=Paper=
{{Paper
|id=Vol-354/paper-10
|storemode=property
|title=Spatial information fusion: Coping with uncertainty in conceptual structures
|pdfUrl=https://ceur-ws.org/Vol-354/p36.pdf
|volume=Vol-354
|dblpUrl=https://dblp.org/rec/conf/iccs/Saint-CyrJP08
}}
==Spatial information fusion: Coping with uncertainty in conceptual structures==
https://ceur-ws.org/Vol-354/p36.pdf
Spatial information fusion: Coping with
uncertainty in conceptual structures.
Florence Dupin de Saint-Cyr1 bannay@irit.fr, Robert Jeansoulin2
robert.jeansoulin@univ-mlv.fr, and Henri Prade1 prade@irit.fr
1
IRIT, CNRS, Univ. de Toulouse, 31062 Toulouse Cedex 09, France.
2
Institut Gaspard Monge, CNRS, Univ. Paris Est, 77453 Marne-la-Vallée, France.
Abstract. A logical formalism associating properties to space parcels in
so-called attribute formulas, is proposed. Properties are related through
the axioms of a taxonomy graph, and parcels through a partonomy graph.
Attributive formulas establish relations between parcels and properties,
and we use them to align different taxonomies, over a compatible parton-
omy, using Formal Concept Analysis. We discuss uncertainty in attribu-
tive formulas, which we extend in a possibilistic logic manner, including
two modalities: true everywhere in the parcel, or at least true somewhere.
Then, we discuss how our formalism can perform a possibilistic fusion
on attributive formulas originating from independent sources, based on
the aligned taxonomy. The issues may come from (a) the uncertainty of
sources, (b) the possible inconsistency of fusion results, (c) the use of dif-
ferent partonomies that may not explicit the somewhere or everywhere
reading associated to the information. Key words: spatial information,
ontology, uncertainty, possibilistic logic, fusion.
Aknowledgements: work funded jointly by Midi-Pyrénées and Provence-Alpes-Côte
d’Azur, through the Inter-Regional Action Project no 05013992 “GEOFUSE.
1 Introduction
The management of multiple sources of information raises many fusion problems
due to the uncertainty and the heterogeneity: geographical information combines
all of them [3, 14, 1], one specific aspect being to deal with geo-located parcels
that are shareable by all sources. The “field model”: (x, y) → f (x, y), though
widely used in applications that involve imagery or gridded data, is much too
limited in situations that deal with non quantitative data, such as landscape
analysis. Spatial information may involve a mix of numeric and symbolic at-
tributes, using different vocabularies, from more or less structured, but never
unstructured, dictionaries. The sources may use different space partitions, and
there may exist several kinds of dependencies, then the spatial fusion must keep
consistent with all of them. After our informal discussion of this issue in [4], we
now provide a logical framework for handling spatial and ontological information.
The novelty is to handle the merging of spatial information in the general
setting of logical information fusion.
Because both numeric and symbolic information may be pervaded by uncer-
tainty and imprecision [11], we must allow for “uncertain attributive formulas”,
to express that for any parcel of a given set, we know at some degree that a
property is true. We can also distinguish between what holds everywhere, or
only somewhere in a parcel. Hence, dealing with spatial data requires relatively
powerful representation languages [12]. Ontology is often used for represent-
ing structured vocabularies [9], and merging geospatial information must face
the problem of heterogeneous ontologies [7]. Therefore, terminology integration,
based on learning data, and information fusion, based on multiple space parti-
tions, are two classical steps in many geographical applications.
Following [18], we use a logical framework for processing ontologies, and
“attributive formulas” that link sets of parcels to set of properties. Only three
conditions are required: 1) a label can be a sub-label of another label, 2) a label
is the reunion of its sub-labels, 3) labels referring to the most specific classes
are mutually exclusive two by two. This representation language can express
both ontological information and attributive formulas. But spatial information
may vary in spatial extent even within a parcel. Indeed, we show that while
inheritance relations can safely be integrated by attributive formulas, termino-
logical mutual exclusion cannot, unless under an explicit and precise reading:
everywhere, or somewhere.
