“TimeOntoImperfection”: An OWL 2 ontology to representing temporal data imperfection using imperfection theories

Abstract

Dealing with temporal data imperfection is a crucial issue in several application domains. In fact, failure to handle these imperfections can have significant consequences and lead to incorrect analysis and decision-making. This is particularly true when handling imperfect temporal data inputs in applications for Alzheimer’s patients as a real example. In this context, there is a need for a global ontology that provides a semantic representation of temporal data imperfection. In the literature, there is a big number of ontologies that represent data. Some represent only perfect temporal data. Some others represent imperfect data but not temporal ones. To the best of our knowledge, there is no ontology that represents temporal data imperfection. In this paper, we represent “TimeOntoImperfection”, a usable global ontology that represents four types of imperfection: imprecision, uncertainty, both uncertainty and imprecision and conflict. We describe the structure of “TimeOntoImperfection”, then we conduct a case a study in which we illustrate the usefulness of our ontology. Finally, we introduce the validation part in the context of CAPTAIN MEMO - an ontology based memory prothesis dedicated to alzheimer patients- and we discuss the encouraging results derived from the evaluation step.

Keywords

Ontology temporal data imperfection temporal reasoning uncertainty imprecision conflict possibilistic ontology fuzzy ontology probabilistic ontology

1 Introduction

Bringing up the subject of data imperfection comes from the need to have “the right to make mistakes”, from inaccuracy, from doubt, from the desire to say “maybe” to things that we are not sure about and of speaking imprecisely. All data may be affected by imperfection. Temporal data is no exception. Indeed, the order relation established on these data can help to detect errors. Temporal data touch all life domains. For example, in the context of an application intended for people suffering from Alzheimer’s disease, specific features add complexity to the processing of data collection. Indeed, patients with Alzheimer’s disease suffer from memory discordance, changes in attention and concentration and a decrease in their abilities [1]. As a result of these changes and disabilities, the data provided by patients with Alzheimer’s disease are mostly subject to different types of imperfections. For example, an alzheimer patient can say “I think it was from 2013 to 2020”. In this information, “uncertainty” as a kind of imperfection figures. Hence, the level of reliability of a user can be very fluctuating.

In the litterature, different ontologies have already been proposed in order to describe perfect temporal data such as OWL-Time ontology 1 . These ontologies can be used or merged with other domain ontologies which need to be extent to represent temporal aspect. However, to our knowledge, there is no ontology that represents imperfect temporal data.

This paper presents and describes “TimeOntoImperfection”, a top-level ontology for temporal data imperfection. It is published and can be downloaded at the “TimeOntoImperfection home page” https://cedric.cnam.fr/isid/ontologies/files/TimeOntoImperfection.html. It reduces to four types of imperfection: imprecision, uncertainty, uncertainty and imprecision at the same time and conflict. It can be merged with other ontologies that need to be extent to dealing with imperfect temporal data. We describe all the component of our ontology: concepts and properties, as well as the reasoning and inference via the SWRL rules. The paper also presents an evaluation of “TimeOntoImperfection” in the context of CAPTAIN-MEMO [2], a memory prothesis dedicated to Alzheimer patients to palliate their mnesic problems. It is based on personLink ontology 2 [3] that describes family relationship.

The paper is structured as follows: In Section 2, a state of the art in the context of temporal data, typology of temporal data imperfection and ontology is reviewed. In section 3, we describe “TimeOntoImperfection” ontology. Section 4 describes a validation of the ontology via an implemented prototype. Section 5 presents an evaluation of the presented work in the context of CAPTAIN-MEMO memory prothesis. Finally, Section 6 summarizes the conclusions of the study as well as future lines of work.

2 State of the art

The present work is closely related to the three following research areas: (i) temporal data, (ii) data imperfection and (iii) ontology. In this section, we present preliminary concepts and related work regarding the mentioned areas as well as the intersection between them.

2.1 Temporal data

Temporal data has an order relation. This can help to detect errors. Temporal data can be represented using time points or time intervals. They can also be conceptualized as quantitative (precisely defined, for example using dates) or qualitative (designates qualitative temporal relationships as “before”). In the following, we explain temporal elements, temporal relations as well as temporal ontologies.

2.1.1 Temporal elements

Intervals or points? The most fundamental question in any temporal model is the choice of the basic temporal primitive. The literature mentions two usual primitives [4]: time instants (or time points, chronos) and time intervals. A temporal model can be based on one of the two primitives or on both at the same time. In the literature, there is an ongoing debate as to which primitive is the most appropriate, with no clear winner. While in the temporal database research community, time instant is more commonly used [5], in the artificial intelligence community time interval or mixed approaches are most popular [4]. In [4], it was shown that the choice of basic primitive mainly depends on the application requirements. [6] believe that time intervals are closer to human intuitive perception of time.

According to the survey presented by [7] and the W3C working group that designed the TIME ontology, the time element is an entity used as a constituent element of a theory of time. These items could be “Time Points” and “Time Intervals”.

TimePoints. A timeline is made up of points in time or instants. The value of a moment reflects its position on the timeline. According to [8], if the instant is the present then the value is equal to zero, if the instant is strictly before the present, the value is negative and if the instant is strictly after the present then the value is positive.

Time intervals. [9] considers time intervals as primitive entities. It considers these elements as those that have a duration by referring to a “during” relationship allowing to define the hierarchies of intervals. Virtually all other proposed approaches consider intervals not as their primitive elements, but as point-based structures. According to [7], intervals could be convex (co) or non-convex (n), open (o) or closed (cl), bounded (b) or unlimited (u) and fuzzy (f) or crispy (cr). For example, [10] admit that time intervals are open, half-open or closed; bounded or unlimited - i.e. having one of the following forms: [a, b], [a, b), [a, ∞), (a, b], (a, b), (a, ∞), where a ≤ b for a, b ∈ R≥ 0.

2.1.2 Temporal relations.

Temporal relations could be: Points-based relation, intervals-based relation, Point-interval relation and Interval-point relation.

In what follows, we present the two most cited qualitative temporal formalisms: Allen’s interval algebra [9] and Vilain and Kautz’s point algebra [11].

