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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

.... expressed genes, i.e. expression signatures [1] Finally, we try to directly classify di erent types of lymphoma (a multiclass problem) using MLP and parallel non linear dichotomizers (PND) i.e. ensembles of learning machines based on output coding decomposition of a multiclass problem [36]. The paper is structured as follows. In the next section a brief overview of DNA microarray technology is presented. In Sect. 3 we outline the character2 istics of SVM needed to understand their application to the analysis of gene expression data and the basics about output coding decomposition ....

....with discrete outputs, or an inner product or one of the L 1 or L 2 norm distances for dichotomizers with continuous outputs. In particular in our experimentation we use parallel non linear dichotomizers (PND) i.e. OC decomposition ensembles where each dichotomizer is implemented through a MLP [36]. 4 Experimental setup We apply the proposed methods to three classi cation tasks related to the recognition of human lymphoma using DNA microarray gene expression data. 4.1 Data Data used in our experiments are taken from [1] and consist of 96 tissue samples from normal and malignant ....

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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

....single output (MISO) MLP or dichotomic decision trees. We call the resulting learning machines Parallel Linear Dichotomizers (PLD) if the dichotomizers used for implementing the dichotomizers are linear (as in [3] or Parallel Non linear Dichotomizers (PND) if the dichotomizers are non linear [17, 38]. Parallel Non linear Dichotomizers (PND) are multiclassi ers based on the decomposition of polychotomies into dichotomies, using dichotomizers solving their classi cation tasks independently from each other [38] In the decomposition unit each dichotomizer is implemented by a separate ....

.... or Parallel Non linear Dichotomizers (PND) if the dichotomizers are non linear [17, 38] Parallel Non linear Dichotomizers (PND) are multiclassi ers based on the decomposition of polychotomies into dichotomies, using dichotomizers solving their classi cation tasks independently from each other [38]. In the decomposition unit each dichotomizer is implemented by a separate nonlinear learning machine, and learns a di erent and speci c dichotomic task using a training set common to all the dichotomizers. The decision unit can use a L 1 norm or another similarity measure between codewords to ....

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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

....single output (MISO) MLP or dichotomic decision trees. We call the resulting learning machines Parallel Linear Dichotomizers (PLD) if the dichotomizers used for implementing the dichotomizers are linear (as in [3] or Parallel Non linear Dichotomizers (PND) if the dichotomizers are non linear [17, 38]. Parallel Non linear Dichotomizers (PND) are multiclassifiers based on the decomposition of polychotomies into dichotomies, using dichotomizers solving their classification tasks independently from each other [38] In the decomposition unit each dichotomizer is implemented by a separate ....

.... or Parallel Non linear Dichotomizers (PND) if the dichotomizers are non linear [17, 38] Parallel Non linear Dichotomizers (PND) are multiclassifiers based on the decomposition of polychotomies into dichotomies, using dichotomizers solving their classification tasks independently from each other [38]. In the decomposition unit each dichotomizer is implemented by a separate nonlinear learning machine, and learns a di#erent and specific dichotomic task using a training set common to all the dichotomizers. The decision unit can use a L 1 norm or another similarity measure between codewords to ....

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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

....and subsequently the outputs of the L dichotomizers are combined to predict the class label. For this task we used One Per Class Parallel Non linear Dichotomizers (OPC PND) and Error Correcting Output Coding Parallel Non linear Dichotomizers (ECOC PND) ensembles based on output coding methods [9] and a multi class MLP as reference. Each MLP of the ensemble was independently trained to learn each individual bit of the codeword coding the classes. The decision unit was implemented using the L norm distance between the outputs of the decom position unit (that is the vector of the continuous ....

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN000, Int. Joint Conf. on Neural Networks, Como, Italy, 2000.

....decomposition methods [9, Polychotomies are performed recombining parallel independent dichotomizers based upon perceptrons. These At present, some of these classes are not included in the on line distribution of NEURObjects. 8 classifiers are called Parallel Non linear Dichotomizers (PND) [25] and are implemented in specific template classes whose parameter is the type of decomposition [28] Using single layer perceptrons as dichotomizers, we obtain classifiers similar to that proposed by Alpaydin and Mayoraz [1] while using MLP we obtain the PND studied in [25] Performance ....

.... Dichotomizers (PND) 25] and are implemented in specific template classes whose parameter is the type of decomposition [28] Using single layer perceptrons as dichotomizers, we obtain classifiers similar to that proposed by Alpaydin and Mayoraz [1] while using MLP we obtain the PND studied in [25]. Performance statistics classes. Classes handling performance statistics such as accuracy, confidence levels, rejection curves, learning rates and confusion matrices, and classes implementing estimations of dependence among output errors in learning machines, using measures based on mutual ....

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

....of learning machines. 3 Estimating the Dependence between Output Errors in ECOC Learning Machines In this section we analyze the dependence among output errors of monolithic Error correcting Output Coding [4, 8] ECOC monolithic for short) and ECOC Parallel Non linear Dichotomizers [9] (ECOC PND for short) learning machines, using the proposed mutual information based measures. 3.1 The problem ECOC is a two stage classification method, that consists in decomposing a multiclass problem in a number of two class (dichotomic) subproblems and then combining them to achieve the ....

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000.

