Complex concept lattices for simulating human prediction in sport

COMPLEX CONCEPT LATTICES FOR SIMULATING HUMANPREDICTION INSPORT∗ ARANDA-CORRAL, Gonzalo A, BORREGO-D´IAZ, Joaqu´ın, GAL´AN-P ´AEZ Juan Abstract In order to address the study of complex systems, the detection of patterns in their dynamics could play a key role in understanding their evolution. In particular, global patterns are required to detect emergent concepts and trends, some of them of a qualitative nature. Formal concept analysis (FCA) is a theory whose goal is to discover and extract knowledge from qualitative data (organized in concept lattices). In complex environments, such as sport competitions, the large amount of information currently available turns concept lattices into complex networks. The authors analyze how to apply FCA reasoning in order to increase confidence in sports predictions by means of detecting regularities from data through the management of intuitive and natural attributes extracted from publicly available information. The complexity of concept lattices -considered as networks with complex topological structureis analyzed. It is applied to building a knowledge based system for confidence-based reasoning, which simulates how humans tend to avoid the complexity of concept networks by means of bounded reasoning skills. Key words Bounded rationality, complex networks, formal concept analysis, sport forecasting. 1 Introduction Sports data analysis, and the study of the complex systems (CS) behind sport competitions, have experienced rapid growth in the recent years. The increasing interest in models of prediction both for betting markets and for optimizing player/team performance, can be explained by the amazing growth of business in the field. Much evidence about the predictability in sport betting encourages many projects and research[1−4]. This paper concerns human reasoning with basic qualitative data for predicting results. The understanding and simulation of forecasting ability allow estimation of global trends for betting markets and eventually to improve betting among the general public. Sport fans make predictions by reasoning with features of qualitative nature (or those that they make qualitative by estimating thresholds). ARANDA-CORRAL Gonzalo A Universidad de Huelva. Department of Information Technology; Crta. Palos de La Frontera s/n. 21819 Palos de La Frontera. Spain. Email: [email protected]. BORREGO-D´IAZ Joaqu´ın · GAL ´AN-P ´AEZ Juan Universidad de Sevilla. Department of Computer Science and Artificial Intelligence; Avda. Reina Mercedes s/n. 41012 Sevilla. Spain. Email: [email protected]; [email protected]. This work is supported by the TIN2009-09492 project of the Spanish Ministry of Science and Innovation, and Excellence project TIC-6064 of Junta de Andaluc´ıa co-financed by FEDER funds. Sport forecasting represents one of the most increasing research topics in this field, because of its economic impact in betting markets as well as for its potential application to problems with similar behavior (other markets)[1]. Because of its economic impact, a considerable number of predictive models have been developed by companies and sport franchises (cf. Chapter 10 in [3]). Particularly interesting is the estimation of betting distribution in crowd betting markets and services (for example, the Spanish national soccer betting service Quiniela, http://www.loteriasyapuestas.es/quiniela). In this case, the aim is to simulate and detect betting trends based on users’ reasoning about the short-term evolution of the competition system. Like other CS, the competition in sport is an open system influenced by an astonishing amount of factors and subsystems. However, humans usually make a quick selection of a few (subjective) features from a limited amount of information (often sport magazines and TV sport news) as well as from their own experience. Therefore, for predicting popular bets it’s more precise to adopt rational strategies for reasoning instead of pure statistical methods. Two basic tasks are involved in the process: attribute selection and how to apply them to the reasoning. Actually, the second task attempts to find relationships between the attributes and the results. Formal concept analysis is the theory used in this paper for discovering these relationships. Formal concept analysis (FCA)[5] is a mathematical theory for data analysis using formal contexts and concept lattices as key tools. Domains can be formally modeled according to the extent and the intent of each formal concept. In FCA, the basic data structure is the formal context (with a qualitative nature) which represents a set of objects and their properties and is useful both in detecting and describing regularities and structures of concepts. It also provides a sound formalism for reasoning with such structures, mainly stem basis and association rules. Therefore, it is interesting to consider its application for reasoning with temporal qualitative data in order to find temporal trends[6]. In this paper, the scope of the FCA application is the challenge of sports betting, specifically, the forecasting of soccer league results. Soccer leagues represent a challenging system with a huge amount of knowledge, available through the WWW and several commercial products. Their