An Adaptation of the GAIA Visualization Method for
Cartography
Using the HSV color system for the representation and comparison of multicriteria profiles
Karim LIDOUH, Yves DE SMET, Esteban ZIMÁNYI
Computer and Decision Engineering (CoDE) Department
Université Libre de Bruxelles (ULB)
Brussels, Belgium
[email protected]
Abstract—Dimensionality reduction has always been important
within disciplines that focus on visual representation of
multivariate information. In the case of cartography this has
been achieved several times by the use of statistics charts and
diagrams, but these are limited in the number of components that
can be combined in a single glyph. On the other hand,
Multicriteria Decision Aid (MCDA) has developed tools to
visually represent multidimensional information, yet these cannot
be directly applied on geographical maps.
In this paper we present a way to adapt the GAIA visualization
tool to the representation and comparison of multicriteria
profiles on maps. The process involves the use of a Principal
Component Analysis (PCA) and the conversion of its result by
applying a HSV color chart. We illustrate this process by
applying it to two case studies: an evaluation of the Human
Development Index (HDI) and of the Environmental
Performance Index (EPI) of the European countries.
Visualization; MCDA; PROMETHEE; GAIA; Cartography;
Dimensionality reduction; Color models
I.
INTRODUCTION
One of the main difficulties in Multicriteria Decision Aid
(MCDA) consists in the geometrical or graphical representation
of its intermediate and final results. Since several years, visual
MCDA has started gaining interest because of the ease with
which the decision makers can interact or understand the
workings of the methods [1, 2]. This trend also present in other
disciplines has led to several dimensionality techniques being
tested and compared [3, 4]. Within the PROMETHEE
methodology, the Principal Component Analysis (PCA) has led
to the development of the GAIA visualization method [5], a
complementary tool of the PROMETHEE methodology [6, 7].
In our previous attempts at integrating MCDA
methodologies with Geographical Information Systems (GIS),
we considered converting the information from the GAIA
visualization into a glyph in order to represent it on
geographical maps [8]. Even though MCDA researchers were
very receptive to this idea, it did not get the same
acknowledgement from geographers who found the tool to be
too complex to interpret. This paper presents a new visual tool
designed to avoid the shortcomings of the previous one and that
allows the comparison of alternative profiles on geographical
maps. This tool makes use of the HSV color model as defined
by Joblove et al. [9] in order to convert geometrical coordinates
into colors.
Section 2 of this paper presents the basics of the
PROMETHEE method that will be used and adapted, while
Section 3 describes the HSV color model and the way the
visual tool is designed. Section 4 then illustrates the tool by
applying it to two case studies. Finally, Section 5 concludes the
paper and presents future perspectives for this concept.
II.
PROMETHEE-GAIA METHODOLOGY
A. The PROMETHEE II ranking method
The PROMETHEE II method allows a decision maker to
rank alternatives based on several criteria. It is based on three
global steps, which are (1) the establishment of an evaluation
table for the alternatives, (2) the computation of preference
degrees for each pair of alternative for each criterion, and (3)
the computation of a net flow score which represents the global
value of each alternative.
The evaluation table will contain rows for all the
alternatives and columns for all the criteria. The elements fj(ai)
will represent the evaluation of alternative i according to
criterion j. We will suppose without loss of generality that
these criteria have to be maximized.
Using these evaluations we compute differences between
each pair of alternatives.
d j (a, b) = f j (a ) − f j (b)
(1)
These differences are then converted into preference
degrees by applying a non-decreasing preference function to
them. These preference degrees only take values between 0 and
1: 0 indicating no preference and 1 indicating a strong
preference for the first alternative compared to the second.
Pj (a, b) = Pj [d j (a, b)]
(2)
Finally, one computes an aggregated score as follows:
k
1
φ (a) =
w j [ Pj (a, x) − Pj ( x, a)]
∑
∑
n − 1 x∈A j =1
interested reader to Brans et al. [5] for a detailed description of
the GAIA tool.
III.
(3)
ADAPTATION TO CARTOGRAPHY
In order to keep as much information as possible from the
GAIA plane we first thought of representing it for each
alternative individually using a glyph called “decision clock”
[8]. An example of such representation is given in Fig. 2.
