A FORMAL CONCEPTUAL MODEL AND DEFINITIONAL FRAMEWORK FOR SPATIAL DATACUBES

Transcription

1 G E O M A T I C A A FORMAL CONCEPTUAL MODEL AND DEFINITIONAL FRAMEWORK FOR SPATIAL DATACUBES Mehrdad Salehi, Yvan Bédard and Sonia Rivest Département des sciences géomatiques, Université Laval, Québec Spatial datacubes extend the datacube concept underlying the field of Business Intelligence (BI) into the realm of spatial analysis, geographic knowledge discovery, and spatial decision-support. The traditional computer science community has defined spatial datacubes and their fundamental components (e.g., spatial dimension and spatial measure) through formal models limiting spatial data as only those data that has a geometric representation. The geomatics community has pursued spatial datacube models with a much richer view of spatial data. However, the proposed models by the geomatics community have not yet been formalized using precise mathematical languages. This paper, for the first time, integrates the rigor of mathematical languages with the richer view of spatial data to provide a formal model and precise definitions of spatial datacubes and their fundamental components. The proposed definitions provide the scientific community with a common and precise terminology for the concepts involved in spatial decision-support databases. Mehrdad Salehi Les cubes de données spatiales étendent le concept de cube de données sous-jacent au domaine de l informatique décisionnelle aux domaines de l analyse spatiale, de la découverte des connaissances géographiques et du soutien aux décisions spatiales. La communauté traditionnelle de l informatique a défini les cubes de données spatiales et leurs composantes fondamentales (p. ex., la dimension spatiale et la mesure spatiale) au moyen de modèles formels limitant les données spatiales seulement à celles pouvant avoir une représentation géométrique. La communauté de la géomatique a approfondi les modèles de cubes de données spatiales avec une vision beaucoup plus étoffée des données spatiales. Toutefois, les modèles proposés par la communauté de la géomatique n ont pas encore été officialisés en utilisant des langages mathématiques précis. Le présent article intègre, pour la première fois, la rigueur des langages mathématiques à la vision plus étoffée des données spatiales afin de présenter un modèle formel et des définitions précises des cubes de données et de leurs composantes fondamentales. Les définitions proposées offrent à la communauté scientifique une terminologie commune et précise des concepts impliqués dans les bases de données qui appuient les décisions. mehrdad.salehi.l@ulaval.ca Yvan Bédard 1. Introduction Strategic decision makers (analysts, executives, and managers) need to analyze and compare summarized data extracted from very large volumes of data. Indeed, it is more efficient to use aggregated and consolidated data covering a certain period of time rather than detailed individual records of transactional databases for strategic decision making. The difficulty in supporting both daily transactions and decision-support needs within a single database requires using a dual-database approach. This forms the typical backbone of data warehouses [Bédard and Han 2008]. A data warehouse is a subject-oriented, integrated, time varying, nonvolatile collection of data that is used primarily in organisational decision making [Chaudhuri and Dayal 1997]. Data warehouses are typically modeled using the datacube (or multidimensional, in the sense of business intelligence) paradigm [Gray et al. 1997; Abelló et al. 2006]. In the datacube structure, analysis is performed along a combination of axes of analysis called dimensions (e.g., categories of products, administrative regions, periods), and hence the structure is termed multidimensional. Each dimension includes one or several hierarchies, each composed of different analysis levels (e.g., cityprovince-country hierarchy and city-county-regioncountry hierarchy which may compose a spatial dimension labelled administrative regions ). The hierarchical structure allows users to view and analyze data at different levels of detail. An instance of a level is a member (e.g., Montreal is a member of the level city ). Measures (e.g., population) are measurable quantities; these are analyzed against the members of different levels of dimensions. For instance, one may be interested in analyzing the measure population with respect to different levels of administrative regions and time dimensions. Sonia Rivest GEOMATICA Vol. 64, No. 3, 2010 pp. 321 to 332

2 Values resulting from unique combinations between members of different dimension levels, along with their measure, are known as facts (e.g., the number of sport articles sold in Montreal in the first quarter of 2006 is 1,800,000 is a fact). A datacube is composed of a number of facts. In order to speed up query answering, a datacube usually includes a number of precomputed facts. A tool called On-Line Analytical Processing (OLAP), which includes data exploration operators such as roll-up and drill-down, are used to interactively query a datacube [Bédard et al. 2004]. It is estimated that about 80% of the data in enterprise databases have a spatial reference [Franklin 1992]. Such a reference is composed of diverse types such as civic addresses, names of places, coordinates, etc. In order to derive the maximum profit from the spatial data and the efficiency of the datacube structure in decision making, the first definitions of spatial datacubes were proposed at the end of the 1990s. This took place in the geomatics and computer science communities by pioneering works at the universities of Simon Fraser (Jiawei Han s team), Minnesota (Shashi Shekhar s team), and Laval (Yvan Bédard s team) [Bédard et al. 2007]. Spatial datacubes provide capabilities not inherent to transaction-oriented systems such as geographic information systems (GIS) and spatial database engines (universal servers) and aim at supporting interactive complex analysis involving spatial and temporal data. The early investigations into spatial datacubes and spatial OLAP (SOLAP) by the geomatics community were published in the Geomatica journal [see Rivest et al. 2001]. These characteristics were then refined by the geomatics community, and models for spatial datacubes were proposed [Rivest et al. 2005; Bédard and Han 2008]. These models consider spatial data as any data that is used to localize a phenomenon on the Earth (e.g., street addresses and