behavior is exhaustively analyzed every week by journalists, betting companies, and supporters. This analysis can involve an exhaustive study of previous matches[7−8] and the performance of players. With the aim of processing the information for predicting the evolution of the system, three dimensions have been considered in existing systems: 1) Those which analyze information on teams (endogenous) versus those which analyze results (exogenous); 2) Those which exploit quantitative data versus those which exploit qualitative knowledge, and finally, 3) Statistic-based ones versus other methods. It is usual to find hybrid models in literature, but rarely pure qualitative and exogenous reasoning systems, although their use is considered for experiments (for example, frugal methods[9] and based on recognition heuristic[10])oraspart of hybrid systems (see, e.g., [11]). There are two reasons that may justify this point. On one hand, when transforming a large quantitative dataset to a qualitative problem one is faced with the selection of an acceptable threshold and the discovery of better relationships between the properties (see, e.g., [12]). On the other hand, a qualitative dataset must be achieved with some amount of information based on confidence, trust, or probability of these data sets. This paper describes the research for selecting and computing attribute sets to be used with FCA in the prediction of soccer match results. This research is previous to building an expert system for advising sport betting which could detect some kind of regularities in data. Concept lattices, which are computed from attribute values, represent a mathematical structure of relationships among the concepts which are involved in selected sport events to study. Since this method is bet-oriented, its performance is evaluated within a confidence-based reasoning system. This system increases the number of hits in forecasting of soccer matches, discovering trends by means of data mining, and reasoning using association rules. The analysis of attributes was used in [6] to describe a confidence-based (and contextual) reasoning system for predicting sports betting. Actually the aim is twofold. On one hand, the complexity of the qualitative representation of soccer competition is investigated. On the other hand, we analyze the attribute selection problem as a feature selection problem, which would shape the behavior of the CS that represents professional soccer leagues. The theoretical framework, on which this model is based, is presented in [13]. Due to the huge amount of information, attribute selection advised by experts is mandatory. In fact, the system can be considered a reasoning model based on bounded rationality and recognition heuristics, and focused on features which were considered important by human experts. Therefore, the system aims to forecast results, but it is designed based on bounded rationality models, instead of pure statistical models. However, in the future hybrid models will be considered. 1.1 Bounded Reasoning on Complex Systems As it has already been mentioned, in order to predict betting trends, attributes have to be selected from publicly available information (Web sites, sport journals, etc.). Sport betters extract the information from these sources that they eventually use to argue a prediction. In this way the aim of the system is to simulate (and to improve if possible) better’s behavior. The system must analyze qualitative features of CS, using FCA. The idea is to isolate qualitative attributes from (past) local interactions among components of complex systems and to apply FCA tools in order to predict properties of system behavior in the near future. Bounded rationality (BR) is intimately related to the human capacity for making inferences under limited time and knowledge[14]. From the viewpoint of the artificial intelligence (AI), BR comprises reasoning techniques that facilitate, for example, context and temporal reasoning. Psychological research on specific heuristics in human inference processing reveals a complex framework where classic (computational) logic is not sound for explaining the success of several of them, for example, recognition heuristic (RH)[10]. A number of experiments show that cognitive mechanisms capable of successful performance in the real world do not need to satisfy the classical norms of rational inference (cf. [15]). In fact, an intriguing question from ecological rationality analysis is: How could more knowledge be no better- or worsethan significantly less knowledge[10]? Several experiments show that bounded rationality is a successful human inference. One of the key features in BR heuristics is that inference process is concentrated on a limited set of experiences in which objects, properties, and actions are selected. The aim is to model this feature with formal concept analysis (FCA). 