This net flow whose value is comprised between –1 and +1
can then be used to rank the alternatives from best to worst.
B. The GAIA visualization method
Using the preference degrees defined previously, one can
compute unicriterion net flows for each alternative and each
criterion as
φ j (a) =
1
∑ [ Pj (a, x) − Pj ( x, a)]
n − 1 x∈A
(4)
These describe the way all the alternatives are ranked
according to each individual criterion [5]. By applying a
Principal Component Analysis on them we obtain a
representation as in Fig. 1, called the GAIA plane.
Figure 2. Decision clocks used to represent GAIA information on a map
This attempt however was not approved unanimously by
the members of the several communities involved. In
particular, geographers found the tool to be too complex to be
interpreted easily. Furthermore, this representation made the
pairwize comparison of alternatives difficult. We therefore
designed a new tool while taking into account all the remarks
that were made. The result is a concept based on several rules
of practice in cartography or graphical representations [10, 11].
Figure 1. GAIA plane
In this plane, the axes represent the criteria and the dots
represent the alternatives whose coordinates in the criteria
space are given by the unicriterion net flows. The additional
axis π , called the decision axis, is the representation of the
normalized weights vector and gives the direction of the best
compromise solution.
A. The HSV color model
To represent the positions of the alternatives on a
geographical map, and thereby indicate the type of profile of
the alternatives, we will make use of the HSV color model [9,
12]. This model is used to represent the coordinates of color
points under the form of a solid cylinder. It stands for “hue”,
“saturation”, and “value”, which are the three coordinates used
to define all colors as can be seen in Fig. 3.
The GAIA plane can be used to identify the best or worst
alternatives. Indeed, good alternatives will be located in the
direction of the decision axis (e.g. a4 and a3 on Fig. 1) and bad
alternatives will be in the opposite direction (e.g. a5 on Fig. 1).
The position of the alternatives also gives us an insight on
the type of profile we are dealing with. An alternative that is
positioned in the direction of a given criterion will have a
strong evaluation on that particular criterion.
Finally the relative position of the alternatives shows us
which ones are similar. This representation is however
imperfect as there can be a loss of information. We refer the
Figure 3. HSV color cylinder [13]
This model presents the advantage of being more intuitive
for human interaction [14]. This characteristic will thus allow
the decision maker to compare the profiles more easily and to
quickly identify their position in the GAIA referential.
B. Converting the GAIA plane into glyphs
By forcing the value component to always have its
maximum value, we reduce the cylinder to a 2-dimensional
circle which we will use to represent the GAIA plane. The
positions of the alternatives will thus be represented by the
angle at which they are positioned (i.e. the hue) and their
distance from the center of the plane (i.e. the saturation). The
corresponding color chart is given in Fig. 4.
By superposing the GAIA plane on the color chart given in
Fig. 4, we can thus identify the color associated with each
A. The Human Development Index in European countries
This index uses four criteria:
•
Life expectancy at birth (in years)
•
Adult literacy rate (in %)
•
Combined gross enrolment ratio in education (in %)
•
GDP per capita (in purchasing power parity US$)
While using the same weights as for the HDI, we will apply
the PROMETHEE method on this set of criteria. The
preference functions used will be semi-linear functions with 0
and the largest difference as the two thresholds. This allows
every difference to be taken into account in our calculations.
This model is rather simple as it wasn’t the scope of this paper.
The result we obtain is a ranking very similar to the HDI
ranking, the only differences being due to the different nature
of the aggregation procedure. The objective of this paper is
however not to cover these differences [15]. The obtained
scores and ranks are given in Table 1.
TABLE I.
Figure 4. HSV color chart used to convert the positions into colors
alternative. Alternatives that are close on the GAIA plane, and
therefore have a similar profile, will end up having a similar
color.
Once these colors have been chosen for each alternative,
they will have to be displayed on the geographical map. But
instead of using it to color entire areas, we chose to display the
colors within circles of variable diameters (within given
thresholds). This allowed us to add another piece of
information to the representation which is the net flow obtained
with PROMETHEE II. A big diameter will correspond to a
high net flow value and a small one to a low value. The result
is a symbol that adequately combines a 2-dimensional result
with the global score of the analysis.
IV.
