An evolutionary algorithm to discover quantitative association rules in multidimensional time series

An evolutionary algorithm to discover quantitative association rules in multidimensional time series M. Martı´nez-Ballesteros • F. Martı ´nez-A´ lvarez • A. Troncoso • J. C. Riquelme Abstract An evolutionary approach for finding existing relationships among several variables of a multidimensional time series is presented in this work. The proposed model to discover these relationships is based on quantitative association rules. This algorithm, called QARGA (Quantitative Association Rules by Genetic Algorithm), uses a particular codification of the individuals that allows solving two basic problems. First, it does not perform a previous attribute discretization and, second, it is not necessary to set which variables belong to the antecedent or consequent. Therefore, it may discover all underlying dependencies among different variables. To evaluate the proposed algorithm three experiments have been carried out. As initial step, several public datasets have been analyzed with the purpose of comparing with other existing evolutionary approaches. Also, the algorithm has been applied to synthetic time series (where the relationships are known) to analyze its potential for discovering rules in time series. Finally, a real-world multidimensional time series composed by several climatological variables has been considered. All the results show a remarkable performance of QARGA. Keywords Time series Quantitative association rules  Evolutionary algorithms Data mining 1 Introduction It is usual to find natural phenomena correlated to some other variables. Thus, real-world processes can be modeled by inferring knowledge from other associated variables that definitively have an effect on the original process. For instance, the existence of acid rain cannot be understood without the existence of other pollutant agents, such as monoxide carbon or sulfur dioxide. In other words, the knowledge of how some variables could affect other ones may be useful to obtain accurate behavior models. Quantitative association rule (QAR) extraction in time series can be of the utmost usefulness for predictive purposes (Shidara et al. 2008; Wang et al. 2008). Thus, it could be interesting to find relationships among several time series to determine the range of values for a particular time series in a given time interval depending on the values of others for the same interval. For instance, rules such as hour 2½10;12^demand 2½12;000;15;000)price 2 ½3:2;4:5can provide useful knowledge for forecasting the electric energy price at peak hours (from 10 am to 12 pm) depending on the values of the energy demand during these hours. This information could help to obtain different models adjusted to different intervals or to develop a family of models for every rule. Hence, QAR are introduced in a new time series framework with the means of obtaining relationships among correlated time series that help to model their behavior. Evolutionary algorithms (EA) have been extensively used for optimization and model adjustment in data mining tasks. In fact, the use metaheuristics in general, and of EA 2.1 Quantitative association rules Association rules (AR) were first defined by Agrawal et al. (1993) as follows. Let I¼fi1;i2;...;ingbe a set of nitems, and D¼ftr1;tr2;...;trNga set of Ntransactions, where each trjcontains a subset of items. Thus, a rule can be defined as X)Y;where X;YIand X\Y¼;. Finally, X and Yare called antecedent (or left side of the rule) and consequent (or right side of the rule), respectively. When the domain is continuous, the association rules are known as QAR. In this context, let F¼fF1;...;Fngbeaset of features, with values in R.LetAand Cbe two disjunct subsets of F, that is, AF;CF;and A\C¼;. A QAR is aruleX)Y;in which features in Abelong to the antecedent X;and features in Cbelong to the consequent Y;such that X¼^ Fi2A Fi2½li;uið1Þ Y¼^ Fj2C Fj2½lj;ujð2Þ where liand ljrepresent the lower limits of the intervals for Fiand Fj;respectively, and the couple uiand ujthe upper ones. For instance, a QAR could be numerically expressed as F12½12;25^F32½5;9)F22½3;7^F52½2;8ð3Þ where F1and F3constitute the features appearing in the antecedent and F2and F5the ones in the consequent. 2.2 Quality parameters This section provides a description of the support, confidence and lift indices (Brin et al. 1997) used to measure the interestingness of rules and of a new index, called recovered;to ensure that the full search space is explored. The support of an itemset Xis defined as the ratio of transactions in the dataset that contain X. Formally: supðXÞ¼#X N¼PðXÞð4Þ where #Xis the number of times that Xappear in the dataset, and Nthe number of transactions forming such dataset. Other authors prefer naming the support of X simply as the probability of X;PðXÞ. Let Xand Ybe the itemsets that identify the antecedent and consequent of a rule, respectively. The confidence of a rule is expressed as follows: conf ðX¼) YÞ¼supðX¼) YÞ supðXÞð5Þ and it can be interpreted as the probability that transactions containing X, also contain Y. In other words, how certain is the rule subjected to analysis. in particular, to deal with data mining-based problems is a hot topic of research nowadays (Alcala´-Fdez et al. 2009a, 2010; Chen et al. 2010; del Jesu ´s et al. 2009; Yan et al. 2009). Also, EA have been used to build rule-based systems (Aguilar-Ruiz et al. 2007; Berlanga et al. 2010; Orriols-Puig and Bernado´-Mansilla 2009). Real-coded genetic algorithms (RCGA) are very important within EA due to the increasing interest in solving real-world optimization problems. The main problem of RCGA, in which many researchers have focused their works, is the definition of adequate genetic operators (Herrera et al. 2004; Kalyanmoy et al. 2002). In particular, a new RCGA, henceforth called QARGA (Quantitative Association Rules by