Visualising Multi-objective Data: From League Tables to Optimisers, and back

Visualising Multi-objective Data:
From League Tables to Optimisers, and back
David Walker
College of Engineering, Mathematics and Physical Sciences
University of Exeter
D.J.Walker@exeter.ac.uk
8th March 2017 – University of Plymouth
David Walker Visualising Multi-objective Data 8th March 2017 1 / 17

League Tables
Times Good University Guide, 2009
8 KPIs – NSS, research quality, student-staﬀ ratio, services and
facilities spend, entry standards, completion, good honours, graduate
prospects
Uni NSS RAE
Student
staﬀ
ratio
£/
student
Entry
Reqs.
Compl-
etion
1/
2:1
Pros-
pects
Ox. 0.840 6.200 11.600 2884.000 502.000 98.600 90.100 83.900
Camb. - 6.500 12.200 2299.000 518.000 97.900 85.400 88.400
Imp. 0.760 5.800 10.400 3218.000 473.000 96.000 69.100 89.300
LSE 0.740 6.300 12.600 1562.000 469.000 96.900 75.200 87.700
Warw. 0.760 5.600 13.600 1881.000 448.000 96.700 79.400 74.900
UCL 0.760 5.500 9.100 1702.000 434.000 94.300 75.100 81.500
Dur. 0.780 5.200 15.400 1375.000 447.000 96.400 78.800 75.900
York 0.770 5.500 13.100 1313.000 423.000 95.200 74.700 70.500
Bristol 0.750 5.200 14.700 1535.000 430.000 95.800 78.400 81.500
King’s 0.770 4.700 11.900 1696.000 406.000 93.200 72.100 80.400

Visualisation
Visualisation is a useful alternative to presenting data in a table – human
beings are well suited to understanding information visually
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.20.40.60.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8

Visualisation
Visualisation is a useful alternative to presenting data in a table – human
beings are well suited to understanding information visually
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
0.20.40.60.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Unfortunately people can generally only think in three dimensions

High-dimensional Visualisation
−1.5 −1.0 −0.5 0.0 0.5 1.0
−1.0
−0.5
0.0
0.5
1.0
1.5
f1,f2 f1,f3 f1,f4 f1,f5
f2,f3 f2,f4 f2,f5
f3,f4 f3,f5
f4,f5
f1 f2 f3 f4 f5
0.0
0.5
1.0
1.5
2.0

Evolutionary Many-objective Optimisation
Evolutionary algorithms generate solutions to many-objective
optimisation problems – comprising M = 4 (or more) conﬂicting
objectives
The quality of a solution p is evaluated using a set of objective
functions:
y = (f1(p), . . . , fM(p))
Compare pairs of solutions using dominance:
f(p) f(q) ⇔ ∀m(fm(p) ≤ fm(q)) ∧ ∃m(fm(p) < fm(q))

Evolutionary Many-objective Optimisation
Evolutionary algorithms generate solutions to many-objective
optimisation problems – comprising M = 4 (or more) conﬂicting
objectives
The quality of a solution p is evaluated using a set of objective
functions:
y = (f1(p), . . . , fM(p))
Compare pairs of solutions using dominance:
f(p) f(q) ⇔ ∀m(fm(p) ≤ fm(q)) ∧ ∃m(fm(p) < fm(q))
Visualise individuals according to their dominance relationships

Pareto Shells
Non-dominated Sorting
1 Set k = 1
2 Identify all of the
non-dominated individuals
and assign them to shell k
3 Increment k
4 If individuals remain, return
to step 2
Shell 1
Shell 4
Oxford
St Andrews
Warwick
Durham
York
Bristol
King's
Leicester
Nottingham
Southampton
Edinburgh
Lancaster
Glasgow
Aberdeen
Manchester
Strathclyde
Cambridge
Imperial
LSE
UCL
SOAS
Sheffield
East Anglia
Cardiff
Reading
Liverpool
Kent
Sussex
Essex
Hull
Royal Holloway
Bradford
Bedfordshire
Abertay
Bath
Newcastle
Surrey
Keele
Birmingham
Aston
Queen's Belfast
Queen Mary
Dundee
Heriot-Watt
City
Robert Gordon
N'ham Trent
Bournemouth
Brighton
Napier
UWIC Cardiff
Stirling
Brunel
Ulster
B'ham City
Glamorgan
Hertfordshire
Roehampton
Leeds
Oxford Brookes
Staffordshire
Coventry
Aberystwyth
Bangor
Swansea
Goldsmiths
Construct a graph
Arrange individuals (nodes) into columns according to Pareto shell
Place edges between individuals in adjacent shells where one
dominates the other