2 Geographic ontologies and attributive formulas
In geographic information we should distinguish the geo part, the info part, and
the association that links them (the what, the there and the is, of Quine[15]):
1) the (attributed) space: one space for all applications, but many different
ways to split it into parts. We limit our study to parcels that have a spatial extent,
and to the finite case where, after intersection, the most elementary parcels form
a finite partition of the space. This is often referred to as a partonomy structure.
2) the (attribute) properties: many property domains, more or less indepen-
dent, can serve different purposes. A taxonomy structure can represent a hierar-
chy of properties, reflecting a partial order. A consistent fusion of partial orders
may help to detect, and to remove errors when mixing such structures.
3) the attribution: in a complex observation process, associations are multiple
in general, and largely pervaded by uncertainty on both parcels and properties.
A similar, but informal approach was proposed in [13]: an ontology is sug-
gested building on three main concepts: (1) a partonomy of physical objects of
which the attributes represent most of the relevant information, (2) a simple tax-
onomy of informational objects, (3) a relation between the informational objects
and those physical objects they inform about. Hence the “relational model” is
more appropriate than the “field model”, to represent the property-parcel link.
There are two other basic links that the relational model can satisfactory encode:
property-property (from the knowledge encoded in a property taxonomy), and
parcel-parcel (from a partonomy).
Handling fusion requires further combination. Let { hset of nodesi, ⊆} be a
poset: nodes are concepts, and edges are specialization/subsumption relations.
Let L a propositional logic language built on a vocabulary V with the usual
connectives: ∧, ∨, →.
Definition 1 (poset definition of an ontology). An ontology is a directed
acyclic graph (dag) G = (X, U ). X ⊆ L is a set of formulas (one per concept);
U is a set of directed arcs (ϕ, ψ) denoting that ϕ is a subclass of ψ.
An ontology admits one single source, ⊥, and one single sink >.
Definition 2 (leaves and levels in an ontology). Levels are defined induc-
tively: L0 is the set of formulas that have no predecessor: (⊥, ϕ) ∈ U , called
leaves, Li is the set of formulas that have no predecessor in G \ (L0 ∪ . . . Li−1 ),
etc. Let Γ + (x) and Γ − (x) be the set of successors and predecessors of x.
Moreover, we impose: (a) G: to be a lattice, (b) all the sub-classes of a class:
to appear in the ontology, (c) all the leaves: to be mutually exclusive two by two.
Proposition 1. Providing that:
(1) we add the appropriate formulas and arcs that turn a dag into a lattice;
(2) we add to each not-leave formula ϕ, a sub-formula “other elements of ϕ”;
(3) we split leaves, wherever necessary, to make them mutually exclusive;
then, we can insure conditions (a), (b) and (c), because the operations (1), (2)
and (3) can always be done in the finite case.
Hence, an ontology will be encoded in the following way.
Definition 3 (logical encoding of an ontology). Any dag G = (X, U ) rep-
resenting an ontology can be associated to a set LG of formulas that hold:
1. ∀(ϕ, ψ) ∈ U , it holds that ϕ → W ψ.
2. ∀ϕ ∈ X \ {L1 ∪ L0 }, it holds that ϕ → ϕi ∈Γ − (ϕ) ϕi .
3. ∀ϕ, ψ ∈ L1 , it holds that ϕ ∧ ψ → ⊥.
4. ∀(ϕ, ψ) ∈ X × X, s.t. ϕ ` ψ, it exists a directed path from ϕ to ψ in G.
Rule 1 expresses that an inclusion relation holds between two classes, 2 is a
kind of closed world assumption version of property (b), 3 expresses property
(c), 4 expresses completeness, as follows: if all the inclusion relations are known
in the ontology, hence all corresponding
V paths must exist in G. From this, it
follows that: ∀ϕ ∈ X, ϕ → ϕi ∈Γ + (ϕ) ϕi . and ∀ϕ ∈ X, ϕ → >.