Genetic Algorithm) is proposed in this work. It is worth noting that QARGA does not perform previous variable discretization, that is, it handles numeric data during the whole rule extraction process, in contrast with many other approaches that perform data discretization to discover rules (Agrawal et al. 1993; Aumann and Lindell 2003; Vannucci and Colla 2004). Furthermore, the approach allows several degrees of freedom in specifying the user’s preference regarding both of the number of attributes and structure of the rules. On the other hand, besides the well-known support and confidence measures, the accuracy of the rules is also obtained with a measure called lift due to its usefulness in the specific area of time series analysis (Ramaswamy et al. 1998). First, QARGA has been applied to datasets from the Bilkent University Function Approximation (BUFA) repository (Guvenir and Uysal 2000). These datasets have been chosen because the literature offers multiple EA applied to them (Alatas and Akin 2006; Alatas et al. 2008; Mata et al. 2002). Later, time series have been synthetically generated to determine the suitability of applying QARGA to temporal data. Finally, multidimensional realworld time series have been used to extract QAR. In particular, climatological time series have been analyzed to discover the factors that cause high ozone concentration levels in atmosphere. The remainder of the paper is divided as follows: Sect. 2 provides a formal description of QAR, as well as introduces the quality indices applied to QARGA. Section 3 presents the most relevant related works found in literature. Section 4 describes the main features of QARGA used in this work. The results of applying the proposed algorithm to different datasets are reported and discussed in Sect. 5. Finally, Sect. 6 summarizes the conclusions. 2 Preliminaries This section is devoted to formally describe QAR and to introduce the quality measures used in this paper. Finally, the interest or lift of a rule is defined as liftðX¼) YÞ¼supðX¼) YÞ supðXÞsupðYÞð6Þ Lift means how many times more often Xand Yare together in the dataset than expected, assuming that the presence of Xand Yin transactions are occurrences statically independent. Lifts greater than one are desired because this fact would involve statistical dependence in simultaneous occurrence of Xand Yand, therefore, the rule would provide valuable information about Xand Y. For a better understanding of such indices, a dataset comprising ten transactions and three features is shown in Table 1. Also consider an example rule F12½180;189^F22½85;95)F32½33;36ð7Þ In this case, the support of the antecedent is 20%, since two transactions, t2and t9;simultaneously satisfy that F1and F2belong to the intervals [180, 189] and [85, 95], respectively (two transactions out of ten, supðXÞ¼0:2). As for the support of the consequent, supðYÞ¼0:2 because only transactions t6and t9satisfy that F32½33;36. Regarding the confidence, only one transaction t9satisfies all the three features (F1and F2in the antecedent, and F3in the consequent) appearing in the rule; in other words, supðX)YÞ¼0:1. Therefore, conf ðX)YÞ¼0:1=0:2¼ 0:5;that is, the rule has a confidence of 50%. Finally, the lift is liftðX¼) YÞ¼0:1=ð0:20:2Þ¼2:5;since supðX) YÞ¼0:1;supðXÞ¼0:2 and supðYÞ¼0:2;as discussed before. Finally, the measure recovered is defined for finding rules covering different regions of the search space. An example eis covered by the rule rif the values of attributes of ebelong to the intervals defined by the rule r. That is, covðe;rÞ¼ 1ifeis covered by r 0 otherwise ð8Þ Given a set of rules r1;...;rnthe measure recovered for the rule rnþ1is defined by recovðrnþ1Þ¼1 NX e2DX n i¼1 covðe;riÞð9Þ Thus, this index provides a measure of the number of instances which have already been covered by a set of previous rules. 3 Related work A thorough review of recently published works reveals that the extraction of AR with numeric attributes is an emerging topic. Mata et al. (2001) proposed a novel technique based on evolutionary techniques to find QAR that was improved in Mata et al. (2002). First, the approach found the sets of attributes which were frequently present in database and called frequent itemsets, and later AR were extracted from these sets. Following with this topic, the optimization of the confidence—avoiding the initial threshold for the minimum support—was the main contribution of the work introduced in Yan et al. (2009). The authors used fitness function and non-threshold requirements for the minimum support. The fitness function is a key parameter in EA and the authors just used the relative confidence as fitness function. Recently, Alcala ´-Fdez et al. presented a study about three algorithms to analyze their effectiveness for mining QAR. In particular, EARMGA (Yan et al. 2009), GAR (Mata et al. 2002) and GENAR (Mata et al. 2001) were applied to two real-world datasets, showing their efficiency in terms of coverage and confidence. On the other hand, data mining techniques for discovering AR in time series can be found in Bellazzi et al. (2005). The authors successfully mined temporal data retrieved from multiple hemodialysis sessions by applying preprocessing, data reduction and filtering as a previous step of the AR extraction process. Finally, AR were obtained by following the well-known Apriori itemset generation strategy (Venturini 1994). An algorithm to discover frequent temporal patterns and temporal AR was introduced in Winarko and Roddick (2007). The algorithm extends the MEMISP algorithm (Lin and Lee 2002) which discovers sequential patterns by using a recursive find-then-index technique. Especially remarkable was the maximum gap time constraint included to remove insignificant patterns and consequently to reduce the number of temporal association rules. Usually, the sliding window concept has been successfully applied to forecast time series (Martı ´nez–A ´lvarez et al. 2011; Nikolaidou and Mitkas 2009). However, this concept has been recently used in (Khan et al. 2010) with the purpose of obtaining a low use of memory and low Table 1 Illustrative dataset Transaction F1F2F3 t1178 75 24 t2186 93 37 t3167 60 22 t4199 112 30 t5154 47 42 t6173 83 33 t7177 91 63 t8159 53 48 t9183 88 35 t10 178 93 58 proposed the use of some new operators such as rounding, repairing or filtrating. Finally, QAR have also been used in the bioinformatics field. Thus, microarray data analysis by means of QAR was addressed in Georgii et al. (2005). The main novelty proposed by the authors was the definition of an AR as a linear combination of weighted variables, against a constant. Also in this context, the authors in Gupta et al. (2006) introduced a multi-step algorithm devoted to mine QAR for protein sequences. Once again, an Apriori-based methodology was used in Nam et al. (2009) to discover temporal associations from gene expression data. 4 Description of the search of rules In a continuous domain, it is necessary to group certain sets of values that share same features and therefore it is required to express the membership of the values to each group. Adaptive intervals instead of fixed ranges have been chosen to represent the membership of such values in this work. The search for the most appropriate intervals has been carried out by means of QARGA. Thus, the intervals are adjusted to find QAR with high values for support and confidence, together with other measures used in order to quantify the quality of the rule. In the population, each individual constitutes a rule. These rules are then subjected to an evolutionary process, in which the mutation and crossover operators are applied and, at the end of the process, the individual that presents the best fitness is designated as the best rule. Moreover, the fitness function has been provided with a set of parameters so that the user can drive the process of search depending on the desired rules. The punishment of the covered instances allows the subsequent rules, found by QARGA, trying to cover those instances that were still uncovered, by means of an iterative rule learning (IRL) (Venturini 1993). The following subsections detail the general scheme of the algorithm as well as the fitness function, the representation of the individuals, and the genetic operators. 4.1 Codification of the individuals Each gene of an individual represents the upper and lower limit of the intervals of each attribute. The individuals are represented by an array of fixed length n;where nis the number of attributes belonging to the database. Furthermore, the elements are real-valued since the values of the attributes are continuous. Two structures are available for the representation of an individual: computational cost of the Apriori-based algorithm presented for discovering itemsets whose support increase over time. In the non-supervised classification domain, the authors in Wan et al. (2007) made use of clustering processes to discretize the attributes of hydrological time series, as a first step of the rules extraction, which were eventually obtained by means of the Apriori algorithm. Following with clustering techniques, fuzzy clustering was used in Chen et al. (2010) to speed up the calculation for requirement satisfaction with multiple minimum supports, enhancing thus the results published in its initial work (Chen et al. 2009). The work introduced in Huang et al. (2008) mined ocean data time series in order to discover relationship between salinity and temperature variations. Concretely, the authors discovered spatio-temporal patterns from the aforementioned variables and reported QAR using Prefix- Span and FITI algorithms (Pei et al. 2001; Tung et al. 2003). Different models to forecast the ozone concentration levels have been recently proposed. Hence, the authors in Agirre-Basurko et al. (2006) developed two multilayer preceptron and a linear regression model for this purpose and prognosticated eight hours ahead for the Spanish city of Bilbao. They concluded that the insertion of extra seasonal variables may improve the general forecasting process. On the other hand, an artificial neural network model was presented in Elkamel et al. (2001). The authors also predicted the ozone concentrations by considering the analysis of additional climatological time series. Finally, temporal variations of the tropospheric ozone levels were analyzed in four sites of the Iberian Peninsula (Adame- Carnero et al. 2010) by means of statistical approaches. Alternatively, the application of QAR can also be found in the data streams domain. In fact, the authors in Orriols- Puig et al. (2008) developed a model capable to classify on-line generated data for both continuous and discrete data streams. MODENAR is a multi-objective pareto-based genetic algorithm that was presented in Alatas et al. (2008). In this approach, the fitness function aimed at optimizing four different variables: Support, confidence, comprehensibility of the rule and the amplitude of the intervals that constitutes the rule. A similar issue was addressed in Tong et al. (2005), in which the authors conducted research on the determination of existing conflicts when minimum support and minimum confidence are simultaneously required. In Alatas and Akin (2008), the use of rough particle swarm techniques as an optimization metaheuristic was presented. In this work, the authors obtained the values for the intervals instead of frequent itemsets. Moreover, they •Upper structure. All the attributes included in the database are depicted in this structure. The limits of the intervals of each attribute are stored, where liis the inferior limit of the interval and uithe superior one. •Lower structure. Nevertheless, not all the attributes will be present in the rules that describe an individual. This structure indicates the type of each attribute, ti;which can have three different values: – 0 when the attribute does not belong to the individual, – 1 when the attribute belongs to the antecedent, and – 2 when it belongs to the consequent. Figure 1shows the codification of a general individual of the population. With the proposed codification, if an attribute is wanted to be retrieved for a specific rule, it can be done by modifying the value equal to 0 of the type by a value equal to 1 or 2. Analogously, an attribute that appears in a rule may stop belonging to such rule by changing the type of the attribute from values 1 or 2 to 0. An illustrative example is depicted in Fig. 2. In particular, the rule X12½20;34^ X32½7;18)X42½12;27is represented. Note that attributes X1and X3appear in the antecedent, X4in the consequent, and X2is not involved in the rule. Therefore, t1¼t3¼1;t2¼0 and t4¼2. 