 75 90 105

Heatmaps are a useful way of visualising a many-objective population They can be unclear because of the arbitrary ordering of individuals and objectives We can order the population so that like individuals are close together in the permutation. The population can be seriated to

• rder individuals or objectives

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 13 / 17

Seriation of Objectives

Similarity between objectives: Amn = 1 − 1 N

N

• i=1

(rm

i

− rn

i )2

Distance summation: For a permutation πn of the nth objective, minimise g(π) =

• m,n

Amn(πm − πn)2

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 14 / 17

Seriation of Objectives

Relax π to a continuous variable z and minimise h(z) =

• m,n

Amn(zm − zn)2 subject to

n zn = 0 and n z2 n = 1

The solution is the Fiedler vector, the smallest non-zero eigenvector, of the graph Laplacian L of A L = D − A where D is a diagonal matrix, Dii = σout

i

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 15 / 17

Seriation of Objectives

1 2 3 4 5 6 7 8 Universities 1 2 3 4 5 6 7 8 KPIs 4 8 3 7 5 6 2 1 15 30 45 60 75 90 105

g(π) = 2.65 × 109 g(π) = 2.23 × 104

Walker et al. Ordering and Visualisation of Many-objective Populations 19 July 2010 16 / 17

Summary

Analysis of performance by key performance indicators can be considered in a multi-objective context Dominance coupled with graph theory provides an intuitive 2D representation A probabilistic interpretation of dominance allows us to represent the population as a graph The power index, which ranks a population based

• n the quality of individuals dominated, was

applied to a multi-objective population ranking problem Heatmap clarity can be improved by seriating

• U. Arts (65)

1 2 3 4 5 6 7 8 Universities 1 2 3 4 5 6 7 8 KPIs 4 8 3 7 5 6 2 1 15 30 45 60 75 90 105

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