Probabilistic Relational Models for Operational Risk: A New Application Area and an Implementation Using Domain Ontologies

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1 Probabilistic Relational Models for Operational Risk: A New Application Area and an Implementation Using Domain Ontologies Marcus Spies Abstract The application of probabilistic relational models (PRM) to the statistical analysis of operational risk is presented. We explain the basic components of PRM, domain theories and dependency models. We discuss two real application scenarios from the IT services domain. Finally, we provide details on an implementation of the PRM approach using semantic web technologies. 1 Introduction In statistical models for operational risk, we often need to combine information from multiple entities. An example would be to build a model combining: Observations related to failures of technical or other service components. Observations related to customer contract related data, like customer claims in the scope of a service contract. Both classes of observations can be described by suitable theoretical distributions whose parameters can be estimated given actual empirical data. Typically, failure probabilities can be assessed using distributions from reliability theory, while customer claims will follow some loss sizes distribution that is typically taken from one of the fat tail distribution families like the Weibull distribution. In operational risk analysis, combination of these distributions is performed using convolution in order to estimate losses as they accumulate over time and compute meaningful scores like value at risk (VaR), see Giudici (2010). M. Spies Chair of Knowledge Management, Department of Computer Science, LMU University of Munich, Germany marcus.spies@ieee.org This work has been supported by the European Commission under the umbrella of the MUSING project, contract number , A. Di Ciaccio et al. (eds.), Advanced Statistical Methods for the Analysis of Large Data-Sets, Studies in Theoretical and Applied Statistics, DOI / , Springer-Verlag Berlin Heidelberg

2 386 M. Spies In the present service provider scenario, however, the combination of the information from both data sets gets more complicated if we want take into account the service providers who are operating the technical components and also appear as contracting party in customer service contracts. This implies that only component failures that fall in the service domain of a given service contract can be the possible causes of certain customer claims. This dependency can be formulated conveniently using database schemes as usual in relational database modelling (see Sect. 2). As a consequence, for an integrated statistical model of multivariate observations in a multi-entity setting, we must combine the sampling spaces for both observation classes by considering a suitable constraint structure on the product space. A formal language is required in which we can declare the constraint on possible dependencies between observations and perform computations related to parameter estimation in accordance with the factual entity dependencies underlying the real data at hand. This requirement for a relational modelling enabled formulation of observation based variable dependency has been noted in several application fields, notably genome analysis, web link analysis, and natural language based information extraction. As a consequence, the new field of probabilistic relational modelling (PRM) and learning is being established by a network of cooperating groups, see Getoor and Taskar (2007). PRM are closely related to Markov random fields, but the approach to defining variables and estimating parameters is different. PRMs can be introduced as extensions of Bayesian networks, which are nothing but single-entity PRMs. Therefore, the PRM approach nicely specializes to well known methods for statistical analysis of graphical models, see Cowell et al. (2007). Several approaches related to PRM with essentially the same basic assumptions have been published, they build on: Using an entity relationship model (ERM) for encoding the relational dependencies in Getoor et al. (2007). Using a suitable subset of first order logic (FOL) and model-theoretic semantics to describe entities, attributes and dependencies in Kersting and De Raedt (2007). Combining ERM and FOL assertions for capturing constraints in the Directed Acyclic Probabilistic Entity Relationship approach (DAPER) (Heckerman 2007). The presentation in the paper is motivated by applications and presents examples from the context of the EU integrated project Multi-Industry Semantics based Next Generation Business Intelligence (MUSING, EU contract number ). In Sect. 2, we present the basic approach to relational modelling of an application domain, in Sect. 3 we explain dependency models in PRMs and their transformation to Bayesian networks, and outline two case studies. In Sect. 4 we discuss an implementation of the model definition and data collection steps using domain ontologies.

