870 Slava Kisilevich, Florian Mansmann, Mirco Nanni, Salvatore Rinzivillo 44.4 Open Issues Spatio-temporal properties of the data introduce additional complexity to the data mining pro- cess and to the clustering in particular. We can differentiate between two types of issues that the analyst should deal with or take into consideration during analysis: general and appli- cation dependent. The general issues involve such aspects as data quality, precision and un- certainty (Miller and Han(2009)). Scalability, spatial resolution and time granularity can be related to application dependent issues. Data quality (spatial and temporal) and precision depends on the way the data is generated. Movement data is usually collected using GPS-enabled devices attached to an object. For example, when a person enters a building a GPS signal can be lost or the positioning may be inaccurate due to a weak connection to satellites. As in the general data preprocessing step, the analyst should decide how to handle missing or inaccurate parts of the data - should it be ignored, tolerated or interpolated. The computational power does not go in line with the pace at which large amounts of data are being generated and stored. Thus, the scalability becomes a significant issue for the analysis and demand new algorithmic solutions or approaches to handle the data. Spatial resolution and time granularity can be regarded as most crucial in spatio-temporal clustering since change in the size of the area over which the attribute is distributed or change in time interval can lead to discovery of completely different clusters and therefore, can lead to the improper explanation of the phenomena under investigation. There are still no gen- eral guidelines for proper selection of spatial and temporal resolution and it is rather unlikely that such guidelines will be proposed. Instead, ad hoc approaches are proposed to handle the problem in specific domains (see for example (Nanni and Pedreschi(2006))). Due to this, the involvement of the domain expert in every step of spatio-temporal clustering becomes essen- tial. The geospatial visual analytics field has recently emerged as the discipline that combines automatic data mining approaches including spatio-temporal clustering with visual reasoning supported by the knowledge of domain experts and has been successfully applied at differ- ent geographical spatio-temporal phenomena ( (Andrienko and Andrienko(2006), Andrienko et al(2007)Andrienko, Andrienko, and Wrobel,Andrienko and Andrienko(2010))). A class of application-dependent issues that is quickly emerging in the spatio-temporal clustering field is related to exploitation of available background knowledge. Indeed, most of the methods and solutions surveyed in this chapter work on an abstract space where loca- tions have no specific meanings and the analysis process extracts information from scratch, instead of starting from (and integrating to) possible a priori knowledge of the phenomena under consideration. On the opposite, a priori knowledge about such phenomena and about the context they take place in is commonly available in real applications, and integrating them in the mining process might improve the output quality (Alvares et al(2007)Alvares, Bogorny, Kuijpers, de Macedo, Moelans, and Vaisman,Baglioni et al(2009)Baglioni, Antonio Fernan- des de Macedo, Renso, Trasarti, and Wachowicz, Kisilevich et al(2010)Kisilevich, Keim, and Rokach). Examples of that include the very basic knowledge of the street network and land usage, that can help in understanding which aspects of the behavior of our objects (e.g., which parts of the trajectory of a moving object) are most discriminant and better suited to form ho- mogeneous clusters; or the existence of recurring events, such as rush hours and planned road maintenance in a urban mobility setting, that are known to interfere with our phenomena in predictable ways. Recently, the spatio-temporal data mining literature has also pointed out that the rele- vant context for the analysis mobile objects includes not only geographic features and other physical constraints, but also the population of objects themselves, since in most application 44 Spatio-temporal clustering 871 scenarios objects can interact and mutually interfere with each other’s activity. Classical ex- amples include traffic jams – an entity that emerges from the interaction of vehicles and, in turn, dominates their behavior. Considering interactions in the clustering process is expected to improve the reliability of clusters, yet a systematic taxonomy of relevant interaction types is still not available (neither a general one, nor any application-specific one), it is still not known how to detect such interactions automatically, and understanding the most suitable way to integrate them in a clustering process is still an open problem. 