Containment and Complementarity Relationships in Multidimensional Linked Open Data

Abstract

The Linked Open Data (LOD) cloud can act as a source of remote multidimensional datasets which are seemingly disparate, but are modeled under common directives and thus often share a common meta-model, dimensions and measures, as well as external codelists. This gives them a latent measure of relatedness that is independent of the publishers' initial intentions, but a derivative of the motivations behind LOD. In this paper we identify the constituents of relatedness between multidimensional LOD data points (observations) modeled with the Data Cube vocabulary, that often exhibit overlapping values both at the schema and at the data level. Treating hierarchies as first-class citizens, we consider observation relatedness in two aspects, namely containment and complementarity, for which we provide formal definitions and representational semantics. Finally, we present a methodology for computing these types of relatedness and we provide an evaluation over real-world datasets.

Keywords

• Linked Data
• Data Cube
• Multidimensional Data
• Data Enrichment

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Recently, more and more bodies such as governments, statistical authorities, public and private organizations, research and health centers publish information in the form of multidimensional Linked Open Data (LOD) [ 2 ][ 3 ] in very different domains, such as census and statistical data, socioeconomic and demographic indicators, clinical trials and health data, environmental and finance data. The abundance of such datasets enable third parties to have access, exploit and combine published data eventually leading to the generation of new quantifiable insights and knowledge, capable of influencing policy-building and decision-making.

Modelling-wise, multidimensional data are traditionally represented as cubes of observations that are instantiated over a fixed set of dimensions and measures. In the LOD paradigm, W3C has proposed the Data Cube Vocabulary [ 2 ], a recommendation for modelling and publishing multidimensional datasets. However, one difference between closed-world multidimensional data stores, such as OLAP databases, and LOD datasets is that proper LOD publishing techniques across remote data providers will lead to the existence of common or shared terms between seemingly independent multidimensional datasets, given that reusability and interoperability are prime drivers in the semantic web. Practically, this means that ontologies, codelists and hierarchies that are commonly used in the LOD cloud are likely to appear across different and disparate LOD multidimensional datasets. provide extended/enriched information. More specifically, we extend the notion of schema complement, defined in [ 15 ] and apply it at the instance level of observations in order to annotate whether two observations can complement each other’s information. Then, we provide a technique for computing these properties as follows: first all observations are placed in an occurrence matrix that represents them as data vectors in a multidimensional feature space encoding both schema and data information along with dimension hierarchies. The occurrence matrix is used to compute the complementarity matrix, which enables us to derive complementarity between observations. Then, the occurrence matrix is transformed to a set of k |O|×|O| containment matrices, where k is the distinct number of dimensions. Full and partial containment for all pairs of observations are given by adding all matrices together.

Contributions. In short, the contributions of this work are as follows: (a) We define new relationships between individual observations and extend the Data Cube terms with properties for representing: full and partial observation containment between two observations as derivatives of the hierarchical relationships between their dimension values, and observation complementarity as a means of comparison and correlation of different measures; (b) we provide an efficient technique of computing these properties based on occurrence and similarity matrices; finally (c) we evaluate our techniques over real-world multidimensional datasets.

This paper is structured as follows: section 2 provides background knowledge and related work, section 3 defines the new properties, section 4 provides the techniques for computing them, while section 5 proposes possible extensions to Data Cube Vocabulary. Finally, section 6 presents an evaluation of our approach and section 7 concludes the paper and presents future directions of this work. 2

Related Work

Generally, the problem of finding related multidimensional observations has been addressed within the contexts of Linked Data and Online Analytical Processing (OLAP) [ 10 ] [ 11 ]. Data mining techniques in OLAP are known as Online Analytical Mining (OLAM) [ 7 ]. OLAM research works study problems such as clustering/classifying OLAP cubes [ 9 ], detecting outliers [ 8 ], performing intelligent aggregations [ 5 ] and building recommender systems for OLAP sessions based on either query formulation or observation similarity [ 1 ][ 16 ]. Applications of these approaches aim to enable discovery of latent knowledge, promote exploratory analysis [ 6 ], improve OLAP query efficiency [ 9 ] and so on. In this context, Aligon et al. [ 1 ] study the problem of finding similarities between OLAP sessions, i.e sequences of queries that are applied online. To this end they define similarity functions by conducting user-based analysis and they compute similarity of sessions by decomposing the session queries’ features and then using the Levenshtein distance, Dice’s coefficient, term frequencyinverse document frequency (tf-idf) and the Smith-Waterman algorithm. They find the latter to be the best performing measure for their purposes.