3 Probabilistic Relational Models for Operational Risk Relational Domain Models The first step towards a probabilistic relational model is to define an explicit formal data theory of the given application domain in a declarative way. This theory contains non-logical (factual) axioms about classes, their generic relationships and attributes. In a statistical setting, the attributes correspond to random variables. The theory has models in terms of real-world scenarios allowing for experiments. These experiments consist of drawing samples of objects instantiating the classes. These objects preserve the relationships as stated in theory. What we observe is the values of the attributes corresponding to the class attributes in the theory. More formally, using concepts of the unified modelling language UML (Object Management Group 2010a,b), let X DfX 1 ;:::;X n g denote a set of classes. Each class X i is further described by attributes A.X i /. These attributes are assumed to take values in discrete domains, i.e. they are considered as nominal scaled variables. In practice, it is often assumed that the attributes are valued in one element of the set of atomic datatypes in XML Schema and are restricted to a suitably enumerated finite subset in case of an infinite value set. The classes in X are further allowed to be related by associations contained in a set R. A direct association is a two-place relation i;j W X i! X j on classes.x i ;X j /. If a direct association is a mapping, it is also called a reference. Thus, in case of a reference, j.x i /j D 1 8x i 2 X i. References can be seen as generalizations of has-a relations in artificial intelligence. An m W n association is modelled by an explicit association class that contains references to associated members in each participating class. Association classes can combine more than two classes. The definition of a suitable set of classes, their attributes and associations is a theory of the application domain in question. Such a theory can be formulated according to different modelling approaches in different specific formal languages. One common choice is UML class diagrams (Object Management Group 2010a,b). Another relevant modelling language is entity relationship modelling (ERM) that allows to derive a relational database schema from the theory. Basically, an ER model interprets classes by schemas of relational tables, references by foreign key relationships, and association classes by explicit join tables containing foreign keys to the tables participating in the association. ERM is based on relational database normalization and schema theories (Beeri et al. 1977, 1983). A particular requirement against an ERM is that references in a domain theory form directed acyclic graph (DAG). Finally, a class based domain data theory may be formulated using description logic by defining a domain ontology, most commonly based on the web ontology language OWL (Motik et al. 2006). In ontology engineering, the attributes of classes are modelled by datatype properties, associations by object properties, which are declared functional for references. Possible transformations and incompatibilities between domain theories from these different modelling languages have been studied extensively and formalized

4 388 M. Spies in the Object Management Group s Ontology Definition Metamodel specification (Object Management Group 2009). 3 Probabilities in Relational Domain Models The second step in setting up a PRM is specifying a probability model covering attributes of classes. The most obvious way is to consider the class attributes as (discrete) random variables. This leads to a family of statistical models studied by Heckerman (2007) andgetoor et al. (2007), and also, in a stronger predicatelogic driven approach, by Laskey (2006). As usual in graphical statistical models, the probability model is stated in terms of statistical dependency or independence assertions (LauritzenandSpiegelhalter1988; Cowell et al. 2007). An attribute dependency assertion is of the form Pr.A j.x i /jpa.a j.x i // with A j.x i / 2 A.X i /,wherepa.a j.x i // denotes a set of parent attributes. Thus, dependency assertions are stated for subsets of attributes of one or several classes, possibly with constraints in first-order logic added, see Heckerman (2007). A set of non-circular dependency assertions for j 2 J is also called a dependency model, denoted by D J. If a dependency model contains only attributes and parents from one class, the model translates directly into a conventional Bayesian network. The interpretation I (in the usual sense of first-order-logic) of a set of classes and associations from a relational domain model is described as I.X / [ I.R/. A generic element of I.X i / is written x i, with attribute value variable x i.a j / for the attribute subscripted j in A.X i /. It is of key importance to observe that different interpretations of a domain theory usually involve different references. Therefore, if a dependency model contains attributes from several classes, different interpretations usually lead to different entities being related. As a consequence, the sample space for a dependency model restricted set of class attributes depends on the interpretation of the domain theory. This is the key statistical issue when moving from single-entity Bayesian networks (BN) to PRMs. Formally, if Pa.A j.x i // is the matrix fy 1.a y1 /;:::;y m.a ym /g, thesample space is the set of possible observations fx i.a j /; y 1.a y1 /;:::;y m.a ym /g 8x i 2 I.X i /; y 1 2 I.Y 1 /;:::;y m 2 I.Y m / s.t..x i ;y 1 ;:::y m / 2 I.R/ it contains all observable cases of class instances in the sample and their corresponding attribute value combinations constrained by the specific references relating x i to y j in the given interpretation. One basic result from Heckerman (2007), Getoor et al. (2007) isthatanyd transforms to exactly one BN for each partial interpretation that fixes class instances and associations only, comprising references. Such a partial interpretation is referred to as skeleton. The random variation is then confined to the attribute values. The practical use of this result is in multi-entity repeated measures analysis. Asa consequence, the usual computations known from Bayesian network algorithms can be applied (Lauritzen and Spiegelhalter 1988; Cowell et al. 2007).