geographic coordinates) regardless of the representation method (i.e., by geometries or by text). This view of spatial data is consistent with international standards in geomatics such as ISO/TC211 (2003). The definition given by these authors to spatial dimensions is not limited to dimensions whose levels members have geometric representations. However, these models do not provide explicitly precise definitions for all components of spatial datacubes. For instance, they do not give precise definitions for the spatial fact and spatial datacube. Moreover, the model and concepts defined by the geomatics community have not yet been supported by formal definitions, i.e., they are not defined using a precise and unambiguous mathematical language. In the middle of the present decade, a number of formal models for spatial datacubes were also proposed by the computer science community (see for example Damiani and Spaccapietra [2006] and Bimonte [2007]). Although the formalisation of these models is valuable, they do not provide common definitions for every fundamental component of the spatial datacube, such as the spatial dimension, spatial measure, and spatial fact. In particular, the suggested formal definitions are based on a restrictive perspective that considers spatial data as only those data that have a geometric representation. Based on such a restricted view of spatial data, which is not in sync with ISO and OGC international standards, the subsequent definitions given to the fundamental components of spatial datacubes does not reveal the entire power of spatial data within these datacube components. For instance, the dimension administrative regions, with the three levels city, province, and country whose members (e.g., Montreal, Quebec, and Canada, respectively) do not have a geometric representation, would not be considered a spatial dimension. Although members of this dimension may not have a geometrical representation, they can still be used to refer to geographic locations and to locate a phenomenon in space (e.g., population of the city Montreal in 2008). Therefore, this dimension is intrinsically spatial. Likewise, measures such as road length and region area are not considered as spatial measures. While not having a geometric representation, these measures convey a spatial property of features that can be used to make thematic maps and perform spatial analysis, (such as number of kilometers of roads per city, county, and province). Hence these measures are inherently spatial. As is clearly demonstrated by Caron [1998], OLAP has powerful potential for spatiotemporal analysis even if spatial data are not represented geometrically. The above discussions illustrate the need for an enhanced model for spatial datacubes that, for the first time, integrates the rigor of formal models with the richer notion of spatial data, to provide a formal model and precise definitions for the fundamental components of spatial datacubes such as the spatial dimension, spatial measure, and spatial fact. In this paper, we make the following contributions: 1. We review and analyze the existing key models for spatial datacubes by especially focusing on the definitions suggested by these models for the fundamental components of spatial datacubes (Section 2). 2. We present a formal model for spatial datacubes at the conceptual level with a primary 322

3 focus on defining a spatial datacube and its components. In order to achieve this, after defining the general components of a datacube (i.e., attribute, level, dimension, measure, hyper-cell, and datacube), the spatial equivalent for these components are defined at both the schema and instance levels (Section 3). The definitions of the spatial components are based on a broader perspective for the spatial data aligned with the international standards for geomatics, i.e., any data that provides a means to localize a phenomenon. Finally, we discuss a number of characteristics of the proposed spatial datacube model (Section 4). 2. State-of-the-Art on Spatial Datacube Modeling Bédard [1997], Bédard et al. [2001], Rivest et al. [2005], Bédard et al. [2007], and Bédard and Han [2008] describe the structure of spatial datacube models and define and categorise spatial dimensions and spatial measures. These definitions rely on a rich perspective for spatial data that involve any data used to localize phenomena on the Earth (e.g., place names, roads addresses, and coordinates). Consequently, the definition given by these authors to spatial dimensions is not limited to dimensions whose members have geometric representations. Rivest et al. [2005] categorize spatial dimensions into three types: non-geometric, geometric, and mixed. In a non-geometric spatial dimension, the spatial reference is nominal (e.g., place names), and no geometric representation is associated with the members of this dimension. The other two types of spatial dimensions include geometric data on a map, and allow the members of the dimension to be visualized and queried graphically. In a geometric spatial dimension, all the members of all the levels have geometric representations, while in a mixed spatial dimension, some members have no geometric representation. Similarly, two types of spatial measures are recognized by the geomatics community Rivest et al. [2005]. A geometric spatial measure is defined as the set of geometries representing spatial objects, such as accident locations. Numeric spatial measures such as distance and area are numeric values that are the results of using spatial operators. Recently, a third type of spatial measure, called a complete spatial measure, was introduced by Bédard and Han [2008]. This type of spatial measure is specific to raster datacubes, and is a combination of a numeric and a geometric spatial measure. It encompasses, for example, pairs consisting of a raster cell position and its associated value. Rivest et al. [2005] introduce different SOLAP navigation operators, such as spatial drill-down, spatial roll-up and spatial drill-across. These operators have been implemented on JMap SOLAP, the first commercialized SOLAP software product [KHEOPS Technologies 2005]. We should add that the above research works do not provide precise definitions for some other fundamental components of spatial datacubes, such as the spatial fact. Moreover, the model and concepts defined by the geomatics community have not yet been presented using an unambiguous and precise mathematical language. Han et al. [1998] introduce