towards a flexible visualization tool for dealing with temporal data
TRANSCRIPT
Towards a flexible visualization tool for dealing
with temporal data
Guy de Tre(1), Nico Van de Weghe(2), Rita de Caluwe(1), andPhilippe De Maeyer(2)
(1) Department of Telecommunications and Information Processing, Ghent University,Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium
(2) Geography Department, Ghent University, Krijgslaan 281 (S8), B-9000 Gent,Belgium
Abstract. Time plays an important role in our everyday’s life. For alot of observations we make and actions we perform, temporal informa-tion is relevant. The importance of time is reflected in the developmentof information systems as, e.g., (temporal) database systems and datawarehouse systems, which have facilities to cope with temporal data andusually manage huge collections of historical data. It is a challenge todevelop intuitive user interaction tools that allow users to fully explorethese collections of temporal data. With this paper, we want to con-tribute to the development of such a tool. The presented approach isbased on a visualization of time intervals as points in a two-dimensionalspace and on temporal reasoning based on this visualization. Flexibilityis provided by allowing to cope with imperfections in both the modellingof time and the temporal reasoning.Keywords: Time modelling, user interfaces, fuzzy set theory.
1 Introduction
The central role of temporality in many computer applications makes the rep-resentation, manipulation and visualization of temporal information highly rel-evant.
Related to database systems and data warehouse systems, several time mod-els for the representation and manipulation of time have been proposed, eachhaving its own applicability and limitations. Research on temporal models is ingeneral motivated by the observation that most databases and data warehousescontain substantial amounts of temporal data, what makes that specific mod-elling and management techniques for temporal data are a requirement for anefficient exploitation of these data collections [7]. An interesting bibliographicoverview of older work can be found in [13]. More recent work and state of theart are summarized in [12, 4]. There have also been some efforts to bring intoline the resulting diversity of concepts and terminology [6].
Almost of equal importance, is research on the visualization of temporal data.As time is usually modelled using a one-dimensional time space, visualization
of large amounts of temporal data tends to result in an overwhelming image of(overlapping) intervals, which lacks clarity and does not provide the user withthe insights and overviews that are necessary to fully explore the data. For thesake of illustration of the problem, consider for example the one-dimensionaltime lines used by historian to represent important dates and periods in history.Such time lines quickly tend to become overloaded with data when more factsare registered.
In this paper, we propose a flexible, alternative approach for the visualizationof temporal data. The approach allows for a better human-computer interactionby providing users with a more compact visualization of the temporal data andthe temporal relationships that exist between these data. By using the approach,the user should better understand the data and gain new insights, which on itsturn should be supportive for the further exploration of the database or datawarehouse. Moreover, the presented approach provides some flexibility and al-lows to deal with possible imperfections of the data (e.g., imprecision, incom-pleteness, or uncertainty), both at the level of the temporal data modelling andat the level of temporal reasoning. The presented approach is inspired by thework of Z. Kulpa [9, 10] in which a diagrammatic representation for intervalarithmetic is presented. Flexibility is provided by applying fuzzy set theory [14].
The remainder of the paper is organized as follows. In Section 2, the basicframework of the visualization approach is described. Both the structural aspectsand behavioral aspects are dealt with. The basic framework is generalized inSection 3 in order to be able to cope with imperfections of the data. In Section 4,an illustrative example demonstrating some of the potentials of the approach ispresented. Finally, some conclusions are given in Section 5.
2 A framework for visualizing temporal data
2.1 Some preliminaries
Within the presented framework it is assumed that time is modelled as beingdiscrete, linear and finite. This means first of all that the restriction is acceptedthat time can only be observed using a limited precision, say ∆. This is notreally a limitation, on condition that the precision is chosen sufficiently accurate.Indeed, this restriction results from the way observations and measurementsare made and it conforms to the way data can be stored in computers. Thediscretization is necessary to circumvent the density problem (i.e., the fact thatbetween any two distinct points, there always exists at least one other point). Theneed for non-linear structures with topologies such as branching time, paralleltime, circular time etc. as suggested by some authors [11], is not supported in themodel. Linearity implies a total order over the time points. Finally, our modelis chosen to be finite in view of a computer representation. This implies that allvalues that exceed the determined upper and lower bound will be handled byintroducing two special values (−∞ and +∞).