Allen’s Interval Algebra. Allen defined 13 basic relations which describe all the possible ways of ordering the endpoints of two precise time intervals. These relations, called Allen relations, are usually denoted by: Before, After, Meets, Met-by, Overlaps, Overlapped-by, During, Contains, Starts, Started-by, Ends, Ended-by, and Equals. Let I = [I^-, I⁺] and J = [J^-, J⁺] be two precise time intervals not reduced to a point. Allen’s relations can be defined from the three ordinary relations between the limits of the two intervals to be positioned relative to each other. For example, the statement I Starts J corresponds to: (I^- = J^-) ∧ (I^- < J⁺) ∧ (I⁺ > J^-) ∧ (I⁺ < J⁺). Table 1 summarizes the 13 Allen relations between the two precise time intervals I and J.

Vilain and Kautz. The second fairly widely used qualitative temporal model is the one introduced by Vilain and Kautz. In this model, three qualitative temporal relations make it possible to compare two instants T1 and T2: T1 Precedes (Precedes) T2, T1 Follows (Follows) T2 and T1 is Same (Equal to) T2, as shown in Table 2.

Table 1
Allen’s relations between two precise time intervals

Relation Inverse Definition

A Before B B After A A⁺ < B^-

A Meets B B MetBy A A⁺ = B^-

A Overlaps B B OverlappedBy A (A^- < B^-) ∧ (A^- < B⁺)

∧ (A⁺ < B⁺)

A Starts B B StartedBy A (A^- = B^-) ∧ (A⁺ < B⁺)

A During B B Contains A (B^- < A^-) ∧ (A⁺ < B⁺)

A Ends B B EndedBy A (B^- < A^-) ∧ (A⁺ = B⁺)

A Equals B B Equals A (A^- = B^-) ∧ (A⁺ = B⁺)

Relation	Inverse	Definition
A Before B	B After A	A⁺ < B^-
A Meets B	B MetBy A	A⁺ = B^-
A Overlaps B	B OverlappedBy A	(A^- < B^-) ∧ (A^- < B⁺)
		∧ (A⁺ < B⁺)
A Starts B	B StartedBy A	(A^- = B^-) ∧ (A⁺ < B⁺)
A During B	B Contains A	(B^- < A^-) ∧ (A⁺ < B⁺)
A Ends B	B EndedBy A	(B^- < A^-) ∧ (A⁺ = B⁺)
A Equals B	B Equals A	(A^- = B^-) ∧ (A⁺ = B⁺)

Table 2

Vilain and Kautz points algebra

Relation	symbol	Inverse
T1 Proceed T2	<	T2 Follows T1
T1 Follows T2	>	T2 Proceed T1
T1 Equals T2	=	T2 Equals T1

2.1.3 Temporal ontologies.

OWL-TIME [12] propose an ontology of time called “owl-time” which describes temporal content, including instants, intervals, durations and calendar terms. Several researchers like [13] adopt an approach that reuses the concepts of time defined in OWL-Time. However, this representation does not create a temporal model to consistently describe all the temporal information of a system. Furthermore, it does not specify how these concepts can be used to represent object properties changing over time, nor does it specify how to reason about the qualitative relationship of time intervals and instants.The Owl-Time ontology is based on the “TemporalEntitiy” concept, which includes only two subclasses: “Instant” and “Interval”. Intervals are considered as things whose range has interior points. An instant is considered as a point in time having no interior points and is also considered as an interval with zero extent having the same beginning and the same end; - The “begins” and “ends” predicates are relations between instants and temporal entities; - There is a "before" relation on temporal entities, which gives directionality to time. If the temporal entity T1 is before the temporal entity T2, then the end of T1 is before the beginning of T2. Thus, “before” can be considered as fundamental for the instants and derived for the intervals; - Ways to specify date and time formats in OWL-Time include the following properties of a "DateTimeDescription": "unitType, year, month, week, day, dayOfWeek, dayOfYear, hour, minute, second, and timeZone”. The “unitType” property specifies the time unit type of the datetime description.

SWRL-Temporal [14]. is a temporal model for representing interval-based information in OWL. SWRL-Temporal defines the “Granularity” class, which has “Years, Months, Days, Hours, Minutes, Seconds and Milliseconds” instances, to express the granularity of the temporal representation. It also defines durations using the “Duration” class, with two functional properties: “hasCount” with an integer range and a “granularity” property with a granularity range. The valid time of the facts and is represented using the “ValidTime” class, which in turn has a “hasGranula-rity” property with a range of granularity. “ValidTime” has two subclasses which are “ValidInstant” and “ValidPeriod”, representing instants and intervals respectively. It also has a “hasTime” data type property with a “dateTime” range, while ValidPeriod has “hasStart” and “hasFinish” data type properties with a dateTime range representing the start and end of the period. corresponding interval. Additionally, SWRL-Temporal contains SWRL rules for temporal reasoning. These predefined rules are intended to check whether predicates comparing a specific duration to that of a validity interval (e.g. durationLessThan, durationEqualTo) are true. Similar predicates are used to verify whether or not an Allen relation holds between two intervals. However, SWRL-temporal does not support qualitative defined intervals.

SOWL [15]. The ontology attempts to overcome the problems of some previous approaches, such as TOWL [16] (i.e. TOWL was not compatible with OWL and did not support supports representation of qualitatively defined intervals), offering a W3C solution for temporal representation. In SOWL, dates are represented using the “dateTime” data type and SWRL rules are used for temporal reasoning. Time concepts are defined using the OWL-time ontology. 4D fluent [17] is the model used to represent dynamic objects through the use of the TimeSlice class and the tsTimeSlice property. An alternative version based on n-ary relations is also proposed in [18]. Allen relations present object properties between intervals. In addition to this, a path consistency based reasoning mechanism has been implemented using SWRL to infer implicit relationships and check the consistency of temporal assertions. The reasoning is sound, complete and tractable for the supported set of interval relations [15]. In addition to the reasoning support mentioned above, query support to SOWL by adding temporal operators on top of SPARQL is proposed in [15]. More precisely, the authors develop a query language based on SPARQL which proposes temporal operators such as “at” (time-instance) with the corresponding temporal representation.

