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Nikhil Pal

Indian Statistical Institute, Calcutta, ECSU, Faculty Member

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Jadavpur University

NIT Durgapur

University of Texas at San Antonio

Alexis Tsoukiàs

Universidad de Guadalajara

Shubhangi Deshpande

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Papers by Nikhil Pal

Executive Advisory Board

by Nicolaos Karayiannis and Nikhil Pal

ieeexplore.ieee.org

... ANCA RALESCU ECE &amp;amp;amp;amp;amp;amp; CS Dept. Univ. of Cincinnati ML 0030 Cincinnat... more

Robust Consensus: A New Measure for Multicriteria Robust Group Decision Making Problems Using Evolutionary Approach

Lecture Notes in Computer Science, 2014

In fuzzy group decision making problems, we often use multiobjective evolutionary optimization. T... more In fuzzy group decision making problems, we often use multiobjective evolutionary optimization. The optimizers search through the whole search space and provide a set of nondominated solutions. But, sometimes the decision makers express their prior preferences using fuzzy numbers. In this case, the optimizers search in the preferred soft region and provide solutions with higher consensus. If perturbation in the decision variable space is unavoidable, we also need to search for robust solutions. Again, this perturbation affects the degree of consensus of the solutions. This leads to search for solutions those are robust to their degree of consensus. In this work, we address these issues by redefining consensus and proposing a new measure called robust consensus. We also provide a reformulation mechanism for multiobjective optimization problems. Our experimental results show that the proposed method is capable of finding robust solutions having robust consensus in the specified soft region.

Discovering biomarkers from gene expression data for predicting cancer subgroups using neural networks and relational fuzzy clustering

by Nikhil Pal and K. Aguan

BMC bioinformatics, 2007

The four heterogeneous childhood cancers, neuroblastoma, non-Hodgkin lymphoma, rhabdomyosarcoma, ... more The four heterogeneous childhood cancers, neuroblastoma, non-Hodgkin lymphoma, rhabdomyosarcoma, and Ewing sarcoma present a similar histology of small round blue cell tumor (SRBCT) and thus often leads to misdiagnosis. Identification of biomarkers for distinguishing these cancers is a well studied problem. Existing methods typically evaluate each gene separately and do not take into account the nonlinear interaction between genes and the tools that are used to design the diagnostic prediction system. Consequently, more genes are usually identified as necessary for prediction. We propose a general scheme for finding a small set of biomarkers to design a diagnostic system for accurate classification of the cancer subgroups. We use multilayer networks with online gene selection ability and relational fuzzy clustering to identify a small set of biomarkers for accurate classification of the training and blind test cases of a well studied data set. Our method discerned just seven biomark...

A Multiobjective Genetic Programming-Based Ensemble for Simultaneous Feature Selection and Classification

by Kaustuv Nag and Nikhil Pal

IEEE transactions on cybernetics, Jan 6, 2015

We present an integrated algorithm for simultaneous feature selection (FS) and designing of diver... more We present an integrated algorithm for simultaneous feature selection (FS) and designing of diverse classifiers using a steady state multiobjective genetic programming (GP), which minimizes three objectives: 1) false positives (FPs); 2) false negatives (FNs); and 3) the number of leaf nodes in the tree. Our method divides a c-class problem into c binary classification problems. It evolves c sets of genetic programs to create c ensembles. During mutation operation, our method exploits the fitness as well as unfitness of features, which dynamically change with generations with a view to using a set of highly relevant features with low redundancy. The classifiers of $i${th} class determine the {net belongingness} of an unknown data point to the $i${th} class using a weighted voting scheme, which makes use of the FP and FN mistakes made on the training data. We test our method on eight microarray and 11 text data sets with diverse number of classes (from 2 to 44), large number of featur...

A Review on Image Segmentation Techniques

Pattern recognition, Jan 1, 1993

Many image segmentation techniques are available in the literature. Some of these techniques use ... more

On cluster validity for the fuzzy c-means model

Fuzzy Systems, IEEE Transactions on, Jan 1, 1995

Some new indexes of cluster validity

Systems, Man, and Cybernetics, Part B: …, Jan 1, 1998

We review two clustering algorithms (hard c-means and single linkage) and three indexes of crisp ... more We review two clustering algorithms (hard c-means and single linkage) and three indexes of crisp cluster validity (Hubert's statistics, the Davies-Bouldin index, and Dunn's index). We illustrate two deficiencies of Dunn's index which make it overly sensitive to noisy clusters and propose several generalizations of it that are not as brittle to outliers in the clusters. Our numerical examples show that the standard measure of interset distance (the minimum distance between points in a pair of sets) is the worst (least reliable) measure upon which to base cluster validation indexes when the clusters are expected to form volumetric clouds. Experimental results also suggest that intercluster separation plays a more important role in cluster validation than cluster diameter. Our simulations show that while Dunn's original index has operational flaws, the concept it embodies provides a rich paradigm for validation of partitions that have cloud-like clusters.

