Trajectory Data Warehouses - Data Warehouse Systems: Design and Implementation - page 469

. We will explain the Trajectory operation when we
discuss the temporal spatial types below.
Another set of operations allows the interaction with the domain and
range .The IsDefinedAt predicate is used to check whether the temporal
function is defined at an instant or is ever defined during a given set of
intervals. Analogously, the predicate HasValue checks whether the function
ever assumed one of the values given as second argument. The operations
AtInstant and AtPeriod restrict the function to a given time or set of time
intervals. The operations InitialInstant and InitialValue return, respectively,
the first instant at which the function is defined and the corresponding value.
The operations FinalInstant and FinalValue are analogous. The Operation At
restricts the temporal type to a value or to a range of values in the range
of the function. The operations AtMin and AtMax reduce the function to the
instants when its value is minimal or maximal, respectively.
For example, IsDefinedAt(SalaryJohn, 2012-06-15) and HasValue
(SalaryJohn, 25) result, respectively, in the Boolean values true and false .
Furthermore, AtInstant(SalaryJohn, 2012-03-15) and AtInstant(SalaryJohn,
2012-07-15) return, respectively the value 20 and '

} { }

2013-01-01) and 20,30

⊥

', because John's salary
is undefined at the latter date. Similarly, AtPeriod(SalaryJohn, [2012-04-01,
2012-11-01)) results in a temporal real with value 20 at [2012-04-01, 2012-
07-01) and 30 at [2012-10-01, 2012-11-01), where the periods have been
projected to the intervals given as parameter of the operation. Further,
InitialInstant(SalaryJohn) and InitialValue(SalaryJohn) return 2012-01-01 and
20 which are, respectively, the initial time and value of the temporal value.
Moreover, At(SalaryJohn, 20) and At(SalaryJohn, 25) return, respectively, a
temporal real with value 20 at [2012-01-01, 2012-07-01) and '

⊥

', because
there is no salary with value 25 whatsoever. Finally, AtMin(SalaryJohn) and
AtMax(SalaryJohn) return, respectively, a temporal real with value 20 at
[2012-01-01, 2012-07-01) and a temporal real with value 30 at [2012-10-01,
2013-01-01).
An important property of any temporal value is its rate of change ,
computed by the Derivative operation, which takes as argument a temporal
integer or real and yields as result a temporal real given by the following
expression:

f ( t + δ ) − f ( t )

f ( t ) = lim
δ→ 0

.

δ

For example, Derivative(SalaryJohn) results in a temporal real with value 0 at
[2012-01-01, 2012-07-01) and [2012-10-01, 2013-01-01). The other operations
of this class will be described in the context of temporal spatial types later
in the chapter.
There are three basic temporal aggregation operations that take
as argument a temporal integer or real and return a real value. Integral
returns the area under the curve defined by the function, Duration returns the