4.2 Generation of the initial population The individuals of the initial population are randomly generated. In other words, the number of attributes appearing in the rule and the type and interval for each attribute are randomly generated. To assure that the individuals represent sound rules when the genes are generated, the following constraints are considered: •Limits of the interval: – The lower limit of the interval has to be less than the upper limit of the interval. If the randomly generated values do not fulfill this requirement, the limits are swapped. – The lower and upper limits of the interval have to be greater and less than the lower and upper limits of the domain of the attribute, respectively. Otherwise, the corresponding limits of the domain of the attribute are assigned. •Type of the attribute: – The number of attributes of the rule has to be greater than a minimum number of attributes defined by the user depending on the desired rule. – The number of attributes belonging to the antecedent of the rule has to be greater than 1. – The number of attributes belonging to the consequent of the rule has to be greater than 1, and less than a maximum number of allowed consequents, which is a parameter defined by the user depending for the desired rule. 4.3 Genetic operators This section describes the genetic operators used in the proposed algorithm, that is, selection, crossover and mutation operators. 1. Selection. An elitist strategy is used to replicate the individual with the best fitness. By contrast, a roulette selection method is used for the remaining individuals rewarding the best individuals according to the fitness. Note that the tournament selection was also used in preliminary studies, showing similar performance to that of roulette selection. 2. Crossover. Two parent individuals xand y, chosen by means of the roulette selection, are combined to generate a new individual z. Formally, let ½lx i;ux i;½ly i;uy iand ½lz i;uz ibe the intervals in which the attribute aivary for the individuals x;yand z; respectively, and let tx i;ty iand tz ibe the type of the attribute aifor the individuals x;yand z;respectively. Then, for each attribute aitwo cases can occur: •tx i¼ty i: The same type is assigned to the descendent and the interval is obtained by generating two random numbers among the limits of the intervals of both parents, as shown in Eqs. 10 and 11. l2 u2 l1 u1 t2 t1 ln un ... tn ... Fig. 1 Representation of an individual of the population 1 520 34 01 12 27 2 7 18 1 X1X2X3X4 Fig. 2 Example of an individual tz i¼tx ið10Þ ½lz i;uz i¼½randomðlx i;ly iÞ;randomðux i;uy iÞ ð11Þ •tx i6¼ ty i: One of the two types is randomly chosen between both of the parents, without modifying the intervals of such attribute, as shown in Eqs. 12 and 13. lz i;uz i  ¼lx i;ux i  ;if tz i¼tx ið12Þ lz i;uz i  ¼ly i;uy i ½;if tz i¼ty ið13Þ The limits and types of the attributes of the offspring are checked, as described in Sect. 4.2, to assure that it represents sound rules. If any attribute does not fulfill the required constraints regarding the type of attributes the individual is discarded and a new individual is obtained from the same parents. The crossover process is depicted in Fig. 3. 3. Mutation. The mutation process consists in modifying according to a probability the genes of randomly selected individuals. The mutation of a gene can be focused on •Type of the attribute. Two equally probable cases can be distinguished: – Null Mutation. The type tiof the selected attribute is different to null, and eventually changed to null. – Not Null Mutation. The type tiof the selected attribute is null and changed to antecedent or consequent. •Intervals of the attribute. Three equally probable cases are possible: –Lower Limit. A random value is added or subtracted to the lower limit of the interval. –Upper Limit. A random value is added or subtracted to the upper limit of the interval. –Both Limits. A random value is added or subtracted to both limits of the interval. For all the three cases, the random value is generated between 0 and a percentage (usually 10%) of the amplitude of the interval and it will be added or subtracted according to a certain probability. The choice between the mutation of the type or the mutation of the interval depends on a given probability. The limits and types of the attributes of the offspring are checked, as described in Sect. 4.2, to assure that it represents sound rules. If any attribute does not fulfill the required constraints regarding of the type of attributes, the individual is discarded and a new mutation is obtained from the same original individual. Some examples of all the kind of mutations are illustrated in Figs. 4,5,6,7and 8. 