=Paper=
{{Paper
|id=None
|storemode=property
|title=Fuzzy Relational Visualization for Decision Support
|pdfUrl=https://ceur-ws.org/Vol-710/paper36.pdf
|volume=Vol-710
|dblpUrl=https://dblp.org/rec/conf/maics/ZierI11
}}
==Fuzzy Relational Visualization for Decision Support==
https://ceur-ws.org/Vol-710/paper36.pdf
Fuzzy Relational Visualization for Decision Support
Brian Zier and Atsushi Inoue ∗
Eastern Washington University
Cheney, WA 99004 USA
Abstract On the other hand, many studies indicate effectiveness
of sharing various visualization among a group in order to
A study on fuzzy relational visualization in system develop- study extensive exploitation and in-depth understanding of
ment aspects is presented. The front-end enables dynamic
data sets (Heer 2006; Heer, Viegas, and Wattenberg 2007;
and scalable changes in visualization according to user’s ex-
pertise and inspiration. Integrative management of various Viegas et al. 2007; Wattenberg and Kriss 2006). In this
data and knowledge is handled by the back-end at any scale framework, group consensus is made as a result of sharing
in cloud computing environment. Extended Logic Program- different interpretations through various visualizations. This
ming is used as the core of fuzzy relational management in suggests necessity of a general visualization platform that is
the back-end, and is capable of consistent uncertainty man- capable of visualizing subjects with a dynamic and scalable
agement among probabilistic reasoning and fuzzy logic while change of configuration (Shneiderman 1996) via interaction
maintaining asymptotically equivalent run-time with the ordi- with users (Shneiderman 1998; Zhang 2008).
nary Logic Programming. A multi-view relational visualiza- Our general application framework for intelligent systems
tion is being implemented and important graphical features deploys Extended Logic Programming (ELP) and a multi-
are highlighted in this paper.
view visualization scheme, and its efficiency and effective-
Keywords: Visualization, Probabilistic Reasoning, Fuzzy ness have been demonstrated throughout a showcase of var-
Logic, Logic Programming. ious applications (Springer, Henry, and Inoue 2009). In this
paper, we discuss the system development of fuzzy rela-
Introduction tional visualization for decision support within this applica-
tion framework. First, the specification and progress are re-
Given the rapid advancement and penetration of informa- ported. Then the management of fuzzy relations (back-end)
tion technologies, visualization becomes more significant in and their visualization scheme (front-end) are described re-
many domains. In sciences, this is utilized in many aspects spectively.
such as populations, evolutions, radiations, transformations
and structures (Wattenberg and Kriss 2006). In business, Specifications and Progress
this is often found instrumental in various decision making,
e.g. sales charts, change of markets and customer relational The ultimate goal of this development is a fuzzy relational
charts. In mathematics, modern education demands visual- visualization with the following general specification:
ization as an essential element, e.g. demonstration of three S1. Various relations are dynamically visualized in various
dimensional functions as free surfaces. In engineering, this aspects.
is a very critical, mandatory tool for system design, e.g. fluid S2. Various types of data are visualized, e.g. tabular, texts,
analysis for nuclear power plants, aerodynamics for aircraft images, multimedia streams, diagrams, etc.
and heat radiation for CPU units to list a few.
Unfortunately, the majority of conventional visualization S3. Relations can be uncertain, i.e. probabilistic, possibilis-
tends to be application-specific and its analytical model is tic, and perceptual (subjective).
rather static in terms of their relational representation and S4. Knowledge and data are integratively managed through-
visualization configuration. For example, MS Excel limits out a canonical representation and process.
its visualization within limited dimensions (1, 2 or 3), its S5. This is scalable in cloud computing environment.
analytical model is limited to statistical, and it only accepts
tabular data. While this indeed serves in many applications, In this specification, the first two (S1 and S2) are consid-
there is a fatal limit in critical decision making support, e.g. ered as matters of the front-end that provide graphical inter-
infrastructure assurance where integrated leverage of knowl- faces and interactions with users, and the rest (S3, S4 and
edge concerning policies and factual (sensory) data is essen- S5) as matters of the back-end that manage relations among
tial (Inoue 2010). all data and knowledge.