Proposition 2. Given any pair of formulas (ϕ, ψ) ∈ X ×X, the logical encoding
of the ontology G = (X, U ) allows us to decide if {ϕ ∧ ψ} ∪ LG is consistent or
not; and if ϕ ∪ LG ` ψ or not.
This formalization of an ontology [16] can be applied to parcels, to provide
a partonomy, and to properties to provide a taxonomy. Their leaves are named
respectively partons, and taxons. Since we need binary links, our language is
built on ordered pairs of formulas of Li × Ls , here denoted (ϕ, p). Such formulas
should be understood as formulas of Li reified by association with a set of parcels
described by a formula of Ls . In other words, to each formula is attached a set
of parcels, where this formula applies.
Definition 4 (attributive formula). An attributive formula f , denoted by a
pair (ϕ, p), is a propositional language formula based on the vocabulary Vi ∪ Vs
where the logical equivalence f ≡ ¬p ∨ ϕ holds and p contains only variables of
the vocabulary Vs (p ∈ Ls ) and ϕ contains only variables of Vi (ϕ ∈ Li ).
The intuitive meaning of (ϕ, p) is: for the set of elementary parcels that sat-
isfy p, the formula ϕ is true. Observe that there exist formulas built on the
vocabulary Vi ∪ Vs which cannot be put under the attributive form, e.g., a ∧ p1
where a is a literal of Vi and p1 a literal of Vs . The introduction of connectives
∧, ∨ and ¬ does make sense, since any pair (ϕ, p) is a classical formula. From the
above definition of (ϕ, p) as being equivalent to ¬p ∨ ϕ, several inference rules
straightforwardly follow from classical logic:
Proposition 3 (inference rules on attributive formulas).
1. (¬ϕ ∨ ϕ0 , p), (ϕ ∨ ϕ00 , p0 ) ` (ϕ0 ∨ ϕ00 , p ∧ p0 )
2. (ϕ, p), (ϕ0 , p) ` (ϕ ∧ ϕ0 , p); 3. (ϕ, p), (ϕ, p0 ) ` (ϕ, p ∨ p0 )
4. if p0 ` p then (ϕ, p) ` (ϕ, p0 ); 5. if ϕ ` ϕ0 then (ϕ, p) ` (ϕ0 , p)
¿From these rules, we can deduce the converse of 2: (ϕ ∧ ϕ0 , p) ` (ϕ, p), (ϕ0 , p)
and that (ϕ, p), (ψ, p0 ) ` (ϕ ∨ ψ, p ∨ p0 ) and (ϕ, p), (ψ, p0 ) ` (ϕ ∧ ψ, p ∧ p0 ).
Remark: the reification allows us to keep inconsistency local.
3 Fusion of properties as an ontology alignment problem
The vocabulary is often insufficient for describing taxons in a non-ambiguous
way. Conversely there may be no proper set of parcels that uniquely satisfies a
given set of properties. Therefore, only many-to-many relationships are really
useful for representing geographic information. Then, between the parcels of a
given subset Pi of the partonomy, and the properties of a given list Lj excerpted
from the taxonomy, we need classically to build three relations:
- Rs that distributes the subset Pi over its parcels;
- Rp that distributes the subset Lj over its properties;
- Ra made of the attributive formulas: pairs from Rs × Rp (learning samples).