....MLP and PND CC. The error on the best classifier drops from 25 of a single MLP to 10 . The confusion matrix obtained for the best ECOC classifier (15 bit code) is displayed in Table 3. The error recovering capabilities of ECOC codes [14; 19] and the independence among the dichotomizers [16] explain the good performances of PND ECOC ensembles of learning machines. From Table 3 the results on specific subtasks can be extracted by adding the number of misclassified patterns for every class contained in a certain super class. In Table 4 we present the results obtained for the same ....

F. Masulli, G. Valentini, Parallel Non linear Dichotomizers, in: IJCNN2000.

....the factors a ecting the e ectiveness of ECOC decomposition methods. In that work we outlined that we would expect an higher dependence among codeword bits in monolithic Error Correcting Output Coding [15, 22] ECOC monolithic for short) compared with ECOC Parallel Non linear Dichotomizers (PND) [22, 23, 21] (ECOC PND for short) learning machines, 3 considering that ECOC monolithic share the same hidden layer of a single MLP, while PND dichotomizers, implemented by a separate MLP for each codeword bit, have their own layer of hidden units, specialized for a speci c dichotomic task. An open problem ....

....a ect the dependence among output errors and their performances. We apply measures based on mutual information for comparing dependence among output errors between ECOC learning machines implemented through a single multi layer perceptron (MLP) and an ECOC Parallel Non linear Dichotomizers (PND) [23]. An extensive experimental comparison is performed using synthetic and UCI data sets [26] and a speci c test of hyphotesis [24] is applied for evaluating if a signi cant statistical di erence among output errors between the two ECOC learning machines does exist. The experimental results allow us ....

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F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000, The IEEE-INNS-ENNS International Joint Conference on Neural Networks, volume 2, pages 29-33, Como, Italy, 2000.

....Error Correcting Output Coding (ECOC) methods; Ensembles of learning machines; Multiclass SVM. 2 1 Introduction Output Coding (OC) decomposition methods [21, 2] have been used in last years to improve the performances of di erent classi ers such us decision trees [11] multilayer perceptrons [19], Naive Bayes [6] and k nearest neighbours classi ers [1] They split a complex multiclass problem (polychotomy) into a set of separated two class problems (dichotomies) and then recompose the outputs of the dichotomizers in order to solve the original polychotomy (see [17] for an overview of ....

.... non linear separation of two classes in the input space, working as a linear machine in an high dimensional feature space [28] Moreover they are theoretically well founded, implementing the structural risk minimization inductive principle [28] OC SVM joins the good properties of OC methods [15, 2, 19, 25] with the accuracy of SVM, powering the strong points of both. Dichotomies induced by OC decomposition methods can induce two class classi cation problems with di erent level of complexity: SVM learning algorithms generate dichotomizers that can (semi)automatically adapt to the complexity of the ....

[Article contains additional citation context not shown here]

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000, The IEEE-INNS-ENNS International Joint Conference on Neural Networks, volume 2, pages 29-33, Como, Italy, 2000.

....Moreover, decompositions based on error correcting output codes can sometimes produce very complex dichotomies. For these reasons, in this paper we propose to implement decomposition schemes generated via error correcting output codes using Parallel Non linear Dichotomizers (PND) model [21, 14] that is a learning machine based on decomposition of polychotomies into dichotomies making use of dichotomizers nonlinear and independent on each other. In this way we can combine the error recovering capabilities of ECOC codes with a high accurate dichotomizers. In the next section we introduce ....

....separated and independent dichotomizers. In this case, we call the resulting learning machines Parallel Linear Dichotomizers (PLD) if the dichotomizers used for implementing the dichotomies are linear (as in [1] or Parallel Non linear Dichotomizers (PND ) if the dichotomizers are non linear [21, 14]. Parallel Non linear Dichotomizers (PND) are multiclassi ers based on the decomposition of polychotomies into dichotomies, using dichotomizers solving their classi cation tasks independently from each other [21, 14] Each dichotomizer is implemented by a separate non linear learning machine, ....

[Article contains additional citation context not shown here]

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000, The IEEE-INNS-ENNS International Joint Conference on Neural Networks, Como, Italy. (in press).

.... it through the gamma distribution [29] 16 3 Numerical experiments In this section we exemplify an application of the mutual information based methods described in the previous sections to the evaluation of the dependence among output errors of Error correcting Output Coding monolithic [9, 10, 25, 26] (ECOC monolithic for short) and ECOC Parallel Non linear Dichotomizers [25, 26, 24] ECOC PND for short) learning machines, using a synthetic data set. 3.1 Experimental setup ECOC is a two stage classi cation method, that consists in decomposing a multiclass problem in a number of two class ....

.... section we exemplify an application of the mutual information based methods described in the previous sections to the evaluation of the dependence among output errors of Error correcting Output Coding monolithic [9, 10, 25, 26] ECOC monolithic for short) and ECOC Parallel Non linear Dichotomizers [25, 26, 24] (ECOC PND for short) learning machines, using a synthetic data set. 3.1 Experimental setup ECOC is a two stage classi cation method, that consists in decomposing a multiclass problem in a number of two class (dichotomic) subproblems and then combining them to achieve the class label. Both ....

F. Masulli and G. Valentini. Parallel Non linear Dichotomizers. In IJCNN2000, The IEEE-INNS-ENNS International Joint Conference on Neural Networks, volume 2, pages 29-33, Como, Italy, 2000.

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