Two major works on the front-end are visualization
∗
E-mail: inoueatsushij@gmail.com scheme and human-computer interaction. There are two
major progresses on the visualization scheme: first, the
Logic Programming (LP) visualizer for educational pur- Table 1: Fuzzy Relations in ELP
pose (Henry and Inoue 2007), then the visualization scheme
for reasoning under uncertainty (Springer and Inoue 2009).
For this fuzzy relational visualization, we further specify this
scheme in order to realize dynamic and scalable visualiza-
tion as a result of developing an interactive graphic interface.
Studies on more sophisticated human-computer interaction,
including integration of this relational visualization scheme
with various conventional visualization, e.g. geographical,
spacial, statistical and functional, are planned and upcom-
ing.
The back-end consists of ELP and extraction function-
alities, and they are placed in a cloud computing envi-
ronment. ELP is developed with Support Logic Program-
ming (SLP) (Baldwin, Martin, and Pilsworth 1995) and a
simple extension of fuzzy probability (Inoue 2008). Ex- in-between correspond to partial truth. Currently we do not
traction functionalities consists of translators from vari- consider cases that fuzzy terms (i.e. fuzzy sets) appear in
ous types of data into ELP and utilities to manage mas- its arguments. They are rather unorthodox in Fuzzy Logic
siveness and high dimensions (Codd 1970; Nugues 2006; framework and, if necessary, can be translated into fuzzy
Moore and Inoue 2008; Yager 1982). Parallelization of predicates pfi (x0 i ), where fi represents the i-th fuzzy term
ELP in a cloud computing environment is currently under- appearing in the arguments, to be properly inserted into the
way (Joxan and Maher 1994). original Horn clause.
Management of Fuzzy Relations: Back-End Query Processing
This section describes management of fuzzy relations from The most critical advantage of ELP is its computational ef-
computational aspects: representation and query processing ficiency, that is asymptotically equivalent with that of LP
of ELP, as well as how various types of data are extracted while the extensions of uncertainty management are indeed
and translated into this representation, i.e. extended Horn in a part of its computation. Consider the following simple
clauses. extended Horn clauses, with query a and unification of some
fuzzy predicates such as a and a0 as well as c and c0 , in order
Representation to demonstrate this efficiency:
Table 1 shows how fuzzy relations can be represented in h1: a ← b ∧ c ∧ d : p1 h6: b : p6
ELP. Like ordinary LP, Horn clauses are used to represent h2: a ← e : p2 h7: d : p7
relations in general. Two extensions are made in those Horn h3: c ← e ∧ f : p3 h8: e
clauses. One is the various probability annotation such as a h4: c ← d : p4 h9: f
point (e.g. 0.62), an interval (e.g. [0.6, 0.68]) and a fuzzy h5: a0 ← c0 ∧ d : p5
(i.e. linguistic) (e.g. ’very low’ and ’high’). The inter-
pretation of those annotated Horn clauses, i.e. probabilistic
events, is P (h) ∈ [0, 1], where P is the annotated probabil-
ity and h is the Horn clause. We interpret P (h) = 1 when no
probability is annotated. The other is use of fuzzy predicates
such as ’Tall’, ’BigFeet’, ’ProceedAtPace’ and ’Level’ in this
table. This is simply a matter of fuzzy predicates observed
in the Horn clause h, and such predicates are specially pro-
cessed in their unification.
Fuzzy probability is formally defined as a normal, con-
vex fuzzy set defined over interval [0, 1] (i.e. a fuzzy num- Figure 1: Snapshot of processing query a in LP
ber), s.t. µp (x ∈ [0, 1]). In addition, a linguistic label is
associated with such a fuzzy set for our advantage, i.e. the First, we consider ordinary LP in order to process query a
linguistic extension of annotated probability in ELP. with the assumption of all annotated probabilities p1, ..., p7
Fuzzy predicate is formally expressed s.t. to be 1 (i.e. the equivalence of no annotations). Figure 1 in-
pf (x1 , . . . , xn ) and its semantics is determined by a dicates the snapshot of this query processing in an AND-OR
corresponding fuzzy set s.t. µpf (x1 , . . . , xn ), where tree. In general, the query processing in LP is Depth-First
(x1 , . . . , xn ) ∈ U , the universe of discourse for this fuzzy Search starting from node a. In LP, we only consider sym-
set. Truth values of such a predicate, by its nature, are bolic unification so that there is no partial unification such as
partial, i.e. τ ∈ [0, 1] where τ = 0 corresponds to false a and a0 , as well as c and c0 . Further, we only need one Horn
and τ = 1 corresponds to true, as well as other values clause to be proven true (so-called an existential query) – ei-
ther h1 or h2 (together with either h3 or h4 in order to prove of multiple predicates, s.t. h = h1 ∧ . . . ∧ hn , we compute
sub-query c) in this query. Sometimes, we need to prove p0 = P 0 (h) = P (h1 ) · . . . · P (hn ).