1.2 Structure of the Paper Section 2 reviews basic features of FCA. In Section 3, the reasoning model for CS based on FCA is described. Section 4 is devoted to analysing the problem of attribute selection to model bounded reasoning for soccer. In Section 5, the scale-free residue of the concept lattice is presented. Section 6 describes the main issues in attribute selection. Section 7 presents the results obtained by the system and how to estimate the reasoning performance from subnetworks of the concept lattice. In Section 8, some conclusions and insights about future work are made. 2 Background: Formal Concept Analysis According to Wille, FCA[5] mathematizes the philosophical understanding of a concept as a unit of thoughts composed of two parts: the extent and the intent. The extent covers all objects belonging to this concept, while the intent comprises all common attributes valid for all the objects under consideration. It also allows the computation of concept hierarchies from data tables. In this section, we succinctly present basic FCA elements (the main reference is [5]). A formal context M=(O,A, I) consists of two sets, O(objects) and A(attributes) and arelationI⊆O×A. Finite contexts can be represented by a 1-0-table (identifying Iwith a boolean function on O×A). See Figure 1 for an example of formal context about living beings. The FCA main goal is the computation of the concept lattice associated with the context. Given X⊆Oand Y⊆Ait defines X:= {a∈A|oIa for all o∈X}and Y:= {o∈O|oIa for all a∈Y}. A (formal) concept is a pair (X,Y ) such that X=Yand Y=X. For example, concepts from formal context about living beings are depicted in Figure 1 (left), right. Each node is a concept, and its intension (or extension) can be formed by the set of attributes (or objects) included along the path to the top (or bottom), e.g., the node tagged with the attribute Legs represents the concept ({Legs, Mobility, Need Water},{Cat, Frog}). Frog Legs Cat Mobility Need water Corn Aquatic Fish Leech AB CDE Need water Aquatic Mobility Legs Cat Leech Frog Corn Fish 1 <5> { } ==> Need water; 2 <3> Need water Aquatic ==> Mobility, 3 <2> Need water Legs ==> Mobility, Figure 1 Formal context, associated concept lattice, and stem basis 2.2 Implication and Association Rules Between Attributes Logical expressions in FCA are implications between attributes. An implication is a pair of sets of attributes, written as Y1→Y2, which is true with respect to M=(O, A,I) according to the following definition. A subset T⊆Arespects Y1→Y2if Y1⊆ Tor Y2⊆T.Itsaysthat Y1→Y2holds in M(M|=Y1→Y2) if for all o∈O,theset{o}respects Y1→Y2.Inthat case, it is said that Y1→Y2is an implication of M. Definition 1 Let Lbe a set of implications and Lbe an implication. 1) Lfollows from L(L|=L) if each subset of Arespecting Lalso respects L. 2) Lis complete if every implication of the context follows from L. 3) Lis non-redundant if for each L∈L,L\{L} |=L. 4) Lis a (implication) basis for Mif Lis complete and non-redundant. It can obtain a basis from the pseudo-intents called stem basis (SB)[16].AnSBforthe formal context on living beings is provided in Figure 1 (down). It is important to remark that SB is only an example of a basis for a formal context. In this paper no specific property of the SB is used, so it can be replaced by any implication basis. In order to work with formal contexts, stem basis, and association rules, the ConExp software (http://sourceforge.net/projects/conexp/) has been selected. It is used as a library to build the module which provides the implications (and association rules) to the reasoning module of our system. The reasoning module is a production system (designed for [17]). Initially, it works with SB, and entailment is based on the following result. Theorem 1 Let Sbe a basis for Mand {A1,A 2,···,A n}∪Y⊆A. The following conditions are equivalent: 1) S∪{A1,A 2,···,A n} pY(pis the entailment with the production system). 2) M|={A1,A 2,···,A n}→Y. Stem basis is an adequate knowledge base (KB) for the production system to reason about attributes and concepts. However, stem basis is designed for entailing true implications only, without any exceptions to the object set nor implications with a low number of counterexamples in the context. Another more important question arises when working on predictions. In this case the goal is to obtain methods for selecting a result among all entailed conclusions (eventually they are mutually incoherent), and Theorem 1 does not provide such a method. Therefore, it is better to consider association rules (with confidence) instead of true implications and the initial production system must be revised for working with confidence. Researching on logical reasoning methods for association rules is a relatively recent promising line of research[18]. In FCA, association rules are implications between sets of attributes. Confidence and support are defined as usual. Recall that the support of X, supp(X)ofaset of attributes X is defined as the proportion of objects which satisfy every attribute of X,and the confidence of an association rule is conf(X→Y) = supp(X∪Y)/supp(X). Confidence can be interpreted as an estimate of the probability P(Y|X), the probability of an object satisfying every attribute of Yunder the condition that it also satisfies every one of X. Conexp software provides association rules (and their confidence) for formal contexts. 