Baikousi et al. [ 16 ] provide distance functions categorized over their relation with the hierarchy space. Similarly to our approach, they consider hierarchies to be of prime importance in the problem and base their distance functions on hierarchies. Finally, they summarize the hierarchical distances with two approaches, simple summation and the Hausdorff distance and they find that both approaches are equally effective. Hsu et al [ 4 ] apply multidimensional scaling methods (MDS) and hierarchical clustering (HC) in order to find similarity between OLAP reports of the same cube. They formally define the problem and its constraints, such as identifying when two OLAP reports are comparable, and conclude that a combination of MDS and HC yields the best results.

In the context of Linked Data, similarity or relatedness between entities has been a main component of entity resolution, record linkage and interlinking [ 17 ][ 18 ][ 19 ]. These approaches deal with discovering links between RDF nodes from different datasets in efficient ways by using distance-based techniques. Statistical linked open data have been addressed by [ 13 ] in the context of online analysis and exploration, and in [ 14 ] as a use case scenario for data source contextualization. To the best of our knowledge, this is the first work that addresses the definition, representation and computation of relationships between individual multidimensional LOD observations. 3

Problem definition

We consider that the problem space is composed by n datasets modeled and validated by the integrity constraints imposed by the Data Cube Vocabulary. A dataset is composed of its schema (i.e. dimensions, measures and attribute definitions), and its data (i.e. observations). The values in dimensions are provided by a fixed set of coded lists (code-value pairs) that are hierarchically structured in levels. Flat coded lists, i.e., simple enumerations, are considered to be hierarchies with exactly one level. These are presented formally in the following:

Definition 1 (Cube Structure): Let D={D1, …, Dn} be the set of all input datasets. A dataset Di D is composed by the set of observations, Oi, and the set of schema definitions, Si, and O={O1, …, On} and S={S1, …, Sn} are the sets of all observations and schema definitions in D. Furthermore, a schema Si consists of the sets of dimension Pi and measure properties Mi defined in Di, i.e., Si={Pi,Mi}. Let P=⋃ and M=⋃ be the set of all k distinct dimensions and l measure properties in D. Any pj P, mj M can belong to more than one Si, as dimension and measure properties are reused among sources. An observation o Oi is an entity that instantiates all dimension and measure properties defined in Si. The value that observation oi has for dimension pj is .

Definition 2 (Coded list terms): Each dimension property pj P takes values from a fixed coded list, i.e. a set of code – value pairs, C(pj)={c(pj)1, … c(pj)m}, j=1..k, (for simplicity we write cji instead of c(pj)i). The coded list defines a hierarchy such that when cji ⥼ cjm, then cji is an ancestor of cjm. Furthermore, we define cjroot as the top concept in the code list of pj , i.e., an ancestor of all other terms in the coded list, such that ⥼ . Every coded list term is an ancestor of itself, i.e. ⥼ .

In Figure 2, we present sample coded lists for the dimensions present in motivating example of Fig. 1.

Americas

USA Texas Austin ) sex Total Male

Female refPeriod

All 2001 2011 refArea

Europe Greece Italy

..

Feb2011 Jan2011

Athens

Rome Next we provide the definitions for containment and complementarity properties. Similarly to [ 14 ] [ 15 ], we apply the notion of complementarity between two observations for denoting whether these are comparable, i.e., they have the same dimension values but measure different phenomena. This is represented by the following.

Definition 3 (Observation Complement): Given two observations oa and ob and their dimensions Pa and Pb, oa is observation complement to ob when: ( ) ( ) ( We denote this relationship with Compl(oa, ob) or equivalently oa Compl ob. Def. 1 states that the dimensions in ob must be a superset of the dimensions in oa and the common dimensions must have the same values. All other dimension values of ob must be equal to the root of the dimension hierarchy, thus providing no further specialization. For example, an observation measuring poverty in Greece per year is observation complement with an observation measuring the population in Greece per year for all genders.

Furthermore, a containment relationship captures whether an observation measure is an aggregation of the measures of the contained observations. For example, an observation measuring the population of Greece implicitly contains all observations measuring the population of sub-regions of Greece. We distinguish between full and partial containment. The former denotes that all contained observations can be combined in a roll-up operation for being observation complement with the containing one, while the latter denotes that both contained and containing observation must be rolled-up on their disjoint dimensions for being observation complement. These are presented in the following definition.