all possible cases (so-called a universal query), i.e. both h1 Partial truth between fuzzy predicates f and f 0 (e.g. dot-
and h2 (together with both h3 and h4) in this query. The ted lines between a and a0 , and between c and c0 in Figure 2)
selection between existential and universal queries depends is determined by applying Mass Assignment Theory, the
on applications. LP assumes close world assumption (i.e. conditional mass assignment mf |f 0 that yields an interval of
negation as failure). Since recent knowledge representation probability1 (Baldwin, Martin, and Pilsworth 1995). This is
technologies often deploy open world assumption such as so-called semantic unification as opposed to symbolic uni-
Web Ontology Language (OWL), this is often considered as fication. Importantly this is not symmetric unlike symbolic
a shortcoming. unification, i.e. mf |f 0 6= mf 0 |f . In the query processing as-
pect of ELP, this is considered as insertion of Horn clauses
f ← f 0 : p and f ← ¬f 0 : p̄, where p = P (f |f 0 ) = mf |f 0
and p̄ = P (f |¬f 0 ) = mf |¬f 0 . Note that neither close world
assumption nor open world assumption holds in any query
process with semantic unification. This is indeed consistent
with Fuzzy Logic.
Computing partial truth adds a few simple arithmetic to
unification as shown in Table 2. While this may increase
a coefficient of run-time, its asymptotic complexity still re-
mains the same. Similarly to semantic unification in com-
parison with symbolic unification, its computation depends
on the shape of fuzzy sets (i.e. #pivotal points) but not on the
Figure 2: Snapshot of processing query a in ELP number of Horn clauses or that of predicates within those.
Furthermore, this computation even becomes less as those
Query processing in ELP remains in Depth-First Search fuzzy sets are more simply represented (e.g., trapezoidal–
starting from node a as indicated in Figure 2. In order to only 4 points).
process query a in ELP, we need to disjunctively combine
all partial truth of h1 and h2, as well as h5 (i.e. the case of Extraction
fuzzy predicates) s.t. Ph1 (a) ∪ Ph2 (a) ∪ Ph5 (a). Therefore,
all Horn clauses need to be proven, i.e. the equivalence of Extraction functionalities translate various types of data into
the universal query in LP. a collection of extended Horn clauses. Following the con-
The partial truth in ELP is represented as a probability: cept of deductive databases, any data in tabular forms are
either one of point, interval and fuzzy, and this is com- translated into a collection of unconditional Horn clauses,
puted according to Jeffreys’ rule (Jeffrey 1965) s.t. P (c) = i.e. facts, and any relational queries, e.g. SQL, are
P (c|h) · P 0 (h) + P (c|¬h) · P 0 (¬h). Therefore, proof by sat- translated into a collection of Horn clauses (Codd 1970;
isfying sub-queries in ELP is a matter of computing such Ceri, Gottlob, and Tanca 1990). Unstructured texts are to
probabilities in a chain reaction. Let a conditional Horn be translated into a collection of Horn clauses as a result of
clause be c ← h : p = P (c|h) and the result of a sub- applying Natural Language Processing (NLP) such as tag-
query (or simply a fact) be h : p0 = P 0 (h). Then the partial ging, syntax parsing and semantic processing in LP (Nugues
truth of query c, i.e. P (c), is computed depending on the 2006). Semi-structured data such as XML, E-mail and
type of annotated probability according to Jeffreys’ rule as Electric Data Interchange (EDI) have a high compatibility
shown in Table 2. Note that P (c|¬h) = 0 (i.e. false) is with Horn clauses (Almendros-jimnez, Becerra-tern, and j.