University League Tables
Colour nodes according to
average rank
Rank the individuals m
times (once for each
KPI) giving rim – the
rank of individual i on
KPI m
Average these ranks
¯ri =
1
M
M
m=1
rim
Shell 1
Shell 2
Shell 3
Shell 4
Shell 5
Shell 6Oxford (1)
St Andrews (7)
Warwick (4)
Durham (9)
York (11)
Bristol (10)
King's (8)
Loughborough (24)
Exeter (17)
Leicester (16)
Nottingham (12)
Southampton (13)
Edinburgh (15)
Lancaster (21)
Glasgow (17)
Aberdeen (29)
Manchester (22)
Strathclyde (36)
Cambridge (6)
Imperial (2)
LSE (5)
UCL (3)
SOAS (27)
Sheffield (20)
East Anglia (35)
Cardiff (26)
Reading (34)
Liverpool (31)
Kent (37)
Sussex (38)
Essex (43)
Hull (49)
Royal Holloway (33)
Bradford (42)
Bedfordshire (91)
Abertay (99)
Bath (14)
Newcastle (19)
Surrey (39)
Keele (40)
Birmingham (23)
Aston (30)
Queen's Belfast (25)
Queen Mary (28)
Dundee (41)
Heriot-Watt (44)
City (50)
Robert Gordon (55)
N'ham Trent (56)
Bournemouth (59)
Brighton (58)
Napier (71)
UWIC Cardiff (86)
Stirling (47)
Brunel (46)
Ulster (52)
B'ham City (60)
Glamorgan (69)
Hertfordshire (70)
Roehampton (80)
Leeds (32)
Oxford Brookes (53)
Staffordshire (72)
Coventry (68)
Aberystwyth (48)
Bangor (54)
Swansea (45)
Goldsmiths (51)
Portsmouth (60)
Plymouth (57)
Central Lancs (64)
West England (63)
Winchester (65)
Glasgow Cal (66)
Lampeter (81)
Bath Spa (75)
Northumbria (67)
U. Arts (72)
S'field Hallam (74)
De Montfort (78)
Canterbury CC (82)
Sunderland (84)
Salford (77)
Chester (87)
Huddersfield (89)
York St John (95)
Manchester Met (89)
Leeds Met (93)
Anglia Ruskin (103)
Bucks New (105)
QM Edinburgh (79)
Chichester (76)
Gloucestershire (62)
Derby (89)
West Scotland (100)
Edge Hill (106)
Cumbria (101)
Teesside (91)
Middlesex (98)
East London (104)
Worcester (85)
Northampton (83)
Kingston (94)
Soton Solent (110)
Wolverhampton (109)
London S Bank (112)
Liverpool JM (96)
Greenwich (108)
Thames Valley (113)
Westminster (97)
Bolton (111)
UWCN (107)
Lincoln (102)
D. Walker, R. Everson and J. Fieldsend, Visualisation and Ordering of Many-objective Populations. In Proc. IEEE Congress on
Evolutionary Computation (CEC 2010), pp3664–3671, 2010.

Water Quality Indicators
D. Walker, D. Jakovljevicć, D. Savić and M. Radovanović,
Multi-criterion Water Quality Analysis of the Danube River
in Serbia: A Visualisation Approach. Water Research 79
(158–172), 2015.