All these ontologies that represent the temporal aspect in a precise way and do not consider the imperfections that can affect the temporal data.

2.2 Temporal data imperfection

Data can be perfect, which means data without errors. It can also be subject to several types of imperfections. According to [19], “imperfection” is used as the most general label. The imperfection may be due to errors or semantic imperfections. We consider the word "imperfection" to be more general than the word "error". An error is a type of imperfection. [20] consider as “errors” any typing or spelling error when data entry is manual or associates values to the wrong instances due to software misbehavior (in the case of the use of applications or software). An “error” represents any syntactical error or input error in an application or software used. A "semantic imperfection" means any inaccuracy, incompleteness, confusion, uncertainty, etc. that touches the data. We classify data into data entered by users and other data from applications.

Uncertainty, according to [21] refers to the “quality” of the information reflecting the reality of its realization. Information is said to be uncertain if it is impossible to judge the binary truth (true/false) of this information. Data may contain uncertainties. This means that the exact value is (partially) unknown, however, usually some knowledge is present anyway, perhaps describing the value. Vagueness is a type of imperfection that refers to the "content" of information. Thus, when we do not have variables, characteristics, etc. to clearly describe the content of the information, the imperfection is called imprecision. The ambiguity of information is the fact that it leads to two or more interpretations, because its limits are not known. Incompleteness represents the lack of information provided by the source. It can be measured by the difference between the amount of information actually provided by the source and the amount of information the source should provide. Conflict characterizes at least two informations leading to contradictory and therefore incompatible interpretations.

2.2.1 Typology of temporal data imperfection

There are a large number of data imperfection typologies. Authors such as [22] and [23] propose generic typologies while others, such as [24, 25] and [26], offer typologies for specific domains. Some of these typologies correspond better to a reality than others. In addition, there is no fixed definition to qualify imperfect data, such as uncertainty and imprecision. We also note that the majority of these typologies share three common concepts which are imprecision, uncertainty and incompleteness. The types of imperfections are interdependent. According to [23], incompleteness is a source of uncertainty and imprecision can also be associated with incompleteness. According to [27], imprecision always refers to incompleteness. Imprecision can be a source of uncertainty, but not necessarily [27].

Temporal data may have more imperfections than those offered in the generic typologies due to the complexity of this type of data and the specificity it contains (i.e. it may be numerical or based on natural language). They may also depend on the general context and may be subject to several factors likely to interfere in the specification of the type of imperfection. Thus, generic typologies cannot be adapted to temporal data (i.e. they are inadequate). For example, if we have the information "I forget the last time I visited my uncle who lives in Japan", the temporal data refers to a “missing”, which is a type of imperfection that none of the existing generic typologies includes. Another example is “On the first day of the week we will have a meeting”. In this example, the temporal data indicates a "circumlocution", which is another type of imperfection that none of the generic typologies contain. Moreover, to our knowledge, there is no typology of temporal data imperfections.

In our previous work [28], we proposed a typology of temporal data imperfection. It is classified into direct imperfections of both numeric temporal data and natural language based temporal data, indirect imperfections that can be deduced from the direct ones and granularity (i.e., context - dependent temporal data) which is related to several factors such as person’s profile and multiculturalism. The direct imperfections can be deduced directly from the given data: uncertainty, typing error, conflict, imprecision, missing, circumlocution and uselessness. The Indirect imperfections are those that can be generated from the direct ones. We distinguish three types of indirect imperfections: (i) The incoherence can be generated from the uncertainty and typing error. (ii) The incompleteness can be generated from the imprecision and the missing. (iii) The redundancy can be generated from the uselessness. Granularity is the general context of the given temporal data which can determine the type of imperfection. Figure 1 depicts our typology.

Fig. 1

Our typology of temporal data imperfection.

2.2.2 Ontologies of data imperfection

A classical ontology is a perfect ontology that does not model any imperfection [29]. However, the world includes inaccuracies and imperfections that cannot be represented by classical ontologies.

To enable agents to handle imperfection, an extension of ontologies that has the ability to support imperfect knowledge is mandatory. These imperfections can be uncertainty, imprecision, conflict, etc.

Probabilistic Ontology

Several definitions of probabilistic ontologies have been proposed. [30] defines probabilistic ontology as “an explicit formal representation of disclosed knowledge of knowledge about a domain of application that includes: the types of entities existing in the domain; the properties of these entities; relationships between entities; the processes and events that occur with these entities; the statistical regularities that characterize the domain; inconclusive, ambiguous, incomplete, unreliable and dissonant knowledge; uncertainty regarding all forms of reported knowledge; where the term entity designates any concept (real, fictitious, concrete or abstract) that can be described and reasoned in the field of application”. According to [31], probabilistic ontologies provide a structured, shareable, and principled way to comprehensively describe knowledge about a domain and the uncertainty surrounding it. They also expand the possibilities of standard ontologies by introducing the requirement for a correct representation of statistical regularities and uncertain proofs about entities in an application domain. Ideally, the representation is in a format that can be read and processed by a computer. According to [32] probabilistic ontologies are used to describe in detail the knowledge of a domain and the uncertainty associated with this knowledge in a reasoned, structured and shareable way. [33] define probabilistic ontology simply as a classical ontology containing uncertain knowledge. In other words, an ontology is only a probabilistic ontology if it contains at least one probabilistic component [33].

Possibilistic Ontology

In the literature, there are not many works defining possibilistic ontology. In the temporal domain, there is no possibilistic temporal ontology. [34] proposes a possibilistic ontology based on WordNet in the information retrieval process. According to [35], in a possibilistic ontology, the relations between the concepts are each endowed with a degree of possibility and a degree of necessity which respectively reflect to what degree is this relation possible? And how certain is this relationship [36] propose a possibilistic extension of OWL 2, called Poss-OWL 2, to deal with uncertain geographic ontology. [37] propose a possibilistic extension of the SROIQ(D) description logic called π-SROIQ(D) by incorporating a level of certainty for different elements of the SROIQ(D) DL to deal with uncertain geographic information. [38] propose a possibilistic ontology based on a case-based reasoning (CBR) approach to perform a possibilistic semantic retrieval algorithm that handles ambiguity and uncertainty problems.

