E. Boros, P. L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. Rutcor Research Report RRR 22-96, Rutgers University, New Brunswick, N.J., July 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Learning systems have been proposed by many researchers in the fields of Artificial Intelligence and Statistics; more recently, have appeared in the literature several approaches that adopt mathematical and combinatorial optimization models and techniques for learning systems, such as in [1], 2] 3] 4] Methods based on neural networks ( 5] 6] have also been used to formalize and solve several learning problems. Lsquare is a particular type of learning system that operates in the logic domain. In this type of learning, the objects to be recognized are described by a number of ....

Boros E., Hammer P.L., Ibaraki T., Kogan A., Mayoraz E., and Muchnik I. An Implementation of Logical Analysis of Data. RUTCOR Research Report 29-9. Rutgers University, NJ, July 1995

....1 X i=1 s h X j=i 1 (v(x i ) v(x j ) 2 for all pairs of points at Hamming distance h apart where s h is the number of points and x 1 ; x 2 ; x s h are the points involved in such pairs. We did this for several data sets in the repository at RUTCOR, Rutgers University (see [2]) In the following we present results for one of the data sets which samples economic conditions in China. Two les associated with this data set are 7 cf.dat which is the full data set of 280 points with dimension 44, and ct.dat which is a training set of 138 points (see Section 4 for the ....

E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. RUTCOR Research Report RRR 22-96, RUTCOR, Rutgers University, 1996.

.... see the books by White (1992) and Gallant (1993) nearest neighbor methods (for example, see the book by Nadler and Smith (1993) and logic domain methods (for example, see Kamath, Karmarkar, Ramakrishnan, and Resende (1992) Triantaphyllou, Allen, Soyster, and Kumara (1994) Felici (1995) and Boros, Hammer, Ibaraki, Kogan, Mayoraz, and Muchnik (1996)) For details on arcing, bagging, boosting, randomization, and stacking, see, for example, Wolpert (1992) Breiman (1996a, 1996b, 1997) Drucker and Cortes (1996) Quinlan (1996) Freund and Schapire (1997) Maclin and Optiz (1997) and Dietterich (1998) For an evaluation of combinations of ....

....83.5 (86.5 predicted) Breast Cancer: 97.1 (96.0 predicted) Congressional Voting: 95.8 (96.8 predicted) Diabetes: 73.3 (77.7 predicted) Heart Disease: 80.7 (79.9 predicted) These average accuracy results essentially match those of the best prior methods. For a comparison, see Boros, Hammer, Ibaraki, Kogan, Mayoraz, and Muchnik (1996). The computing effort for constructing the family C of classification methods for a given pair A and B of the six data sets is reasonable. In the worst case, it takes less than 20 minutes on a Sun UltraSparc 1 to process completely any one of the data sets. The description of the entire ....

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Boros, E., Hammer, P. L., Ibaraki, T., Kogan, A., Mayoraz, E., and Muchnik, I. (1996), An implementation of logical analysis of data, RUTCOR Research Report 22-96, Rutgers University, NJ.

....de nes a K partition of the input space into sets F 1 (k) called classes and denoted k . The collection of learning algorithms available to solve classi cation problems originate in di erent domains such as: statistics (e.g. Bayesian classi ers, see [8] logic (e.g.logical analysis of data [4, 1]) neural networks (e.g. perceptron algorithm [16] backpropagation [19] arti cial intelligence (e.g. decision trees [2, 14] Among these, only those capable of handling multiclass problems are applied, in general, to solve problems where the number of classes exceeds two. It is possible, ....

....the two kinds of reconstruction: arg max 1 i K p i ; p i = 2 K(K 1) X j 6=i ( p ij ) 4 IDIAP RR 97 09 where takes the form of a threshold function at 0:5 for the rough reconstruction and the identity function for the soft reconstruction. Note that whenever is symmetric on [0,1], i.e. 1 x) 1 (x) then the p i s sum to 1 and can thus be considered as probability estimates. The two schemes presented explore the available information di erently and about this a remark can be made. The following example follows from an observation made in [9] Consider the matrix P ....