4.4 The fitness function The fitness of each individual allows deciding which are the best candidates to remain in subsequent generations. In order to make this decision, it is desirable that the support would be high, since this fact implies that more samples from the database are covered. Nevertheless, to take into consideration only the support is not enough to calculate the fitness because the algorithm would try to enlarge the amplitude of the intervals until the whole domain of each attribute would be completed. For this reason, it is 34 4738 5718 32 211 ytneraPxtneraP 23 3720 4418 32 211 Child z 13 3619 4011 25 210 ])75,04(modnar,)83,91(modnar[ [random (13, 34) , random (36, 47)] [18,32] Fig. 3 Crossover for the individuals xand y necessary to include a measure to limit the growth of the intervals during the evolutionary process. The chosen fitness function to be maximized is f¼wssupþwcconf wrrecovþwnnAttribwaampl ð14Þ where sup;conf and recov are defined in Sect. 2.2,nAttrib is the number of attributes appearing in the rule, ampl is the average size of intervals of the attributes that compose the rule, and ws;wc;wr;wnand waare the weights to drive the search, and will vary depending on the required rules. The support rewards the rules fulfilled by many instances and the weight wscan increase or decrease its effect. The confidence together with the support are the most widely measures used to evaluate the quality of the QAR. The confidence is the grade of reliability of the rule. High values of wcmay be used when rules without error are desired, and viceversa. The number of recovered instances is used to indicate that a sample has already been covered by a previous rule. Thus, rules covering different regions of the search space are preferred. The process of punishing the covered 34 4738 5718 32 211 Individual x 34 4738 5718 32 201 Individual x’ Fig. 4 Scheme of Null Mutation 1 Individual y’ Individual y 13 3619 4011 25 210 1 13 3619 4011 25 2 Fig. 5 Scheme of Not Null Mutation 34 4738 5718 32 201 Individual x 36 4738 5718 32 201 Individual x’ Fig. 6 Scheme of Lower Limit Mutation 34 4738 5718 32 201 Individual x 34 4538 5718 32 201 Individual x’ Fig. 7 Scheme of Upper Limit Mutation 34 4738 5718 32 201 Individual x 36 4538 5718 32 201 Individual x’ Fig. 8 Scheme of Lower and Upper Limits Mutation instances is now described. Every time the evolutionary process ends and the best individual is chosen as the best rule, the database is processed in order to find those instances already covered by the rule. Hence, each instance has a counter that increases its value by one every time a rule covers it. The number of attributes of a rule can be adjusted by means of the weight wn. Thus, when wnis set to a value close to 0, few attributes are obtained and, on the other hand, when wnis set to a value close to 1, many attributes appear in rules. The amplitude controls the size of the intervals of the attributes that compose the rules and those individuals with large intervals are penalized by means of the factor wa; which allows the rules be more or less permissive regarding the amplitude of the intervals. Hence, the user can model the behavior of the rules that can be obtained by varying the weights in the fitness function. Therefore, the user can obtain rules according to their needs without a previous data discretization. 4.5 The IRL approach The proposed algorithm is based on the iterative rule learning (IRL) process, whose general scheme is shown in Fig. 9. The EA is applied in each iteration obtaining one rule per iteration, which is precisely the best individual discovered. While the number of desired rules is not reached, IRL allows penalization in already covered instances, with the aim of finding rules that cover those instances that have not been covered yet in subsequent iterations. The main advantage of the approach is that attempts at covering every region in the solutions domain, that is, the set of rules will cover all the consequent domain. The iterative process ends when it finds the desired number of rules. Figure 10 provides the scheme of the EA, which is the main step of the IRL process depicted in Fig. 9. First, the rules population is initialized and evaluated. All rules are evaluated according to (14). Thus, in each iteration the selection operator is applied to select the best rules on the basis of the fitness function. Then, the crossover operator is applied to the selected rules while the population size is not completed. Individuals are randomly selected in order to apply the mutation operator. Finally, the new population is again evaluated by the fitness function and the evolutionary process restarts. Note that the process will be repeated as many times as the maximum number of preset generations indicates. 5 Results In this section the results obtained from the application of the proposed approach to different datasets are presented. First, Sect. 5.1 provides a detailed description of all used datasets. A summary of the key parameters configuration used for all the algorithms can be found in Sect. 5.2. Finally, the results are gathered and discussed in Sect. 5.3. The approach has been initially tested on several widely studied datasets from the public BUFA repository, and the accuracy of QARGA has been compared with that of the algorithms introduced in Yan et al. (2009) and Mata et al. (2001) in Sect. 5.3.1. On the other hand, two different kind Data set Evolutionary Algorithm Rule Nº Rules > Max_Rules END no yes punishment The best Individual Fig. 9 Scheme of IRL Initialize Rules Population Evaluate Rules with (1) Generations < Max Generations END no yes Evaluate new rules with (1) Select Rules by fitness Crossover and Mutation of Rules Replace with new rules Fig. 10 EA scheme of time series are analyzed: synthetically generated and real-world multidimensional temporal data. Sect. 5.3.2 is devoted to evaluate the accuracy of QARGA when it is applied to synthetically generated multidimensional time series. Likewise, the real-world case is reported in Sect. 5.3.3, where QAR are obtained to discover relationships between the tropospheric ozone and other climatological time series. 