This paper continues a
series of publications of authors’ research materials in the field of
visualization of cognitive models based on fuzzy cognitive maps (FCM). A FCM
reflects researcher’s subjective idea of a system in the form of a set of
semantic categories (called factors or concepts) and a set of causal
relationships between them [1, 2]. Thus, a FCM can be graphically represented
in the form of a weighted directed graph, the vertices of which correspond to
concepts, and edges – to cause-and-effect relationships.
One of the conditions for
effective work with a cognitive model is ensuring its visual representation. In
[3], the authors proposed an approach to FCM visualization based on using the
concept of visualization metaphor and its two components – spatial metaphor and
representation metaphor [4]. FCM visualization metaphor is based on graph
visualization algorithms [5, 6] and cognitive clarity concept, which characterizes
the ease of intuitive understanding of information [7] and takes into account
the problem of human’s limited cognitive abilities when reading graphs (a
detailed analysis of this problem can be found, for example, in [8]). Thus, a
link has been discovered between the quality of FCM visualization metaphor and
the level of cognitive clarity of the resulting visual image: the higher the
level of cognitive clarity provided by the visualization metaphor, the simpler
is the process of expert understanding of the cognitive model in its visual
analysis. To assess the level of cognitive clarity, a set of criteria is
proposed. It is concluded that cognitive clarity criteria are the means of the
most natural evaluation of visualization metaphor quality.
The present work is
devoted to the development of FCM visualization metaphor in the direction of
automating the construction of FCM visual image, which is optimal in terms of
cognitive clarity criteria. The previous part of the study focused on the
representation metaphor, whereas this work focuses on the spatial metaphor,
which is the basic component of the visualization metaphor and serves as the
foundation for the subsequent formation of the representation metaphor.
In [3], it was noted that
many of cognitive clarity criteria contradict each other, and it is impossible
in the general case to ensure that FCM visual image meets all the criteria at
the same time from an algorithmic point of view.
However, a number of
features of FCM visualization process are noteworthy:
1) existing graph
visualization algorithms [5, 6] ensure the formation of sufficiently acceptable
(from the point of view of individual cognitive clarity criteria) visual images
of an FCM;
2) building FCM visual
image using such algorithms does not require large computational and time
resources. Thus, it is possible to generate a large number of visual images of
the selected FCM within a feasible period of time (including application of parallel
computing technologies);
3) with a number of
constructed visual images of an FCM available, in the general case, an image can
be chosen from them that most fully meets cognitive clarity criteria.
These features formed the
basis for the development of FCM visualization metaphor to ensure the
construction of FCM visual image that is optimal in terms of cognitive clarity
criteria. The proposed FCM visualization algorithm using metaphor is shown in
Fig. 1.
Fig.
1. FCM visualization algorithm using metaphor
Let us describe the stages
of this algorithm.
At the first stage, graph
visualization algorithms are applied to construct N different tilings of
a cognitive graph (where number N can be quite large and depends on the
system performance and the allowed time spent at this stage). Graph tiling refers
to the collection of coordinates of all its vertices and edges. Thus, tiling
uniquely sets the location of all elements of the graph in space (in the
two-dimensional case under consideration, on the plane).
The possibility of
obtaining a large number of different tilings of the same graph is based on the
following factors, as well as their combinations:
1) at this stage, a set of
different graph visualization algorithms can be applied, those operation is
based on various principles and therefore leads to the formation of different tilings
with the same input data;
2) as a rule, graph
visualization algorithms provide for a number of customizable parameters (e.g.,
formulas coefficients), changing of which affects the results of the
algorithms, i.e. the resulting tilings;
3) as a rule, graph
visualization algorithms use as input data the initial coordinates of vertices
and edges generated randomly at each new start of the algorithm. Thus,
stochastic elements are introduced into the results of their work.
It should be noted that tilings
built at this stage, due to their large number, are not displayed on the
screen, but just stored in RAM.
The second stage of the
algorithm involves assessing the degrees of conformity of the generated tilings
to the criteria of cognitive clarity. Note that these criteria are formulated
at a qualitative level using a natural language. At the same time, the proposed
metaphor, that targets the automation of the construction of an optimal FCM
visual image, implies the implementation of all the main stages of the
algorithm without human participation. Therefore, the stage of assessment of
the resulting tiles according to the cognitive clarity criteria needs the
algorithm to be designed. For this purpose, it is necessary to obtain a
formalized representation of these criteria. This issue will be discussed in
more detail below.