a model for implementing spatial datacube applications. This model relies on the well-known implementation models of datacubes, i.e., star/snowflake schemas, and consists of dimensions and measures. In this model, three types of dimensions are recognized. Non-spatial dimensions do not include geometric data. In spatialto-spatial dimensions, all levels have geometric data associated with their members. When the lowerlevels of a dimension include geometric data but the levels above a certain level do not, the dimension is said to be spatial-to-nonspatial. According to these definitions, we can deduce that the spatiality of a dimension depends on having at least one geometric member. Therefore, a dimension that does not involve a geometric representation but addresses a spatial phenomenon (e.g., by names of cities, provinces, and countries) is not considered a spatial dimension in this model. In addition, Han et al. [1998] distinguish two types of measures for spatial datacubes: numerical measures and spatial measures. A numerical measure contains only numerical data. In order to be qualified as spatial, a measure should contain one or a collection of pointers to geometries. We should note that the categorization of measures, based on the storage format rather than the nature of the data, is at the implementation level not the conceptual level. Accordingly, new definitions are required to extend the definition of spatial dimensions and to provide a categorization and a definition for spatial measures at the conceptual level. In order to extend the concept of datacube to the spatial domain, Shekhar et al. [2001] introduced the map cube operator. This operator accepts a base map and a table associated with the map and generates a set of maps for analysis and comparison. The output maps are produced using OLAP operations on hierarchies and measures. The authors also propose a formal classification for geometric aggregation functions. Within this research, no definitions for specific components of spatial datacubes such as spatial dimension, spatial measure, and spatial fact have been provided. 323

4 At the conceptual level, Jensen et al. [2004] introduce a spatial datacube model for use in location-based services. This model is an n-dimensional fact schema consisting of a fact type and a set of dimension types. A dimension type includes a set of category types (or levels) and a partial and a total containment relationship between category types. This model has two distinctive characteristics compared with other proposed models: (1) it accepts partial containment relationships between two geometric levels of a hierarchy, and (2) it handles the imprecision of aggregation paths. An algebra is also proposed to serve as a basis for the model s query language. In this work, however, no definition is given for the specific components of spatial datacubes. Another framework for implementing spatial datacubes based on the star schema, called GeoDWFrame, is proposed by Fidalgo et al. [2004]. They classify and informally define different types of spatial dimensions in terms of the approach that is used to implement the dimensions, i.e., the technique used to normalize and to store the geometric and descriptive data. As stated by the authors, the principal idea underlying this classification is to reduce geometric data redundancy in implementing spatial datacubes. Obviously, this classification is not suitable for distinguishing spatial dimensions at the conceptual level. To allow the modelling of spatial measures at multiple levels of geometric granularity, Damiani and Spaccapietra [2006] introduce a formal model at the conceptual level, called Multi-granular Spatial Data warehouse (MuSD). For navigating in MuSD, this model is integrated with an algebra, which includes a number of spatial and non-spatial operators. In this mode, like the spatial-to-spatial dimension defined by Han et al. [1998], a spatial hierarchy is presented as a hierarchy where all the levels have a geometrical representation. A spatial measure is considered, like a spatial dimension, as a hierarchy of levels with a geometric representation. As a result, we conclude that labelling a hierarchy, a dimension, and a measure as spatial requires them to have a geometric representation. Further, a spatial fact is defined as a fact describing an event that occurred on the Earth in a position that is relevant to know and analyze. In addition, this model explicitly defines a spatial datacube. The necessary condition for a datacube to be labelled as spatial is to have at least one measure with a geometric representation. However, we suggest that a spatial datacube whose only dimensions have geometric representations can be considered a spatial datacube. The reason being that even when the dimensions have geometric representations, the user can still interactively explore and visualize the maps of the dimensions provided in the datacube. At the conceptual level, Bimonte [2007] presents a formal model, called GeoCube, with an algebra that supports spatial data within datacubes. The formal representation of GeoCube s general components and operations is valuable, and is explained in detail through various examples. In this model, a geographic entity is considered as an entity with a geometric attribute. According to the international standards in geomatics, however, the definition of a geographic (or spatial) entity is not limited to entities with a geometric representation, but can also include non-geometric attributes. Based on this perspective, the definition given by Bimonte [2007] to a geographic dimension, i.e., a dimension whose members include geographic entities, is limited. He also introduces three types of hierarchies, i.e., descriptive, spatial, and generalization. A descriptive hierarchy is defined by descriptive attributes of objects. A spatial hierarchy is defined as a hierarchy whose levels are related by the topological relationships of inclusion and intersection. We should note that even with a geometric representation, the levels of a spatial hierarchy can be related based on semantic rather than topological relationships. For example, consider the spatial hierarchy financial institution, with the two levels branch and headquarter. While the members of these two levels have geometric representations on the map and are very likely disjoint, a branch is semantically associated to its headquarters. Finally, Bimonte defines a generalization hierarchy as a hierarchy where