F2.2. Añua

As mentioned in the previous section, the notion of a discrete bounded time domain is used to define how the transformations affect time. In order to describe these transformations in detail, the following formalisation of the time domain is used:

where s and e are the starting and ending instants of the temporal domain, labeled by granularity G; E is the number of instants in the domain (or the temporal extent of the domain), and P “ tp1, . . . , pnu is the set of time points such that any p “ ttj, . . . , tku,

with sG ď tj ă tk ď eG, where i P Z, ti “ Gpiq is an instant labeled by granularity

G. Additionally, the following constraints guarantee that the time domain is discrete and every instant within the bounds exist in the abstraction, ensuring that the time domain is complete as discussed in chapter 4: p: n ď E, 1 ď |p| ď E and E “ ÿ

pPP

|p|. These properties of the abstraction correspond directly to the properties of time used to categorise the interactions, as summarised in table 6.2. The starting and ending instants correspond to the bounds of the time domain, while the extent correspond to the time extent, or the number of instants in that dataset. Granularity (G) reflects the current granularity used to label the time domain, including the order of the labels. The time domain object itself can correspond to the whole time domain or segments of it, with the possibility of multiple time domains T , T1_{, T}2_{, . . . being defined. Although}

some of these properties are redundant because they can be derived from each other, they are used in this chapter to describe how some operators can directly modify a property through various types of interactions, with the changes then propagated to other properties.

Example Consider the following example:

T “ t1999YEAR, 2015YEAR, YEAR, 5, t2011, 2012, 2013, 2014, 2015uu

This instance of the time domain has a lower bound 1999, upper bound 2015, gran- ularity YEAR, extent 5 and the set of time points P corresponding to the instants 2011 and 2015. As it is discussed in the description of the operators, time points can contain aggregations of instants. The following is a valid alternative to P : tr2011, 2012, 2013s, r2014, 2015su. In this case, the time domain was modified to contain two representable time points.

Parameters of transformations

In order to modify the properties of the abstraction, the proposed operators can receive different parameters, illustrated in fig. 6.1. The basic variation of parameters is based on two types of time, instants ti, . . . , tn and duration d, as well as granularities G, H, I, . . ..

t

2011 2012 2013 2014 2015

1

T

Fig. 6.1 Illustration guide for the description of the operators. The instant highlighted in red was passed as a parameter; the blue square bracket with a number on top indicates that it is a duration of length 1 that was passed as parameter; the green colour used in the labels indicates the granularity that was used – additional granularities have different colours, as seen in fig. 6.5.

The parameters allow the transformations to have the same result through different inputs and behaviours. Passing instants as parameters to a transformation may be more suitable for one kind of interaction mechanism rather than passing a duration, for example. Besides these three parameters, certain transformations require different parameters based on their functionality – these other parameters are introduced along with the operators.

In the following sections, each transformation is described along with a function

signature that summarises the transformation along with the input parameters. Due

to the complexity of the results of the transformations and the fact that they do not illustrate the varying behaviours by the use of different parameters, the formalisation of the output is not included. Instead, illustrations with before, during and after figures are used to describe the operators in detail. Additionally, pseudo-code algorithms are included in appendix C in order to facilitate the understanding of the operators.

Preconditions and ambiguity

Applying the operators on the time domain with multiple types of parameters requires a series of decisions on the validity and interpretation of the relationship between them. This includes deciding whether it is logical to use a certain operator in combination with a configuration of time domain or checking if a parameter is temporally valid. In this thesis, the conceptual and pseudo-code descriptions do not include such tests of validity; this is assumed to be decided when the operators or the framework are implemented, when such tests would be embedded in the application. This also includes the use of optional and default parameters; for certain operators, default functionality is suggested to complete the space of possibilities.

The second issue regards the ambiguity that results from matching parameters to the time domain. One simple example is scanning for an instant in the time domain in order to divide it in two at that point: after the instant is found, it can be included in the first subdivision of the time domain or the second subdivision. While this has no direct bearing on the conceptual definition of the operator, which is to divide the time domain, the functionality and results can be radically different. Two possible interpretations are possible: matching instants through lower than or lower than or

equal to relations, as in figure 6.2. Given the division example, in the first case, the

first subdivision would include all instants lower than the parameter instant. In the second case, the first subdivision would include all instants lower or equal than the instant passed as parameter. In this chapter, a lower than interpretation is used for consistency in the descriptions.

lower than lower or equal t t t1 t2 t3 t4 t5 t t1 t2 t3 t4 t5 t1 t2 t3 t4 t5

Fig. 6.2 Different relationship types when matching instants in operators. The upper part of the figure shows that t3 was passed as parameter. In each example, the box

outlines the instants that were matched being lower than or lower than or equal to t3,

respectively.