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Temporal, Spatial, and Spatio-temporal Granularities Gabriele - - PowerPoint PPT Presentation

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions Temporal, Spatial, and Spatio-temporal Granularities Gabriele Pozzani Department of Computer Science, University of Verona, Italy 27th

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal, Spatial, and Spatio-temporal Granularities

Gabriele Pozzani

Department of Computer Science, University of Verona, Italy

27th March, 2009

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Outline

1

Introduction

2

Temporal granularity

3

Spatial granularity

4

Spatio-temporal granularity

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Outline

1

Introduction

2

Temporal granularity

3

Spatial granularity

4

Spatio-temporal granularity

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Starting with an example

Diffusion of avian influenza

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal granularity + . . .

Years:

T

2003 2004 2005

112 cases of avian influenza

is a temporal granularity representing years. a temporal granularity is a partition of the time line each element of the partition is called granule each granule can be used to provide information with a time qualification

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

. . . + spatial granularity = . . .

Nations: is a spatial granularity representing world nations. a spatial granularity is a partition of a space a granule in the granularity represents a region of the partition each granule can be used to provide information with a spatial qualification

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

. . . = spatio-temporal granularity

1997 2003 2004 2005 2006

A spatio-temporal granularity represents changes in time of a spatial granularity: it associates a space to time it can be used to provide information with a spatio-temporal qualification

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Outline

1

Introduction

2

Temporal granularity

3

Spatial granularity

4

Spatio-temporal granularity

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal granularity

Formal definition A time domain is represented as a pair (T, ≤) where T is a set of time instants ≤ is a total order over T A time granularity G is defined as a mapping from an index set I to the power set of the time domain T such that: if i < j and G(i) and G(j) are non-empty, then each element of G(i) is less than all elements of G(j); if i < k < j and G(i) and G(j) are non-empty, then G(k) is non-empty. This definition was developed mainly by Bettini et al. since the last years of 1990’s [BettiniDESW97] it is well-knows it is accepted by whole research community

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal granularity

Granules A granule is a set of instants perceived and used as an indivisible entity. A granule can represent single instants, a time interval or a set of non-contigous instants. WorkingDays: ij ij+1 ij+2 Note: granules are total ordered, like time instants, hence navigation among granules is totally and uniquely defined.

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal granularity

Related notions Origin: (of granularity G) is a specially designated granule, e.g., G(0). Anchor: is the greatest lower bound of the set of time domain elements corresponding to the origin. Image: of a granularity is the union of the granules in the granularity. Extent: of a granularity is the smallest interval of the time domain that contains the image of the granularity. Formally, it is the set {t ∈ T|∃a, b ∈ Im, a ≤ t ≤ b} where T is the time domain and Im is the image of the granularity.

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal granularity

Relationships (I) Several relations can subsist between two different granularities: GroupsInto FinerThan Partitions

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Temporal granularity

Relationships (II) Subgranularity

H G

7 2 6 8 3 9 10 4

ShiftEquivalent

H G

3 4 7 8 5 9 2 6

CoveredBy

H G

7 6 8 9 10 3 4 5 2

GroupsPeriodicallyInto

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Related notions (II)

By using relations between granularities, we can define the following notions: Bottom granularity (w.r.t. a relationship g-rel): given a set of granularities having the same time domain, a granularity G in the set is a bottom granularity with respect to g-rel, if G g-rel H for each granularity H in the set Lattice (w.r.t. a relationship g-rel): a set of granularities s.t. for each pair of granularities in the set there exists a least upper bound and a greatest lower bound w.r.t. g-rel Calendar: a set of granularities over a single time domain that includes a bottom granularity with respect to GroupsInto

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Temporal multi-granularity (I)

An example This is a temporal multigranular system. It refers to: years months days weeks working days

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Temporal multi-granularity (III)

Granule conversions

January 2009

Month Day

  • 2. Down conversion

Week

....

  • 4. Next conversion
  • 1. Find the GLB

....

  • 3. Up conversion

image from [NingWJ02]

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Temporal multi-granularity (II)

Information conversion

T T

8 cases 3 cases 12 cases 1 case 4 cases 2 cases 2 cases 2 cases 7 cases 8 cases 9 cases 12 cases

GroupsInto Months: Years:

70 cases

Multigranularity and relationships allow one: to transfer information from a granularity to a related one to integrate information associated to different granularities and coming from different sources

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Calendar algebra

In order to complete the framework for temporal granularity some

  • perations was defined [NingWJ02].

This framework is called calendar algebra. These operations allow one to build new granularities from other

  • nes:

grouping-oriented operations combine the granules of a given granularity to form the granules of a new granularity granule-oriented operations construct a new granularity choosing some granules from a given one set operations are based on the viewpoint that each granularity corresponds to a set of granules mapped from the labels. They extend over time granularities the usual set operation.