expected for close world assumption (i.e. negation as fail- Enciso-baos 2008). As a consequence of this, anything that
ure) and P (c|¬h) = [0, 1] (i.e. unknown) for open world can be represented in XML is translated, e.g. diagrams.
assumption. Multimedia data such as images, audio and video are han-
dled through their summarization, e.g. color histograms,
edge and shape extractions and any other image process-
Table 2: Partial truth of query c, i.e. P (c) ing. Tagged information and attributes are straightforwardly
translated into Horn clauses. Texts such as captions are
translated by applying NLP. Their contents can be efficiently
summarized and, in a sense, compressed by applying Gran-
ular Computing and linguistic summary (Moore and Inoue
2008; Yager 1982). This can also be applied to any other
data that are massively large and highly dimensional.
Overall, a rich set of extraction functionalities serves as a
strong interface because Horn clauses are considered rather
1
The conditional mass assignment, i.e. semantic unification,
may also yield a point probability (Baldwin, Martin, and Pilsworth
When h in the conditional Horn clause c ← h : p consists 1995). However, we do not consider this in ELP.
as a pivotal language (thus, users do not have to be exten- a smaller rectangular bar is displayed representing the pos-
sively exposed to ELP). In knowledge management for in- sible range of probabilities (from 0.0 to 1.0) using a color
frastructure assurance, various factual (sensory) data can be spectrum or gradient. Therefore, the color at the left end
entered in tabular forms and XML. Knowledge such as poli- of the bar represents a probability of 0, and the color at the
cies and scheduling rules can be entered in texts. Then, mi- other end represents 1. Any color in between is then easily
nor modification and refinement are to be made as deemed seen as representing a probability somewhere between these
necessary through some human-computer interaction for vi- possible values. The outer box is then filled with the color
sualization in decision making. representing the point probability for that particular event.
(See figure 3).
Visualization Scheme: Front-End The second method involves representing an interval
probability. This method is very similar to the last, in that we
The concepts of creating a visualization with various views
still have a rectangular box with a smaller bar with the spec-
represent the data at different levels of detail. We chose to
trum representing the range of probabilities. The difference
implement a global view and a local view. The global view
is that the outer box is filled also with a gradient over the
displays the relations in a wide range. This view allows a
probability interval for that event. So if the probability was
user to gain a broad understanding of the various compo-
[0.1, 0.4], then the outer box would be filled with a gradi-
nents and relationships as a whole. Additionally, it is im-
ent ranging over the colors represented within the spectrum
portant for the user to be able to more closely understand
between those values. (See figure 4).
particular subsets of the whole, particularly when the visu-
alization is large. This necessitates the local view, which
allows the user to drill down to a particular subsection of the
global visualization and view the details of the relations.
We designed and implemented the prototype front-end ap- Figure 4: Interval probability
plication with several things in mind. This included the ca-
pacity to utilize the program on various platforms, leaving
the door open for future expansion. For example, we wanted Both of these methods are very good visual representa-
to ensure that this application was independent of any spe- tions of the probability. These visualizations make it quick
cific operating system. We also kept in mind that the fu- to easily identify the probability of a particular event. They
ture (or even the current) trend of technology is moving to are also easy to compare, even between the single point
service-based applications in cloud computing. Because the probability and the interval probability. The challenge which
potential for this visualization system could grow to very we faced was determining an equally good method of vi-
large applications, having a powerful back-end system per- sualizing a fuzzy probability. In this case, the probability
forming the operations and calculations could be beneficial, of a particular event is represented by a fuzzy set. This
requiring only minimal processing power of the front-end means that each probability will have a membership value
system. Additionally, the system would be universally avail- based on the membership function which defines the fuzzy
able and accessible regardless of where the user is. Due to set. After some discussion, we came up with three feasi-
these future possibilities, we designed the input specifica- ble representations of fuzzy probability for this particular
tion around XML and implemented the visualization com- project. There obviously could be many more ways to rep-
ponents in the Java programming language. resent fuzzy probabilities; however, we needed ways that
would be easy to directly compare with the other two rep-
Input file format design resentations.