3 Bounded Reasoning on Complex Networks Our general approach to qualitative reasoning on CS is based on considering local interactions as objects, which have several features (attributes). Therefore, using FCA, a complex network representing some features of the CS is obtained as a concept lattice. Complex networks are a widely used representation of selected features from a CS. The topological structure of the network aids in understanding a considerable number of characteristics of the associated CS. When the goal is to reason with qualitative features, it can be interesting to extract emergent concepts from these interactions. It’s here where FCA can play a relevant role. The selection of FCA as basic mathematic machinery for processing the qualitative information about complex systems lies in the fact that human reasoning (BR) about the dynamics and organization of a complex system has a qualitative nature. Therefore, human reasoning and conjectures about the CS can be expressed in qualitative terms (possibly choosing thresholds and multivalued attributes). Once qualitative hypothesis are presented, non-symbolic mechanisms and reasoning are useful to validate the conjectures. The qualitative reasoning process, in FCA would be depicted as in Figure 2. Once the observer has a collection of relevant attributes to study on the system, he can consider local interactions or nodes as objects of a formal context. This context (often a huge formal context) is built by means of data extraction, database processing, expert observations, data mining, etc. The observer has to select attributes and objects he considers relevant to determine CS dynamics, and the reasoning focuses on the associated subcontext (contextual selection). It is expected that reasoning with the contextual selection gives some information about the CS. The complex nature of the contexts associated to CS suggests studying its topological structure when there is not a suitable BR method to achieve the aim. When experience and knowl- edge about the CS suggests a reasoning method, an FCA-based reasoning task is defined and results are obtained. In [6], this general approach was applied using argumentative reasoning on contextual selections. The structure of contextual selection showed some insights on attribute selection[13]. Figure 2 FCA-based model for qualitative reasoning with complex systems 3.1 Reasoning Under Contextual Selection: Logical Foundations The model (described in [13]) is composed of events (objects) which have a number of properties (attributes). They constitute a universal formal context M(which we call monster context following the tradition in model theory). Thus, Mcan be considered as the global memory from which subcontexts are extracted. Once the specific context is considered, it is also possible to consider background knowledge Δ (in the form of propositional logic formulas) which would be combined with the knowledge extracted from the formal context (stem basis or association rules). Definition 2 Let M=(O,A,I) be the monster context, and let O⊆O. 1) A context on Ois a context M=(O1,A,I)whereO⊆O1⊆O. 2) A contextual selection on Oand Mis a map s:O→P(O1)×P(A). 3) A contextual KB for an object o∈Ow.r.t. a selection swith confidence γis a subset of association rules with confidence greater or equal to γof the formal context associated with s(o)=(s1(o),s 2(o)), that is, to the context M(s(o)) := (s1(o),s 2(o),I s1(o)×s2(o)) (note that when confidence is 1 the contextual KB is a implicational basis). Contextual KB is useful for entailing attributes on an object. The reasoning model is based on arguments extracted from subcontexts[13]. Definition 3 Let Lbe an implication and Δ a background knowledge. It is said that Lis a possible consequence of Munder Δ, M|=Δ ∃L,ifthereexistsMa non-empty subcontext of Msuch that M|=Δ∪{L}. Note that by Theorem 1, when Δ is a set of implications, it holds that |=∃is equivalent to the relation ∃defined by: M∃Lif there exists M|= Δ a non-empty subcontext of Msuch that SpL(where Sis a stem basis for M). Particularly interesting is the case of inferring properties about future events when the monster model represents attributes of past events. The inference process for Δ ∃consists of the three steps shown below (see Figure 3). 1) A question on whether a new event (object) has a property (attribute) is raised. Some properties on the new object are known (attribute values) {A1,A 2,···,A n}. 2) The contextual selection outputs a subcontext of M. A contextual KB L(for some confidence threshold) is computed for the subcontext. Selection provides a relatively small set of attributes, selected from own experience and beliefs. 3) The production system is executed on L∪{A1,A 2,···,A n}. The results obtained are the attributes inferred about the new object. Figure 3 FCA-based reasoning based on ∃ If Ais inferred by the production system, then M|=Δ ∃{A1,A 2,···,A n}→{A}. Note that it does not compute all valid implications, only those entailed from the set of attributes selected by the user. Therefore, we need to understand the topology of the lattice to properly choose the attributes to reason. Attributes are essential components in the contextual selection. Association rules are extracted from the contexts and used by the production system. From these association rules and some initial facts, based on the match we want to forecast, the production system infers the confidence (probability) for each one of the three possible results of the match, home win, draw or away win. Thus attributes constitute one of the most important and sensitive parts of the system. They are sensitive because the accuracy of the inferred results will depend on how they represent the behavior of the teams. Lastly, attribute selection allows isolating a piece of the monster context where argument reasoning is based. 