Heatmaps
A heatmap is a graphical
representation of a dataset –
rows indicate individuals and
columns indicate KPIs
“Warm” colours indicate large
values
“Cool” colours indicate small
values
1 2 3 4 5 6 7 8
Criteria
0
20
40
60
80
100
Individuals 15
30
45
60
75
90
105

Seriation of Heatmaps
Reorder the rows of the heatmap so that similar individuals are placed
together and patterns can be identiﬁed
Seriation is a procedure for permuting items based on their similarity
Aij = 1 −
1
M(N − 1)2
M
m=1
(rim − rjm)2
g(π) =
N
i=1
N
j=1
Aij (πi − πj )2
D. Walker, R. Everson and J. Fieldsend, Visualisation Mutually Non-dominating Solution Sets in Many-objective Optimisation.
In IEEE Transactions on Evolutionary Computation 17(2)165–184, 2013.

0 20 40 60 80 100
0
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

0 20 40 60 80 100
0
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
0 20 40 60 80 100
0
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

Seriation of Heatmaps: University League Tables
1 2 3 4 5 6 7 8
Criteria
0
20
40
60
80
100
Individuals
15
30
45
60
75
90
105

Seriation of Heatmaps: University League Tables
1 2 3 4 5 6 7 8
Criteria
0
20
40
60
80
100
Individuals
15
30
45
60
75
90
105
1 2 3 4 5 6 7 8
Criteria
72
68
49
5
53
9
Individuals
15
30
45
60
75
90
105

Seriation of Heatmaps: Radar Waveform Design
Seriate according to individuals then KPIs to reveal further
information
1 2 3 4 5 6 7 8 9
Criteria
0
20
40
60
80
100
120
140
160
180
Individuals
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9
Criteria
77
28
142
185
104
114
147
65
76
32
Individuals
20
40
60
80
100
120
140
160
180
200

Seriation of Heatmaps: Radar Waveform Design
Seriate according to individuals then KPIs to reveal further
information
1 2 3 4 5 6 7 8 9
Criteria
0
20
40
60
80
100
120
140
160
180
Individuals
20
40
60
80
100
120
140
160
180
200
1 2 3 4 5 6 7 8 9
Criteria
77
28
142
185
104
114
147
65
76
32
Individuals
20
40
60
80
100
120
140
160
180
200
4 9 2 8 6 5 7 1 3
Criteria
77
28
142
185
104
114
147
65
76
32
Individuals
20
40
60
80
100
120
140
160
180
200

Treemaps
Visualise data represented as a tree
using space to illustrate the importance
of a node
Additional degrees of freedom (e.g.,
colour)
Many diﬀerent algorithms for arranging
a treemap
Classiﬁcation of the top 100
websites visited in 2010
(UK, France, Germany,
Italy, Spain, Switzerland,
Brazil, US and Australia)

Dominance trees
Step 1: Pareto sorting
Construct a partial ordering of individuals using Pareto sorting – this
results in a graph
Set 2: Prune edges using dominance distance
Remove edges such that each node has exactly one parent node (retain
the parent with the smallest dominance distance) and insert an artiﬁcial
“root” using the global best
A
B
C
D
E
F
D. Walker, Visualising Multi-objective Populations with Treemaps. In Proc. Genetic and Evolutionary Computation Conference
(GECCO 2015) Companion Volume, 963–970, 2015.

Dominance trees
Step 1: Pareto sorting
Construct a partial ordering of individuals using Pareto sorting – this
results in a graph
Set 2: Prune edges using dominance distance
Remove edges such that each node has exactly one parent node (retain
the parent with the smallest dominance distance) and insert an artiﬁcial
“root” using the global best
A
B
C
D
E
F
nr
A
B
C
D
E
F
D. Walker, Visualising Multi-objective Populations with Treemaps. In Proc. Genetic and Evolutionary Computation Conference
(GECCO 2015) Companion Volume, 963–970, 2015.

Circular Treemaps
Good University Guide
Oxford
SOAS
Water Quality Analysis

Summary
Performance data
Performance data is ubiquitous
University league tables, hospital performance, quality of life, water
quality, optimisation. . .
By visualising it we can better understand and make use of this data
Visualisation Methods
Pareto shells
Seriation
Treemaps

Visualising Multi-objective Data: From League Tables to Optimisers, and back

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