ILLUSTRATIONS
This section presents two case studies on which we have
applied our representation tool. Both cases use the 27 member
states of the European Union (EU). They study two wellknown indices:
•
The Human Development Index (HDI) as defined and
computed on the UNDP website (http://hdr.undp.org).
•
The Environmental Performance Index (EPI) as
presented by the Yale Center for Environmental Law
& Policy (http://epi.yale.edu).
A demo application with the data of these cases will be
available online on the website: http://mcda-gis.ulb.ac.be
Austria
Belgium
Bulgaria
Cyprus
Czech Republic
Denmark
Estonia
Finland
France
Germany
Greece
Hungary
Ireland
Italy
Latvia
Lithuania
Luxembourg
Malta
Netherlands
Poland
Portugal
Romania
Slovakia
Slovenia
Spain
Sweden
United Kingdom
HDI
0,9506
0,9484
0,8339
0,9121
0,8972
0,9537
0,8727
0,9557
0,9547
0,9401
0,9487
0,8766
0,9607
0,9444
0,8650
0,8706
0,9927
0,8939
0,9582
0,8751
0,8999
0,8257
0,8723
0,9247
0,9490
0,9586
0,9423
RESULTS FOR THE HDI CASE STUDY
Net flow
0,2997
0,1798
-0,5368
-0,2407
-0,1943
0,1178
-0,1684
0,2420
0,3867
0,1847
0,0198
-0,3668
0,1872
0,1818
-0,1624
-0,1301
0,3874
-0,2002
0,3631
-0,1373
-0,3634
-0,6640
-0,2961
0,0880
0,2266
0,4279
0,1680
HDI rank
8
11
26
16
18
7
22
5
6
14
10
20
2
12
25
24
1
19
4
21
17
27
23
15
9
3
13
Flow rank
5
11
26
22
20
13
19
6
3
9
15
25
8
10
18
16
2
21
4
17
24
27
23
14
7
1
12
The resulting GAIA plane keeps most of the information
since there are only four criteria. The delta value (i.e. the
percentage of information kept) is equal to 84%.
Our aim in applying these two methods is not only to give a
ranking to the decision maker, but also to justify this ranking
by allowing comparisons of the alternatives, identification of
similar profiles, and detection of spatial correlations if any.
Fig. 5 gives the GAIA plane for this case, which will be
converted and displayed on the geographical map in Fig. 6.
In this representation, we have chosen to rotate the
referential in order to give a particular meaning to the colors. In
this case, the color green indicates the best alternatives as it lies
in the direction of the decision axis. This allows us to verify
that countries with well-positioned profiles obtain a high score
(identifiable by the bigger size of their circle):
•
Sweden, France, Netherlands, Luxembourg, Finland,
Ireland, Denmark, Belgium and Spain all obtain a
green color and a circle with reasonable size.
•
Some countries such as Italy, Germany, and Spain
obtain a blue color but still have a high position in the
ranking.
•
The lowest positions are taken by the countries in
purple and pink such as Romania, Bulgaria, and
Hungary whose symbols have the smallest sizes.
We see that the profile indeed confirms the rank that is
obtained by the countries. As seen in the previous examples,
there are color clusters of similar profiles on the map. These
clusters not only share similar profiles but are also close to
each other geographically. This type of result is typical of
Figure 5. GAIA plane for the HDI case study
spatial decision problems where entities close in space usually
display a similar behavior.
Figure 6. Geographical view for the HDI case study
B. The Environmental Performance Index in European
countries
For this second case study we chose an index with a higher
number of criteria with the purpose of triggering a significant
loss of data in the process of applying the Principal Component
Analysis. The EPI uses an entire hierarchy of criteria with as
much as 37 nodes. The first level separates the criteria into two
categories: (1) Ecosystem vitality and (2) Environmental
health. For the sake of this exercise we will consider only the
10 criteria present at the second level of the hierarchy:
•
Climate Change
•
Agriculture
•
Fisheries
•
Forestry
•
Biodiversity and Habitat
•
Water (Effects on Ecosystem)
•
Air Pollution (Effects on Ecosystem)
•
Environmental Burden of Disease
•
Air Pollution (Effects on Humans)
•
Water (Effects on Humans)
All of these criteria are indices collected from several
sources or aggregated by the UNDP with values ranging from 0
to 100.