3.2 Confidence-Based Reasoning System The system works on facts (a, c), where ais an attribute and cis the estimated probability of the trueness of a, which we also call confidence (by similarity with the same term for association rules). See [6] for a more detailed description of the system. It has a module for a confidencebased reasoning system (Figure 4). Figure 4 FCA-based reasoning system Its entries for a match Team1-Team2are: the contextual KB for a threshold, given as a rule set, and attribute values for the current match (except 1, X, 2) as facts, all of them with a confidence (see below). After the execution of the production system, the output is a triple (1,c 1),(X,cx),(2,c 2)of (attribute, confidence) for the match. The attribute with greater confidence is selected as the prediction. Production system execution is standard, with several modes for confidence computing based in uncertain reasoning in expert systems[19].Anyfact (attribute) ais initialized with confidence conf(a):=|{o:oIa}| +1 |O|+1 . 4 Attributes and Formal Contexts for Soccer Leagues For both selecting data and building contexts, some assumptions on forecasting in soccer league matches have been considered. Reconsiderations of such decisions can be easily computed in the system. First, we consider that the regularity of a team’s behavior only depends on the contextual selection that has been considered. This contextual selection is obtained by taking matches from the last Xweeks backwards, starting from the week just before the one we want to forecast. Second, FCA methods are used to discover regularity features, thus it does not consider forecasting exceptions (unexpected results). Therefore, the model can be considered a starting point for a betting expert who would adjust attributes, in order to obtain more personalized criteria. These attributes have to be computed and used to entail a forecasting. In order to select the most interesting attributes for the system, starting from an initial configuration, the user can compute the associated concept lattice and check it. In this way, attributes’ goodness (and thresholds) can be evaluated to reconsider current attribute selection. For example, in Figure 5, the concept lattice associated to a contextual selection for M´alaga-Sevilla match is shown. This contextual selection is based on a given attribute selection and the matches of the last 38 weeks, just before the queried match date. In this concept lattice, the attribute ID 1T16 is defined by: ’the budget of team2is greater than γ1times the budget of team1’, where γ1is the threshold the expert must estimate. In the concept lattice we can observe that the biggest concept containing the attributes team2wins and ID 1T16 covers about the 10% of the objects owned by the first attribute, therefore it is suggested to use the second attribute for reasoning with association rules to get a prediction. Figure 5 Concept lattice for the match M´alaga-Sevilla (week 31, season 2009-10) The system computes the value of an amount of attributes of objects. Experimentally a boolean combination of attributes is possible. Once the temporal context has been computed, the system can build contextual selections by selecting the match and the attribute set. The selection of attributes was made by considering diverse kinds of factors (as for example the difference between team budgets). Seventeen relevant attributes were selected. The attribute set has three special attributes, Team1wins (1), Team2wins (2), and draws (X). Data are automatically extracted from The RSSSF Archive (http://www.rsssf.com). Objects are matches and attribute set is a list of features, including timestamp (week, year). Data was collected for the past twelve years. 4.1 Attribute Selection As it was remarked before, the attribute selection is based on users’s attribute selection, which have to be computed from publically available data (provided by the monster context). Discussion on the soundness of each selection is a human process based on experience. In this section a representative set of attributes is presented. Criteria and expected correlation with match results are also discussed. There are many approaches and studies about the problem of selecting relevant features for soccer matches and competitions (cf. [7, 8]), but in our case we need those associated with bounded rationality. Therefore, attributes have to be computed from publicly available data and well-known relevant features of teams. Criteria for selecting attributes. A set of attributes with many possibilities through customizable parameters was selected. When the parameters are set up with proper values, the attribute set will represent the team’s logical behavior. Recall that formal concept analysis works with qualitative attributes and all teams’s information we work with have quantitative nature. Thus it is necessary to convert quantitative attributes into qualitative ones. This task is left to users by choosing a proper threshold for each attribute. Before choosing the base