Fuzzy Ontology

Fuzzy ontology is an extension of precise ontology according to [40, 41] and [39]. It is based on the integration of fuzzy logic in the definition of precise ontology to represent and reason about imprecise data [42]. It is very useful in many areas. According to [43], a fuzzy ontology has nine components: (1) precise concepts, (2) fuzzy concepts, (3) precise semantic relations, (4) fuzzy semantic relations, (5) precise conceptual relations, (6) fuzzy conceptual relations, (7) fuzzification relations, (8) axioms and (9) instances. The precise components keep the same definitions. [53] presents a fuzzy-based ontology, called “MemoFuzzyOnto” to represent and reason about imprecise temporal data.

Evidential Ontology

In the literature, very few works define an evidential ontology. [44] propose “BeliefOWL” which is a new approach to represent uncertainty in an OWL ontology. They only considered the case of the inclusion of uncertainty in the classes. This uncertainty is modeled via the Dempster-Shafer theory of evidence. They extended the classes of OWL ontologies with masses of beliefs, then they applied structural translation rules in order to obtain a DAG (Directed acyclic graph).

2.3 Discussion

Most of the cited ontologies of data imperfection do not consider the temporal dimension. Besides, most the cited temporal ontologies do not consider the imperfections that my affect temporal data. Some of the cited ontologies consider only one type of imperfection, such as imprecision, ant not all of them.

In our previous works [45 , 52] and [49], we extent the 4D-fluent approach, the OWL-ontology and the Allen’s interval algebra to represent and reason about many defined types of imperfections defined in the presented typology, that may affect temporal data, using probability, possibility and evidence theories. Based on these approaches, we proposed many ontologies of respectively, imprecise, uncertain, both imprecise and uncertain, and conflicting temporal data [50].

Based on the proposed ontologies, we create an ontology, named “TimeOntoImperfection”, that encompasses all the treated types of imperfections of temporal data, as shown in Figure 2.

Fig. 2

“TimeOntoImperfection” ontology composition

3 Ontology description

The temporal data imperfection ontology, named “TimeOntoImperfection” 3 , is an OWL 2 and a top-level ontology. It can be merged with other ontologies of domain that must be extended to represent and reason about imperfect temporal data. It is based on the extension of the 4D-fluent approach with predefined elements of the OWL-Time ontology, elements that we have defined to represent the targeted imperfections, and the extension of Allen’s interval algebra to reason about imperfect temporal data.

3.1 “ TimeOntoImperfection” ’s components

In the literature, there is not, to our knowledge, an ontology of temporal data imperfection. We create “TimeOntoImperfection” ontology using the Protégé ontology editor 4 . It contains 5 classes, 201 object properties and 44 data type properties. They represent time interval bounds, time points, dates and time clocks as well as all measures of the different types of imperfections that are the measures of certainties, measures of possibilities, measures of necessities and belief masses.

The proposed ontology is based on 4D-fluents approach, new ontological components and components based on OWL-Time ontology to represent imperfect quantitative temporal data and associated qualitative temporal relations in OWL2.

The 4D-?uents approach introduces two classes named “TimeSlice” and “TimeInterval”, and four properties named “tsTimeSliceOf”, “tsTimeIntervalOf”, “HasBeginnig”, and “HasEnd”.

We reduce to imprecision, uncertainty, both uncertainty and imprecision at the same time and conflict as kinds of temporal data imperfections. We clarify each part of the ontology in Figure 3.

Fig. 3

“TimeOntoImperfection” representation

3.2 “TimeOntoImperfection” ontology: Concepts

The proposed representation is based on extending the 4D-fluents approach, components defined in OWL-Time ontology as well as new ontology components that we define. “TimeSlice” is the class domain for entities representing temporal parts.

“time:TemporalEntity” is a subclass of the “TimeSlice” class. “time:TimeInterval” and “time:TimeInstant” are respectively the classes representing intervals and time points. “time:DateTimeDescription” is the class representing dates and time clocks. “time:TimeInterval”, “time:TimeInstant” and “time:DateTimeDescription” are subclasses of the “time:TemporalEntity” class. Figure 4 shows the concepts of the ontology in the protégé tool.

Fig. 4

The concepts of “TimeOntoImperfection” ontology

3.3 “TimeOntoImperfection” ontology: Datatype properties

We present the datatype properties of the ontology related to each type of imperfection. Figure 5 presents the datatype properties of the ontology in the protégé tool.

Imprecise Temporal Data: Dates and Time Clocks. We represent precise time points (dates and time clocks). For the dates, let D, Mo and Y be, respectively, precise day, month and year. We use three datatype properties from OWL-Time named “time:day”,“time:month” and “time:year” to relate, respectively, the “time:DateTimeDescription”class and D, Mo and Y. Similarly, for the time clocks, let S, Mi and H be, respectively, precise seconds, minutes and hours. We use three datatype properties from OWL-Time named “time:second”, “time:minute” and “time:hour” to connect the “time:DateTimeDescription” class, respectively, with S, Mi and H. We represent imprecise time points (dates and time clocks). For the dates, let D, Mo and Y be, respectively, imprecise day, month and year. We represent them by disjunctive ascending sets D⁽¹⁾ . . . D^(d), Mo⁽¹⁾ . . . Mo^(mo) and Y⁽¹⁾ . . . Y^(y). We define for each of D, Mo and Y, respectively, two datatype properties: “HasDayFrom” and “HasDayTo”, “HasMonthFrom” and “HasMonthTo”, “HasYearFrom” and “HasYearTo”. They are all connected to the “time:DateTimeDescription” class. Similarly, for the time clocks, let S, Mi and H be, respectively, imprecise seconds, minutes and hours, represented by disjunctive ascending sets S⁽¹⁾ . . . S^(s), Mi⁽¹⁾ . . . Mi^(mi) and H⁽¹⁾ . . . H^(h). We define for each of S, Mi and H, respectively, two datatype properties: “HasSecondsFrom” and “HasSecondsTo”, “HasMinutesFrom” and “HasMinutesTo”, “HasHoursFrom” and “HasHoursTo”.