5 Probabilistic Relational Models for Operational Risk 389 However, in practical terms, a scenario with specific and fixed instances of all classes, references and other associations is often not useful. Therefore, in terms of statistical learning, we generalize over possible skeletons by allowing only as many parameters as the generic dependency assertions require and tie parameters from training on different skeletons accordingly. In general, the log-likelihood function for a PRM can be expressed within each partial interpretation or skeleton as sum across all classes, then across all attributes appearing on the LHS (left hand side) of a dependency assertion, of the conditional log-probabilities of the elements in the skeleton. More formally, given a dependency model D J, then, for one skeleton, we have (see Getoor et al. (2007), p. 162, adapted to our notation, and, in particular, letting S.x i.a j // D fy 1.a y1 /;:::;y m.a ym /g if.x i ;y 1 ;:::;y m / 2 I.R/, and else D;), and A j the range of the attribute subscripted j in A j.x i /,andn the count of classes in the domain theory) L. DJ ji ; S / D log Pr.I js ; DJ / D X nx X X log Pr.x i.a j /js.x i.a j //; DJ / j 2J id1 a j 2A j x i 2I.X i / 3.1 Case Studies Customer Claims Risk in IT Services A promising application of the approach is based on the MUSING cooperation with the Italian bank MPS (Monte dei Paschi di Siena). The purpose is to evaluate the risk (in terms of probability and cost) of customer claims following failures in financial IT services providers infrastructures. While the exact relationship between IT failures and customer claims is only known in special cases, investments in network infrastructure and software can be prioritized if probable causes of high cost or high frequency claims can be inferred. Building on the example of the introduction, a simplified domain theory focussing on corporate customers and network servers can be set up as follows: 1. The relevant classes in this application domain are customers, (IT) service providers, services, server operators, servers, service contracts. 2. Each service contract is between one service provider and one customer for one service. 3. Each claim refers to one service contract. 4. Each service has one provider, failures are summarized in an attribute called downtime average. 5. Each service is deployed on some service components, an m W n relationship, since components can provide different services. 6. A service component is managed by a service operator. The operator defines maintenance policies etc.

6 390 M. Spies Fig. 1 A database scheme diagram for operational risk analysis in IT services customer relationships, produced with Microsoft (R) Visual Studio 2010 This domain theory (with some extras) is depicted as database scheme diagram in Fig. 1. Note that the overall structure of the diagram is an acyclic directed graph. In this situation, a PRM based on a fixed partial interpretation for a multi-entity repeated measures analysis is realistic, because entities are linked by contracts for some period of time during which failures, claims, maintenance problems etc can be recorded and analyzed. To illustrate this, Fig. 2 shows an overlay of a partial interpretation with a suitable ground Bayesian network, simplified (and with a little bit of humour added) from MUSING. In a database setting, the computation of the log-likelihood can be conveniently implemented in SQL (for a simplified example, see Getoor et al. (2007)). Note that the Bayesian network structure would change with variations in the objects associations, even if the generic dependencies were the same, e.g. those from services downtimes to claims. Corporate Customers Risk in Services Provisioning A second application case study is the MUSING pilot for assessing operational risks at a virtual network operator (VNO) with a large customer base in Israel. The key practical advantage of the PRM based solution is the ability to integrate corporate customer data with IT operational services protocols into predictions and assessments of likely operational

7 Probabilistic Relational Models for Operational Risk 391 MI1 VNO1 MI2 VNO2 FS1 sun1 FS2 ibm2 FS3 dell3 LossT topscout LossF flopscout ClaimA DT2 flight Booking c1 c3 DT1 webhosting Claim4 c2 ClaimB ClaimC Fig. 2 Overlay of a partial interpretation (dashed items) and the ground Bayesian network (elliptic nodes and directed arcs) for the domain model from Fig. 1. In the partial interpretation, c1 to c4 denote contract instances linking a service (like webhosting), a provider (like flopscout) and a customer (like Tom), etc. A ground Bayesian network is provided for some attributes of different classes (MI for maintenance policy by a server operator, DT for downtime of a service, FS for failure severity of a server). The Bayesian network exhibits instance specific probabilistic attribute value dependencies c4 losses and their economic impact on the VNO. This ability has been used to deploy a decision support dashboard application prototype at the VNO. The application integrates indicators from the public part of the corporate balance sheets with operational data for the services the customer requests and computes a qualitative overall riskiness score. We now turn to the implementation details for both case studies. 4 An Implementation Using a Domain Ontology Development and Runtime Environment In this section, we introduce and discuss the EU MUSING implementation of the PRM domain modelling and data integration based on domain ontologies using the web ontology language OWL, see Motik et al. (2006). For the MUSING project, the specific need arose to have a PRM framework available that could easily be combined with existing domain ontologies as they were built for several risk management domains throughout the project using the web ontology language, see Motik et al. (2006). The key motivation for an ontology based representation of data for analytic processing is the combination