Formal Concept Analysis (FCA [17, 10]) uses Ra to build a Galois lattice,
with all the pairs (extension, intention), named concepts, whose components are
referring to each other bi-univoquely. A partonomy of parcels, and a taxonomy of
properties, can be computed by FCA, from a specific Ra . More interesting is to
discover if some additional knowledge emerges from the fusion of two information
sources: (Rs1 ,Rp1 ,Ra1 ) and (Rs2 ,Rp2 ,Ra2 ). The fusion of partonomies is easy, if
we can neglect data matching issues: the geometric intersections between parcels
of Rs1 and Rs2 , become leaves of the fusion Rs . The fusion of taxonomies is more
difficult: an important literature (semantic web, etc.) converges now to the notion
of ontology alignment [8]. We distinguish: (a) the concatenation Ra = Ra1 +Ra2 ,
(b) the structural alignment that identifies candidate concepts for attributive
formulas, and their partial order (FCA); (c) the labeling of concepts, either from
T1 or T2 , or by coupling (sign &) concepts from both; (d) the decision to keep
or discard these candidate nodes, according to one or several criteria.
In land cover analysis, when experts from two disciplines build a domain
ontology that reflects their respective knowledge, often it results in concurrent
taxonomies, as in Fig.1: taxonomy T1 seems broader than taxonomy T2 , which
focuses on moorlands, and T1 accepts multi-heritage, while T2 doesn’t.
Fig. 1. an example of two taxonomies
One approach -“mutual exclusion”- is to concatenate the taxonomies, un-
der the assumption that they are disjoint, and that only one label is allowed,
from whatever vocabulary: it is the smallest one, but isn’t practicable, e.g.:
agriculture and herbaceous aren’t necessarily exclusive. Another approach -
“cross-product”- is to consider as equally possible, every couple of labels com-
patible with both original partial orders: it doesn’t impose anything, hence, it
doesn’t provide any new information.
Better solution -“aligned taxonomy”- : to use the relation Ra , built for each
p, by concatenating all the attributive formulas (ϕ1 i , p) on T1 , with all (ϕ2 i , p)
on T2 for the same p. A regular FCA algorithm can compute Fig. 2: this more in-
formative solution filters only the concepts that fit with the actual observations,
i.e.: the original nodes plus only 4 new cross-product nodes.
Fig. 2. corresponding aligned taxonomy (solution 3).
4 Representing uncertain geographical information
When uncertainty takes place, attribute values of objects may become ill-known,
and should be represented by distributions over possible values:
-In a relational database, the distributions are defined on attribute domains.
-In formal concept analysis only boolean values can refer to the fact that the
object has, or not, the property.
-In the logical language, formulas are associated to certainty levels that together
define constraints on underlying distributions over interpretations. It allows to
represent disjunctions, and that some alternatives are more likely than others.
We want also to detail the behaviour of a property within a parcel that has
a spatial extent: it can apply either to the whole parcel, or only to a sub-part.
Our attributive language is extended in a possibilistic logic manner, by al-
lowing uncertainty on properties. Let us recall that a standard propositional
possibilistic formula [5] is a pair made of a logical proposition (Boolean), associ-
ated with a certainty level. The semantic counterpart of a possibilistic formula
(ϕ, α) is a constraint N (ϕ) ≥ α expressing that α is a lower bound on the ne-
cessity measure N [6] of logical formula ϕ. Possibilistic logic has been proved to
be sound and complete with respect to a semantics expressed in terms of the
greatest possibility distribution π underlying N (N (ϕ) = 1 − supω|=¬ϕ π(ω)).
This distribution rank-orders interpretations according to their plausibility [5].
Note that a possibilistic formula (ϕ, α) can be viewed at the meta level as
being only true or false, since either N (ϕ) ≥ α or N (ϕ) < α. This allows us
to introduce possibilistic formula instead of propositional formula inside our
attributive pair, and leads to the following definition.
Definition 5 (uncertain attributive formula). An uncertain attributive
formula is a pair ((ϕ, α), p) meaning that for the set of elementary parcels that
satisfy p, the formula ϕ is certain at least at level α.
The inference rules of possibilistic logic [5] straightforwardly extend into the
following rules for reasoning with uncertain attributive formulas:
Proposition 4 (inference rules on uncertain attributive formulas).