the members of different levels represent the same geographic information at different scales. He defines a geographic (or spatial) measure, in a similarly limited way, as an object with a geometric attribute. In this model, no definition is given for spatial facts. In order to represent conceptual models of spatial datacubes visually, Malinowski and Zimányi [2008] introduce MultiDim, a spatially extended entity-relationship model. Based on this work, a conceptual model is created in terms of dimensions and the relationships between levels of dimensions (i.e., entities), which is modeled by a fact relationship. While the fact relationship includes measures, the dimensions consist of a number of hierarchies of levels. The authors define the concepts spatial level, hierarchy, dimension, and measure. According to their definitions, a spatial level includes at least one attribute with a geometric representation. They require that a spatial hierarchy include at least one spatial level and that a spatial dimension include at least one spatial hierarchy. Although Malinowski 324

5 and Zimányi [2008] define the spatiality of a level as having a level with a geometric attribute, on the one hand, we notice that their definitions of the spatial hierarchy and the spatial dimension are different from the definition suggested by Han et al. [1998] and Damiani and Spaccapietra [2006]. On the other hand, we can recognize that Malinowski and Zimányi [2008], like Han et al. [1998], consider a spatial measure as a measure represented geometrically. Although in a previous paper, the authors considered a measure that holds a numeric value calculated using metric or topological operators as a spatial measure [Malinowski and Zimányi 2004], in this recent work, a measure calculated using spatial operators such as road length is considered a conventional measure. In summary, we can conclude that based on this model, in order to label a level, a hierarchy, a dimension, or a measure as spatial these components should involve geometric data. In this section, we reviewed two categories of models for spatial datacubes. The first category of these models, principally proposed by the geomatics community, has considered a richer and more comprehensive view of spatial data and subsequently has defined some of the components of spatial datacubes. However, the provided definitions by these models have not yet been formally defined. The second category includes a number of formal models for spatial datacubes proposed by the computer science community traditionally dealing with the frequent but simplest cases of spatial data. These models consider a limited notion of spatial data, restricted to features (members) with geometric representations, and typically not involving other types of spatial reference. These latter models ignore a huge amount of data that are inherently spatial but are not represented geometrically to be considered spatial. Based on such limited assumptions, the definitions given to the components of spatial datacubes (e.g., spatial dimension, spatial measure, and spatial fact) do not correctly convey the power of spatial data integrated with these components. Among all the proposed models, we did not find a common and formal definition for all fundamental components of the spatial datacubes. 3. A Model for Spatial Datacubes at the Conceptual Level In this section, we will define a model for spatial datacubes at the conceptual level that includes formal definitions for spatial datacubes and their various constituents. These definitions are based on a broader view for spatial data that is consistent with the international standards in geomatics. The proposed model explicitly distinguishes between the schema (i.e., the intentional representation), which defines the structure of a datacube element, and the instance (i.e., the extensional representation), which is the value associated to a constituent. This section is organized so that after defining an element of the schema (i.e., level, dimension schema, measure, hyper-cell, and datacube schema), the definition of its instance (i.e., member, dimension instance, measure value, fact, and datacube) is provided. The definitions of the elements of the model are followed by a number of examples. For this purpose, we consider a running example: the spatial datacube fire disaster for analyzing fire losses and injuries for different classes of fire in different administrative regions of Canada and the USA and at different epochs. The fire disaster datacube consists of three dimensions and three measures. The first dimension, administrative regions, has the following levels: city, county, province/territory (in Canada), state (in the USA), country, and all. The second dimension, called time, includes three levels: day, month, and year. Fire class is the third dimension and includes two levels: fire class and all fire classes. The levels of the three dimensions have a number of attributes. For example, the level city of administrative regions dimension has the attribute location. The three measures of this datacube are fire zone, which geometrically represents the location of fire zones, surface of destroyed residential area, which expresses the area in km 2 of the residential zones that were destroyed by the fire, and number of injuries, which states the number of people injured by the fire. Definition 1: In order to describe a level, we need to define the level s attributes. The level attribute a i (attribute, for short) is defined by the triple a i = (type nature domain) where: type is the data type associated to the attribute a i. nature refers to the spatial, temporal, or thematic nature of the attribute a i. domain is the domain of attribute s values. The type of an attribute can be numeric (e.g., real and integer), textual, date (e.g., instant and interval), or geometric (e.g., point, line, polygon, and a set of these geometries). The nature of an attribute indicates whether that attribute describes a phenomenon in space, in time, or in a theme. Its nature, i.e., what the attribute represents, such as 325