Fig. 5

The datatype properties of “TimeOntoImperfection” ontology

Uncertain Temporal Data. We represent uncertain time intervals and time points and associated qualitative relations. We use probability and possibility theories to estimate certainty degrees. Let J = [J^-, J⁺] be an uncertain time interval. J^- and J⁺ are respectively the beginning and ending bounds; and c^- and c⁺ are, respectively,the associated certainty degrees. We introduce two datatype properties named “HasBeginnigCertainty” and “HasEndCertainty”. They are associated to the predefined class “time:TimeInterval”. “HasBeginnigCertainty” and “HasEndCertainty” represent, respectively, the certainty degrees associated to c^- and c⁺. Let P be a certain time point. We introduce a datatype property named “HasTimePoint”. It connects the “time:TimeInstant” class and the corresponding value of P. Let Q be an uncertain time point,where“q” is the associated certainty degree to the corresponding value of Q. We use the datatype property “HasTimePoint” to connect the “time:TimeInstant” class and Q. To represent the certainty degree associated to Q, we propose a datatype property named “PointCertainty” associated to “time:TimeInstant”.

Uncertain and Imprecise Temporal Data. we consider composite imperfection types, precisely simultaneously uncertainty and imprecision. We represent both uncertain and imprecise temporal data and associated qualitative relations using possibility theory. Let I = [I^-, I⁺] be a simultaneously uncertain and imprecise time interval; where I^- = I^-(1) . . . I^-(N) and I⁺ = I⁺⁽¹⁾ . . . I^+(N)associated with possibility and necessity degrees related to imprecision and possibility and necessity degrees related to uncertainty. We introduce two datatype properties. the first one is “PossibilityImprecisionInterval”and the second one is “NecessitiyImprecisionInterval”. It represents respectively possibility and necessity degrees associated to I. We define two datatype properties “PossibilityUncertaintyInterval”and “NecessityUncertaintyInterval” to represent both possibility and necessity degrees related to uncertainty. It relates “time:TimeInterval” with, respectively, the possibility and necessity degrees associated to I. Similarly, for the time points, let P = [P⁽¹⁾, P⁽ⁿ⁾] be a time point that is both uncertain and imprecise associated with degrees of possibility and necessity related to imprecision and degrees of possibility and necessity related to uncertainty. We add two datatype properties“PointBegins” and “PointEnds”; where “PointBegins” has range the value of the beginning bound of the time point and “PointEnds” has range the value of the ending bound of the time point. We define two data type properties named“PossibilityImprecisionPoint” and “NecessityImprecisionPoint” to represent respectively the possibility and degree of necessity associated with P. We also introduce two datatype properties named “PossibilityUncertaintyPoint” and “NecessityUncertaintyPoint” to represent both the degrees of possibility and necessity related to uncertainty.

Conflicting Temporal Data. We represent conflicting time intervals and time points using evidence theory. We define two datatype properties named “BeginningMass” and “EndingMass” to represent the belief degrees associated to J^- and J⁺. We define a datatype property “PointMass” to represent the belief degree associated to “P” and bind the class “time:Instant” to “mq”.

3.4 “TimeOntoImperfection” ontology: Object properties

The object property “TsTimeSliceInstance”, defined in “TimeOntoImperfection, links the class “TimeSlice” and the class “time:TemporalEntity” classes.

”Is_a” connects the “time:TemporalEntity” class with the “time:TimeInterval”, “time:TimeInstant” class and the “time:DateTimeDescription” class. In addition, we define object properties that represent qualitative relationships between two intervals (Interval-Interval), a time interval and a time point (Interval-Point), a time point and a time interval (Point -Interval) and two time points (Point-Point).

We instantiate object properties based on our extensions of Allen’s algebra for all the types of the treated imperfections. For example, “RelationIntervals” can be one of Allen’s relations. In other words, 13 object properties are associated: “BeforeIntervals”, “MeetsIntervals”, “OverlapsIntervals”, “StartsIntervals”, “DuringIntervals”, “EndsInterva-ls”, “AfterIntervals”, “MetByIntervals”, “Overla-ppedByIntervals”, “StartedbyIntervals”, “Contai-nsIntervals”, “EndedbyIntervals”, and “EqualsIntervals”. Figure 6 presents the object properties of the ontology in the protégé tool.

Fig. 6

The object properties of “TimeOntoImperfection” ontology

3.4.1 Reasoning about temporal data imperfection in OWL 2

We extend the Allen’s algebra to:(1) reason about imperfect quantitative temporal data to infer qualitative temporal relations and (2) to reason about the qualitative temporal relations to infer new ones.

We extend the Allen’s interval algebra to reason about imperfect time intervals. When considering perfect time intervals, our approach reduces to Allen’s interval algebra. We redefine the 13 Allen’s relations to propose temporal relations between imperfect time intervals.

For example, for uncertain time intervals, let A = [A-_ca-, A+_ca+]and B = [B-_cb-, B+_cb+]be two uncertain time intervals. For instance, we redefine the relation “Before (A, B)” as: “Before_ c (A, B)”; where “c” is the certainty degree associated to the relation “Before” between A and B. This means that the uncertain ending bound of the interval A is less than the uncertain beginning bound of B. Table 3 presents Allen’s relations between uncertain intervals. ${Before}_{c} (A, B) \Rightarrow A_{ca -}^{+} < B_{cb +}^{-}$ (1)

Table 3

Temporal relations between two uncertain time intervals A and B

Relation(A,B)	Relations between interval bounds	Inverse(B,A)
Before_c(A,B)	$A_{ca +}^{+} < B_{cb -}^{-}$	After_c(B,A)
Meets_c(A,B)	$A_{ca +}^{+} = B_{cb -}^{-}$	MetBy_c(B,A)
Overlaps_c(A,B)	$(A_{ca -}^{-} < B_{cb -}^{-})$
	$\land (A_{ca +}^{+} > B_{cb -}^{-})$
	$\land (A_{ca +}^{+} < B_{cb +}^{+})$	OverlappedBy_c(B,A)
Starts_c(A,B)	$(A_{ca -}^{-} = B_{cb -}^{-})$
	$\land (A_{ca +}^{+} < B_{cb +}^{+})$	StartedBy_c(B,A)
During_c(A,B)	$(B_{cb -}^{-} < A_{ca -}^{-})$
	$\land (A_{ca +}^{+} < B_{cb +}^{+})$	Contains_c(B,A)
Ends_c(A,B)	$(B_{cb -}^{-} < A_{ca -}^{-})$
	$\land (A_{ca +}^{+} = B_{cb +}^{+})$	EndedBy_c(B,A)
Equals_c(A,B)	$(A_{ca -}^{-} = B_{cb -}^{-})$
	$\land (A_{ca +}^{+} = B_{cb +}^{+})$	Equals_c(B,A)

The certainty degree “c” is inferred from the certainty degrees “c_a+” and “c_b-” using a Bayesian Network [51, 52].