44.5 Conclusions In this chapter we focused on geographical spatio-temporal clustering. We presented a classi- fication of main spatio-temporal types of data: ST events, Geo-referenced variables, Moving objects and Trajectories. We described in detail how spatio-temporal clustering is applied on trajectories, provided an overview of recent research developments and presented possible sce- narios in several application domains such as movement, cellular networks and environmental studies. References Agrawal R, Faloutsos C, Swami AN (1993) Efficient Similarity Search In Sequence Databases. In: Lomet D (ed) Proceedings of the 4th International Conference of Founda- tions of Data Organization and Algorithms (FODO), Springer Verlag, Chicago, Illinois, pp 69–84 Alon J, Sclaroff S, Kollios G, Pavlovic V (2003) Discovering clusters in motion time-series data. In: CVPR (1), pp 375–381 Alvares LO, Bogorny V, Kuijpers B, de Macedo JAF, Moelans B, Vaisman A (2007) A model for enriching trajectories with semantic geographical information. In: GIS ’07: Proceed- ings of the 15th annual ACM international symposium on Advances in geographic in- formation systems, pp 1–8 Andrienko G, Andrienko N (2008) Spatio-temporal aggregation for visual analysis of move- ments. In: Proceedings of IEEE Symposium on Visual Analytics Science and Technol- ogy (VAST 2008), IEEE Computer Society Press, pp 51–58 Andrienko G, Andrienko N (2009) Interactive cluster analysis of diverse types of spatiotem- poral data. ACM SIGKDD Explorations Andrienko G, Andrienko N (2010) Spatial generalization and aggregation of massive move- ment data. IEEE Transactions on Visualization and Computer Graphics (TVCG) Ac- cepted Andrienko G, Andrienko N, Wrobel S (2007) Visual analytics tools for analysis of movement data. SIGKDD Explorations Newsletter 9(2):38–46 Andrienko G, Andrienko N, Rinzivillo S, Nanni M, Pedreschi D, Giannotti F (2009) Inter- active Visual Clustering of Large Collections of Trajectories. VAST 2009 Andrienko N, Andrienko G (2006) Exploratory analysis of spatial and temporal data: a sys- tematic approach. Springer Verlag Ankerst M, Breunig MM, Kriegel HP, Sander J (1999) Optics: ordering points to identify the clustering structure. SIGMOD Rec 28(2):49–60 872 Slava Kisilevich, Florian Mansmann, Mirco Nanni, Salvatore Rinzivillo Baglioni M, Antonio Fernandes de Macedo J, Renso C, Trasarti R, Wachowicz M (2009) Towards semantic interpretation of movement behavior. Advances in GIScience pp 271– 288 Berndt DJ, Clifford J (1996) Finding patterns in time series: a dynamic programming ap- proach. Advances in knowledge discovery and data mining pp 229–248 Birant D, Kut A (2006) An algorithm to discover spatialtemporal distributions of physical seawater characteristics and a case study in turkish seas. Journal of Marine Science and Technology pp 183–192 Birant D, Kut A (2007) St-dbscan: An algorithm for clustering spatial-temporal data. Data Knowl Eng 60(1):208–221 Chan KP, chee Fu AW (1999) Efficient time series matching by wavelets. In: In ICDE, pp 126–133 Chen L, ¨ Ozsu MT, Oria V (2005) Robust and fast similarity search for moving object trajec- tories. In: SIGMOD ’05: Proceedings of the 2005 ACM SIGMOD international confer- ence on Management of data, ACM, New York, NY, USA, pp 491–502 Chudova D, Gaffney S, Mjolsness E, Smyth P (2003) Translation-invariant mixture models for curve clustering. In: KDD ’03: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, New York, NY, USA, pp 79–88 Ciaccia P, Patella M, Zezula P (1997) M-tree: An efficient access method for similarity search in metric spaces. In: Jarke M, Carey M, Dittrich KR, Lochovsky F, Loucopoulos