1. ((¬ϕ ∨ ϕ0 , α), p), ((ϕ ∨ ϕ00 , β), p0 ) ` ((ϕ0 ∨ ϕ00 , min(α, β)), p ∧ p0 )
2. ((ϕ, α), p), ((ϕ0 , β), p) ` ((ϕ ∧ ϕ0 , min(α, β)), p)
3.A. ((ϕ, α), p), ((ϕ, β), p0 ) ` ((ϕ, min(α, β)), p ∨ p0 )
3.B. ((ϕ, α), p), ((ϕ, β), p0 ) ` ((ϕ, max(α, β)), p ∧ p0 )
4. if p ` p0 then ((ϕ, α), p0 ) ` ((ϕ, α), p); 5. if ϕ ` ϕ0 then ((ϕ, α), p) ` ((ϕ0 , α), p)
Rules 3.A-B correspond to the fact that either i) we locate ourselves in the
parcels that satisfy both p and p0 , and then the certainty level of ϕ can reach
the maximal upper bound of the certainty levels known in p or in p0 , or ii) we
consider any parcel in the union of the models of p and p0 and then the certainty
level is only guaranteed to be greater than the minimum of α and β.
Still, attributive information itself may have two different intended meanings,
namely when stating (ϕ, p) one may want to express that:
– everywhere in each parcel satisfying p, ϕ holds as true, denoted by (ϕ, p, e).
Then, for instance, (Agriculture, p, e) cannot be consistent with (F orest, p, e)
since “Agriculture” and “Forest” are mutually exclusive in taxonomy 1.
– somewhere in each parcel satisfying p, ϕ holds as true, denoted by (ϕ, p, s).
Then, replacing e by s in this example is no longer inconsistent, since in each
parcel there may exist “Agricultural” parts and “Forest” parts.
Note that these two meanings differ from the case where two exclusive labels
such as “Water” and “Grass” might be attributed to the same parcel because
they are intimately mixed, as in a “Swamp”. This latter case should be handled
by adding a new appropriate label in the ontology.
More formally, for a given parcel p in the partonomy, if p is:
-not a leave, (ϕ, p, s) means: ∀p0 , p0 ` p, (ϕ, p0 , s) holds;
-a leave, but made of parts o, (ϕ, p, s) means that ∃o ∈ p, ϕ(o).
Thus, it is clear that inference rules that hold for “everywhere”, not necessar-
ily hold for “somewhere”. Indeed, the rule 2.2 (ϕ, p), (ψ, p) ` (ϕ∧ψ, p) is no longer
valid since ∃o ∈ p, ϕ(o) and ∃o0 ∈ p, ψ(o0 ) doesn’t entail ∃o00 ∈ p, ϕ(o00 ) ∧ ψ(o00 ).
More generally, here are the rules that hold for the “somewhere” reading:
Proposition 5 (inference rules on attributive formulas).
1’. (¬ϕ ∨ ϕ0 , p ∧ p0 , e), (ϕ ∨ ϕ00 , p0 , s) ` (ϕ0 ∨ ϕ00 , p ∧ p0 , s)
2’. (ϕ, p, s), (ϕ0 , p, e) ` (ϕ ∧ ϕ0 , p, s); 3’. (ϕ, p, s), (ϕ, p0 , s) ` (ϕ, p ∨ p0 , s)
4’. if p0 ` p then (ϕ, p, s) ` (ϕ, p0 , s); 5’. if ϕ ` ϕ0 then (ϕ, p, s) ` (ϕ0 , p, s)
where (ϕ, p, s) stands ∀p0 , p0 ` p ∃o ∈ p0 , ϕ(o), and (ϕ, p, e) for ∀o ∈ p, ϕ(o).