E. Boros, P. L. Hammer, Toshihide Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. RRR 22-96, RUTCOR{Rutgers University's Center For Operations Research, http://rutcor.rutgers.edu:80/~rrr/, Submitted, July 1996.

....this setting, the consistency constraint requires that if F (x) and F (x 0 ) di er, m(x) and m(x 0 ) must have at least one component in which one is 1 and the other is 0. 1. 3 Existing approaches The binarization of supervised data has been addressed in the work on logical analysis of data [1]. In that approach, a list of potential discriminants is rst generated. Then, a subset is extracted from IDIAP RR 99 04 3 this list by the resolution of a minimum set covering problem de ned as follows. Let A be the matrix with one column per possible discriminant and one row per pair of points ....

....procedure to be accomplished in a time less than quadratic in the number of data. This represents an important result, as the algorithm may constitute a precious tool to be used in conjunction with learning methods that deal with data in binary format, as is the case of logical analysis of data [1]. It is an exciting research subject and, in spite of the encouraging results obtain herewith, new challenges are still to come and additional work to be developed. An important step to be taken, is to test the actual quality of the sets of the obtained discriminants, with the application of the ....

E. Boros, P. L. Hammer, Toshihide Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. RRR 22-96, RUTCOR{Rutgers University's Center For Operations Research, July 1996. http://rutcor.rutgers.edu:80/~rrr/ Submitted.

.... data, Boolean functions, decomposable functions, computational learning theory, random graphs, probabilistic analysis 1 Introduction Extracting knowledge from given data has been studied in such fields as knowl edge engineering, data mining, artificial intelligence and database theory (e.g. [4, 6, 8]) Logical analysis of data (LAD) is one of the methodologies for knowledge discovery. LAD is based on Boolean logic, that is, a given data set is repre sented as a pair of set T of true vectors (examples that cause the phenomenon to occur) and set F of false vectors (examples not causing the ....

E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, IEEE Trans. on Knowledge and Data Engineering 12 (2000) 292-306.

....related to such data sets include classification (i.e. identification of the type of a new observation not included in the data set) determination of characteristic properties of observations of the same type, analysis of the role of various attributes, etc. The logical analysis of data (LAD) [21, 13, 6, 5, 4, 7, 20, 8, 18] is a methodology addressing the above kinds of problems. The mathematical foundation of LAD is in discrete mathematics, with a special emphasis on the theory of Boolean functions. Patterns are the key building blocks in LAD [21, 13] as well as in many other rule induction algorithms (such as ....

....negative. It will be seen below that in spite of this seemingly unproductive role of the selectivity preference, it can become extremely useful in combination with other criteria. While previous theoretical LAD studies employed exclusively the simplicity preference, the implementation of LAD [6] took into account another very natural suitability criterion. This criterion is related to the so called coverage Cov(P ) of a pattern P , i.e. the set of vectors X # T for which P (X) 1. Note that the second condition in the definition of a pattern guarantees that for every pattern P the ....

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E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik "An implementation of logical analysis of data", IEEE Transactions on Knowledge and Data Engineering, Vol. 12, No. 2, April 2000, 292--306.

....should also be included in the list, since DNF is a standard form of representation of Boolean functions. The prime implicants in DNF of an extension are also called association rules in data mining (e.g. 1, 28] and patterns in papers on logical analysis of data [16] and its applications [10]. Unfortunately, real world data might not be complete, adding another dimension of complication. In other words, the values of some elements x j in a given data vector x may not be available for various reasons, such as the test to measure the x j was not conducted because it takes too much time ....

....problems had arisen from real world applications, in which the sizes of such pBmb instances are usually large. Hence it would be important to develop fast heuristic algorithms for all the problems discussed in this paper, particularly for the NP hard cases. An attempt in this direction is found in [10]. Acknowledgment The authors express their gratitude to the referees whose constructive comments helped significantly improve this paper. ....