All the other tables presenting Allen’s relation between imperfect time intervals of the related to imprecision, simultaneously uncertainty and imprecision, and conflict are detailed in our previous works (we can not cite it in the anonymous manuscript).

We infer via a set of SWRL rules our extension of Allen’s algebra. We use Pellet reasoner. For each qualitative temporal relation, we associate an SWRL rule. For example, the SWRL rule for inferring the temporal relation“Meet (I, J)” is as follows:

$\begin{matrix} TimeInterval (I) \land TimeInterval (J) \land \\ HasEndFrom (I, a) \land HasBeginningFrom (J, b) \\ \land Equals (a, b) \land HasEndTo (I, c) \land HasBeginningTo (J, d) \\ \land Equals (c, d) \Rightarrow Meet (I, J) \end{matrix}$ (2)

Figure 7 presents an example of a SWRL rule. It deduces the “ConflictingBeforeIntervals” relation. This one means the relation "Before" that may exist between two conflicting time intervals I and J

Fig. 7

An example of a SWRL rule

4 Prototype based on “TimeOntoImperfection” ontology

We propose a prototype based on our “TimeOntoImperfection” ontology. This prototype has been implemented in Java. The main interface of the prototype allows the user to enter perfect and/or imperfect time intervals and time points and to calculate degrees of certainty, possibility, necessity and degrees of credibility and masses of belief based on our proposed approaches as shown in Figure 8.

Fig. 8

User Interface of our prototype

After each new temporal data entry, the “Add Qualitative Temporal Relation” component is automatically executed to deduce the missing data, in particular the associated qualitative relations and the associated measures based on the SWRL rules.

Our prototype also makes it possible to perform a search on the data entered and saved by using a filter integrated in a choice bar. This is implemented with SPARQL queries.

5 Evaluation

In this section, we present the evaluations carried out within the framework of CAPTAIN MEMO. CAPTAIN MEMO [2] is an "intelligent" memory prosthesis intended for Alzheimer’s patients at the early stage of the disease to alleviate their memory problems. It was developed in collaboration between the CEDRIC laboratory (CNAM, France) and the MIRACL laboratory (University of Sfax, Tunisia), in particular the framework of the thesis of [1]. This prosthesis is based on a fuzzy temporal OWL 2 ontology, called MemoFuzzyOnto [53], and a multilingual and multicultural ontology to represent family relationships, called PersonLink [3], which allows to model and reason about the interpersonal relationships (for example, mother, neighbour). CAPTAIN MEMO aims to offer “intelligent” user interfaces based on semantic data processing, and adapting to the patient. It offers several services.

We integrate “TimeOntoImperfection ontology into CAPTAIN MEMO and we evaluate our approaches dealing with the different types of imperfection. This evaluation involves people with Alzheimer’s disease who stay in the Sfax retirement home in Tunisia.

This section aims to evaluate the effectiveness of our approaches to the imperfections of the temporal data processed. This evaluation is carried out within the framework of CAPTAIN MEMO. We use Precision measures.

A total of 20 patients with Alzheimer’s disease P=P_1...P_20and their associated caregivers C=C_1...C_20were recruited to participate in this study. All caregivers are first-degree relatives, such as a son or wife. All patients with Alzheimer’s disease are at an early stage of the disease. They were between 68 and 75 years old. We asked each patient’s legal sponsor for the consent letter. We excluded participants with overt behavioral disturbances, severe aphasia, and severe hearing and/or visual loss. Each patient was asked to answer the same 10 questions relating to their privacy and background at each testing session. The knowledge bases associated with the participants are structured using the “PersonLink ontology.

Two scenarios are proposed:

First scenario “Without integrating our prototype into CAPTAIN MEMO”: We do not integrate our prototype into “CAPTAIN MEMO memory prosthesis. All the data entered by the Alzheimer’s patient P_iis recorded in KBi/sc1 (knowledge base corresponding to the data entered by the patient Pi on the basis of the first scenario). KBi/s1 are generated after 8 weeks of using CAPTAIN MEMO. Each caregiver C_iis asked to identify only the real data entered by the patient. The answers form the reference knowledge base relating to the first KBi/GS1 scenario.

Second Scenario “We integrate our ontology-based prototype into CAPTAIN MEMO”: We integrate our prototype into “CAPTAIN MEMO. All the data entered by the Alzheimer patient P_iand having a measure of possibility, a measure of necessity and a mass of belief are recorded in KBi/sc2. KBi/sc2 are generated after using CAPTAIN MEMO.

We compare the generated knowledge base of the second scenario to the golden standard of the first scenario. We use the Precision evaluation metric:

>Pi@1 (|KBi/sc1 ∩ KBi/ GS1| / |KBi/sc1|): represents the Precision associated with the patient Pi according to the first scenario.

Pi@2 (|KBi/sc2 ∩ KBi/ GS1| / |KBi/sc2|): represents the Accuracy associated with the patient Pi according to the second scenario.

Table 4 and Figure 9 show that the global average accuracy associated with the second scenario (0.850) is better than the global average accuracy associated with the first scenario (0.775). These results prove the relevance of the proposed approach.