Definition 4 (Partial and full containment): Given two observations oa and ob, their dimensions Pa and Pb and their measures Ma and Mb, partial containment between oa and ob exists when: ( ) ( ) ( ⥼ ) An observation oa partially contains ob when (i) there is one Mi shared between oa and ob, (ii) the dimensions of oa are a subset of the dimensions of ob and (iii) there exists at least one dimension whose value for oa is a hierarchical ancestor of the respective dimension value for ob. We denote this as Contpartial(oa, ob) or equivalently oa Contpartial ob. Similarly, full containment between oa and ob exists when: ( ) ( ) ( ⥼ ) That is, an observation oa fully contains ob when (i) there is one Mi shared between oa and ob, (ii) the dimensions of oa are a subset of the dimensions of ob and (iii) values of all dimensions for oa are hierarchical ancestors of the respective dimension values for ob. We denote this with Contfull(oa, ob) or equivalently oa Contfull ob. Observe by definition that the containment property is not symmetric and that given oa Contfull ob we cannot derive that ob Contfull oa. Based on the above definitions, our problem can be outlined as follows.

Problem: Given a set of source datasets D, and O the set of observations in D, for each pair of observations oi, oj O, i≠j, assess whether a) oi Contfull oj , b) oi Contpartial oj and c) oi OC oj. In the following section, we provide our techniques for computing these properties. 4

Computing containment and complementarity properties

Our technique for computing containment and complementarity properties considers that observations are data vectors in a multidimensional feature space composed of all schema definitions and coded list values in D. In addition, the feature set is enriched with all ancestor values in the hierarchy of each coded list, up to the higher common ancestor of all values in D. This is represented by an occurrence matrix that captures occurrence (1 or 0) of a dimension, measure definition and coded list value in the set of observations. The occurrence matrix is used for calculating complementarity properties between two observations. It is, then, used for constructing containment matrices that are used for calculating containment properties.

4.1 Constructing the Occurrence Matrix

Each oi defines a bit vector oi of |C|+|P|+|M| dimensions and all oi O yield a |O|x|C+P+M| occurrence matrix OM that consists of the following sub-matrices:  OMC is the |O|x|C| matrix defined by the occurrences of coded list values in the respective dimension values of all observations. Each value cji Cj corresponding to dimension pj is treated as a feature, i.e., a column in OMC. Hierarchical containment is encoded into OMC using a bottom-up algorithm that places a value of 1 in column cji if the value of the dimension pj of oa is equal to the feature cji and then gives the value of 1 to all columns corresponding to the parents of cji. Finally, we fill with 1’s the cjroot of all observations that do not contain pj in their schema.  OMP and OMM are the |O|x|P| and |O|x|M| matrices defined by the occurrences of dimension and measure properties in each observation. Each dimension and measure definition is considered as a feature and is marked with 1 if oa contains it and 0 if not. The measure values are not taken into account. Therefore, OM= [OMC, OMP, OMM].

The OCC of the example of Fig. 1, given the hierarchies shown in Figure 2, is shown in Table 2. The sub-matrix OMC can be further broken down in separate sub matrices for each coded list, i.e., OMC = [OMC1, …, OMCk] where OMCi is a sub-matrix that represents occurrences for all values of dimension pi and k=|C|. Therefore OM becomes [OMC1,…, OMCk, OMP, OMM]. obs11 obs12 obs21 obs22 obs31 obs32 obs33

4.2 Pair-wise observation containment

Containment matrices. A containment matrix CMi is a |O|x|O| bit-vector matrix that captures pair-wise containment relationships between observations, for each dimension pi. If cell CMi[oa,ob] 0 then observations oa and ob are hierarchically related for dimension pi P (e.g. oa refers to Greece and ob refers to Europe), while CMi[oa,ob] 0 holds otherwise.

Computation of containment matrices. To compute a containment matrix CMi, we first take each bit array OMCi in OMC and we apply a containment function sf for all pairs of observations oa and ob by considering the rows of oa and ob in OMCi as two bit vectors, a and b, respectively. Then, two observations are hierarchically related if the bit-wise AND operation between their corresponding bit vectors yields one of the two bit vectors as shown in [ 12 ]. Given this, we define sf for a pair of observations oa, ob and their bit-vectors a,b resp. in OMCi as the following conditional function: ( ) { ( ) ( ) means that we apply sf for oa and ob in OMCi and if the AND between a and b gives the bit vector b, then oa is contained by ob. If a=b then the relationship still holds. By applying this function for each sub matrix in OCC we acquire a set of k containment matrices of |O|x|O| dimensions, CM1, …, CMk, each capturing containment information for a given dimension property. Then, addition of all CMi yields the Overall Containment Matrix OCM, which holds full and partial relationships in the form of normalized similarities between pairs of observations as follows: ∑ ∑ OCM retains values in the range of [ 0,1 ]. Full containment is given when two observations have similarity 1, i.e., oa Contfull ob iff cell OCM[oa, ob]=1. Partial containment is given when two observations have similarity between 0 and 1, i.e., oa Contpartial ob iff OCM[oa, ob]>0. Finally, no containment between observations exists when OCM[oa, ob]=0.