P, Jeusfeld MA (eds) Proceedings of the 23rd International Conference on Very Large Data Bases (VLDB’97), Morgan Kaufmann Publishers, Inc., Athens, Greece, pp 426–435 Cohen S., Rokach L., Maimon O., Decision Tree Instance Space Decomposition with Grouped Gain-Ratio, Information Science, Volume 177, Issue 17, pp. 3592-3612, 2007. Ester M, Kriegel HP, Sander J, Xu X (1996) A density-based algorithm for discovering clus- ters in large spatial databases with noise. Data Mining and Knowledge Discovery pp 226–231 Fosca G, Dino P (2008) Mobility, Data Mining and Privacy: Geographic Knowledge Discov- ery. Springer Frentzos E, Gratsias K, Theodoridis Y (2007) Index-based most similar trajectory search. In: ICDE, pp 816–825 Gaffney S, Smyth P (1999) Trajectory clustering with mixtures of regression models. In: KDD ’99: Proceedings of the fifth ACM SIGKDD international conference on Knowl- edge discovery and data mining, ACM, New York, NY, USA, pp 63–72 Giannotti F, Nanni M, Pinelli F, Pedreschi D (2007) Trajectory pattern mining. In: Proceed- ings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, p 339 Grinstein G, Plaisant C, Laskowski S, OConnell T, Scholtz J, Whiting M (2008) VAST 2008 Challenge: Introducing mini-challenges. In: Proceedings of IEEE Symposium, vol 1, pp 195–196 Gudmundsson J, van Kreveld M (2006) Computing longest duration flocks in trajectory data. In: GIS ’06: Proceedings of the 14th annual ACM international symposium on Advances in geographic information systems, ACM, New York, NY, USA, pp 35–42 Hwang SY, Liu YH, Chiu JK, Lim EP (2005) Mining mobile group patterns: A trajectory- based approach. In: PAKDD, pp 713–718 Iyengar VS (2004) On detecting space-time clusters. In: Proceedings of the 10th International Conference on Knowledge Discovery and Data Mining (KDD’04), ACM, pp 587–592 44 Spatio-temporal clustering 873 Jeung H, Yiu ML, Zhou X, Jensen CS, Shen HT (2008) Discovery of convoys in trajectory databases. Proc VLDB Endow 1(1):1068–1080 Kalnis P, Mamoulis N, Bakiras S (2005) On discovering moving clusters in spatio-temporal data. Advances in Spatial and Temporal Databases pp 364–381 Kang J, Yong HS (2009) Mining Trajectory Patterns by Incorporating Temporal Properties. Proceedings of the 1st International Conference on Emerging Databases Kang JH, Welbourne W, Stewart B, Borriello G (2004) Extracting places from traces of locations. In: WMASH ’04: Proceedings of the 2nd ACM international workshop on Wireless mobile applications and services on WLAN hotspots, ACM, New York, NY, USA, pp 110–118 Kisilevich S, Keim D, Rokach L (2010) A novel approach to mining travel sequences us- ing collections of geo-tagged photos. In: The 13th AGILE International Conference on Geographic Information Science Kulldorff M (1997) A spatial scan statistic. Communications in Statistics: Theory and Meth- ods 26(6):1481–1496 Lee JG, Han J, Whang KY (2007) Trajectory clustering: a partition-and-group framework. In: SIGMOD Conference, pp 593–604 Li Y, Han J, Yang J (2004a) Clustering moving objects. In: Proceedings of the 10th Inter- national Conference on Knowledge Discovery and Data Mining (KDD’04), ACM, pp 617–622 Li Y, Han J, Yang J (2004b) Clustering moving objects. In: KDD, pp 617–622 Maimon O., and Rokach, L. Data Mining by Attribute Decomposition with semiconductors manufacturing case study, in Data Mining for Design and Manufacturing: Methods and Applications, D. Braha (ed.), Kluwer Academic Publishers, pp. 311–336, 2001. Miller HJ, Han J (2009) Geographic data mining and knowledge discovery. Chapman & Hall/CRC Nanni M, Pedreschi D (2006) Time-focused clustering of trajectories of moving objects. Journal of Intelligent Information Systems 27(3):267–289 Palma AT, Bogorny V, Kuijpers B, Alvares LO (2008) A clustering-based approach for dis- covering interesting places in trajectories. In: SAC ’08: Proceedings of the 2008 ACM symposium on Applied computing, pp 863–868 Pelekis N, Kopanakis I, Marketos G, Ntoutsi I, Andrienko G, Theodoridis Y (2007) Similar- ity search in trajectory databases. In: TIME ’07: Proceedings of the 14th International Symposium on Temporal Representation and Reasoning, IEEE Computer Society, Wash- ington, DC, USA, pp 129–140 Reades J, Calabrese F, Sevtsuk A, Ratti C (2007) Cellular