6 326 spatial data about the location of a shopping center, is independent of the type, i.e., how the attribute represents, for instance by geometries on the map or by textual address. The independence of the nature of an attribute from its representation method is necessary in order to describe an attribute appropriately at the conceptual level. Accordingly, we distinguish three categories of attributes by referring to their natures, i.e., spatial, temporal, and thematic attributes. Temporal attributes convey temporal information on a phenomenon like age. Non-spatial and non-temporal information is described by thematic attributes such as price. Before describing spatial attributes, we explain spatial references and their categories according to the international standards in geomatics. Spatial references are used to localize spatial features in the geographic space, and are divided into two categories: direct and indirect. Direct spatial referencing is achieved by means of geometries embedded in a coordinate system [ISO/TC ]. However, spatial references are not limited to geometric coordinates. Indeed, indirect spatial references go beyond geometries, and use spatial identifiers such as place names, distances, and postal codes for spatial referencing [ISO/TC ]. For example, a place name such as Montreal refers to the geographic location of the city of Montreal; it can be used alone to find this place on the Earth or it can be linked to geographic coordinates in a gazetteer to position this place on a map. A distance can be used to localize a phenomenon with respect to a linear reference system, such as a distance or a civic address number along a street. A postal code refers to a geographic region that is defined by address blocks or by a place name like a municipality name, allowing one to find it on the Earth. Multiple direct and indirect spatial references can refer to the same place in the real world, and these references are convertible. For instance, using Google Maps, one can enter a place name, such as Montreal, into the gazetteer and get the place s geometric representation as a map on the screen. International standards in geomatics are currently pursuing further investigation to establish a conversion methodology among various spatial references [ISO/TC ]. Definition 1.1: Inspired by the above perspective for spatial references, we consider spatial attributes (at spatial ), beyond only the attributes with a geometric representation. A spatial attribute is any attribute that describes spatial properties of phenomena occurring in geographic space. Examples of these properties include location (e.g., geographic coordinates, address, postal code), shape (e.g., a polygon representing the extent of a city), direction (e.g., direction of a highway), length (e.g., road length), and area (e.g., area of a house). In order to be consistent with the international standards in geomatics, we adopt the same strategy they use for categorizing spatial attributes. Definition 1.2: A geometric spatial attribute (at geo ) is a spatial attribute that is represented by a geometry. More precisely, the type of a geometric spatial attribute in Definition 1 is geometric. A geometric spatial attribute is typically used to represent a direct spatial reference. For instance, a point can represent the position of a feature with a location in space without extent, such as the position of a hotel on a small-scale map. A line can describe the position of a linear feature like a road or river. The positions of two-dimensional features are represented by polygons, such as the extent of a forest stand or a city on a medium-scale map. More complex geometries can also be used, such as the aggregation of a set of lines and a set of polygons, to represent features like hydrological networks. ISO and OGC explicitly support such complex geometries as well as spatial database modeling methods, such as Perceptory [Bédard et al. 2004; Bédard and Larrivée 2008] and MADS [Parent et al. 2006]. Definition 1.3: A non-geometric spatial attribute (at non-geo ) is a spatial attribute that is represented by data types other than the geometric type, such as textual or numeric types. A non-geometric spatial attribute can describe indirect spatial references such as place names and addresses, or other spatial properties of features like the length of a road or the area of a house. Non-geometric attributes convey spatial information that can be used for mapping and spatial analysis using a gazetteer, geocoding, or linear referencing, among other methods. Example 1: Referring to Definition 1, the following attributes for the levels of different dimensions of the fire disaster datacube are defined: location = (geometric: polygon and point, spatial, polygons and points in a plane) name = (textual, spatial, { Gatineau, Montreal, Austin, Quebec, Texas, Canada, North America, }) date = (date: instant, temporal, { , , 2008, }) type = (textual, thematic, {A, B, C, D, E, all_fire_classes}) Location and name are, respectively, geometric and non-geometric spatial attributes that will be used to describe the levels of the administrative

7 regions dimension, such as city. Date is a temporal attribute that describes the levels of the time dimension (e.g., day ). Finally, the attribute type is a thematic attribute that is used to describe the levels of the fire class dimension, such as the fire class level. Definition 2: A level defines the granularity of analysis along a dimension, and is described by l = {a 1,..., a n }, where l is the name of the level and {a 1,..., a n } is its set of attributes. Among a level s attributes, there is at least one distinguished identifier. In the following formal definitions in firstorder logic, connectives are denoted by (logical inclusive or, logical and, logical implica tion), and (logical not). The symbols and are the universal and existential quantifiers. Unary predicates are expressed in the form p(x), stating that x is a p, and the symbol stands for set membership. Definition 2.1: Let l be a level, a spatial level (lspatial) is a member of the following set: l spatial = l a i a i l at spatial a i Example 2: Now, we define the levels of the fire disaster datacube. The identifiers are highlighted in italic. Non-spatial levels are defined as follows: day = {date} month = {date} year = {date} fire class = {type} all fire classes = {type} (1) Both the attributes name and location, as defined in Example 1, are spatial attributes. Referring to the Definition 2.1, the following levels are spatial: city = {name, location} county = {name, location} province/territory = {name, location} state = {name, location} country = {name, location} all = {name, location} Definition 3: An instance of a level is a member of that level. Instantiation is achieved by assigning values to a subset of the level s attributes. Since identifiers are used to uniquely identify