5.1 Dataset description This section presents the number of records and number of attributes of the BUFA repository datasets as well as how synthetic time series were generated and what real-world time series consist of. 5.1.1 Public datasets QARGA has been applied to 15 public datasets: Basketball, Bodyfat, Bolts, Kinematics, Longley, Normal Body Temperature, Plastic, Pollution, Pw Linear, Pyramidines, Quake, Schools, Sleep, Stock Price and Vineyard, which can be found at BUFA repository (Guvenir and Uysal 2000). Relevant information about these datasets is summarized in Table 2. 5.1.2 Synthetic multidimensional time series In this section two different synthetically generated multidimensional time series are described: time series without and with disjunctions, respectively. In particular, multidimensional time series are generated, that is, time series characterized by more than one variable in each time stamp. Or, in other words, two or more time series simultaneously observed that characterize the same phenomenon. Formally, a multidimensional time series MTS can be expressed as MTS ¼½X1ðtÞ;...;XnðtÞT;where each XiðtÞis a variable measured along with the time, t;and nis the number of inter-related time series that identifies the whole MTS. Thus, the goal of applying QAR to MTS is to discover existing relationships among those Xiforming the MTS;along with the time. Regarding the time series with no disjunctions, Table 3 defines a three-dimensional time series, n¼3;in which three variables X1;X2and X3share static relationships in fixed intervals of time. Thus, 100 values for each variable X1;X2and X3were generated and uniformly distributed in four intervals. To obtain these series, values for variables X1;X2and X3were randomly selected for every tiaccording to constraints listed in Table 3, where tivaries from 1 to 100. Finally, the resulting time series are depicted in Fig. 11. With reference to time series with disjunctions, a bi-dimensional time series represented by two variables, X1 and X2, has been generated with, again, 100 values for each time series uniformly distributed in four intervals. However, the main difference regarding the previous situation lies in the fact that now the variables X1and X2can be defined by more than one possible set of values. Table 4shows the constraints considered to generate the time series with disjunctions. The series is generated then as follows: For every tiand Xjone interval is randomly chosen and, then, a value is randomly chosen from the interval previously selected. For instance, X1can indistinctively belong to intervals ½10;20or ½15;35when t2 ½26;50in set #1. Table 2 Public datasets. Dataset Records Attributes Basketball 96 5 Bodyfat 252 18 Bolts 40 8 Kinematics 8192 9 Longley 16 7 Normal Body Temperature 130 3 Plastic 1650 3 Pw Linear 200 11 Pollution 60 16 Pyramidines 74 28 Quake 2178 4 School 62 20 Sleep 57 8 Stock price 950 10 Vineyard 52 4 25 50 75 100 10 20 30 40 50 60 70 80 90 100 Time Values X1 X2 X3 Fig. 11 Time series with no disjunctions conjunction #31and rule #7 is approximately #32. In the same fashion, rule #10 is related to #33and rule #11 to #34. The remaining rules discovered relationships varying among several conjunctions. For rule #6, note that the X2 time series has values ranging in an interval formed from the union of rules #31and #32. Last, rule #9 is a rule resulting from the four possible conjunctions in t2 ½76;100;that is, it combines the intervals for both X1and X2. Therefore, it would be a rule shared by #31;#32;#33 and #34. In general, rules in this interval of time share a support close to the expected one as well as a confidence verging on 100% for most cases. Finally, it can be concluded that all the rules discovered by QARGA can be considered interesting since the lift is high for all of them. Moreover, the amplitude of intervals is moderate and intervals limits are very similar to those initially set in Table 5. 5.3.3 Results in ozone time series Now QARGA is applied to ozone time series and other inter-dependant temporal variables. Table 17 shows the support, confidence, number of records, average amplitude and lift of the obtained rules by QARGA when the ozone is imposed to be in the consequent. The climatological variables that most frequently appear are temperature, humidity and hour of the day. Consequently, it can be concluded that the other variables are not as correlated with ozone as the aforementioned ones. Some other interesting conclusions can be extracted from these rules. Hence, when the temperature reaches high values, the ozone concentration in the atmosphere presents high values, even reaching 203 lg/m 3 . Nevertheless, when the temperature is relatively low, the concentration of ozone falls to values around 116 lg/m 3 . That is, there exists a perfect correlation between the ranges of the temperature and the ozone. With reference to the humidity, there exists an inversely proportional relationship to the ozone. Thus, when examining the first rule, in contrast to the temperature, when the humidity falls, the ozone raises, and viceversa, as occurred in the fourth rule (rule #3). From the remaining rules, it can also be observed that the time slot is present in two rules. This fact is due to the close association existing between the temperature and the hour of the day and, possibly, to the traffic, whose density varies along the day and typically generates high concentrations of ozone. Note that during the night and first hours Table 16 Rules found by QARGA for time series with disjunctions ID Rules Conf. (%) Lift Sup. (%) Amp. (%) #0 X12[20, 30] ^X22[61, 79] ¼) t2[1, 25] 100 4.00 19 17.33 #1 t2[28, 50] ^X22[47, 58] ¼) X12[12, 20] 93.3 2.39 14 13.67 #2 t2[26, 46] ^X12[25, 31] ¼) X22[41, 49] 100 2.38 7 11.33 #3 t2[32, 47] ^X22[46, 53] ¼) X12[16, 22] 62.5 2.98 5 9.35 #4 t2[58, 75] ^X22[40, 50] ¼) X12[4, 15] 100 3.23 13 12.67 #5 X12[4, 14] ^X22[60, 70] ¼) t2[54, 72] 100 5.26 8 12.67 #6 t2[79, 95] ^X12[12, 18] ¼) X22[39, 49] 85.7 1.82 6 10.62 #7 t2[76, 91] ^X12[15, 24] ¼) X22[41, 