census: Explorations in urban data collection. IEEE Pervasive Computing 6(3):30–38 Rinzivillo S, Pedreschi D, Nanni M, Giannotti F, Andrienko N, Andrienko G (2008) Visually driven analysis of movement data by progressive clustering. Information Visualization 7(3):225–239 Rokach L. and Maimon O., Feature Set Decomposition for Decision Trees, Journal of Intel- ligent Data Analysis, Volume 9, Number 2, 2005b, pp 131–158. Rokach L., Genetic algorithm-based feature set partitioning for classification prob- lems,Pattern Recognition, 41(5):1676–1700, 2008. Rokach L., Maimon O. and Lavi I., Space Decomposition In Data Mining: A Clustering Ap- proach, Proceedings of the 14th International Symposium On Methodologies For Intel- ligent Systems, Maebashi, Japan, Lecture Notes in Computer Science, Springer-Verlag, 2003, pp. 24–31. 874 Slava Kisilevich, Florian Mansmann, Mirco Nanni, Salvatore Rinzivillo Schilit BN, LaMarca A, Borriello G, Griswold WG, McDonald D, Lazowska E, Balachan- dran A, Hong J, Iverson V (2003) Challenge: ubiquitous location-aware computing and the ”place lab” initiative. In: WMASH ’03: Proceedings of the 1st ACM international workshop on Wireless mobile applications and services on WLAN hotspots, ACM, New York, NY, USA, pp 29–35 Stolorz P, Nakamura H, Mesrobian E, Muntz RR, Santos JR, Yi J, Ng K (1995) Fast spatio- temporal data mining of large geophysical datasets. In: Proceedings of the First Interna- tional Conference on Knowledge Discovery and Data Mining (KDD’95), AAAI Press, pp 300–305 Theodoridis Y (2003) Ten benchmark database queries for location-based services. The Computer Journal 46(6):713–725 Vieira MR, Bakalov P, Tsotras VJ (2009) On-line discovery of flock patterns in spatio- temporal data. In: GIS ’09: Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM, New York, NY, USA, pp 286–295 Vlachos M, Kollios G, Gunopulos D (2002) Discovering similar multidimensional trajecto- ries. In: Proceedings of the International Conference on Data Engineering, pp 673–684 Vlachos M, Hadjieleftheriou M, Gunopulos D, Keogh E (2003) Indexing multi-dimensional time-series with support for multiple distance measures. In: KDD ’03: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, New York, NY, USA, pp 216–225 Wang M, Wang A, Li A (2006) Mining Spatial-temporal Clusters from Geo-databases. Lec- ture Notes in Computer Science 4093:263 Zhang P, Huang Y, Shekhar S, Kumar V (2003) Correlation analysis of spatial time series datasets: A filter-and-refine approach. In: In the Proc. of the 7th PAKDD Zhang T, Ramakrishnan R, Livny M (1996) BIRCH: an efficient data clustering method for very large databases. ACM SIGMOD Record 25(2):103–114 Zheng Y, Zhang L, Xie X, Ma WY (2009) Mining interesting locations and travel sequences from gps trajectories. In: WWW ’09: Proceedings of the 18th international conference on World wide web, pp 791–800 45 Data Mining for Imbalanced Datasets: An Overview Nitesh V. Chawla Department of Computer Science and Engineering University of Notre Dame IN 46530, USA nchawla@cse.nd.edu Summary. A dataset is imbalanced if the classification categories are not approximately equally represented. Recent years brought increased interest in applying machine learning techniques to difficult “real-world” problems, many of which are characterized by imbalanced data. Additionally the distribution of the testing data may differ from that of the training data, and the true misclassification costs may be unknown at learning time. Predictive accuracy, a popular choice for evaluating performance of a classifier, might not be appropriate when the data is imbalanced and/or the costs of different errors vary markedly. In this Chapter, we dis- cuss some of the sampling techniques used for balancing the datasets, and the performance measures more appropriate for mining imbalanced datasets. Key words: imbalanced datasets, classification, sampling, ROC, cost-sensitive measures, pre- cision and recall 45.1 Introduction The issue with imbalance in the class distribution became more pronounced with the appli- cations of the machine learning algorithms to the real world. These applications range from telecommunications management (Ezawa et al., 1996), bioinformatics (Radivojac et al., 2004), text classification (Lewis and Catlett, 1994, Dumais et al., 1998, Mladeni ´ c and Grobelnik, 1999, Cohen, 1995b), speech recognition (Liu et al., 2004), to detection of oil spills in satel- lite images (Kubat et al., 1998). The imbalance can be an artifact of class distribution and/or different costs of errors or examples. It has received attention from machine learning and Data Mining community in form of Workshops (Japkowicz, 2000b, Chawla et al., 2003a,Dietterich et al., 2003, Ferri et al., 2004) and Special Issues (Chawla et al., 2004a). The range of papers in these venues exhibited the pervasive and ubiquitous nature of the class imbalance issues faced by the Data Mining community. Sampling methodologies continue to be popular in the research work. However, the research continues to evolve with different applications, as each application provides a compelling problem. One focus of the initial workshops was primarily O. Maimon, L. Rokach (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed., DOI 10.1007/978-0-387-09823-4_45, © Springer Science+Business Media, LLC 2010 876 Nitesh V. Chawla the performance evaluation criteria for mining imbalanced datasets. The limitation of the ac- curacy as the performance measure was quickly established. ROC curves soon emerged as a popular choice (Ferri et al., 2004). The compelling question, given the different class distributions is: What is the correct distribution for a learning algorithm? Weiss and Provost presented a detailed analysis on the effect of class distribution on classifier learning (Weiss and Provost, 2003). Our observations agree with their work that the natural distribution is often not the best distribution for learn- ing a classifier (Chawla, 2003). Also, the imbalance in the data can be more characteristic of “sparseness” in feature space than the class imbalance. Various re-sampling strategies have been used such as random oversampling with replacement, random undersampling, focused oversampling, focused undersampling, oversampling with synthetic generation of new sam- ples based on the known information, and combinations of the above techniques (Chawla et al., 2004b). In addition to the issue of inter-class distribution, another important probem arising due to the sparsity in data is the distribution of data within each class (Japkowicz, 2001a). This problem was also linked to the issue of small disjuncts in the decision tree learning. Yet an- other, school of thought is a recognition based approach in the form of a one-class learner. The one-class learners provide an interesting alternative to the traditional discriminative approach, where in the classifier is learned on the target class alone (Japkowicz, 2001b, Juszczak and Duin, 2003, Raskutti and Kowalczyk, 2004, Tax, 2001). In this chapter 1 , we present a liberal overview of the problem of mining imbalanced datasets with particular focus on performance measures and sampling methodologies. We will present our novel oversampling technique, SMOTE, and its extension in the boosting proce- dure — SMOTEBoost. 45.2 Performance Measure A classifier is, typically, evaluated by a confusion matrix as illustrated in Figure 45.1 (Chawla et al., 2002). The columns are the Predicted class and the rows are the Actual class. In the con- fusion matrix, TN is the number of negative examples correctly classified (True Negatives), FP is the number of negative examples incorrectly classified as positive (False Positives), FN is the number of positive examples incorrectly classified as negative (False Negatives) and TP is the number of positive examples correctly classified (True Positives). Predictive accuracy is defined as Accuracy =(TP+ TN)/(TP+ FP+ TN+ FN). However, predictive accuracy might not be appropriate when the data is imbalanced and/or the costs of different errors vary markedly. As an example, consider the classification of pixels in mammogram images as possibly cancerous (Woods et al., 1993). A typical mammography dataset might contain 98% normal pixels and 2% abnormal pixels. A simple default strategy of guessing the majority class would give a predictive accuracy of 98%. The nature of the application requires a fairly high rate of correct detection in the minority class and allows for a small error rate in the majority class in order to achieve this (Chawla et al., 2002). Simple predictive accuracy is clearly not appropriate in such situations. 