The method which came to mind first was to represent
We had to develop a format which would include all neces- the shape of the fuzzy set. This was quickly modified to
sary information about the reasoning processes to be visu- include the gradient of color to enhance this representation.
alized. Because the reasoning process can easily be repre- Inside the rectangular box used for the other two methods,
sented in a tree structure, we chose an XML format for the the shape of the fuzzy set would be drawn and filled with
input file. the portion of the gradient which fit within that shape. (See
figure 5).
Fuzzy probabilities in the local view
Figure 5: Fuzzy probability using the shape of the fuzzy set
Figure 3: Point probability
The second method which we consider is to represent the
In the aforementioned research, two methods for pre- probability with color gradients layered based on particular
senting probabilities in the local view are offered (Springer α-cuts of the fuzzy set. After some experimentation, we
and Inoue 2009). One of these methods represents a crisp, discovered that the most effective method for representing
single-point probability; for example: 0.8. The paper sug- in this way was to use the maximum number of α-cuts based
gests that a rectangular box is displayed. Inside this box, on the height of the containing rectangular box (in pixels).
So, if the containing rectangular box was 24 pixels tall, we
take 24 α-cuts of the fuzzy set and paint that row of pixels
with the gradient representing the probability interval at that
α-cut. This results in a color pattern which we will describe
as a two dimensional gradient. (See figure 6). Figure 9: Comparison among shape of the fuzzy set, α-cuts
and decreasing brightness
Layout The local view was developed to be a box-in-box
Figure 6: Fuzzy probability using interval gradients for each style layout. There is a top panel, a left panel, and a bottom
α-cut right panel. The top panel is used for displaying informa-
tion about the node, currently just the node’s name. The left
The third and final method we considered was to repre- panel is used for displaying the calculated probability panel
sent the fuzzy set by changing the color value of the gra- underneath the name of the node. The bottom right panel is
dient based on the fuzzy set. For example, decreasing the then used as a container for any children of the node. All
saturation or the brightness of the color based on the mem- nodes with dependencies and children are given this same
bership value of the particular point. For this representation, three-part layout. This layout is then added to the parent’s
we developed three variations, one which decreased the sat- bottom right panel, creating an embedded box-in-box style
uration, another the brightness, and the other a combination as specified. For the leaf nodes with no children, we sim-
of brightness and saturation. (See figure 7). After compar- ply display a single panel which contains the name of the
ing these three options, we found that the method which de- node and a given probability panel to the right. The given
creased the brightness was the most clear and intuitive (see probability panels for the deepest leaf nodes are drawn to
figure 8). touch the right border of their enclosing box, while other
leaf nodes that are not as deep are indented to the left to al-
low quick vertical comparison between different levels. Ac-
cording to the input file specification, we can have several
different ’branches’ (i.e. separate Horn clauses) or depen-
dencies grouped together by ’and’s (i.e. conjunctively con-
nected predicates within a Horn clause). This is represented
Figure 7: Comparing variations of color value in the local view by a slightly thicker border between the
different children. Figure 10 shows a complete local view.
Figure 8: Fuzzy probability with decreasing brightness
After examining all three of these methods (shape of the
fuzzy set, two dimensional gradients, and color value), we
determined that certain people will view each of these with
different degrees of usefulness. One person may find the
first option the most intuitive. However, others may find
the second or third options most intuitive. We decided that
it would be most beneficial to include all three representa-
tions of fuzzy probability (see figure 9) in the local view Figure 10: Complete local view
program and allow the user to toggle between them depend-
ing on their personal preference or intuition. This will allow
the user to choose an option which suits their eye and allows
Global view development
them to easily compare between point, interval, and fuzzy
probabilities. The global view is conceptually straight-forward, and sim-
pler than the local view. However, implementation turns out
Local view development to be more challenging. The global view is a representa-
tion of the entirety of the reasoning process. Ideally with
For this development, we chose to use the Java programming this visualization application, a user will be able to view
language for several reasons. First and most importantly, the whole reasoning process in the global view with little
Java is platform independent. Java GUI programs can also to no detail and, in order to see more detail, look at a par-
easily be converted to web applets, which could make the ap- ticular subsection of the reasoning process in the local view.