Observing time using a maximum precision ∆ and with respect to a givenorigin t0 involves a crisp discrete (countable) set of time points, given by:
T0,∆ = {tk ∈ T |tk = t0 + k∆, k ∈ Z}, where Z = {0, 1,−1, 2,−2, . . . } (1)
which is a proper subset of the set T , which represents the continuum of thephysical time points. The discretization can be described as a surjective mappingfrom T on T0,∆, which maps each τ ∈ T on the element tk ∈ T0,∆ that lies inthe interval γ0,k, k ∈ Z, which is defined as:
[t0 + k∆, t0 + (k + 1)∆[
or, alternatively:
[t0 + (2k − 1)∆
2, t0 + (2k + 1)
∆
2[
The intervals γ0,k with ∆ as length are usually called ‘chronons’ in literature(cf. [6]). The introduction of chronons eliminates the necessity to make a dis-tinction between points and intervals, because even a chronon, the smallest unitin the model, is essentially an interval.
2.2 Structural aspects
Time is a very complex notion, which is used mostly rather intuitively within acontext of physical phenomena and established conventions. Generally, temporalinformation can be subdivided in three categories.
1. Durational information is unrelated to a particular point in time and de-scribes a duration: e.g. 5 days, 1h 35min.
2. Positional temporal information is related (directly or indirectly) to eitherthe origin or the current time point (now): e.g. in the 18th century, yesterday.
3. Repetitive temporal information describes positional temporal informationthat repeats itself with a given period: e.g. every hour, monthly.
In this paper, only time intervals are dealt with. Time intervals allow tomodel positional temporal information [1] and are the basic building blocks ofmost time models supporting database and data warehouse systems. To handledurational and repetitive temporal information, other structures are necessary[3]. Such structures are not dealt with in this paper.
Visualizing time intervals A time interval
[a, b]
is defined by a starting time point a and a length l, which defines the distance(number of chronons) from a to the ending time point b, i.e., l = b − a. A timepoint is equivalent to a time interval with l = 0 (or b = a). Note that the length
of an interval is not the same as the duration of the interval: the length of a timepoint is 0, its duration is 1 chronon (∆).
At the basis of the visualization approach is the visualization of intervals.Instead of representing time intervals on a one-dimensional time axis, time in-tervals are visualized as points in a two-dimensional space. This allows for a morecompact representation. The two-dimensional space is defined by two orthogonalaxes (cf. Figure 1):
– A horizontal axis X = Tposition that is used to represent positional infor-mation about the starting time points and ending time points of the timeintervals and
– a vertical axis Y = Tlength that is used to represent length information aboutthe time intervals.
Tposit ion
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a
p[a,b]
a b
Fig. 1. Visualization of a time interval [a, b].
A time interval [a, b] is visualized by a time point p[a,b], which position isobtained by rotating [a, b] counterclockwise around a over a fixed angle α asdepicted in Figure 1. This implies that the (x, y) coordinates of p[a,b] are givenby:
x = a + d(a, b). cos α
y = d(a, b). sin α
where d(a, b) = |b−a| is the distance between a and b, which corresponds to thelength of the interval [a, b]. Because time points a = [a, a] have a length 0, theyare represented on the X-axis with coordinates (a, 0).
For the sake of symmetry, the angle α will be chosen to be equal to 60◦ inthe remainder of the paper (cf. Figure 1). In that case the (x, y) coordinates ofp[a,b] become:
x = a +d(a, b)
2y = d(a, b). sin 60◦
Special choices for α are:
– α = 0◦: in this case we obtain the classical one-dimensional representationof time where the X-axis represents the continuous time axis.
– α = 90◦: in this case the X-coordinate represents the real starting point ofthe interval (x = a), whereas the Y-coordinate represents the real length ofthe interval (y = d(a, b)).