50] 85.7 1.75 6 10.95 #8 t2[77, 95] ^X12[16, 22] ¼) X22[31, 39] 100 33.33 3 10.62 #9 t2[76, 99] ^X12[18, 40] ¼) X22[31, 50] 100 1.79 18 21.33 #10 X12[28, 34] ^X22[36, 43] ¼) t2[83, 98] 60.0 4.29 3 9.35 #11 t2[76, 99] ^X12[30, 38] ¼) X22[40, 50] 100 1.89 13 13.67 Table 17 Association rules found by QARGA with high confidence ID Rule Sup. (%) Conf. (%) #r Ampl. Lift #0 temp. 2[32, 42] and hum. 2[19, 41] and hour 2[13, 19] ¼) O32[104, 203] 14 84 213 34 2.27 #1 hour 2[2, 11] ¼) O32[16, 97] 37 90 557 45 1.64 #2 hour 2[13, 22] ¼) O32[88, 189] 35 84 522 55 1.67 #3 temp. 2[16, 22] and hum. 2[75, 90] ¼) O32[22, 110] 16 100 234 36 1.43 #4 temp. 2[18, 29] and speed 2[0, 10] ¼) O32[23, 116] 41 93 613 38 1.28 28.6 (12.6) 90.2 (6.7) 427.80 (±189.5) 41.6 (8.6) 1.7 (0.4) of the day, the ozone is relatively low, reaching values similar to that of low temperatures. However, from midday to the nightfall—the rushing hours—the amount of ozone increases considerably, reaching values near to 200 lg/m 3 as it happened with high temperatures. Also note that in one rule the speed of the wind appears indicating that when it is low the ozone also is. However, this rule is not conclusive and the authors do not dare to state that the speed of the wind is directly proportional to the ozone. With the aim of comparing the results and evaluating the quality, the Apriori algorithm has been applied to these time series. The most remarkable feature of this algorithm is that is based on a previous or a priori knowledge of the frequent itemsets in order to reduce the space of search and, consequently, increase the efficiency. Besides, the user has to establish the constraints for minimum support and confidence. It is also worth mentioning that Apriori does not work with real values directly and it performs a previous discretization of all continuous variables. Hence, Table 18 collects the results provided by Apriori when discretizing the continuous variables with three intervals. In this case, the temperature and the humidity appear again but, by contrast, the hour of the day does not seem to be an important variable. The speed and the direction of the wind also appear in the antecedent. It can also be observed that low temperatures also involve low ozone concentrations, and viceversa, as it happened with the rules shown in Table 17. With regard to the humidity the same situation is reported: it is inversely proportional to the ozone. However, when analyzing the direction of the wind in some rules, the results are not conclusive. Actually, for equal values of the direction, different ranges of ozone are mined, which means that this variable presents no proportional (neither direct nor inverse) relationship with the ozone and, therefore, it does not contribute with meaningful information. Finally, the speed of the wind presents the same behavior shown in Table 17, that is, low values involve low ozone concentrations. The comparison between Tables 17 and 18 reveals that the support reached by QARGA is much greater in three rules whereas two rules present slightly lower supports. The confidence for the majority of the rules found by QARGA overcomes 90%, even reaching 100% in the fourth rule. This fact highlights the small errors committed by QARGA, providing exact rules in the majority of cases. Furthermore, the number of covered instances is higher than the ones by Apriori due to the direct relation existing with the support. The average amplitude for the rules provided by QARGA is much smaller, ranging from 38 to 55, while the intervals found by Apriori varies from 32 to 98. The lift is very similar in QARGA and Aprori, and for both algorithms it is greater than 1. Last, Apriori has just found rules in which the values of the ozone varied only in two of the three possible intervals associated with the labels previously generated during the discretization process. Furthermore, it is unable to find rules with ozone concentrations higher than 183 lg/m 3 .On the contrary, QARGA obtained rules for concentrations of ozone higher than 200 lg/m 3 . To sum up, from this kind of experimentation it can be concluded that QARGA obtains better results compared with Apriori, since support and confidence are higher and amplitude is smaller, which involves less errors in rules. Table 19 shows the rules obtained by QARGA when the target is to find rules with the highest number of attributes, the highest confidence and the smallest amplitude possible, even if this fact may lead to lower supports. It can be observed that the majority of rules have a large number of attributes. The variables that most frequently appear, therefore the most meaningful ones, are the temperature, the humidity and the hour of day. From the extracted rules, several conclusions can be drawn. First, note that the selected rules are those that present high concentrations of ozone in the consequent, since this is the situation that really involves environmental concerns. As with the first experimentation, a directly proportional relation between the temperature and the ozone has been discovered. In other words, when the Table 18 Association rules found by Apriori (three intervals used for discretization) ID Rule Sup. (%) Conf. (%) #r Ampl. Lift #0 temp. 2[16, 25] and hum. 2[65, 91] and speed 2[0, 9] ¼) O32[14, 99] 20 90 297 32 1.59 #1 temp. 2[16, 25] and dir. 2[120, 240] and speed 2[0, 9] ¼) O32[14, 99] 16 85 239 56 1.49 #2 dir. 2[240, 360] and speed 2[0, 9] ¼) O32[14, 99] 16 79 241 71 1.38 #3 hum. 2[14, 40] and dir. 2[120, 240] ¼) O32[99, 183] 18 73 261 77 1.80 #4 temp. 2[25, 35] and dir. 2[120, 240] ¼) O32[99, 183] 20 70 296 98 1.70 18.0 (2.0) 79.4 (8.3) 266.8 (28.5) 66.8 (24.6) 1.6 (0.2) temperature reaches values of almost 40 C the ozone level raises up to values greater than 200 lg/m 3 . Moreover, the humidity also presents an inversely proportional relationship with the ozone. Thus, when it reaches high values, the concentration