E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, RUTCOR Research Report RRR 22-96, Rutgers University, 1996.

....by T the set of positive examples, and by F the set of negative examples. It is further assumed that each example is given as a binary n dimensional vector. This is a typical problem setting studied in such elds as knowledge discovery, data mining, learning theory and logical analysis of data [3, 1, 7, 13, 15, 26, 30, 31]. Let B = f0; 1g n . A pair (T; F ) T; F 2 B n , is called a partially de ned Boolean function (pdBf) where a 2 T (resp. b 2 F ) are called true vectors (resp. false vectors) Usually the sets T and F are assumed to be disjoint, i.e. T F = and fully speci ed Boolean functions f that ....

E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, RUTCOR Research Report RRR 22-96, Rutgers University, 1996; IEEE Trans. on Knowledge and Data Engineering, to appear.

....about subjects other than cats. YES NO YES NO Figure 1: A binary decision tree Logical Analysis of Data LAD is another technique to nd logical rules characterizing the di erence between relevant and irrelevant documents, based on a set of training documents (Crama et al. 1988; Boros et al. 1999). The distinguishing feature of LAD is that it searches in an exhaustive manner for all highly relevant terms, and then separately for all reasonable rules formulated with those terms, and applies optimization models to select the best ones. In a typical problem arising from the TREC collection ....

Boros, E., P.L. Hammer, P., Ibaraki, T., Kogan, A., Mayoraz, E., and Muchnik, I. (1999). An implementation of logical analysis of data. IEEE Transactions on Knowledge and Data Engineering. In print.

....606, email: ibaraki kuamp.kyoto u.ac.jp) x Department of Accounting and Information Systems, Faculty of Management, Rutgers University, Newark, NJ 07102, U.S.A. 1 data (LAD) and its applicability to oncology, psychiatry, oil exploration, economic analysis, and other elds are described in [7, 16, 19]. Preliminary results reported in [7] already indicate that the LAD approach not only has high classi cation power for distinguishing between positive and negative examples, but also is quite successful in extracting underlying structural information from the given data, which is useful in ....

....x Department of Accounting and Information Systems, Faculty of Management, Rutgers University, Newark, NJ 07102, U.S.A. 1 data (LAD) and its applicability to oncology, psychiatry, oil exploration, economic analysis, and other elds are described in [7, 16, 19] Preliminary results reported in [7] already indicate that the LAD approach not only has high classi cation power for distinguishing between positive and negative examples, but also is quite successful in extracting underlying structural information from the given data, which is useful in understanding why and how phenomena occur. ....

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E. Boros, P. L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, RUTCOR Research Report RRR 22-96, RUTCOR, Rutgers University, 1996.

.... of Data (LAD) which is a general approach for knowledge discovery and automated learning proposed in the mid eighties [Hammer, 1986] Classi cation is one particular usage of this theory, which was extensively developed and implemented in the mid nineties and which showed great potentialities [Boros et al. 1996]. Thus, besides data compression, there is a need in data binarization in view of mining, where the most relevant information for further processing has to be maintained (Sect. 2) In Sect. 3, the binarization problem is stated and some classical approaches are brie y presented. Section 4 ....

.... If D denotes the initial number of discriminants and if there are n examples in X , the construction of the constraint matrix A is in O(n 2 D) A naive implementation of this greedy heuristic has a complexity in O(n 2 Dd) and has demonstrated its limitations in the experiments reported in [Boros et al. 1996]. A very nice solution proposed in [Almuallim and Dietterich, 1994] denoted Simple Greedy in Sect. 5) consists in resolving this minimum set covering problem using the same greedy heuristic, but without enumerating any column of A. A clever data structure is used that allows to determine the ....

E. Boros, P. L. Hammer, Toshihide Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. RRR 22-96, RUTCOR{Rutgers University's Center For Operations Research, http://rutcor.rutgers.edu:80/~rrr/, July 1996. To appear in IEEE Trans. on Knowledge and Data Engineering.