the members of a level, they should have a unique existing value for each member of the level. Formally, a member m of a level is defined by the triple m = ( AT m,v,:). In this equation, AT m = {a 1,..., a k } is the set of the member s attributes, and includes a subset of the level s attributes. V = {v 1,..., v k } is the set of values of the domain of the attributes AT m, and : is a function from elements in AT m to elements in V. Definition 3.1: Let m = (AT m,v,:) be a member of a spatial level. A geometric member (m geo ) is defined below: m geo = AT m, V,: a i a i AT m at geo a i Definition 3.2: Similarly, a non-geometric member (m non -geo ) of a spatial level is formally defined as: m non geo = AT m,v,: a i a i AT m at non geo a i Example 3: A number of members for the levels of the fire disaster datacube, defined in Example 2, are presented below. These members will be used to define the other components of this datacube. city : geometric member (name: Austin, location: ct_loc) and the non-geometric members (name: Gatineau), county : non-geometric member (name: Hull), geometric member (name: Travis, location: cnt_loc), province/territory : geometric members (name: Quebec, location: p_loc), state : geometric members (name: Texas, location: s_loc), country : geometric members (name: Canada, location: c_loc), all : geometric member (name: North America, location: NA_loc) day : members (date: ), month : members (date: ), year : members (date: 2006), fire class : members (type: A),, (type: E) all fire classes : member (type: all_fire_classes) For the sake of simplicity, we recognize members by the values given to their identifiers. For example, the member (name: Gatineau, location: ct_loc) is recognized as Gatineau, and the member (date: ) is referred to as In the above members, ct_loc, cnt_loc, p_loc, s_loc, c_loc, and NA_loc are, respectively, the polygons representing the location of members Austin, Travis, Quebec, Texas, Canada, and North America. (2) a j a j AT m at geo a j (3) 327

8 Definition 4: A dimension schema (dimension, hereafter) describes an axis of analysis or a theme of interest for a user, under which the data analysis is to be performed [Rafanelli 2003]. A dimension d includes a number of related levels. These levels are ordered from detailed to general, and form a hierarchy of abstraction levels. Formally, a dimension is defined as a pair d = (Ld, <), which forms a lattice on levels L d = {l 1 l 2,..., l n }. The set l d has two distinct levels, which are the lower-bound (leaf) and upper-bound (root, typically named all) of the lattice (dimension), and < is a partial-order (roll-up) relation, on the levels in L d. For two levels l 1,l 2 of a dimension, if l 1 < l 2, we say that l 1 (the lower-level) rolls-up to l 2 (the higher-level), and l 1 and l 2 are two consecutive levels of the dimension. Definition 4.1: Let d = (L d, <) be a dimension. A spatial dimension (d spatial ) is defined as follows: d spatial = { (L d, <) l (l L d l spatial (l))} (4) Like a spatial attribute, a spatial dimension is often incompletely described as a dimension whose levels involve geometric attributes. But as we stated earlier, spatial attributes are more than simply attributes with a geometric representation. Hence, spatial dimensions may include spatial levels whose attributes are non-geometric. Figure 1: The graphic representation for the spatial dimension administrative regions. within the dimension. For example, the dimension administrative regions includes two hierarchies: Canadian division = (city < county, county < province/territory, province/territory < country, country < all) and USA division = (city < county, county < state, state < country, country < all). These two hierarchies represent administrative divisions within two countries, Canada and the USA. The graphic representation of these two hierarchies is represented in Figure Example 4: In the following, we define the three dimensions administrative regions, time, and fire class of the fire disaster datacube using the levels defined in Example 2. administrative regions = (city < county, county < province/territory (in Canada), county < state (in the USA), province/territory < country (in Canada), state < country (in the USA), country < all) time = (day < month, month < year) fire class = (fire class < all fire classes) As defined in Example 2, all the levels that appear in the administrative regions dimension are spatial. Consequently, referring to Definition 4.1, administrative regions is a spatial dimension. The dimension fire class is a non-spatial dimension. The graphic representation of the dimension administrative regions is shown in Figure 1 by a directed acyclic graph where the arrows show the order between the levels. In some cases, a dimension can include several hierarchies h 1 = (L h1, <), h 2 = (L h2, <),, where each hierarchy represents an analytic perspective Figure 2: The graphic representation for two hierarchies: (a) Canadian division and (b) USA division. We can see from the structure of these two hierarchies that the end-user needs to distinguish between provinces/territories and states, because they are not considered equivalent for the purposes of the end-user s analysis. However, for cities, counties, and countries, they are considered to be the same in this example.

9 Definition 5: Like Bimonte [2007], we define a dimension instance. An instance (di) for a dimension d = (L d, <) is a pair (L, ) (where L = {m 1,..., m n } is a set of members for levels in L d, and is an order (or roll-up) relation between these members), such that if m i and m j are respectively members of the two levels l i and l j in L d and l i < l j, the follow ing condition is met: m L m m j m j L. Instances of spatial dimensions are of three types: non-geometric, geometric, and mixed. Definition 5.1: Let di = (L, ) be an instance of a spatial dimension. A geometric dimension instance (di geo ) is a member of the following set: di geo = { (L, ) m (m L m geo (m)) } (5) Definition 5.2: A non-geometric dimensions instance is defined as: di non-geo = { (L, ) m (m L m non-geo (m)) } (6) Definition 5.3: Finally, a mixed dimensions instance is a member of the following set: di mixed = L, m 1,m 2 m 1 L m geo m 1 m 2 L m non geo m 2 } (7) Example 5: Referring to the members defined in Example 3, an instance of the spatial dimension administrative regions is (Gatineau Hull, Hull Quebec, Quebec Canada,, Austin Travis, Travis Texas, Texas USA,, Canada North America, USA