At the third stage of the
algorithm, on the basis of the obtained criterial estimates of the tilings and
previously specified priorities of the cognitive clarity criteria, an optimal tiling
is selected. In this case, it is necessary to apply decision rules that model
various forms of compromise among the criteria. One of the admissible rules is
proposed further in this paper.
The goal of the
transformation stage of the selected optimal tiling is to further increase its
cognitive clarity by performing one or more of the following types of
operations: rotation by a certain angle, reflection relative to the horizontal
or vertical axis, compression or extension along a certain direction. Thus,
this stage performs the function of post-processing of the resulting tile and
is generally not mandatory.
Finally, the last stage of
the algorithm involves displaying FCM visual image based on the selected
optimal tiling. Herewith, as a rule, representation metaphor is applied that
corresponds to the current stage of construction or analysis of a cognitive
model.
Obviously, the efficiency
of the proposed algorithm depends on number of the processed tilings N:
the more tilings have been generated, the higher the probability is that a tiling
with a high level of cognitive clarity will be found among them. At the same
time, in practice, the value of N must be limited in order to satisfy
the specified time constraints on the process of constructing a visual image of
an FCM.
It should be also noted
that the stages of the algorithm surrounded by dashed lines (i.e., the
construction of tilings and their assessment by criteria) can be performed simultaneously.
In the case of using parallel computing technologies, this can significantly
increase number N, which will lead to an increase in the efficiency of
the algorithm.
We define the
formalization of a certain cognitive clarity criterion as the developing
methods, techniques and algorithms that allow determining a numerical score for
a visual image of an arbitrary cognitive map characterizing the extent to which
this image complies with the selected criterion. Formalization of most criteria
(e.g. such as minimizing edge lengths, minimizing edge crossing, minimizing the
number of curved edges) is trivial, and its description is of no interest as it
reduces to solving simple problems of computational geometry. Let us consider
possible ways of formalizing several nontrivial cognitive clarity criteria.
This criterion is based on
the observation that laying out edges in the directions “from top to bottom”
and “from left to right” helps to accelerate “reading” of a FCM in comparison
with the orientation of edges in the opposite directions. We shall call the
directions that facilitate faster “reading” of FCMs as well as edges having
such directions convenient. As an example, we can compare two visual images of
a cognitive graph in Figure 2.
Fig. 2.
Examples of visual images with convenient (a) and not convenient (b) directions
of edges
Apparently, convenient
directions coincide with the direction of reading, adopted in a particular
language culture. Therefore, other conditions being equal, preference should be
given to visual images containing a greater number of convenient edges. It should
be borne in mind that the described property is inherently fuzzy. So, edge
orientation “from top to bottom” and “from right to left” can be considered
partially convenient, since one of the usual directions of reading is
preserved. Therefore, the mathematical apparatus of the fuzzy set theory can be
used to formalize the criterion in question.
Let A be a fuzzy
set formalizing the concept of a “convenient edge direction”. In order to set
its membership function, let us define the edge direction as angle α
between the vector drawn from the beginning of the edge to its end and the
positive direction of the horizontal axis OX. Then membership function 
must satisfy the following requirements:
1) 
when 
2) 
when 
3) 
when 
è 
4) increases monotonically on
the interval
5) decreases monotonically on
the interval 
Given these requirements, 
can be accepted on the interval , and 
can
be accepted on the interval 
.
Having determined for each
FCM edge the degree of its membership to set A, we can obtain a value
characterizing the overall score of the entire visual image by this criterion –
for example, as the average value of membership degrees of all edges.
Influence intensities
should also be taken into account in the final assessment, since providing
convenient directions for more significant influences is more important than
for less significant ones. Therefore, absolute values of influence intensities
can be used as weighting coefficients and membership values of the
corresponding edges can be multiplied by them when calculating the average
value.
This criterion is based on
the idea that "reading" a FCM will be faster if gaze direction has to
be changed as little as possible during the process of viewing paths and cycles
of a graph.
We will call two edges
consecutive if one of them enters the vertex from which the other one starts.
Thus, any path and cycle of a graph consists of pairs of consecutive edges.
Therefore, in accordance with this criterion, preference should be given to
visual images with a greater number of pairs of consecutive edges depicted
unidirectionally. For example, let us compare two visual images of a fragment
of some FCM (Fig. 3).