Comparison with related work The presented approach is related to thediagrammatic representation for interval arithmetic as presented by Z. Kulpa[9, 10], where p[a,b] is defined to be the top of the isosceles rectangle with basis[a, b] and fixed basis angles α = 45◦. Furthermore, common aspects with theapproach presented by G. Ligozat et al. [8, 2] are obtained when considering thespecial case where α = 90◦. The main advantage of the presented approach isdue to its generality and its ability to work with different choices for the angleα. This supports the construction of more dynamical interfaces where the usercan chose the angle that responds best to his/her needs.
2.3 Behavioral aspects
In order to be useful, the compact representation of time should allow users tohave a better overview and better insights in temporal data. Moreover the repre-sentation should support temporal reasoning in a rather intuitive way. Thereforeit is a requirement that the thirteen possible relations between two time inter-vals, as defined by Allen in his interval-based temporal logic [1] also have anintuitive visual interpretation.
The thirteen Allen relations are depicted in Figure 2
relation inverse relation
before after
meets met-by
overlaps overlapped-by
starts started-by
during contains
finishes finished-by
equal
Fig. 2. The 13 Allen relations between two time intervals.
Depending on the choice of the angle α, a number of Allen relationshipscan be visualized. In this paper, only the symmetrical case where α = 60◦ isdescribed. The visualization of the 13 Allen relations is given in Figure 3. For
Tposit ion
Tlengt h
a
p[ a, b]
a bTposit ion
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a
p[ a, b]
a b
[ , ] [ , ]a b before c d[ , ] [ , ]a b after c d
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a bTposit ion
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p[ a, b]
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[ , ] [ , ]a b met-by c d[ , ] [ , ]a b meets c d
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p[ a, b]
a bTposition
Tlength
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[ , ] [ , ]a b overlapped-by c d[ , ] [ , ]a b overlaps c d
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p[ a, b]
a bTposit ion
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a
p[ a, b]
a b
[ , ] [ , ]a b started-by c d[ , ] [ , ]a b starts c d
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a
p[ a, b]
a bTposition
Tlength
p[a,b]
a b
[ , ] [ , ]a b during c d[ , ] [ , ]a b contains c d
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a
p[ a, b]
a bTposit ion
Tlengt h
a
p[ a, b]
a b
[ , ] [ , ]a b finished-by c d[ , ] [ , ]a b finishes c d
Tposition
Tlength
a
p[a,b]
a b
[ , ] [ , ]a b equals c d
Fig. 3. Visualization of the 13 Allen relations.
each relation, the locations of the points p[c,d] that represent intervals [c, d] thatsatisfy the relation are visualized (marked).
3 Generalizing the framework: coping with imperfect
data
The visualization approach described in the previous section can be made moreflexible by allowing imperfections at the levels of temporal data modelling andtemporal reasoning. In order to cope with imperfections, fuzzy set theory isapplied [14].
3.1 Structural aspects
It might be the case that at least one of the start and end dates of a time intervalis ill-known. If this is the case, the time interval is called a fuzzy time interval.Fuzzy time intervals can be modelled by a fuzzy set with a convex membershipfunction, as illustrated at the left side of Figure 4. For more information aboutthe interpretation and operations on fuzzy time intervals, we refer to [5]. In gen-
Tposition
Tlength
a
psupport
Ta1 a4a2 a3
1 pcore
m[ a, b]
a1 a4a2 a3
m[a,b]
[a,b]
Fig. 4. Visualization of fuzzy time intervals.
eral, any adequate convex membership function can be used for the modelling.In practice, normalized trapezoidal membership functions are frequently used asapproximation, because these can be easily implemented. The normalized trape-zoidal membership function µ[a,b] of the fuzzy time interval [a, b] is characterizedby four time elements a1, a2, a3 and a4 of which a1 and a4 define the supportof [a, b] and a2 and a3 define the core of [a, b]. In the remainder of the paperthe special case of fuzzy time intervals with normalized trapezoidal membershipfunctions is studied.