Fig. 9

Evaluation Results

Table 4

Precision of the results of the evaluation of our approach

Early stage Alzheimer’s patients	Precision according to the first scenario: Pi@1	Precision according to the second scenario: Pi@2
P1	0.864	0.986
P2	0.834	0.882
P3	0.895	0.935
P4	0.724	0.731
P5	0.822	0.954
P6	0.671	0.701
P7	0.735	0.902
P8	0.642	0.714
P9	0.821	0.931
P10	0.771	0.777
P11	0.878	0.954
P12	0.723	0.765
P13	0.699	0.798
P14	0.815	0.932
P15	0.798	0.874
P16	0.801	0.899
P17	0.865	0.945
P18	0.645	0.712
P19	0.705	0.765
P20	0.801	0.862

6 Conclusion

In this paper, we propose our ontology of temporal data imperfection named “TimeOntoImperfection”. The crucial point is that there is no ontology of temporal data imperfection. As already presented in the state of the art, the ontologies which represent the temporal aspect do not deal with imperfections. We implemented a prototype based on “TimeOntoImperfection” which we then integrated into the CAPTAIN MEMO memory prosthesis.

We evaluated our work on people with Alzheimer’s disease who stay in the retirement home of Sfax in Tunisia. In this evaluation, we proposed two scenarios: the first without integration of the prototype and the second by integrating it. The precision measurements obtained show the contribution of our proposed ontology. It proves the usefulness of the approach in the small amount of the entered data as all the inferences are well established and the precision results are promising.

As for continuity of this work, we plan to work on handling larger data and more Alzheimer patients to improve the experiments process and to make evaluations from a user perspective. Besides, we plan to handle other types of imperfection such as redundancy.

Footnotes

https://www.w3.org/TR/owl-time/

References

Ghorbel

Dialogue graphique intelligent, fondé sur une ontologie, pour une prothèse de mémoire, Doctoral dissertation, Paris, CNAM, 2018.

Métais

, Ghorbel

, Herradi

, Hamdi

, Lammari

, Nakache

and soukane

, Memory prosthesis, Nonpharmacological Therapies in Dementia 3(2) (2012), 177.

Herradi

Représentation sémantique multilingue, multiculturelle et temporelle des relations interpersonnelles, appliquée à une prothèse de mémoire, Paris: Doctoral dissertation, Conservatoire national des arts et metiers-CNAM, 2018.

Vila

, A survey on temporal reasoning in artificial intelligence, AICOM (Artificial Intelligence Communications) (7) (1994), pp. 4–28.

Etzion

Temporal databases: research and practice, s.l.:Springer Science & Business Media, 1998.

Nagypál

, Motik

A Fuzzy Model for Representing Uncertain, Subjective, and Vague Temporal Knowledge in Ontologies. s.l., Springer, (2003), pp. 906–923.

Ermolayev

, Batsakis

, Keberle

, Tatarintseva

and Antoniou

, Ontologies of Time: Review and Trends, International Journal of Computer Science & Applications 11(3), 2014.

Ermolayev

, Keberle

, Matzke

W.E.

, Sohnius

Fuzzy Time Intervals for Simulating Actions, s.l., LNBIP (2008), pp. 429–444.

Allen

Maintaining Knowledge about Temporal Intervals, Commun. ACM, (1983), pp. 832–843.

10.

Alur

, Feder

and Henzinger

T.A.

, The benefits of relaxing punctuality, Journal of the ACM (JACM) 43(1) (1996), pp. 116–146.

11.

Vilain

, Kautz

H.A.

Constraint Propagation Algorithms for Temporal Reasoning, Philadelphia, s.n., (1986), pp. 377–382.

12.

Hobbs

J.R.

and Pan

, An Ontology of Time for the Semantic Web,pp, ACM Transactions on Asian Language Processing (TALIP): Special issue on Temporal Information Processing 3(1) (2004), 66–85.

13.

Tao

, Wei

W.Q.

, Solbrig

H.R.

, Savova

, Chute

C.G.

CNTRO: a semantic web ontology for temporal relation inferencing in clinical narratives, In AMIA annual symposium proceedings, American Medical Informatics Association (2010), pp. 787.

14.

O’Connor

, Das

A Method for Representing and Querying Temporal Information in OWL, Biomedical Engineering Systems and Technologies (2011), pp. 97–110.

15.

Batsakis

, Petrakis

SOWL: A Framework for Handling Spatio-Temporal Information in OWL 2.0, International Workshop on Rules and Rule Markup Languages for the Semantic Web (2011), pp. 242–249.

16.

Frasincar

, Milea

, Kaymak

tOWL: Integrating Time in OWL, Semantic Web Information Management: A Model-Based Perspective (2010), pp. 225–246.

17.

Welty

, Fikes

A Reusable Ontology for Fluents in OWL, 4th International Conference on Formal Ontology in Information Systems (2006), pp. 226–336.

18.

Anagnostopoulos

, Batsakis

, Petrakis

CHRONOS: A Reasoning Engine for Qualitative Temporal Information in OWL, Procedia Computer Science (2013), pp. 70–77.

19.

Smets

Imperfect Information: Imprecision and Uncertainty, uncertainty management in information systems: from needs to solutions, Springer, 1997.

20.

Vetrò

, Canova

, Torchiano

, Minotas

C.O.

, Iemma

and Morando

, Open data quality measurement framework: Definition and application to Open Government Data, Quarterly Government Information 33 (2016), pp. 325–337.

21.

Ammar

Analyse et traitement possibiliste de signaux ultrasonores en vue d’assistance des non voyants, Sfax: Doctoral dissertation, Ecole Nationale d’Ingenieurs de Sfax (ENIS), 2014.

22.

Niskanen

V.A.

Introduction to imprecise reasoning, Uncertainty, decision-making and knowledge engineering, Finland, Publications of the Society for Artificial Intelligencen, 1989.

23.

Bouchon-Meunier

La logique floue et ses applications, 1995.

24.

Fisher

, Comber

, Wadsworth

Nature de l’incertitude pour les données spatiales, Qualitée de l’information géeographique (2005), pp. 49–64.

25.

Desjardin

, Nocent

, deRunz

Prise en compte de l’imperfection, Actes des 2émes Journées d’Informatique et Archéologie de Paris –JIAP (2012), pp. 385396.

26.

Sta

H.B.