Fig. 3.
Examples of visual images with unidirectional (a) and bidirectional (b)
consecutive edges
Obviously, the
unidirectional property is fuzzy. Suppose B is a fuzzy set formalizing
the concept of unidirectional edges. The membership degree of a pair of
consecutive edges to set B is determined by angle 
between these edges. We assume that
changing gaze direction by 90 degrees or more slows down the process of viewing
the path in the graph significantly. Accordingly, the following requirements
are imposed on the membership function 
:
1)
when 
2)
when 
3)
and decreases monotonically when 
Given these requirements,
on the interval 
we accept 
By analogy with the
previous criterion, the score of the entire visual image by the criterion under
study can be found as the average value of membership degrees of all pairs of
consecutive edges to set B. Influence intensities can also be taken into
account in a similar way.
Due to the fact that FCM
structure reflects the structure of a simulated system it is important to
ensure the symmetry of FCM visual image to increase its cognitive clarity.
Thus, symmetries of a graph image help to detect symmetries inherent in the
system itself.
Let us consider various
aspects of determining degree of image symmetry in relation to FCM visual
image.
Firstly, the following
types of symmetries are the simplest to perceive and, therefore, of greatest
practical interest:
1) axial symmetry with
respect to the horizontal axis of an image;
2) axial symmetry with
respect to the vertical axis of an image;
3) central symmetry with
respect to the geometric center of an image.
Secondly, in the case of
an FCM, as well as any digraph, the following levels of symmetry can be
distinguished (Fig. 4 considers the case of symmetry about the vertical axis):
1) lack of symmetry at the
level of any elements of the graph (Fig. 4, a);
2) at the level of vertices
excluding edges (Fig. 4, b);
3) at the level of edges
excluding their directions (Fig. 4, c);
4) at the level of edges
including their directions (Fig. 4, d).
Fig.
4. Examples of different symmetry levels of FCM visual image (about the
vertical axis)
The above example allows
for the conclusion that symmetry at the vertex level does not bring any
tangible effect to increasing cognitive clarity of an FCM visual image. Thus,
only symmetry at the level of edges is of practical interest.
Thirdly, it is obvious
that in addition to strict symmetry (Fig. 5, a), we can also speak of
approximate symmetry (Fig. 5, b), which can be represented as a certain
deviation from the strict one.
Fig. 5.
Examples of strict (a) and approximate (b) symmetry of FCM visual image
With this in mind, the
degree of symmetry of an FCM visual image can be defined as a measure of its
proximity to a strictly symmetric image. Thus, it is necessary to develop an
algorithm that is able to determine the degree of symmetry for an arbitrary
image taking into account a given type and level of symmetry.
The main idea of the
proposed algorithm is as follows. For each element of an FCM visual image, the
position of its “reflection” relative to a given axis or center is calculated.
Next, for each of the “reflections”, the element closest to it (in the sense of
the chosen metric, for example, Euclidean distance) is selected from among all
the elements of the image. Distances (in the selected metric) from all
“reflections” to their nearest elements are added up. The resulting value
characterizes the degree of symmetry of the visual image in question and has
the following properties:
1) it is equal to 0 if the
image has strict symmetry of a given (or stronger) level and type;
2) it is greater than 0 in
all other cases;
3) it increases as the image
becomes less and less symmetrical;
4) it does not have an upper
bound since there is no “maximally asymmetric” image.
In the context of the
problem under consideration, of primary interest is the class of decision rules
based on various types of criteria aggregations, primarily, sum and product
ones. At the same time, there is reason to believe that the structure of
relationships among cognitive clarity criteria is quite complex and is
characterized by the following features:
1) criteria may exist that
determine quality of a metaphor not separately but in combination with some
other criteria;
2) in the whole set of
criteria, there may be several “bundles of criteria” affecting metaphor quality
independently of each other.
To formalize the described
assumption, we shall accept that set of criteria 
can
be divided into disjoint subsets 
. Further, we
will also assume that FCM visual image scores by all criteria take their values
in the interval [0,1].
For each criteria subset Gi,
we shall introduce value 
– visual image
score for this subset. We shall specify the following requirements for such a score:
1) if the image score by at
least one criterion from subset Gi is 0, then 
;
2) 
if and only if the image score by
all criteria from subset Gi is 1;
3) if the image score
according to all criteria from subset Gi is a, then 
(idempotency).