With such a complex problem we are bound to lose a high
amount of information when applying the GAIA method and
indeed, we obtain a GAIA plane with a delta value of 47% (see
Fig. 7). This means that even if some positions on the plane
might seem good or bad, we will not be sure until we check the
complete ranking obtained by PROMETHEE II. This ranking,
Figure 7. GAIA plane for the EPI case study
as well as the one from the EPI index site, is given in Table 2.
TABLE II.
Austria
Belgium
Bulgaria
Cyprus
Czech Republic
Denmark
Estonia
Finland
France
Germany
Greece
Hungary
Ireland
Italy
Latvia
Lithuania
Luxembourg
Malta
Netherlands
Poland
Portugal
Romania
Slovakia
Slovenia
Spain
Sweden
United Kingdom
EPI
78,1
58,1
62,5
56,3
71,6
69,2
63,8
74,7
78,2
73,2
60,9
69,1
67,1
73,1
72,5
68,3
67,8
76,3
66,4
63,1
73,0
67,0
74,5
65,0
70,6
86,0
74,2
RESULTS FOR THE EPI CASE STUDY
Net flow
0,133
-0,068
-0,213
-0,093
0,018
0,058
-0,217
0,126
0,152
0,094
-0,06
-0,05
0,076
0,108
-0,116
-0,17
0,028
0,136
0,021
-0,209
0,06
-0,207
0,014
-0,043
0,057
0,267
0,097
EPI rank
3
26
24
27
12
14
22
5
2
8
25
15
18
9
11
16
17
4
20
23
10
19
6
21
13
1
7
Flow rank
4
20
26
21
15
11
27
5
2
8
19
18
9
6
22
23
13
3
14
25
10
24
16
17
12
1
7
Once again we see that there are only slight differences
between the two rankings, however the loss of information on
the GAIA plane might greatly affect the representation on the
geographical map (see Fig. 8). In this case we chose to orient
the referential so that the red color indicates bad alternatives.
We can see that the worst alternatives in the ranking appear
in red, orange or purple, such as Romania, Bulgaria, Estonia,
Poland, etc. However, red does not indicate the worst
alternative which in this case is Estonia. Instead, because of the
loss of data on the GAIA plane, Romania’s seems to be the
lowest score. The global score is actually given to us by the
size of their circle. Even though it might be misleading, the
user should see the colors as an indication of a country’s strong
or weak points instead of an indication of how good or bad they
are:
•
Bulgaria, for example, has one of the highest
evaluations in fisheries and the highest in agriculture.
•
Estonia has the lowest evaluations in environmental
health and forestry combined with one of the highest in
agriculture and air pollution effects on humans.
•
Romania has one of the worst scores in environmental
health and the worst in water effects on humans
combined with one of the best in agriculture and
forestry.
Figure 8. Geographical view for th EPI case study
In general, we can notice that the high number of criteria
has not hindered our ability to identify good or bad alternatives
on the map since the result from GAIA has been completed
with information from the complete ranking. The colors still
allow us to make interesting observations provided that these
are compared to the initial data set. Once again the
identification of geographical areas with similar profiles is thus
made possible.
V.
CONCLUSION AND PERSPECTIVES
In this paper, we have presented a new way of representing
multi-dimensional information using a single glyph. The HSV
color system allows us to convert a set of coordinates into a
color which can then easily be associated to certain
characteristics by the user. We apply this tool to the GAIA
visualization method for ranking problems and thereby display
the results of a MCDA analysis on a geographical map. By
comparing the alternatives to each other, the user can identify
areas of alternatives with similar profiles or detect the best
alternatives by appreciating their global score as the size of the
symbol.
Even though we have limited its application to problems
with a finite set of alternatives where outranking methods could
be used, we might consider applying it to other problem types
(e.g. choice, sorting …). Provided that we use the proper
methods to solve them, the results obtained could be displayed
using the HSV color system. Furthermore, while we introduced
a glyph to apply the tool to problems with vector data, it is also
possible to apply it to problems involving raster data by using
the colors computed to color the pixels of a map. The use of
other types of methods would then be more relevant because of
the high number of comparisons to be made. This tool can
indeed easily be adapted to multiattribute utility problems.
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