set of attributes, we have carried out an analysis of information about soccer results. The aim has been to discover which factors are more influential in teams’ behavior and which ones are less influential. First of all, we have collected every interesting factor found, and after analyzing each one, individually, we have chosen the most suitable ones. Examined factors can be classified in four different categories (Table 1): those related to league ranking, those related to previous teams’s results, those related to historical direct matches, and any other factors. It is worthy to note that to increase possibilities of the attribute set, and considering the boolean nature of formal context attributes, we have added the option to create new ones by means of logical combinations of these attributes. According to those factors, it considers a base set of 18 attributes, customizable by some parameters. This will let us obtain a diverse set of attributes (Table 2). 1) Threshold: Parameter to be used to translate quantitative attribute values into qualitative ones. 2) Team: Recall that in the formal context considered, objects are matches but attributes refer to team properties. This parameter will set the team from the object (match) by which the attribute will be considered. It has two possible values: {HOME, AWAY}. Thus, usually, we will have each attribute in a context twice, once for the home team and once for the away one. 3) Number of matches: number of past matches to be considered when some attributes are computed, e.g., the ones associated with previous team results. 4) Kind of matches: sets the type of past matches to be taken into account, when computing some attributes, considering home/away team’s condition at matches. It has three possible values: {MATCHES AS HOME TEAM, MATCHES AS AWAY TEAM, ALL MATCHES}. With these parameters, and the possibility of creating compound attributes, it is possible to build a set of detailed attributes. It is worthy to note that experiments showed that the simplest and most logical attributes give a good representation of team behavior. However we consider that a versatile attributes set, as the above described, was necessary because of the huge number of factors that can determine the result of a soccer match. The task of customizing 2009-10 season Experiments with the system show forecasts of about 58.16% by a contextual selection based on the previous 38 matches of each team. Such a percentage of hits for a qualitative reasoning system may be considered as an acceptable result comparable with results conjectured by experts[9,15]. Experiments with other contextual selections show an increase in the number of hits by about 7% in the second half of the season. The reason is that data from the first half provides more recent information on teams and past matches. 2010-11 season According to the idea mentioned above, we have evaluated the system in the second half of the 2010-11 soccer season. A way to evaluate the effectiveness of this forecasting system is to compare the number of successes in our pool with the most popular betting selections. The popular selections are collected from the most voted results for each match, published on a state agency website that controls soccer pools. In Figure 9, both results are compared. The average number of bets per week from which this popular selection is obtained is 19,064,873. Note that Spanish soccer pools are over 15 matches. 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Users System 15 10 5 0 week 22 week 38 Figure 9 Correct predictions on the last 17 weeks of the season 2010-11 7.1 Analysis of the Reasoning Performance and Comparison with Other Methods In order to understand the complexity of logical reasoning in a CLCS, it is possible to analyze the topological structure of concept nodes which are involved in frequent deduction tasks. In Figure 10 the deduction graph of the monster model for sport forecasting is depicted. Its nodes are the concepts of CLCS whose intension contains both the left-hand side of at least 10% of SB rules and the right-hand side of at least 10% of the same kind of rules. The graph contains 715 nodes and 1296 arcs. Its diameter is 7 (considered as a directed graph) which suggests a reasonable (upper bound) estimation of rule firings by the production system. The existence of many connected components is useful in deciding which attributes are relevant to achieving a certain result in prediction (by analyzing the attributes which appear in the same connected component as the targeted attributes). No statistical approaches have been taken into account, because it was not the aim of this paper. However a comparative study of our system against C4.5 classifier has been done. Therefore, two different attribute selections have been considered and used for both C4.5 classifier and our system. The experiment is to forecast all matches (380) in the 2010-11 season. In order to estimate each match result, considering N(weeks) as a timestamp, previous matches are used to build the contextual selection (or training set in C4.5) from weeks N−1toN−19 (190 objects). Figure 11 shows the percentage of correct predictions for our system and C4.5 classifier, using both attribute selections. Other cols are also shown: ‘user’s most