One of the operations
meeting the specified requirements is a weighted product aggregation:
where 
is
image score by the j-th criterion from Gi, 
is relative importance of the j-th
criterion within Gi 
,
l is power of Gi.
To obtain final score, we
shall apply weighted sum aggregation to the scores obtained for all subsets:
where 
–
is relative importance of subset Gi 
.
Moreover, value 
can be interpreted as FCM visual image
score that fully satisfies the subset of criteria Gi and does
not completely satisfy other subsets of criteria.
Thus, for the final score F,
the following properties are guaranteed:
1) 
;
2) F = 1 if scores for all
criteria are 1;
3) F = 0 if scores for all
subsets of criteria are 0 (i.e., at least
one criterion scored with 0 is present in each subset).
It should be noted that
since the proposed decision rule is based on the combined use of sum and product
aggregations of criteria these types of aggregations are its “extreme” special
cases. Thus, a sum aggregation type will be obtained if a separate subset 
is assigned to
each of the criteria 
.
Assignment of all criteria to one subset G1 will result in a product
aggregation.
Selection and
justification of parameters of the decision rule (the number of subsets of
criteria, distribution of criteria by subsets, etc.) is the task of the analyst
performing a visual analysis of an FCM. This problem should be solved from
knowledge of the features of a particular stage of cognitive modeling (in
particular, which cognitive clarity criteria are most significant at this stage),
as well as involving analyst’s intuition and experience. So, the relative
importance of the subsets and criteria within the subsets can be set based on a
pairwise comparison method.
The complex of algorithms providing
the proposed development of FCM visualization metaphor was implemented within
the framework of FCM visualization subsystem, which is part of IGLA DSS based
on fuzzy cognitive models [9].
In order to verify the
operability of the proposed development of FCM visualization metaphor, a series
of experiments was carried out. Within its framework the following metaphor
parameters were varied:
•
graph
visualization algorithms used;
•
number
of generated tilings N;
•
system
of preferences based on cognitive clarity criteria.
Based on the results of the
series of experiments, the following conclusions have been made:
•
all
varied parameters influence the quality of the metaphor, i.e. the level of
cognitive clarity of the resulting visual image;
•
the
most appropriate is the use of graph visualization algorithms ISOM and LinLog;
•
an
acceptable level of metaphor quality is achieved already when N = 100,
while an increase in N leads to an increase in time spent on the visualization
process, which for large values of N may be unacceptable (note that the unit
cost of processing one tiling is 0.03-0.09 seconds depending on the size of FCM;
these data were obtained under the following conditions: i5-2450M processor,
parallel computing technologies were not used);
•
the
decision rule introduced in this paper allows for flexible control of the
relative importance of cognitive clarity criteria and helps to determine the
acceptable forms of compromise among them.
Let us consider in more
detail conditions and results of one of the experiments.
An FCM of analysis and
planning of software projects was selected as an FCM for which a visual image
had to be built [10]. The choice of this FCM was determined by its relatively
small size and simple structure, which positively affected the ease of
interpretation of the visualization results obtained (and, thus, simplified the
verification of the metaphor under study).
At the stage of
constructing FCM tilings, ISOM and LinLog algorithms were implemented. A total
of 100 tilings were generated. Assessment of the degree of compliance of the tilings
with cognitive clarity criteria was performed using the proposed methods of
formalizing these criteria.
The parameters of the
decision rule were set as follows. The set of criteria was divided into two
subsets: G1 and G2. At the same time, the
following criteria were assigned to the subset G1:
•
optimizing
placement area (within the framework of this experiment, tiling had to be placed
in a square-shaped area);
•
minimizing
edge lengths;
•
unifying
edge lengths;
•
maximizing
graph symmetry.
Subset G2
included the rest of the criteria:
•
optimizing
edge directions;
•
minimizing
edge crossing;
•
minimizing
the number of curved edges;
•
maximizing
unidirectionality of consecutive edges;
•
maximizing
angles between incident edges.
The priorities of the
subsets G1 and G2 were set equal to 0.3 and
0.7, respectively. Criteria priorities within the subsets were distributed
evenly, i.e., 0.25 and 0.2 for the criteria from G1 and G2,
respectively.
The tiling shown in Fig. 6
was recognized the best of those generated (hereinafter we will refer to it as Tiling
1). For comparison, let us also consider two other tilings, randomly selected
from among generated ones (Fig. 7-8), which we will refer to as 2 and 3,
respectively.