Moreover, between “somewhere” and “everywhere” formulas, we have:
6’. ¬(ϕ, p,s) ≡ (¬ϕ, p,e)
Taxonomy information and attributive information should be handled sep-
arately, because they refer to different types of information, and, more impor-
tantly, because taxonomy distinctions expressed by mutual exclusiveness of tax-
ons do not mean that they cannot be simultaneously true in a given area: the
taxonomy-formula (a ↔ ¬b), with a, b ∈ Vi coming from the same taxonomy, dif-
fers from the attributive-formula (a ↔ ¬b, >), applied to every parcel (with the
everywhere reading), since it may happen that for a parcel p, we have (a, p)∧(b, p)
(with a somewhere reading). The latter may mean that p contains at least two
distinct parts, and that ∃o ∈ p, ϕ(o) ∧ ∃o0 ∈ p, ψ(o0 ).
However, subsumption properties can be added to attributive formulas with-
out any problem. Indeed ϕ ` ψ means ∀o, ϕ(o) → ψ(o), and if we have (ϕ, p),
implicitly meaning that ∃o ∈ p, ϕ(o), then we obtain ∃o ∈ p, ψ(o), i.e., (ψ, p).
Thus we can write the subsumption property as (ϕ → ψ, >).
5 Conclusion
Fusing consistent knowledge bases merely amounts to apply logical inference to
the union of the knowledge bases. In presence of inconsistency, another combi-
nation process should be defined and used.
Possibilistic information fusion easily extends to attributive formulas: each
given (ϕ, p) is equivalent to the conjunction of the (ϕ, pi ), on the leaves of the
partonomy, such that pi |= p. We can always refine two finite partonomies by
taking the non-empty intersection of pairs of leaves, and possibilistic fusion takes
place for each pi . Clearly, we have four possible logical readings of two labels a
and b associated with an area covered by two elementary parcels p1 and p2 :
i. (a ∧ b, p1 ∨ p2 ): means that both a and b apply to each of p1 and p2 .
ii. (a ∧ b, p1 ) ∨ (a ∧ b, p2 ): both a and b apply to p1 or both apply to p2 .
iii. (a ∨ b, p1 ∨ p2 ): a applies to each of p1 , p2 or b applies to each of p1 , p2 .
iii. (a ∨ b, p1 ) ∨ (a ∨ b, p2 ): we don’t know what of a or b applies to what
of p1 or p2 . This may be particularized by excluding that a label apply to both
parcels: ¬(a, p1 ∨ p2 ) ∧ ¬(b, p1 ∨ p2 ).
When a and b are mutually exclusive the everywhere meaning is impossible
(if we admit that sources provide consistent information).
Another ambiguity is about if the “closed world assumption” (CWA) holds,
e.g.: if a source says that pi contains Conifer and Agriculture, does it exclude that
pi would also contain Marsh ? It would be indeed excluded under CWA. Also,
CWA may help to induce “everywhere” from “somewhere” information. Indeed,
if we know that all formulas attached to p are ϕ1 , . . . ϕn with a somewhere
meaning: (ϕ1 , p, s) ∧ . . . ∧ (ϕn , p, s)), then CWA entails that if there were another
ψ that holds somewhere W in p, it would have been already said, hence we can jump
to the conclusion that ( i=1,n ϕi , p, e).
Our logical framework also allows a possibilistic handling of uncertainty, and
then a variety of combination operations, which may depend on the level of
conflict between the sources, or on their relative priority [2], can be encoded.
After having identified representational needs (references to ontologies, uncer-
tainty) when dealing with spatial information and restating ontology alignement
procedures, a general logical setting has been proposed. This setting offers a
non-ambiguous representation, propagates uncertainty in a possibilistic manner,
and provides also the basis for handling multiple source information fusion.
As discussed along the paper, the handling of spatial information raises gen-
eral problems, such as the representation of uncertainty or the use of the closed
world assumption, as well as specific spatial problems. A particular representa-
tion issue is related to the need of “localizing” properties. First, this requires
the use of two vocabularies referring respectively to parcels and to properties.
Moreover, we have seen that it is often important to explicitly distinguish be-
tween the cases where a property holds everywhere or somewhere into a parcel:
we have detailed this for fusion purpose, it may be present also when learning
the taxonomy alignment (further research).
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