of ozone is low, as it can be determined from the observation of the second rule. Alternatively, when the humidity increases, the ozone level decreases, as listed in the remaining rules. The hour of the day is also present in the majority of the rules. The time slot is similar for all the rules since, as discussed previously, ozone and the hour share a directly proportional relationship. The peaks of ozone are reached during the rushing hours (from midday to nightfall), that is, during the hours in which the temperature is high and the traffic is usually heavy. In some rules, the speed of the wind appears as a crucial factor. However, there does not exist such a higher correlation with the ozone because greater speeds should have be found in the fourth rule (in which the ozone presents the highest concentration). By contrast, in the second rule, in which the ozone is lower, the speed of the wind is slightly superior. The analysis of the direction of the wind reveals that it is not a variable that determines the amount of ozone in the atmosphere. However, when the direction is comprised in an interval from 150and 200;the concentration of ozone increases. Table 20 gathers the results obtained by Apriori when data were discretized in ten intervals. The temperature is, again, the main variable. However, it is worth pointing out that no relevant rules were discovered in which the humidity or the hour of the day appear. One of the most remarkable feature of the extracted rules by Apriori when discretizing with ten intervals is that they all have only one attribute in the antecedent. This situation highlights, once again, that rules provided by QARGA enhance that of Apriori, since they are more expressive and provide more information due to a greater number of attributes in antecedents. With respect to the temperature, two rules with different antecedent but same consequent have been discovered. Note that they could have been fused into one rule as the consequent is the same. Besides, the obtained confidence is quite low which leads to rules with considerably high errors. The case of the direction of the wind is similar to that of the temperature. The third and fifth rules share the same antecedent for the same direction and, however, the consequent for both rules is different even when they could have been fused into just one rule. The confidence hardly reaches 30%, which leads to an almost null reliability. The speed of the wind appears in one rule in which its value is low and the ozone presents medium values. However, the confidence is quite low. Table 19 Association rules found by QARGA with high confidence ID Rule Sup. (%) Conf. (%) #r Ampl. Lift #0 temp. 2[38, 42] and hum. 2[25, 33] and hour 2[15, 18] ¼) O32[140, 206] 3 90 47 20 6.61 #1 temp. 2[26, 33] and hum. 2[29, 46] and dir. 2[149, 231] and speed 2[12, 20] and hour 2[15, 19] ¼) O32[103, 160] 3 93 40 29 2.90 #2 temp. 2[29, 37] and hum. 2[21, 36] and dir. 2[161, 187] and hour 2[14, 19] ¼) O32[109, 180] 5 83 70 25 2.83 #3 temp. 2[34, 42] and hum. 2[22, 32] and dir. 2[152, 189] and speed 2[9, 16] and hour 2[15, 17] ¼) O32[134, 206] 2 87 33 23 4.99 #4 hum. 2[22, 32] and hour 2[13, 18] ¼) O32[144, 198] 7 48 97 23 4.27 4.0 (2.0) 80.2 (18.4) 57.4 (26.1) 24.0 (3.3) 4.32 (1.58) Table 20 Association rules found by Apriori (ten intervals used for discretization) ID Rule Sup. (%) Conf. (%) #r Ampl. Lift #0 temp. 2[27, 30] ¼) O32[90, 115] 5 37 79 14 1.43 #1 temp. 2[24, 27] ¼) O32[90, 115] 6 33 85 14 1.43 #2 dir. 2[144, 180] ¼) O32[90, 115] 10 29 156 31 1.18 #3 speed 2[6, 8] ¼) O32[90, 115] 5 24 75 14 1.03 #4 dir. 2[144, 180] ¼) O32[115, 141] 6 16 88 31 1.27 6.4 (2.1) 27.8 (8.2) 96.6 (33.6) 20.8 (9.3) 1.27 (0.17) The comparison of the Tables 19 and 20 leads to several conclusions. As regards the attributes, QARGA always obtains rules with greater number of them and, consequently, the information brought by these rules is higher than that of Apriori. When taking into consideration the support, QARGA presents low values since there are few instances in the dataset with high ozone values. However, if the confidence of the rules for both algorithms is compared, it can be observed that QARGA has values even greater than 90% while Apriori never overcomes 40%. Unlike the lift values from the first kind of experimentation, where the interest of the rules in QARGA and Apriori was quite similar, it can be observed that, in this case, the results of the lift are very different. Rules found by QARGA present lift values between 3 and 6, while Apriori never exceeds 1.50. This is an important result that indicates that QARGA find more interesting rules than Apriori does. Another relevant remark is that Apriori discovers rules with different intervals for the same variable in the antecedent but equal consequents and viceversa. This fact never occurs in QARGA. The ozone levels obtained by Apriori never exceeds 140 lg/m 3 , while QARGA reached values greater than 200 lg/m 3 . This appreciation is of the utmost relevance, since environment is really concerned by high levels of ozone and, consequently, discovering rules with these values of ozone is useless. 6 Conclusions An evolutionary algorithm has been proposed in this work to obtain QAR from time series. In order to evaluate its performance, the approach has been applied to several datasets and compared with the most recently published results. Thus, a bank of public datasets retrieved from the BUFA repository has been used to test the accuracy of the algorithm. The algorithm has shown to be efficient when mining synthetically generated multidimensional time series. Also, the proposed methodology has successfully obtained meaningful QAR from multidimensional realworld time series. In particular, relevant dependencies between the ozone concentration in the atmosphere and other climatological-related time series have been found. 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