# QB4 MobOLAP: A Vocabulary Extension for Mobility OLAP on the Semantic Web
## 3 Research Methods
### Mobility Operator
For example, AtInstant(Delivery323, \"2019-09-03 07:05:04\") returns 0 and AtPeriod(Delivery323, \"2019-09-03 07:05:04\", \"2019-09-03 07:05:13\") returns a temporal float with a value of 0 at (\"2019-09-03 07:05:04\", \"2019-09-03 07:05:10\") and a value of 20 at (\"2019-09-03 07:05:11\", \"2019-09-03 07:05:13\"). InitialInstant and InitialValue return the first instant at which the function is defined and the corresponding value. For example, InitialInstant(Delivery323) returns \"2019-09-03 07:05:04\" and InitialValue (Delivery323) returns 0. FinalInstant and FinalValue are equivalent. For example, FinalInstant(Delivery323) returns \"2019-09-03 07:05:13\" and FinalValue (Delivery323) returns 20. The At Operation constrains the temporal type to a single value or a range of values within the function's range. The AtMin and AtMax functions return to their minimal and maximal values, respectively. For example, At(Delivery323,20) returns a value of 20 at (\"2019-09-03 07:05:11\", \"2019-09-03 07:05:13\"), AtMin(Delivery323) returns a value of 0 at (\"2019-09-03 07:05:04\", \"2019-09-03 07:05:10\"), and AtMax(Delivery323) returns a value of 20 at (\"2019-09-03 07:05:11\", \"2019-09-03 07:05:13\"). These operations can be summarized as follows:  
AtInstant: moving(\(\alpha\)), instant time \(\rightarrow\)\(\alpha\)  
AtPeriod: moving(\(\alpha\)), period of time \(\rightarrow\) moving(\(\alpha\))  
InitialInstant: moving(\(\alpha\)) \(\rightarrow\) instant time  
InitialValue: moving(\(\alpha\)) \(\rightarrow\)\(\alpha\)  
FinalInstant: moving(\(\alpha\)) \(\rightarrow\)\(\alpha\)  
Figure 2: Examples of temporal-spatial types: delivery Path of trucks. FinalValue: moving(\(\alpha\)) \(\rightarrow\)\(\alpha\)  
At: moving(\(\alpha\)), \(\alpha\)\(\rightarrow\) moving(\(\alpha\))  
AtMin: moving(\(\alpha\)) \(\rightarrow\) moving(\(\alpha\))  
AtMax: moving(\(\alpha\)) \(\rightarrow\) moving(\(\alpha\))  
StartSequence moving(\(\alpha\)) \(\rightarrow\) moving(\(\alpha\))  
FinalSequence moving(\(\alpha\)) \(\rightarrow\) moving(\(\alpha\))  
A critical property of any temporal value is its rate of change function. The rate of change function is computed using the derivation operation, which takes a temporal integer or real as an argument and returns a temporal real. Some examples of rate of change functions are:  
Derivative: moving(\(\alpha\)) \(\rightarrow\) moving(real)  
Speed: moving(\(\alpha\)) \(\rightarrow\) moving(real)  
Temporal Aggregation operations is an operator that takes a temporal integer or real and yields a temporal real. Some temporal aggregation operations are:  
Integral : moving(\(\alpha\)) \(\rightarrow\) real \(\int_{T}f(x)dx\). Duration : moving(\(\alpha\)) \(\rightarrow\) real \(\int_{T}dx\). Length : moving(\(\alpha\)) \(\rightarrow\) real \(\int_{T}\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}dx}\). TWAvg: moving (\(\alpha\)) \(\rightarrow\) real Integral/Duration  
TWVariance : moving(\(\alpha\)) \(\rightarrow\) real \(\int_{T}\sqrt{\dfrac{\left(f(x)-TAvg)^{2}\right)}{Duration}}\)  
TWStDev : moving(\(\alpha\)) \(\rightarrow\) real \(\sqrt{TVariance}\)  
TWMin: moving(\(\alpha\)) \(\rightarrow\) real Min(RangeValue(\(\alpha\)))  
TWMax: moving(\(\alpha\)) \(\rightarrow\) real Max(RangeValue(\(\alpha\)))  
The lifting function is a function that generalizes non-temporal data types to mobility data types. The lifting operation over non-temporal types substitutes its argument types for each time type and returns a temporal type. The operation more than and equal (\(\geq\)) has lifted versions; for instance, one or both of its parameters can be of the temporal type, resulting in a time boolean. Aggregation operations can be lifted in the same way. A lifted Avg operation, for example, merges a set of temporal reals to produce a new temporal real in which the average is computed at each instant. Figure 3 depicts a sample of the TAvg and TSum aggregation of the truckload delivery.