1 The chapter will utilize excerpts from our published work in various Journals and Confer- ences. Please see the references for the original publications. 45 Data Mining for Imbalanced Datasets: An Overview 877 Predicted Negative Predicted Positive TN FP FN TP Actual Negative Actual Positive Fig. 45.1. Confusion Matrix 45.2.1 ROC Curves The Receiver Operating Characteristic (ROC) curve is a standard technique for summarizing classifier performance over a range of tradeoffs between true positive and false positive error rates (Swets, 1988). The Area Under the Curve (AUC) is an accepted performance metric for a ROC curve (Bradley, 1997). Percent True Positive Percent False Positive 0 100 100 original data set increased undersampling of the majority class moves the operating point to the upper right ROC (100, 100) y = x Ideal point Fig. 45.2. Illustration of Sweeping out an ROC Curve through under-sampling. Increased under-sampling of the majority (negative) class will move the performance from the lower left point to the upper right. ROC curves can be thought of as representing the family of best decision boundaries for relative costs of TP and FP. On an ROC curve the X-axis represents %FP = FP/(TN+FP) and the Y-axis represents %TP = TP/(TP+ FN). The ideal point on the ROC curve would be (0,100), that is all positive examples are classified correctly and no negative examples are misclassified as positive. One way an ROC curve can be swept out is by manipulating the balance of training samples for each class in the training set. Figure 45.2 shows an illustra- tion (Chawla et al., 2002). The line y = x represents the scenario of randomly guessing the class. A single operating point of a classifier can be chosen from the trade-off between the 878 Nitesh V. Chawla %TP and %FP, that is, one can choose the classifier giving the best %TP for an acceptable %FP (Neyman-Pearson method) (Egan, 1975). Area Under the ROC Curve (AUC) is a use- ful metric for classifier performance as it is independent of the decision criterion selected and prior probabilities. The AUC comparison can establish a dominance relationship between clas- sifiers. If the ROC curves are intersecting, the total AUC is an average comparison between models (Lee, 2000). The ROC convex hull can also be used as a robust method of identifying potentially opti- mal classifiers (Provost and Fawcett, 2001). Given a family of ROC curves, the ROC convex hull can include points that are more towards the north-west frontier of the ROC space. If a line passes through a point on the convex hull, then there is no other line with the same slope passing through another point with a larger true positive (TP) intercept. Thus, the classifier at that point is optimal under any distribution assumptions in tandem with that slope (Provost and Fawcett, 2001). Moreover, distribution/cost sensitive applications can require a ranking or a probabilistic estimate of the instances. For instance, revisiting our mammography data example, a proba- bilistic estimate or ranking of cancerous cases can be decisive for the practitioner (Chawla, 2003, Maloof, 2003). The cost of further tests can be decreased by thresholding the patients at a particular rank. Secondly, probabilistic estimates can allow one to threshold ranking for class membership at values < 0.5. The ROC methodology by (Hand, 1997) allows for rank- ing of examples based on their class memberships — whether a randomly chosen majority class example has a higher majority class membership than a randomly chosen minority class example. It is equivalent to the Wilcoxon test statistic. 