plication even more portable by making it available through This means that the global view should accommodate a large
a web interface online. number of nodes in a small amount of space, while still pro-
viding significant information regarding the reasoning pro- both nodes know their own positions, so we simply have
cess. Our previous work determined that the global view each node draw a link from its position to its parent.
should be, what we call, a circular tree. It ”combines the
relationship visibility of a standard tree structure with the ’And Arcs’ With both the nodes and links in place, we
radial organization and space efficiency of a tree ring struc- must also draw connecting arcs for the branches which are
ture.” (Springer and Inoue 2009). grouped by ’and’s. These ’and arcs’ must connect the links
from the first child node to the last which are part of the
conjoined dependencies. To implement this, we must know
the point of origin (i.e. the parent node’s position), the angle
of the first child in relation to the parent, and the angle of
the last child in relation to the parent. This was a challenge
as we have the angle for each node from the root node, not
the angle with respect to the parent node. However, because
we can calculate the coordinates of the child node as well as
the parent node, we can calculate the angle from the parent
as follows (equation 2, where As is the start angle, Ae is the
Figure 11: With a small radius, the nodes are too close to- end angle, (xs , ys ) is the start location, (xe , ye ) is the end
gether, but by increasing the radius, we create more space location, and (xp , yp ) is the parent location. For coding, we
between the nodes while maintaining the same angles of assume As , Ae ≥ 0 and make necessary conversion, e.g.
placement adding 360 degrees.).
y −y
Node Positioning In order to develop this view, we needed As = tan−1 xss −xpp
to confront challenging issues. The first and the most critical y −y (2)
Ae = tan−1 xee −xpp
issue is the node placement. We had to determine a method
of calculating the position of each node. We considered a After calculating the angle from the parent to both the first
couple of different options, but decided that it was simplest and last child in the ’and’ group, we can then simply draw
to divide the space among the child nodes evenly. For ex- an arc from the first to the last.
ample, the root node will begin with 360 degrees of space,
which it will divide evenly among its children. Each of those Zooming and Fuzzy Nodes The capability to zoom in or
nodes will then be given a placement angle as well as a cer- out on the global view is very critical in visualization. As
tain number of degrees to allocate to their children. One far as coding was concerned, this is fortunately very simple
issue with dividing the ’arc space’ evenly among the chil- because all of the position calculations are based on the node
dren is that we could have one branch with many children diameter and angles. The zoom feature simply scales the
and descendants and another with very few. However, both node diameter, which effectively scales everything else. It
branches would be given an equal amount of space. This re- has been designed as a slider control in the bottom of the
quires us to ensure that all nodes are given enough space in window, but could easily be changed to be any other type of
their angle on the circumference. If we have nodes of a par- interface control as well.
ticular size and which have been given a certain angle with Lastly for the global view, we added a simple indication of
which to work, the only thing left to manipulate in order to fuzzy nodes. For those nodes (represented in the local view
give enough space is the radius (see figure 11). We decided with italicized names), we drew a dashed white circle just
to use a consistent distance between levels of the tree. This outside the node. This allows the user to easily differentiate
was the simplest to implement, as well as, what we believe between fuzzy and crisp events, but does not detract from the
to be, the most clear visually. So in order to calculate node ability to see the coloring of the node. A completed global
positions, we must determine the required circumference to view is shown in figure 12.
give enough space for the most nodes in the smallest angle
as follows (equation 1, where C is the set of children, d is Coloring
the node diameter, a is the node’s given arc space, and l is The color calculations for the probabilities are made by a
the node level.). simple scale of the two primary color (red and green in our
1.5 · |C| · d case) components by the probability. Because the probabil-
−d (1) ity is between 0 and 1, this scales each value between 0 and
2π · a · (l − 1) 255 in the RGB color space. To calculate the red component,
Once we know the circumference, we can determine the ra- we invert the calculation such that the higher the probability
dius, and since we know the depth level of the nodes, we is, the lower the red value becomes. The combination of the
can also determine the distance between levels, or what we scaled red and green values gives us our color.
call the link length. With this information, we can now eas- For the interval probabilities, the calculations are the
ily calculate the positions of each node because we know its same, except that we just have to do the one for the low
specified angle, its depth level, and the link length (i.e. the end of the interval and the other for the high end. To create
distance between each level). After calculating the place- the gradient, we generate the start color for the low proba-
ment of each node, we must generate the links between bility and the end color for the high probability. This creates
them. Each node had a reference to its parent node, and a gradient from the low to the high.
the ability to look at a particular subsection or branch of the
tree in the global view (from the global view), the ability to
look at a particular subsection or branch of the tree in the
local view (from the global view), the ability to go back to
the complete global view (from the local view), and popup
windows which include more specific information about the
nodes and ’and’ branches in both views (particularly the
probabilities). We will briefly describe each of these ideas
further.