.... is an important ingredient of many approaches in this field [1, 9, 11, 16, 21, 22] The classification power of the Boolean based methodology of the logical analysis of data (LAD) and its applicability to oncology, psychiatry, oil exploration, economic analysis, and other fields are described in [6, 15, 18]. Preliminary results reported in [6] already indicate that the LAD approach not only has high classification power for distinguishing between positive and negative examples, but also is quite successful in extracting underlying structural information from the given data, which is useful in ....

.... in this field [1, 9, 11, 16, 21, 22] The classification power of the Boolean based methodology of the logical analysis of data (LAD) and its applicability to oncology, psychiatry, oil exploration, economic analysis, and other fields are described in [6, 15, 18] Preliminary results reported in [6] already indicate that the LAD approach not only has high classification power for distinguishing between positive and negative examples, but also is quite successful in extracting underlying structural information from the given data, which is useful in understanding why and how phenomena occur. ....

[Article contains additional citation context not shown here]

E. Boros, P. L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, Technical Report RRR 22-96, RUTCOR, Rutgers University, 1996. RRR 04-97 Page 51

....a set of false points, the central question of logical analysis of data (LAD) is the study of those Boolean functions (called extensions of data sets) whose values agree with those of the given points. The basic concepts of LAD are introduced in [9] and an implementation of LAD is described in [6]. A typical data set will usually have exponentially many extensions. In the absence of any additional information about the properties of the data set, the choice of an extension would be totally arbitrary, and therefore would risk to omit the most significant features of the data set. However, ....

E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An Implementation of Logical Analysis of Data, RUTCOR Research Report 22 - 96, Rutgers University, New Brunswick, NJ, 1996.

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E. Boros, P. L. Hammer, Toshihide Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. RRR 22-96, RUTCOR--Rutgers University's Center For Operations Research, http://rutcor.rutgers.edu:80/~rrr/,Submitted, July 1996.

....25 terms selected from the positive and negative examples are merged, to form a list with no more than 50 terms. The MG retrieval system is used (massively) to transform every judged document into a Boolean vector with one component for each distinct classification term. The RUTCOR LAD program (Boros, Hammer, Ibaraki, Kogan, Mayoraz, Muchnik, 1996) is used (twice for each topic) with several modifications, to search exhaustively for Boolean prime implicants which characterize the positive and the negative examples. Due to computer speed limitations, we have limited the search in our official submissions to terms of order three (i.e terms ....

....routing problem. We suspect that several factors combine to produce this discouraging result. 4. The LAD Approach in the Non Official Runs After we submitted our official runs, we continued our experiments . We have implemented a method based on the Logical Analysis of Data, as it is described in Boros, Hammer, Ibaraki, Kogan, Mayoraz, Muchnik, 1996 ( see for further details, Boros, Hammer, Ibaraki Kogan, 1997, Boros, Ibaraki Makino, 1997, and, Boros, Ibaraki Makino, 1998) with several modifications. In these experiments, the algorithm we implemented consists of 4 phases. The first phase is a more or less standard indexing of the ....

Boros, E, Hammer, P. L., Ibaraki, T., Kogan, A., Mayoraz, E. and Muchnik, I. (1996). An Implementation of Logical Analysis of Data. Rutcor Research Report 22-96 RUTCOR, Rutgers University.

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E. Boros, P. L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz, and I. Muchnik. An implementation of logical analysis of data. Rutcor Research Report RRR 22-96, Rutgers University, New Brunswick, N.J., July 1996.

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T. Ibaraki E. Boros, P. L. Hammer et al., An implementation of logical analysis of data, Tech. report, IDIAP, Martigny, Valais, Switzerland, July 1996.

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Peter L. Hammer et al. An implementation of logical analysis of data. RRR 2296, The State University of New Jersey, Rutgers, July 1996.

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T. Ibaraki E. Boros, P. L. Hammer et al., An implementation of logical analysis of data, Tech. report, IDIAP, Martigny, Valais, Switzerland, July 1996.

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