voted results’, ‘the local team always wins’ (All 1) and two that are randomly generated. These ‘randomly generated’ cols were built assuming different weights per result. It means, <1 : 55%,X : 23%,2 : 22% > and <1 : 65%,X : 18%,2 : 17% >were used, where 1,X,2 are the probabilities of forecasting a match with the result: local team wins, draw and away team wins, respectively. The same experiment was repeated for the last five seasons using the same attribute selections and parameters than the former (see Figure 12). It is interesting to remark two features of this experiment: On one hand, the number of correct predictions increases over the years, that is to say, it is more predictable, mostly due to the increase of the team’s budget difference in the present years. On the other hand, system predictions fits well with the popular selection in the last year, but worse in former years. System accuracy is worse in previous years due to the fact that the attribute selection was designed regarding current league trends, to be used to forecast results in the near future. Figure 10 Deduction graph extracted from CLCS of season 2005-06 Figure 11 Comparative of correct predictions on the whole season 2010-11 Figure 12 Comparative of correct predictions on the last five seasons Table 4 Statistical significance of non-parametric tests of similarity between correct predictions of the system and popular selection, and hypothesized median for the one sample test Season Two paired samples One sample Hypothesized median θ(one sample) 2006-07 0.075 0.545 1.5 2007-08 0.174 0.081 1.5 2008-09 0.344 0.945 1.5 2009-10 0.491 0.107 1 2010-11 0.697 0.943 1 7.2 Testing Bounded Rationality Hypothesis Since the system is based on bounded reasoning, it is expected for both popular selection and system predictions to be similar. The system can be considered as a tool for simulating human behavior while betting in a rational way. In order to justify this assertion, another statistical test was performed to measure the similarity between popular selection and system predictions. Due to the nature of the sample, a non-parametric test should be used, in this way, the Wilcoxon signed rank test has been chosen. The test has been applied in two different modes: Wilcoxon signed rank test for two related (paired) samples Recall this test starts from a null hypothesis stating that the median of the difference between the two paired samples is 0. If the statistical significance of the test is greater than 0.05 (significance level) the null hypothesis can be retained. To apply this test, the paired samples to be considered are the number of correct predictions in each week (from 0 to 10) of both, popular selection and system. It is possible that the number of correct predictions of the system and the popular selection are the same, but having different sets of well predicted matches. Therefore another more exhaustive test should be done. Wilcoxon signed rank test for one sample Recall the test starts from a null hypothesis stating that the median of the sample is equal to a hypothesized value θ. If the statistical significance of the test is greater than 0.05 (significance level) the null hypothesis can be retained. To apply this test, the sample to be considered is the symmetric difference between the well predicted matches of the system and the popular selection, for each week. This test is expected to be more exhaustive than the former because it measures the differences individually case by case. In Table 4, the results of both tests for the last five seasons are shown. It can conclude that the predictions of the popular selection and the system are quite similar. It is worthy to note that the similarity increases from past years to the present, reinforcing the assertions made in Subsection 7.1 (see also Figure 12). 8 Conclusions and Future Work The challenge of detecting emergent concepts to reason on complex systems represents an exciting field of research. Concepts with qualitative nature are extracted from data considering only partial features of complex system dynamics, that is to say, a partial understanding of systems. In this paper, FCA is used with this aim in a specific application. The attribute selection problem using an FCA-based reasoning system for sport forecasting is analyzed. In fact, the reasoning system is a computational logic model for bounded rationality. The model is concerned with association rule reasoning and it does not use -in its current formmore sophisticated probability tools (as for example, [8, 11, 23]). As it is stated in [24], the theory of probabilistic mental models assumes that inferences about unknown states of the world are based on probability cues[25]. It can say that confidence of association rules extracted from subcontexts play the role of probability cues. It is worth it to note that, while the classifier achieves the highest performance (58, 68%) when the number of matches increases from 190 to 380, our system reaches the highest performance (59, 74%) using only 190 instances. This can be explained by the fact that our system uses a kind of fast and frugal[24] method, and these are designed to achieve acceptable results using as few as possible resources. We may assert that our model recognises trends in contexts. Trends (represented as association rules) can be considered a kind of recognizing method. Although the system is based on bounded rationality models instead of statistic models, hybrid models will be considered in future work. It could be interesting to use the system to detect rigged results (‘likelihood evidence’ of illegal behavior) in soccer. Recently some work about a similar question has been published (see, e.g., [26]). In the short term, we carry on extending the system to combine the results of two or more different attribute sets. Therefore, the system will return a single, more reliable result. In the long term, we aim to extend the model in order to obtain a general system to detect emergent concepts in complex systems. Finally, from the point of view of CS research, current and future authors’ work on CLCS show that those associated to CS where rational activities are relevant (for example, reasoning and classifying) present similar regularities which do not appear in those without human intervention nor in random contexts. In some way, this phenomenon would be useful to estimate rationality influence in a CS. 9Acknowledgements The authors would like to thank the anonymous reviewers and Prof. Rafael Blanquero- Bravo from University of Seville for their valuable comments and suggestions to improve the quality of the paper. References [1] Why Spain will win ···,Engineering &Technology, 5 June–18 June 2010. [2] Andersson P, Memmert D, and Popowicz E, Forecasting outcomes of the World Cup 2006 in football: Performance and confidence of bettors and laypeople, Psychology of Sport &Exercise, 2009, 10(1): 116–123. [3] Schumaker R P, Solieman O K, and Chen H, Sports data mining, Integrated Series in Information Systems, 2010, 26: 55–63. [4] Vlastakis N, Dotsis G, and Markellos R, How efficient is the European football betting market? Evidence from arbitrage and trading strategies, Journal of Forecasting, 2009, 28(5): 426–444. [5] Ganter B and Wille R, Formal Concept Analysis — Mathematical Foundations, Springer, Berlin, Heidelberg, 1999. [6] Aranda-Corral G A, Borrego-D´ıaz J, and Gal´an J, Confidence-based reasoning with local temporal formal contexts, Lecture Notes in Computer Science, 2011, 6692: 461–468. [7] Carling C, Williams A M, and Reilly T, Handbook of Soccer Match Analysis, Routledge Press, New York, 2005. [8] Oberstone J, Differentiating the top English premier league football clubs from the rest of the pack: Identifying the keys to success, Journal of Quantitative Analysis in Sports, 2009, 5(3): Article 10. [9] Goldstein D G and Gigerenzer G, Fast and frugal forecasting, Int. J. of Forecasting, 2009, 25(4): 760–772. [10] Goldstein D G and Gigerenzer G, Models of ecological rationality: The recognition heuristic, Psychological Review, 2002, 109(1): 75–90. [11] Min B, Kim J, Choe C, Eom H, and McKay R I, A compound framework for sports results prediction: A football case study, Know. Based Syst., 2008, 21(7): 551–562. [12] Imberman S P, Domanskiand B, and Orchard R A, Using booleanized data to discover better relationships between metrics, Proc.Int.CMGConference, 1999, 530–539. [13] Aranda-Corral G A, Borrego-D´ıaz J, and Gal´an-P´aez J, Bounded rationality for data reasoning based on formal concept analysis, Proc. Int. Workshop on Database and Expert Systems Applications, 2011, 350–354. [14] Simon H A, Models of Bounded Rationality, MA: MIT Press, Cambridge, 1982. [15] Andersson P, Ekman M, and Edman J, Forecasting the fast and frugal way: A study of performance and information-processing strategies of experts and non-experts when predicting the World Cup 2002 in soccer, Working Paper Series in Business Administration, 2003, 9Stockholm School of Economics. [16] Guigues V and Duquenne J L, Familles minimales d’ implications informatives resultant d’un tableau de donnees binaires, Math. Sci. Humaines, 1986, 95: 5–18. [17] Aranda-Corral G A and Borrego-D´ıaz J, Reconciling knowledge in social tagging web services, Lecture Notes in Artificial Intelligence, 2010, 6077: 383–390. [18] Balc´azar J L, Redundancy, deduction schemes, and minimum-size bases for association rules, Logical Methods in Computer Science, 2010, 6(2): 1–23. [19] Giarratano J C and Riley G D, Expert Systems: Principles and Programming, Brooks/Cole Publishing Co, Pacific Grove, CA, 2005. [20] Motter A E, de Moura A P S, Lai Y, and Dasgupta P, Topology of the conceptual network of language, Physical Review E, 2002, 65(6): 065102. [21] Clauset A, Shalizi C R, and Newman M E J, Power-law distributions in empirical data, SIAM Review, 2009, 51(4): 661–703. [22] Albert R and Barab´asi A L, Statistical mechanics of complex networks, Reviews of Modern Physics, 2002, 74(1): 47–97. [23] Carmichael F, Thomas D, and Ward R, Team performance: The case of English premiership football, Managerial and Decision Economics, 2000, 21(1): 31–45. [24] Goldstein D G and Gigerenzer G, Reasoning the fast and frugal way: Models of bounded rationality, Psychological Review, 1996, 103(4): 650–669. [25] Brunswik E, Representative design and probabilistic theory in a functional psychology, Psychological Review, 1955, 62(3): 193–217. [26] DiFatta G, Haworth G, and Regan K, Skill rating by Bayesian inference, Proc. 2009 IEEE Symposium on Computational Intelligence and Data Mining, 2009, 89–94.