8 392 M. Spies of statistical analyses with text processing (annotation, text classification etc), see Therefore, a solution was envisioned that would make it very simple to add attribute (datatype property) dependencies to an existing OWL ontology. Moreover, the MUSING ontologies are constantly populated (enriched by new instances) in real application environments. Therefore, simple mechanisms to collect the sufficient statistics of the parametric PRM learning were needed, as well. This could be accomplished using the SPARQL RDF-query language that was extended by SELECT COUNT and UPDATE constructs during the MUSING project lifetime, see Prudhommeaux and Seaborne (2008). This allows to collect counts and store them into appropriate places in the domain ontology itself. The persistent storage of counts is combined with a versioning mechanism in order to handle learning data from various skeletons and from various datasets independently. This was implemented by means of a service oriented architecture offering domain or application specific repositories in addition to the definitory parts (classes, properties, axioms) of the ontologies. In addition, an export utility of the resulting ground Bayesian network an XML representation was implemented. For this, the XML for Bayesian networks format (Microsoft Corp. 2009) was used. Using this utility, applications of MUSING can pass parameter estimates to several commercial or freely available Bayesian network tools for further processing. An import utility for parameter priors could be added easily. In terms of combining PRMs and ontologies, the approach taken was to define a set of classes suitable for enabling a straightforward declarative definition of a dependency model D in any domain ontology. This amounts to defining suitable ontology classes and properties such that a dependency assertion can be stated by defining appropriate instances of additional and restrictions on internal classes of a given domain ontology. These instances shall then be used for collecting counts as sufficient statistics for parameter estimation within a given skeleton. Two generic classes were needed to back up an implementation of this approach. The first, ObservableAttribute, serves mainly as a source of inheritance relationships for other ontology classes appearing as (object) properties in a dependency model. This means that, in our approach, dependency assertions are expressed in terms of object properties of domain classes. In order to ensure finiteness of the implied discrete random variables, the classes used for any dependency assertion were defined by enumerations. The second class for setting up the construction, CartesianProduct, is instantiated for each dependency assertion. It carries object properties referencing the attributes (object properties) participating in the conditional probability distribution being defined. Coming back to our running example of the virtual network operator and the assessment of operational risk, a given scenario of services, customers and providers can readily be modelled as a fixed skeleton. For our prototypes, we only used simple dependency models with one element in the LHS and in the RHS of any dependency assertion. At this point, it should be remarked that, due to the triangulation theorem for Bayesian networks (see Lauritzen and Spiegelhalter (1988)), the maximum

9 Probabilistic Relational Models for Operational Risk 393 ObservableAttribute Severity severity sety setx CartesianProduct setx : ObservableAttribute sety : ObservableAttribute count : int relativefrequency : double BusinessLine businessline ASP provider Customer businessline : BusinessLine owner Service owner : Customer provider : ASP causedby Event causedby : Service severity : Severity Fig. 3 A domain ontology with two additional entities for probabilistic dependencies. The abstract class ObservableAttribute is used to enable enumerative classes (here severity and business line) for use as discrete random variables. The class CartesianProduct is used to express bivariate dependency assertions. Instances of this class correspond to dependency assertions involving two variables. (Generated with TopBraidComposer of TopQuadrant Inc.) number of classes involved in a dependency assertion can be limited to three. The approach is illustrated in the subsequent figures, Figs. 3 and 4. 5 Conclusion This contribution shows a novel application area for probabilistic relational models in risk analysis as implemented in the EU project Multi-Industry Semantics based Next Generation Business Intelligence (MUSING) for operational and credit risks following the Basel II framework (Basel Committee on Banking Supervision 2004). In particular, as MUSING is strongly relying on domain ontologies, we studied an approach to implementing the dependency model definition and data processing

10 394 M. Spies ISP ISP_1 Event Service BusinessLine provider CargoAirTransport Event_9 causedby Service_1 Customer severity Severity owner Customer_3 businessline Middle Fig. 4 An example of ontology population (diamond shapes represent instances) affecting two entities involved in a relationship with a dependency assertion (system failure events affecting business lines). The counts used for parameter estimation are stored in the suitable instance of the ObservableAttribute class, the count updating is implemented in SPARQL code that is executed upon committing the population instances to the ontology repository steps needed for a PRM analysis using ontologies conforming to the web ontology language (OWL) standard. It could be shown that a rather straightforward extension of any domain ontology suffices to enable declarative dependency model definitions and to collect sufficient statistics that are readily exported to standard Bayesian networks tools for further processing. Acknowledgements The author gratefully acknowledges technical exchanges about practical applications related to IT Operational Risk Management and Credit Risk Management within the MUSING consortium. Special thanks go to Prof. P. Giudici, head of Libero Lenti Laboratory at Pavia University, and Prof. Ron Kenett, Torino University and head of the KPA consultancy (Tel Aviv). The author also acknowledges support by MetaWare S.p.A. of Pisa as overall project coordinator. References Basel Committee on Banking Supervision: International convergence of capital measurement and capital standards: A revised framework comprehensive version (2004). URL org/publ/bcbs107.htm Beeri, C., Fagin, R., Howard, J.: A complete axiomatization for functional and multivalued dependencies in database relations. In: Int. Conf. Mgmt of Data, pp ACM (1977) Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. J. ACM 30(3), (1983)

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