4.3 Pair-wise observation complementarity

Following the definition of observation complementarity, we use OM to assess whether the dimension values between pairs of observations are the same, given that holds. To check if is true between two observations oa and ob we apply a bit-wise AND to the bit vectors a, b of oa and ob the same way as when trying to compute CM matrices, by using function sf(oa, ob). This is because values in OMP capture whether an observation has a dimension in its schema. Given this, we want to assess whether two bit vectors in OMP are related via a containment property, which justifies the use of function sf, this time on the matrix of the dimension properties. Then, we check if the dimension values of the two observations are equal: ( ) { ( ( ) ) ( ) ( ) means that we apply sf for oa and ob in OMP, while a and b are the bit vectors in OMC. This results in a |O|x|O| Complementarity matrix that gives the value of 1 for complementary observations and 0 otherwise, i.e., oa Compl ob iff Complementarity [oa, ob]>0.

Observe that the time complexity of computing containment and complementarity matrices for N observations is O(N2), further optimization is left as future work. Two approaches to improve running time are (i) parallelism and (ii) reducing the search space by taking into account characteristics of the incoming schemata. For example, if we can conclude that there can be no containment and/or complementarity between observations of Di and Dj just by examining their schema, we do not need to perform any computations on observation pairs between Di and Dj. 5

Complementarity and containment properties in Data Cube

We propose simple extensions to the Data Cube Vocabulary such that complementarity and containment, full and partial, between observations can be represented. We define three properties, containment, partialContainment and fullContainment, where partialContainment is a sub-property of the generic containment property, and fullContainment is a sub-property of partialContainment, which reflects the fact that full containment is a specialization of partial containment. As an example, a relationship Contpartial(oa, ob) is then modelled as shown in the top part of Figure 3. The containment relationship becomes a blank node of the appropriate type and is reified to include information on ob and other possible metadata on the relationship. Similarly, complementarity is denoted with the property imis:complement, as shown in the bottom part of Figure 3. imis:complement _:bn2 imis:observation imis:observation

ex:obs_b ex:obs_c In this section we present the evaluation of our approach over real-world statistical datasets. Our experiments were performed using Java and Apache Jena for handling the RDF models and creating the matrices, and R for computations on the matrices, application of the functions sf and cf and so on.

Datasets. Four datasets have been used. D1 and D2 measure poverty in two different sets of EU sub-regions and periods, D3 measure population in three EU countries and their first-level sub-regions and D4 measure households with internet access in seven EU countries and their sub-regions. They exhibit an overlap of 5 dimensions (location, time, sex, unit and age) and 3 measures (poverty, population and households with internet access). The mappings between the coded lists for the common dimension values have been manually created. Observe that creating the mappings for different dimension values is an orthogonal work to our approach; many approaches from the fields of entity resolution and LOD interlinking can be applied. The datasets are either downloaded as RDF or converted using a conversion script written in Java. Eurostat Linked Data Wrapper1, the Eurostat database2 and World Bank3 were used as sources. The datasets where pre-processed to include data about EU countries, as well as selected sub-regions based on official EU geo classifications (NUTS4), for various time periods, taken from the Gregorian calendar classification of data.gov.uk5. The dataset structures are summarized in Table 2. #of obs.

D1 (539) D2 (1693) D3 (629) refArea 85 regions,20 countries 293 regions, 33 countries 42 regions, 3 countries refPeriod 2004-2011 2003-2010 2009-2013 sex N/A N/A M, F, Total unit Yes Yes Yes age Yes N/A N/A poverty Yes Yes N/A internet N/A N/A N/A population N/A N/A Yes 1 http://estatwrap.ontologycentral.com/ 2 http://epp.eurostat.ec.europa.eu/portal/page/portal/statistics/search_database 3 http://data.worldbank.org/ 4 http://nuts.geovocab.org/ 5 http://datahub.io/dataset/data-gov-uk-time-intervals D4 (316) 65 regions,7 countries 2009-2013 N/A N/A N/A N/A Yes Table 2: Dimensions and measures of the input datasets. Measures are marked in grey