North America). Since this dimension instance involves both geometric members, such as Austin and USA, and non-geometric members, like Gatineau and Hull, it is a mixed dimension instance. The graphic representation of this mixed instance is shown in Figure 3. In this figure, Gatineau and Hull are respectively non-geometric members of the levels city and county and are represented by their names. The geometric members are shown by their geometries. Definition 6: A measure is an attribute that is analyzed against different levels of the dimensions. Accordingly, a spatial measure (measure spatial ) is a spatial attribute. Two types of spatial measures for a vector spatial datacube are recognized: numeric and geometric [Rivest et al. 2005]. Definition 6.1: A numeric spatial measure (measure spatial-numeric ) is a non-geometric spatial attribute. Figure 3: A graphic representation of a mixed instance for the spatial dimension administrative regions. Definition 6.2: A spatial measure that is represented by a geometric spatial attribute is a geometric spatial measure (measure spatial-geometric ). A geometric spatial measure can be computed, for instance, using topological operators (overlap) on members of different dimension levels or can be an independent geometry, such as the location of fires. Example 6: In the fire disaster datacube, the number of injuries is a non-spatial measure. This measure is described by the thematic attribute number of injuries = (numeric, thematic, natural numbers). The measure surface of destroyed residential area is a non-geometric spatial measure, and is described by the attribute surface of destroyed residential area = (numeric, spatial, positive real numbers) expressing the surface of residences that are destroyed by fire disasters. Finally, fire zone is a geometric spatial measure as it is described by a geometric spatial attribute fire zone = (geometric: polygon, spatial, set of polygons representing the location of fires). The fire zone measure represents the location of fires geometrically as polygons on the map. Definition 7: A datacube schema (dcs) is the triple (D dcs, MS dcs, HC dcs ) where: D dcs is a finite set of dimensions, MS dcs is a finite set of measures, and HC dcs is a finite set of hyper-cells (or cuboids [Han and Kamber 2006]) as defined below. A hyper-cell (hc) consists of a pair (L,MS dcs ), where L is a finite set of dimension levels. The set 329

10 L includes exactly one level from every dimension in D dcs. One should note that we have chosen to use the term hyper-cell instead of hypercube (datacube) (which is common in the literature), for such a cell. From a user s point of view, there is only one datacube model for an application. This datacube model embraces all the dimensions, all the measures, and all the possible hyper-cells. Indeed, we consider a hyper-cell as containing only a set of levels and measures. A hyper-cell describes a model for a number of facts, as we will define later. Analytically, the number of possible hyper-cells for a datacube schema is expressed by the product of the numbers of different dimension levels. Definition 7.1: Let hc = (L, MS dcs ) be a hyper-cell. A spatial hyper-cell (hc spatial ) is defined as follows: hc spatial = L, MS dcs l l L l spatial l ms msø MS dcs measure spatial ms } (8) Example 7: The schema for the datacube fire disaster is defined as: The set of dimensions: D fire accident = {administrative regions, time, fire class} The set of measures: MS fire accident = {number of injuries, surface of destroyed residential area, fire zone} Hyper-cells: The number of hyper-cells for the datacube schema fire disaster is 36, which is the result of 6 (number of administrative regions dimension levels) multiplied by 3 (number of time dimension levels) multiplied by 2 (number of fire class dimension levels). Because of the large number of these hyper-cells, we do not define all of them for this example; instead, we define two hypercells: ({city, month, fire class}, {number of injuries, fire zone, surface of destroyed residential area}) and ({country, day, fire class}, {number of injuries, fire zone, surface of destroyed residential area}). The former hypercell is graphically represented in Figure 4. The hyper-cell in Figure 4 includes the spatial level city and two spatial measures fire zone and surface of destroyed residential area. Referring to Definition 7.1, this hyper-cell is spatial. Such a hyper-cell defines a model for a number of facts. These facts are used to answer to queries such as: What is the number of injuries of fire of class A in the city of Montreal in July 2006? or Where are the fire zones of class B in the city of Toronto in January 2007? Definition 8: A datacube (dc) is an instance for a datacube schema dcs = (D dcs, MS dcs, HC dcs ), and consists of a pair (DI, F), where DI is a set of instances for dimensions in D dcs. In DI there is exactly one instance for every dimension in D dcs. F is a set of facts defined over dimension instances DI. A fact describes an event of interest for a decision-making process within an enterprise, and is an instance of a given hyper-cell hc in HC dcs. Therefore, a fact f is defined by a pair (M,V), where M is a finite set of members of dimension instances in DI (exactly one member from each dimension instance in DI), and V is a finite set of measure values for measures in MS dcs. These measure values are calculated with respect to the members of M. Definition 8.1: As we mentioned earlier, a fact can be modeled by a hyper-cell. A spatial fact (f spatial ) is an instance of a spatial hyper-cell and describes an event of interest for a decision-making process that happened in the space. A spatial fact can be of one of two types, geometric and non-geometric. Definition 8.2: Let f = M,V) be a spatial fact, a geometric fact (f geo ) is defined as follows: f geo = M, V m m M m geo m v v V geometry v } (9) Definition 8.3: If a spatial fact is not geometric, it is a non-geometric fact ( f non-geo ). Figure 4: A graphic representation for the spatial hyper-cell ({city, month, fire class}, {number of injuries, fire zone, surface of destroyed residential area}). The levels are shown as three faces of the cell while the measures are inside the cell. 330 Definition 8.4: A spatial datacube stores spatially referenced facts. However, to be recognized as spatial by the IT community, the datacube must also supply a cartographic representation where the user can exploit the provided maps in a significant way.