45.2.2 Precision and Recall From the confusion matrix in Figure 45.1, we can derive the expression for precision and recall (Buckland and Gey, 1994). precision = TP TP+ FP recall = TP TP+ FN The main goal for learning from imbalanced datasets is to improve the recall without hurting the precision. However, recall and precision goals can be often conflicting, since when increasing the true positive for the minority class, the number of false positives can also be increased; this will reduce the precision. The F-value metric is one measure that combines the trade-offs of precision and recall, and outputs a single number reflecting the “goodness” of a classifier in the presence of rare classes. While ROC curves represent the trade-off between values of TP and FP, the F-value represents the trade-off among different values of TP, FP, and FN (Buckland and Gey, 1994). The expression for the F-value is as follows: F −value = (1 + β 2 ) ∗recall ∗ precision β 2 ∗recall + precision where β corresponds to the relative importance of precision vs recall. It is usually set to 1. 45 Data Mining for Imbalanced Datasets: An Overview 879 45.2.3 Cost-sensitive Measures Cost Matrix Cost-sensitive measures usually assume that the costs of making an error are known (Turney, 2000, Domingos, 1999, Elkan, 2001). That is one has a cost-matrix, which defines the costs incurred in false positives and false negatives. Each example, x, can be associated with a cost C(i, j, x), which defines the cost of predicting class i for x when the “true” class is j. The goal is to take a decision to minimize the expected cost. The optimal prediction for x can be defined as ∑ j P( j|x)C(i, j,x) (45.1) The aforementioned equation requires a computation of conditional probablities of class j given feature vector or example x. While the cost equation is straightforward, we don’t always have a cost attached to making an error. The costs can be different for every example and not only for every type of error. Thus, C(i, j) is not always ≡ to C(i, j,x). Cost Curves (Drummond and Holte, 2000) propose cost-curves, where the x-axis represents of the fraction of the positive class in the training set, and the y-axis represents the expected error rate grown on each of the training sets. The training sets for a data set is generated by under (or over) sampling. The error rates for class distributions not represented are construed by interpolation. They define two cost-sensitive components for a machine learning algorithm: 1) producing a variety of classifiers applicable for different distributions and 2) selecting the appropriate classifier for the right distribution. However, when the misclassification costs are known, the x-axis can represent the “probability cost function”, which is the normalized product of C(−| +) ∗P(+); the y-axis represents the expected cost. 45.3 Sampling Strategies Over and under-sampling methodologies have received significant attention to counter the effect of imbalanced data sets (Solberg and Solberg, 1996, Japkowicz, 2000a, Chawla et al., 2002, Weiss and Provost, 2003, Kubat and Matwin, 1997, Jo and Japkowicz, 2004, Batista et al., 2004, Phua and Alahakoon, 2004, Laurikkala, 2001, Ling and Li, 1998). Various stud- ies in imbalanced datasets have used different variants of over and under sampling, and have presented (sometimes conflicting) viewpoints on usefulness of oversampling versus under- sampling (Chawla, 2003,Maloof, 2003, Drummond and Holte, 2003,Batista et al., 2004). The random under and over sampling methods have their various short-comings. The random undersampling method can potentially remove certain im- portant examples, and random oversampling can lead to overfitting. However, there has been progression in both the under and over sampling methods. (Kubat and Matwin, 1997) used one-sided selection to selectively undersample the original population. They used Tomek Links (Tomek, 1976) to identify the noisy and borderline examples. They also used the Condensed Nearest Neighbor (CNN) rule (Hart, 1968) to remove examples from the majority class that are far away from the decision border. (Laurikkala, 2001) proposed Neighborhood . geographical spatio-temporal phenomena ( (Andrienko and Andrienko (20 06), Andrienko et al (20 07)Andrienko, Andrienko, and Wrobel,Andrienko and Andrienko (20 10))). A class of application-dependent. Conference on Knowledge Discovery and Data Mining (KDD’04), ACM, pp 617– 622 Li Y, Han J, Yang J (20 04b) Clustering moving objects. In: KDD, pp 617– 622 Maimon O., and Rokach, L. Data Mining by Attribute. 35 92- 36 12, 20 07. Ester M, Kriegel HP, Sander J, Xu X (1996) A density-based algorithm for discovering clus- ters in large spatial databases with noise. Data Mining and Knowledge Discovery pp 22 6 23 1 Fosca