Generalized clustering networks and Kohonen's self-organizing scheme

Neural Networks, IEEE …, Jan 1, 1993

Fuzzy Kohonen clustering networks

Pattern recognition, Jan 1, 1994

Abstraet--Kohonen networks are well known for cluster analysis (unsupervised learning). This clas... more Abstraet--Kohonen networks are well known for cluster analysis (unsupervised learning). This class of algorithms is a set of heuristic procedures that suffers from several major problems (e.g. neither termination or convergence is guaranteed, no model is optimized by the learning strategy, and the output is often dependent on the sequence of data). A fuzzy Kohonen clustering network is proposed which integrates the Fuzzy c-Means (FCM) model into the learning rate and updating strategies of the Kohonen network. This yield an optimization problem related to FCM, and the numerical results show improved convergence as well as reduced labeling errors. It is proved that the proposed scheme is equivalent to the c-Means algorithms. The new method can be viewed as a Kohonen type of FCM, but is "self-organizing" since the "size" of the update neighborhood and learning rate in the competitive layer are automatically adjusted during learning. Anderson's IRIS data is used to illustrate this method; and results are compared with the standard Kohonen approach.

A robust self-tuning scheme for PI-and PD-type fuzzy controllers

Fuzzy Systems, IEEE Transactions on, Jan 1, 1999

... results for some typical second-order linear as well as nonlinear processes using both of ...... more

A possibilistic fuzzy c-means clustering algorithm

Fuzzy Systems, IEEE …, Jan 1, 2005

In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated bo... more In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated both membership and typicality values when clustering unlabeled data. FPCM constrains the typicality values so that the sum over all data points of typicalities to a cluster is one. The row sum constraint produces unrealistic typicality values for large data sets. In this paper, we propose a new model called possibilistic-fuzzy c-means (PFCM) model. PFCM produces memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. PFCM is a hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) that often avoids various problems of PCM, FCM and FPCM. PFCM solves the noise sensitivity defect of FCM, overcomes the coincident clusters problem of PCM and eliminates the row sum constraints of FPCM. We derive the first-order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima of the PFCM objective functional. Several numerical examples are given that compare FCM and PCM to PFCM. Our examples show that PFCM compares favorably to both of the previous models. Since PFCM prototypes are less sensitive to outliers and can avoid coincident clusters, PFCM is a strong candidate for fuzzy rule-based system identification.

Entropic thresholding

Signal processing, Jan 1, 1989

... 5(a) and 5(b) represent the input image of Abraham Lincoln and its gray level histogram, resp... more ... 5(a) and 5(b) represent the input image of Abraham Lincoln and its gray level histogram, respectively. ... [3] JN Kapur, PK Sahoo and AKC Wong, &amp;quot;A new method for grey-level picturethresholding using the entropy of the histogram&amp;quot;, Comp. Graphics, Vision and Image Proc. ...

Entropy: a new definition and its applications

… , Man and Cybernetics, IEEE Transactions on, Jan 1, 1991

Shannon's definition of entropy is critically examined and a new definition of classical entropy ... more Shannon's definition of entropy is critically examined and a new definition of classical entropy based on the exponential behavior of information-gain is proposed along with its justification. The concept is then extended to gray tone image for defining its global, local and conditional entropy. Based on these definitions four algorithms for object extraction are developed and implemented. One of these algorithms uses a Poisson distribution-based model of an ideal image. Finally, a concept of positional entropy giving an information regarding the location of an object in a scene is introduced.

A mixed c-means clustering model

Fuzzy Systems, 1997., …, Jan 1, 1997

A Mixed c-Means Clustering Model Nikhil R. Pal and Kuhu Pal Machine Intelligence Unlit Indian Sta... more

Executive Advisory Board

by Nicolaos Karayiannis and Nikhil Pal

ieeexplore.ieee.org

... ANCA RALESCU ECE &amp;amp;amp;amp;amp;amp; CS Dept. Univ. of Cincinnati ML 0030 Cincinnat... more

Robust Consensus: A New Measure for Multicriteria Robust Group Decision Making Problems Using Evolutionary Approach

Lecture Notes in Computer Science, 2014

In fuzzy group decision making problems, we often use multiobjective evolutionary optimization. T... more In fuzzy group decision making problems, we often use multiobjective evolutionary optimization. The optimizers search through the whole search space and provide a set of nondominated solutions. But, sometimes the decision makers express their prior preferences using fuzzy numbers. In this case, the optimizers search in the preferred soft region and provide solutions with higher consensus. If perturbation in the decision variable space is unavoidable, we also need to search for robust solutions. Again, this perturbation affects the degree of consensus of the solutions. This leads to search for solutions those are robust to their degree of consensus. In this work, we address these issues by redefining consensus and proposing a new measure called robust consensus. We also provide a reformulation mechanism for multiobjective optimization problems. Our experimental results show that the proposed method is capable of finding robust solutions having robust consensus in the specified soft region.