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Outline

1

Introduction

2

Temporal granularity

3

Spatial granularity

4

Spatio-temporal granularity

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

The notion of spatial granularity

Temporal vs. spatial granularities There are deep differences between spatial and temporal granularities granules:

usually, temporal granularities are “periodical” spatial granularities are not periodical and their granules may have any possible shape

relations between granules:

elements of the time domain (instants) and time granules are usually ordered the spatial domain supports several relations (topological relations, direction based relations,. . . )

These differences require to represent and manage temporal and spatial granularities in a different way

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The notion of spatial granularity

Layers Spatial granularities have two layers:

1

the spatial domain, over which we identify the regions defining granules

2

the index structure used to access and manage granules

Index structure Domain Spatial

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The notion of spatial granularity

Given a spatial domain, a spatial granularity is made up of: a multidigraph MG

nodes represent granules edges represent spatial relations between granules

a mapping G that associates to each node its spatial extent a mapping DA that defines for each edge label its spatial meaning

the edges reflect the spatial relations

Spatial Domain

N E E SE NE N

Multidigraph

S E

G

l1 l4 l2 g1 g4 g3 l3 g2

Multidigraph MG: nodes: {l1, l2, l3, l4} edges: {(l1, l2)SE, (l2, l3)N, (l2, l3)NE, (l2, l4)NE, (l3, l4)SE, (l3, l4)E} G = {l1 → g1, l2 → g3, l3 → g3, l4 → g4} Mapping DA: N →{(A, B)|somePoint(B) is up north of center(A)} . . .

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

The notion of spatial granularity

An example

  • S. Lucia

Borgo Milano Node Label Labeled Edge Parona Node Mapping Granule’s Representative Spatial Domain S Mezzane di Sotto S.Martino Negrar S.Giovanni Node NE S.Mauro Castel d’Azzano Verona E SE NE N Granule

  • S. Massimo
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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Relations between spatial granularities (I)

Between spatial granularities we can define several relations similar to those defined for temporal granularities: GroupsInto FinerThan Subgranularity Partition CoveredBy Disjoint Overlap Some ones have also a strong version considering also the existing spatial relations between granules.

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Relations between spatial granularities (II)

An example

GroupsInto R

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Operations between spatial granularities

Also operations over spatial granularities have been defined. These

  • perations allow one to build new granularities from existing ones.

Villages CholeraAreas , ) SelectIntersect(

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Using spatial granularities, an example

The John Snow’s study about cholera cases in London, 1894

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Using spatial granularities, an example

The John Snow’s study about cholera cases in London, 1894

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Using spatial granularities, an example

The John Snow’s study about cholera cases in London, 1894

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

Outline

1

Introduction

2

Temporal granularity

3

Spatial granularity

4

Spatio-temporal granularity

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

The notion of spatio-temporal granularity

Spatial evolution We have to represent a spatial evolution: for every time instant in a time domain T, a spatial evolution E maintains the spatial granularity (belonging to a family GF) that is valid at that time ∀t ∈ T : ∃Gk

S ∈ GF : E(t) = Gk S

we will manage the split and merge operations and trace the granules history w.r.t. the previous versions maintain links between the versions of each spatial granule at the different instants

T

t GF

Gk

S

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

The notion of spatio-temporal granularity

A spatio-temporal granularity is made up of a temporal granularity GT and a spatial evolution E over the temporal domain of GT: GST =<GT, E > time instants, and then spatial granularities in the evolution, are “grouped” in temporal granules we can use temporal granules to manage and reason on spatial granularities

it is not simply a spatial granularity versioning system

G

T i+2 i+1 i

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Outline Introduction Temporal granularity Spatial granularity Spatio-temporal granularity Conclusions

The notion of spatio-temporal granularity

Advantages Associating a spatial granularity to each time instant: allows the spatial information to change during a temporal granule represents homogeneously several spatial evolution semantics

discontinuous changes (e.g. administrative divisions changes) “continuous” changes (e.g. pollution areas evolution)

allows one to build, reason on, and manage spatio-temporal granularities without loss of information just partitioning the spatial information associated to instants accordingly to the temporal granularity we are interested in

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Relations between spatio-temporal granularities

Relations are useful in order to perform spatio-temporal reasoning it is possible to translate information expressed by using a granularity into an equivalent information represented by another granularity

AlwaysGroupsInto

Provinces Regions

. . . . . .

t5 t4 t3 t2 t1 t1 t2 t3 t4 t5

. . . . . .

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Operations over spatio-temporal granularities

Operations over spatio-temporal granularities allow one to build new granularities from existing ones. They extend spatial operations to spatio-temporal granularities combining them with time.

i+2 i i+1 i+2 i i+1 T T Subset Spatial

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Thanks for your attention

Questions?

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References

  • A. Belussi, C. Combi, and G. Pozzani.

“Towards a formal framework for spatio-temporal granularities”, Temporal Representation and Reasoning, 2008. TIME ’08. 15th International Symposium on, pp. 49–53, June 2008. P . Ning, X. S. Wang, and S. Jajodia. “An algebraic representation of calendars”,

  • Ann. Math. Artif. Intell, vol. 36, no. 1-2, pp. 5–38, 2002.
  • C. Bettini, C. E. Dyreson, W. S. Evans, R. T. Snodgrass, and
  • X. S. Wang,

“A glossary of time granularity concepts”, LNCS, vol. 1399, pp. 406–413, 1998.

  • E. Camossi, M. Bertolotto, and E. Bertino.

“A multigranular object-oriented framework supporting spatio-temporal granularity conversions”, International Journal of Geographical Information Science,

  • vol. 20, no. 5, pp. 511–534, 2006.