The ability to look at a particular subsection or branch in
the global view could be implemented in such a way that you
click on a particular node in the global view and it makes
this node into the root node in the center of the screen and
repositions each of its children around it in the same circular
fashion. This allows the user to more closely examine a par-
ticular branch without yet having to see all detail associated
with the nodes (as in the local view). For example, assume
Figure 12: Completed global view we have a tree with a root node that has five children, and
each of these branches has hundreds of descendants. The
user could click on one of the five children, which would
The concept of coloring for the local view had already then become the root and take the center position, and its
been well-defined in Springer’s previous work; however, the descendants would then be repositioned all around it, giving
global view had not been specified aside from remaining each more space and hopefully making it clearer to see the
consistent with the coloring in the local view. We deter- dependency links.
mined that the leaf nodes should display their given prob- The ability to look at a particular subsection or branch
abilities and the non-leaf nodes should display their calcu- in the local view would be very similar to the previous
lated probabilities. Then the links between nodes display the idea. However, once a user has found a particular (small)
given probability for their appropriate ’and’ branch. This is branch which a user wishes to examine more closely, the
consistent with the local view in which the leaf nodes display user can choose to view a particular node (and all dependen-
their given probabilities and non-leaf nodes their calculated cies/children) in the local view. This would transition them
probabilities. seamlessly and allow them to see all of the information the
In order to represent point and interval probabilities on a local view presents which the global view does not. Fol-
node, we simply color the node the appropriate solid or gra- lowing this concept is the idea that a user should be able to
dient color in the scheme previously discussed. For fuzzy easily return to the global view after examining a subsection
probabilities, we take the core, which is an α-cut where or branch of the whole tree. Ideally the user should be re-
α = 1–representing the best-case interval. This produces turned to the same subsection from which they came in the
an interval, which we represent with a gradient over the global view to maintain a consistent frame of reference be-
color spectrum discussed above. The links are colored in tween views. However, this could also be accomplished by
the same way, with the gradient going across the width of highlighting or outlining the nodes (in the global view) of
the link rather than down its length (see figure 13). Other- the particular subsection the user was examining in the local
wise it could appear that the probability changes from one view so the user can easily recognize and find the nodes in
node to the next, when we are trying to represent a proba- the scope of the rest of the tree.
bility of the rule or event as a whole, not some transitional Finally, we have had some study about various popups in
probability. each view. In the global view, a user could click a node and
initiate a popup indicating the node’s name together with
calculated or given probability. Likewise, if the user were to
click a link, a popup could indicate the given probability for
that particular branch. Currently the given probability for
an ’and’ group is not shown in the local view, but a popup
could display that probability for its branch. There are many
ways in which a popup-style window could enhance both of
Figure 13: Node and link coloring the views. We plan to study those throughout a challenging
knowledge management case.
We have a few other potential enhancements. As an al-
Transition between local and global views ternative to the popup for displaying given probabilities of
A future goal of this project is to allow a user to easily make ’and’ groups in the local view, we have studied effective-
transition between the local view and global view, and also ness of placing additional probability panels to the left of
to potentially include more detail in the global view as the the children which are a part of the ’and’ branch. These
user zooms and manipulates the view. First ideas include: panels would span the height of the children in the group
and probably be narrower than the regular probability pan- gramming. Theory and Practice of Logic Programming
els. However, this would distinguish these probabilities from 8(3).
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Acknowledgment eon, M. 2007. ManyEyes: a Site for Visualization at In-
This work is conducted for and partially supported by NSF- ternet Scale. IEEE Trans. on Visualization and Computer
IUSTF International Collaboration on Infrastructure Secu- Graphics 13(6):1121–1128.
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