Table 1 presents the
results of tiling assessment according to the cognitive clarity criteria.
Criteria scores of tilings are normalized to the range [0, 1] (normalization
was carried out taking into account the scores of all 100 tilings obtained). Meanwhile,
the initial requirement to minimize a number of criteria was taken into
account. Thus, after normalization, all criteria must be maximized.
Table 1. Results of assessment of generated tilings
|
Criteria and subsets of criteria
|
Tiling number
|
|
1
|
2
|
3
|
|
Subset G1
|
0.860
|
0.715
|
0.467
|
|
Optimizing placement area
|
0.824
|
0.985
|
0.792
|
|
Minimizing edge lengths
|
0.796
|
0.654
|
0.439
|
|
Unifying edge lengths
|
0.952
|
0.696
|
0.299
|
|
Maximizing graph symmetry
|
0.875
|
0.584
|
0.457
|
|
Subset G2
|
0.749
|
0.410
|
0.778
|
|
Optimizing edge directions
|
0.804
|
0.232
|
0.765
|
|
Minimizing edge crossing
|
1
|
0.540
|
1
|
|
Minimizing the number of curved edges
|
1
|
0.673
|
0.694
|
|
Maximizing unidirectionality of consecutive edges
|
0.368
|
0.350
|
0.892
|
|
Maximizing angles between incident edges
|
0.796
|
0.394
|
0.603
|
|
Final score
|
0.782
|
0.502
|
0.685
|
Fig. 6.
Tiling 1
Fig. 7.
Tiling 2
Fig. 8.
Tiling 3
Figure 9 presents the
final version of FCM visual image obtained on the basis of Tiling 1 as a result
of transformation operations performed on it. In this case, the following
sequence of operations was performed: 90 degree clockwise rotation; mirror
reflection with respect to the vertical axis of tiling; vertical compression of
tiling by 30%.
Fig.
9. FCM visual image
The key effect of applying
the proposed FCM visualization metaphor for visual analysis of cognitive models
is a significant reduction in the time spent on building FCM visual image with
a high level of cognitive clarity. So, for the FCM considered, the use of
metaphor made it possible to reduce the time required to build such a visual
image (Fig. 9) by 4 times.
The resulting visual image
can be adapted for visual analysis at any stage of cognitive modeling by using
the appropriate representation metaphor (this issue was discussed in detail in
the authors’ paper [3]). So, the use of the metaphor shown in Fig. 10 allows
the analyst to interpret FCM system indicators at the stage of structure and
target analysis. Meanwhile, the time spent is reduced by an average of 3-4
times compared to the traditional method of interpretation (implying the need
for the analyst to read a large amount of data in numerical representation).
Thus, it can be said that the efficiency of cognitive interpretation of FCM
visual image increases by means of reducing the time spent on it [11].
Fig.
10. The result of applying the metaphor of FCM system indicators
The paper presents the
development of the visualization metaphor of fuzzy cognitive maps in the
direction of automating the construction of a visual image of a fuzzy cognitive
map optimal from the point of view of cognitive clarity criteria. A generalized
FCM visualization algorithm using a metaphor is presented. The work of the
algorithm is based on the use of graph visualization algorithms and taking into
account formalized cognitive clarity criteria. Possible methods of formalizing
several of these criteria, which are nontrivial, are described. A decision rule
is also proposed for choosing the optimal visual image, which allows for
controlling the relative importance of cognitive clarity criteria and
determining the acceptable forms of compromise between them. An example of applying
the proposed metaphor is given, confirming its efficiency and effectiveness.
Let us indicate directions
for further research.
The first one is the
search and formalization of relationships between the spatial metaphor and the FCM
representation metaphor. In particular, identifying situations in which a
change of the representation metaphor requires adjusting preferences according
to cognitive clarity criteria, which leads to a change of the optimal visual
image.
The second one is the
development of recommendations on the selection of parameters of the decision
rule proposed in the work based on identifying the relationships between the
parameters of this rule and its effectiveness in constructing visual images of
various FCMs.
The third one is the
development of new decision rules for choosing the optimal FCM tiling from the
point of view of cognitive clarity criteria, allowing for more flexible
consideration of user's preference characteristics and acceptable forms of
compromise among the cognitive clarity criteria.
The reported study was
funded by RFBR, project number 19-07-00844.
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