11 Thus, as in spatial databases, the ability to produce cartographic outputs and manipulations is a central criterion to determine whether to label a datacube with the term spatial. Such cartographic data navigation capabilities are typically enabled when the datacube has a geometric measure (or a geometric) or mixed dimension instances. Here, one should note that a datacube including only a non-geometric dimension instance (e.g., names of cities, provinces and countries) or numeric spatial measures is not typically considered a spatial datacube. However, this does not remove the spatial characteristics of non-geometric instances of spatial dimensions and spatial facts as well as numeric spatial measures. Based on the above discussion, we define a spatial datacube, (dc spatial ) as a datacube dc = (DI F), where among the facts in F, there is at least one geometric fact: dc spatial = DI, F f f F f geo f (10) Example 8: The datacube fire disaster is defined as: Three dimension instances: An instance for the administrative regions dimension as defined in Example 5: {(Gatineau Hull, Hull Quebec, Quebec Canada,, Austin Travis, Travis Texas, Texas USA, Canada North America, USA North America)} An instance for the time dimension: {( , ,, , , ,, , )} An instance for the fire class dimension: {(A all_fire_classes, B all_fire_classes, C all_fire_classes, D all_fire_classes, E all_fire_classes)}. Based on the above dimension instances, the facts for the spatial hyper-cell ({city, month, fire class}, {number of injuries, fire zone, surface of destroyed residential area}) are presented in Table 1. All these facts are geometric, because they include geometries representing fire zones (each p i represents the location of a fire zone). Referring to the Definition 8.4, the fire disaster datacube is a spatial datacube. The facts in Table 1 represent the number of injuries, the fire zones, and the surfaces of destroyed residential area in different cities, different months, for different fire classes. As stated earlier, the number of hyper-cells for a datacube schema can be large. Furthermore, it is possible to have several thousand facts for each hyper-cell, leading to a very large numbers of facts. In this example, however, we presented a limited number of illustrative facts. 4. Characteristics of the Proposed Spatial Datacube Model 4.1. General Characteristics The proposed model has the necessary and fundamental features that any datacube model should include. These features are the following [Blaschka et al. 1998; Pedersen 2000; Torlone 2003]: Separation between structure and content: This is a fundamental feature of any database model. The proposed model makes a distinction Table 1: The representation of a number of facts for the hyper-cell ({city, month, fire class}, {number of injuries, fire zone, surface of destroyed residential area}). City Month Fire Class Number of injuries Fire zone Surface of destroyed residential area Montreal A 14 P Gatineau B 3 P Sherbrooke C 0 P Austin B 18 P k 8700 Houston A 14 P k

12 332 between the schema, which represents the structure of data (e.g., the level, dimension schema, hyper-cell, and datacube schema), and the instances, which are the data contents (e.g., the member, dimension instance, fact, datacube). Explicit notion of dimension and datacube: The proposed model formally defines the different components of a datacube, (i.e., the level, member, dimension, measure, fact, and datacube). Explicit multiple hierarchies in dimensions: As defined, a dimension can include a number of hierarchies of levels, and different aggregation paths are allowed within a dimension. In Example 4, the administrative regions dimension includes two hierarchies, Canadian division and USA division. Several attributes per level: Within the proposed model, a level can include a set of attributes. Including attributes allows the representation of the descriptions of a level. The level city in Example 2 has two attributes: name and location. Measure sets: This feature indicates that the model should be able to support facts that involve several measures. According to the proposed definition, a hyper-cell, which describes a number of facts, can contain a set of measures. Including a number of measures in hyper-cells allows us to have facts with a set of measures Spatial Characteristics Taking into account that the model describes spatial datacubes, in addition to the above general characteristics, the proposed model has the following two features: Supporting spatial data: A basic requirement of a model for spatial datacubes is to support spatial data. The proposed model supports spatial data within levels, dimensions, measures, facts, and datacubes. This support includes both spatial data represented by geometry, (i.e., geometric spatial attribute as well as spatial data) represented by data types other than the geometric type such as addresses and place name, (i.e., non-geometric spatial data). Explicit and precise definitions for the fundamental components: One of the principal features of the proposed model is to provide a precise definition framework for the different components of spatial datacubes, distinguishing these components from the non-spatial ones. Such a framework is based on a view of spatial data considered as having both geometric and non-geometric representations, as suggested by the international standards in geomatics (i.e., ISO/TC211 and OGC). For each element of a datacube, its spatial equivalent is defined. Two types of spatial attributes are recognized: geometric spatial attribute and non-geometric spatial attribute. We defined a spatial level and two types of members for spatial levels, (i.e., geometric and non-geometric members). A spatial dimension was defined, and instances of spatial dimensions were divided into three types: geometric, non-geometric, and mixed. In addition, two categories for spatial measures are discriminated: numeric and geometric spatial measures. We divided spatial facts into two types, (i.e., geometric and non-geometric facts). Finally, a spatial datacube is defined as a datacube that includes at least one geometric fact. 5. Conclusion and Future Work In this paper, we addressed an important issue in the realm of spatial decision-support databases: the lack of a formal model that correctly and precisely defines fundamental components of spatial datacubes (e.g., spatial dimension, spatial measure, and spatial fact). In order to present this issue and propose a solution for it, we made two strategic contributions. As the first contribution, we reviewed and analyzed the existing models for spatial datacubes by specially focusing on the definitions given by these models to spatial datacube components. The results show that, on the one hand, there are some models that consider a boarder view of spatial data in alignment with the international standard in geomatics. However, these models have not yet been presented in a formalized way. On the other hand, there are a number of formal models for spatial datacubes, but these models consider a limited perspective for spatial data as only those data that have a geometric representation. Consequently, the formal definitions given to spatial datacubes and their fundamental components by the latter models do not correctly reveal the entire power of spatial data integrated with in these datacube components. The second contribution of the present paper was to propose a formal model for spatial datacubes with a primary focus on recognizing and precisely defining its different spatial components at both the schema and instance levels. To achieve this goal, we revisited the definition of the spatial attribute taking into account the international standards in geomatics. A spatial attribute was defined as any attribute describing spatial proper-