A fuzzy time interval [a, b] with normalized trapezoidal membership functionµ[a,b] is visualized by the line
[pcore[a,b], p
support[a,b]]
[ , ] [ , ]a b before c d[ , ] [ , ]a b after c d
[ , ] [ , ]a b met-by c d[ , ] [ , ]a b meets c d
[ , ] [ , ]a b overlapped-by c d[ , ] [ , ]a b overlaps c d
[ , ] [ , ]a b started-by c d[ , ] [ , ]a b starts c d
[ , ] [ , ]a b during c d[ , ] [ , ]a b contains c d
[ , ] [ , ]a b finished-by c d[ , ] [ , ]a b finishes c d
[ , ] [ , ]a b equals c d
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pcore
a1 a4a2 a3
[a,b]
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pcore
a1 a4a2 a3
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[a,b]
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a
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pcore
a1 a4a2 a3
[a,b]
[a,b]
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a
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pcore
a1 a4a2 a3
[a,b]
[a,b]
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a
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pcore
a1 a4a2 a3
[a,b]
[a,b]
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a
psupport
pcore
a1 a4a2 a3
[a,b]
[a,b]
Fig. 5. Visualization of the 13 generalized Allen relations.
which connects the points pcore[a,b] and psupport
[a,b] that respectively representthe core
core([a, b]) = {x|x ∈ Tposition ∧ µ[a,b](x) = 1}
and support
support([a, b]) = {x|x ∈ Tposition ∧ µ[a,b](x) > 0}
of [a, b] (cf. right side of Figure 4). Every other point of [pcore[a,b], p
support[a,b]]
represents an α-cut
{x|x ∈ Tposition ∧ µ[a,b](x) ≥ α}
or a strict α-cut{x|x ∈ Tposition ∧ µ[a,b](x) > α}
of [a, b].Because core([a, b]) ⊆ support([a, b]), the point pcore
[a,b] must be located in
the triangle determined by a1, a4 and psupport[a,b].
The special case of a crisp time interval [a, b] is still visualized by a time pointpcore
[a,b] = psupport[a,b], because in that case the core([a, b]) = support([a, b]).
3.2 Behavioral aspects
In order to be supportive for temporal reasoning with ill-known time intervals,the thirteen Allen relations must be generalized for fuzzy time intervals andvisualized in an intuitive way. The generalization of the Allen relations is dealtwith in [5]. In that paper, it is also illustrated how the degree of certainty thatan Allen relation holds for two given fuzzy time intervals [a, b] and [c, d] can becalculated. In this paper we focus on the visualization of the generalizations ofthe Allen relations. The generalized counterparts of the visualizations in Figure 3are presented in Figure 5.
As can be observed in the visualizations of Figure 5, three kinds of areas canbe distinguished:
– An area A where the relation does not hold (not marked),– an area A where the relation holds (marked with full lines) and– an area A where the relation possibly holds (marked with dotted lines).
With respect to the location of the line [pcore[c,d], p
support[c,d]], which represents
the fuzzy time interval [c, d], the following situations can be distinguished:
– [pcore[c,d], p
support[c,d]] is completely located in A: in this situation, the rela-
tion does not hold, for sure.– [pcore
[c,d], psupport
[c,d]] is completely located in A: in this situation, the rela-tions holds, for sure.
– [pcore[c,d], p
support[c,d]] (partly) overlaps A or is located in A: in this situation,
it is uncertain whether the relation holds or not. The degree of certainty canbe calculated as described in [5].
Beside these general situations, it is interesting to observe the following spe-cial cases:
– If the start date (resp. end date) is perfectly known, then the supportand core intervals start (resp. end) at the same time point. In such cases,the straight lines [a1, p
support[a,b]] and [a2, p
core[a,b]] (resp. [a3, p
core[a,b]] and
[a4, psupport
[a,b]]) coincide and [pcore[a,b], p
support[a,b]] is located on [a2, p
core[a,b]]
(resp. [a3, pcore
[a,b]]), which is completely consistent with the description ofSection 2.
– If the fuzzy interval is perfectly known, i.e. both the start date and the enddate of the interval are perfectly known, then psupport
[a,b] coincides pcore[a,b],
which is again consistent with the perfectly known cases described in Sec-tion 2.
In these special cases, it can occur that [pcore[c,d], p
support[c,d]] is located in both
A and A, without crossing A. In such a situation it is uncertain whether therelation holds or not.