Computing containment and complementarity properties over the observations of the four datasets resulted in the creation of multiple relationships summed up in Table 3. The results do not include self-containing or self-complementing observations. We have defined three metrics, full, partial and compl. For a pair of datasets Di and Dj, they measure the total number of pairs that exhibit full containment, partial containment and observation complementarity respectively, as a percentage of the total possible number of pairs in the given two datasets, minus the diagonal. As can be seen, most new relationships are partial containments, which is a reasonable result given that it is the most weakly defined relationship in terms of its prerequisites. The strictest relationship, observation complementarity, resulted in linking 0.03% of the total possible observation pairs. Sample observations participating in the newly created relationships can be seen in fig 4. The created links are modeled after our proposed vocabulary and uploaded in RDF form in an Openlink Virtuoso store6. D1 D2 D3 D4

D1 647 (0.31%) full 34.3k (16.32%) partial N/A compl 605 (0.02%) full 605k (14.83%) partial 1238 (0.04%) compl N/A full N/A partial N/A compl

D2 N/A full N/A partial N/A compl 3370 (0.14%) full 378k (14.83%) partial N/A (complement N/A full N/A partial N/A compl

D3 N/A full N/A partial N/A compl N/A full N/A partial 204 (0.004%) compl 1k (0.26%) full 261k (65.9%) partial N/A compl

D4 N/A full N/A partial N/A compl N/A full N/A partial N/A compl N/A full N/A partial N/A compl N/A full N/A full N/A full 437 (0.17%) full N/A partial N/A partial N/A partial 22.2k (22.3%) partial 328 (0.05%) compl 218 (0.005%) compl 592 (0.07%) compl N/A compl Table 3: Results of new relationships for test datasets D1...D4. Each cell [i,j] contains information on the total number of pairs exhibiting each relationship, as well as the percentage over all possible pair-wise combinations of observations for the combination

Discussion. The relatedness properties that we have defined yield interesting information on how existing observations can be combined across datasets as well as in the same source dataset. The advantages of this work are two-fold, first it creates new information by combining existing facts (complementarity) and second it creates a containment graph among observations that helps exploration, aggregation and discovery of nearby multidimensional observations. The definitions that we have provided can also be studied in terms of their impact at the dataset level, for example, D3 has 65.9% partial containment in itself, with only 0.26% full containment. This hints that the structure of the dataset’s hierarchies is such that there are a few top level concepts in comparison to lower-level concepts, and that also the depth of the hierarchies is not very large. As a matter of fact, this is true for D3 as it addresses 3 countries and 42 sub-regions, all of them lying on the same level. Figure 4 shows an example of an observation _:b134 that exhibits both complementarity and containment with _:b432 and _:b135 resp. Complementarity shows that we can combine the measures of two observations set on the same dimension values in order to complement their information. Two complementary observations can be joined on their common dimension values and combined into a new observation, whose schema contains the union of their dimension and measure sets. Figure 4 shows how we can combine population and households with internet access because they are both set on region EL1 in 2010 although _:b432 does not involve gender dimension.

_:b134 a qb:Observation ; qb:dataSet _:population ; sdmx-dimension:sex sex:Total ; sdmx-dimension:refArea nuts2008:EL1 ; sdmx-dimension:refPeriod uk:2010 ; sdmx-measure:obsValue 3,590,447 .

observation complement fullContainment _:b135 _:b432 a qb:Observation ; qb:dataSet _:internetHouseholds ; sdmx-dimension:refArea nuts2008:EL1 ; sdmx-dimension:refPeriod uk:2010 ; sdmx-measure:obsValue 31.5 . a qb:Observation ; qb:dataSet _:population ; sdmx-dimension:sex sex:Male ; sdmx-dimension:refArea nuts2008:EL11 ; sdmx-dimension:refPeriod uk:2010 ; sdmx-measure:obsValue 302,855 . In this paper, we have presented a novel approach for identifying and modeling relationships between observations of multidimensional linked open data. We have defined three new properties, namely full and partial observation containment and observation complementarity between two observations as derivatives of the hierarchical relationships between their dimension values, and as a means of comparison and correlation of their different measures. We have proposed a possible extension on the Data Cube terms for representing these properties and we have provided an evaluation of our approach over real-world statistical datasets.

A future direction concerns techniques for grouping and combining observations into new datasets based on their containment and complementarity properties. Another direction is the incorporation of the Data Cube attributes in our algorithms as these often represent valuable information like units of measure, observation status or folded dimensions and can be used for creating dimension value mappings at the preprocessing stage. Finally, we will explore techniques for optimizing the computation time and scale up the overall performance of our approach.

Acknowledgement. This work has been co-financed by the EU project DIACHRON and by EU and Greek national funds through the Operational Program "Competitiveness and Entrepreneurship" (OPCE ΙΙ) of the National Strategic Reference Framework (NSRF) - Research Funding Program: KRIPIS. 8

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