, Quality and the efficiency of data in Smart Cities,pp, Future Generation Computer Systems 74 (2017), 409–416.

27.

Smets

Imperfect information: Imprecision-Uncertainty, Uncertainty Management in Information Systems: From Needs to Solutions, 1999.

28.

Achich

, Ghorbel

, Hamdi

, Metais

, Gargouri

A typology of temporal data imperfection, In 11th International Conference on Knowledge Engineering and Ontology Development, SciTePress-Science and Technology Publications, (2019), pp. 305–311.

29.

Achich

, Bouaziz

, Algergawy

, Gargouri

Ontology visualization: an overview, In Intelligent Systems Design and Applications: 17th International Conference on Intelligent Systems Design and Applications held in Delhi, India, Springer International Publishing, (2017), pp. 880–891.

30.

DaCosta

P.C.G.

Bayesian Semantics for the SemanticWeb, George Mason University, (2005), pp. 312.

31.

Costa

P.C.G.

, Ladeira

, Carvalho

R.N.

, Laskey

K.B.

, Santos

L.L.

, Matsumoto

A First-Order Bayesian Tool for Probabilistic Ontologies, Coconut Grove, Florida, USA, s.n, 2008.

32.

Carvalho

R.N.

, Haberlin

, Costa

P.C.G.

, Laskey

K.B.

, Chang

K.C.

Modeling a probabilistic ontology for maritime domain awareness, In 14th International Conference on Information Fusion (2011), pp. 18.

33.

Hlel

, Jamoussi

, Hamadou

A probabilistic ontology for the prediction of author’s Interests, In : Computational Collective Intelligence: 7th International Conference, ICCCI, Proceedings, Part II. Springer International Publishing, (2015), pp. 492–501.

34.

Loiseau

Recherche flexible d’information par filtrage flou qualitatif, Universite Paul Sabatier, Toulouse: These de doctorat, 2004.

35.

Lamine

R.B.

, Omri

M.N.

Information Retrieval Model Based on Possibilistic Ontology, International Journal of Computer Science, 2012.

36.

Bourai

S.B.

, Mokhtari

, Khellaf

Poss-SROIQ(D): Possibilistic Description Logic Extension toward an Uncertain Geographic Ontology, New Trends in Databases and Information Systems, (2014), pp. 277–286.

37.

Bal-Bourai

, Mokhtari

π-SROIQ (D): Possibilistic Description Logic for Uncertain Geographic Information, Springer, Cham, (2016), pp. 818–829.

38.

Salem

Y.B.

, Idoudi

, Saheb Ettabaa

, Hamrouni

and Solaiman

, Ontology based possibilistic reasoning for breast cancer aided diagnosis, Inpp, Information Systems: 14th European, Mediterranean, and Middle Eastern Conference, Springer International Publishing, Proceedings 14 (2017), 353–366.

39.

Akremi

, Ayadi

M.G.

, Zghal

M.G.

A Fuzzy OWL Ontologies Embedding for Complex Ontology Alignments, In Discovery Science: 25th International Conference, Montpellier, France, Proceedings. Cham: Springer Nature Switzerland, (2022), pp. 394–404.

40.

Kaur

, Kaur

Extension of a crisp ontology to fuzzy ontology, International Journal Of Computational Engineering Research (2012), pp. 201–207.

41.

Zekri

, Bouaziz

, Turki

A fuzzy-based ontology for Alzheimer’s disease decision support, In 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2015), pp. 1–6.

42.

Rodríguez

N.D.

, Cuéllar

M.P.

, Lilius

, Calvo-Flores

M.D.

A fuzzy ontology for semantic modelling and recognition of human behavior, Knowledge-Based Systems (2014), pp. 46–60.

43.

Ghorbel

, Maalej

, Bouaziz

Un framework pour la génération semi-automatique d’ontologies floues : Text2FuzzyOnto, Tech. Sci. Informatiques, (2013), p. 671–699.

44.

Essaid

, Yaghlane

B.B.

BeliefOWL: An Evidential Representation in OWL Ontology, In URSW, (2009), pp. 77–80.

45.

Achich

, Ghorbel

, Hamdi

, Métais

, Gargouri

Representing and Reasoning About Precise and Imprecise Time Points and Intervals in Semantic Web: Dealing with Dates and Time Clocks, Linz, Austria: Springer, Cham, 2019.

46.

Achich

, Ghorbel

, Hamdi

, Métais

, Gargouri

Approach to Reasoning about Uncertain Temporal Data in OWL 2, Verona: Elsevier-Procedia Computer Science, 2020.

47.

Achich

, Ghorbel

, Hamdi

, Métais

, Gargouri

2021, Handling Uncertain Time Intervals in OWL 2: Possibility Vs Probability Theories-based Approaches, In 2021 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2021, pp. 1–7. IEEE.

48.

Achich

, Ghorbel

, Hamdi

, Métais

, Gargouri

Dealing with Uncertain and Imprecise Time Intervals in OWL2: A Possibility Theory-Based Approach, In RCIS, (2021), pp. 541–557.

49.

Achich

, Ghorbel

, Bouhamed

S.A.

, Hamdi

, Métais

, Gargouri

, Kharfia

and Gargouri

, An Evidence Theory-Based Approach to Handling Conflicting Temporal Data in OWL 2, New Generation Computing 40 (2022), 845–870.

50.

Achich

Traitement de l’imperfection des données temporelles saisies par l’utilisateur-Application aux logiciels destinés à des patients malades d’Alzheimer, Doctoral dissertation, HESAM Universite Paris France; L’Universite de Sfax Tunisie, 2021.

51.

Pearl

Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann-Elsevier, 1988.

52.

Achich

, Ghorbel

, Hamdi

, Métais

and Gargouri

, Certain and uncertain temporal data representation and reasoning in OWL 2, International Journal on SemanticWeb and Information Systems (IJSWIS) 17(3) (2021), 51–72.

53.

Ghrobel

, Hamdi

, Métais

, Ellouze

, Gargouri

A Fuzzy-Based Approach for Representing and Reasoning on Imprecise Time Intervals in Fuzzy-OWL 2 Ontology, Paris, Springer, Cham., (2018), pp. 167–178.