From the above cases, it should be clear that the presented approach is infact a generalization of the approach that is described in Section 2.
4 An illustrative example
Consider the visualization of the reign periods of European leaders of statesduring the time period [1900 − 1910]. Furthermore, assume that some of theseperiods were ill-known to the person who entered them in the database. The fullreign periods can be summarized in the following table:
Number Country Name support([a, b]) core([a, b])
1 Belgium Leopold II [1865, 1908] [1865, 1908]2 Belgium Albert I [1908, 1934] [1908, 1934]3 France Loubet [1898, 1907] [1899, 1906]4 France Fallieres [1906, 1913] [1907, 1913]5 Italy Vittorio Emanuele III [1899, 1944] [1900, 1945]6 Spain Maria Cristina [1885, 1902] [1885, 1902]7 Spain Alfonso XIII [1902, 1931] [1903, 1930]8 United Kingdom Victoria [1837, 1901] [1837, 1901]9 United Kingdom Edward VII [1901, 1910] [1902, 1909]10 United Kingdom George V [1910, 1935] [1909, 1936]
If these time intervals are represented on a one dimensional time axis, it isdifficult to obtain an overview and insight in the data because there is a lotof overlap between the intervals. Furthermore, if the visualization is furtherrestricted to the time interval [1900, 1910], as is the case in this example, thevisualization problem is even harder.
Using the presented approach, the same information, restricted to the timeinterval [1900, 1910], is visualized as shown in Figure 6 (a) —only the numbers
are presented in the figure—. In the example there is no longer overlap betweenrepresentations. To illustrate the visualization of the Allen relations, the relation‘overlapped-by’ is shown for the record with number 3 (‘Loubet’). All records,which representations are located in the area marked with full lines representleaders of states whose reign period overlapped with the reign period of Loubet —within the restricted period of [1900, 1910]— (these are the records with number7 and 9), for sure. Records, which representations are located in the area markedwith dotted lines represent leaders whose reign period possibly overlapped withthe reign period of Loubet —within the restricted period of [1900, 1910]— (recordwith number 4). All other records represent leaders whose reign period did notoverlap with the reign period of Loubet, for sure.
Remark that on the basis of the full (unrestricted) temporal information inthe table, the reign period of the records with number 1, 5, 6 and 8 also overlapswith the reign period of record 3. These records are, due to the restricted periodof [1900, 1910], not retrieved by the ‘overlapped-by’ relation, but can be obtainedby additionally considering the relations ‘starts’ and ‘started-by’.
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1
2
3
4
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10
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19001901190219031904190519061907190819091910
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2
3
4
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10
(a) (b)
Fig. 6. Visualization of reign periods, restricted to the time period [1900, 1910].
5 Conclusions
In this paper, we have presented an approach to visualize fuzzy time intervalsand the generalizations of the 13 Allen temporal relations. The approach is ageneralization of a method for visualizing regular time intervals, that is closelyrelated to the diagrammatic representation for interval arithmetic as presentedby Z. Kulpa [9, 10], but has the advantage of providing the user with the abilityto work with different choices for the angle α. This supports the constructionof more dynamical interfaces where the user can chose the angle that respondsbest to his/her needs. As illustrated with the real world example, the presentedapproach allows for a better human-computer interaction by providing userswith a more compact visualization of the temporal data and the temporal Allenrelations that exists between these data. By using the approach, the user should
better understand the data and gain new insights, which on its turn should besupportive for the further exploration of the database or data warehouse. Bysupporting fuzzy time intervals, the approach offers some flexibility and allowsto deal with ill-known temporal data, both at the level of the data modellingand at the level of temporal reasoning.
In this paper it is assumed that fuzzy time intervals are modelled by convexfuzzy sets with a normalized trapezoidal membership function. Furthermore,for the sake of symmetry, a fixed angle α of 60◦ is used for the modelling.In general, more general convex membership functions can be used, as well asdifferent choices for α are allowed. Future work will further explore the facilitiesof the model, including a study of its limitations when different choices for α
are made, the modelling of operators like zooming, translations and stretching,the impact of granularities and the visualization of durational and repetitivetemporal information.
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