Discovering biomarkers from gene expression data for predicting cancer subgroups using neural networks and relational fuzzy clustering

by Nikhil Pal and K. Aguan

BMC bioinformatics, 2007

The four heterogeneous childhood cancers, neuroblastoma, non-Hodgkin lymphoma, rhabdomyosarcoma, ... more The four heterogeneous childhood cancers, neuroblastoma, non-Hodgkin lymphoma, rhabdomyosarcoma, and Ewing sarcoma present a similar histology of small round blue cell tumor (SRBCT) and thus often leads to misdiagnosis. Identification of biomarkers for distinguishing these cancers is a well studied problem. Existing methods typically evaluate each gene separately and do not take into account the nonlinear interaction between genes and the tools that are used to design the diagnostic prediction system. Consequently, more genes are usually identified as necessary for prediction. We propose a general scheme for finding a small set of biomarkers to design a diagnostic system for accurate classification of the cancer subgroups. We use multilayer networks with online gene selection ability and relational fuzzy clustering to identify a small set of biomarkers for accurate classification of the training and blind test cases of a well studied data set. Our method discerned just seven biomark...

A Multiobjective Genetic Programming-Based Ensemble for Simultaneous Feature Selection and Classification

by Kaustuv Nag and Nikhil Pal

IEEE transactions on cybernetics, Jan 6, 2015

We present an integrated algorithm for simultaneous feature selection (FS) and designing of diver... more We present an integrated algorithm for simultaneous feature selection (FS) and designing of diverse classifiers using a steady state multiobjective genetic programming (GP), which minimizes three objectives: 1) false positives (FPs); 2) false negatives (FNs); and 3) the number of leaf nodes in the tree. Our method divides a c-class problem into c binary classification problems. It evolves c sets of genetic programs to create c ensembles. During mutation operation, our method exploits the fitness as well as unfitness of features, which dynamically change with generations with a view to using a set of highly relevant features with low redundancy. The classifiers of $i${th} class determine the {net belongingness} of an unknown data point to the $i${th} class using a weighted voting scheme, which makes use of the FP and FN mistakes made on the training data. We test our method on eight microarray and 11 text data sets with diverse number of classes (from 2 to 44), large number of featur...

A Review on Image Segmentation Techniques

Pattern recognition, Jan 1, 1993

Many image segmentation techniques are available in the literature. Some of these techniques use ... more

On cluster validity for the fuzzy c-means model

Fuzzy Systems, IEEE Transactions on, Jan 1, 1995

Some new indexes of cluster validity

Systems, Man, and Cybernetics, Part B: …, Jan 1, 1998

We review two clustering algorithms (hard c-means and single linkage) and three indexes of crisp ... more We review two clustering algorithms (hard c-means and single linkage) and three indexes of crisp cluster validity (Hubert's statistics, the Davies-Bouldin index, and Dunn's index). We illustrate two deficiencies of Dunn's index which make it overly sensitive to noisy clusters and propose several generalizations of it that are not as brittle to outliers in the clusters. Our numerical examples show that the standard measure of interset distance (the minimum distance between points in a pair of sets) is the worst (least reliable) measure upon which to base cluster validation indexes when the clusters are expected to form volumetric clouds. Experimental results also suggest that intercluster separation plays a more important role in cluster validation than cluster diameter. Our simulations show that while Dunn's original index has operational flaws, the concept it embodies provides a rich paradigm for validation of partitions that have cloud-like clusters.

Generalized clustering networks and Kohonen's self-organizing scheme

Neural Networks, IEEE …, Jan 1, 1993

Fuzzy Kohonen clustering networks

Pattern recognition, Jan 1, 1994

Abstraet--Kohonen networks are well known for cluster analysis (unsupervised learning). This clas... more Abstraet--Kohonen networks are well known for cluster analysis (unsupervised learning). This class of algorithms is a set of heuristic procedures that suffers from several major problems (e.g. neither termination or convergence is guaranteed, no model is optimized by the learning strategy, and the output is often dependent on the sequence of data). A fuzzy Kohonen clustering network is proposed which integrates the Fuzzy c-Means (FCM) model into the learning rate and updating strategies of the Kohonen network. This yield an optimization problem related to FCM, and the numerical results show improved convergence as well as reduced labeling errors. It is proved that the proposed scheme is equivalent to the c-Means algorithms. The new method can be viewed as a Kohonen type of FCM, but is "self-organizing" since the "size" of the update neighborhood and learning rate in the competitive layer are automatically adjusted during learning. Anderson's IRIS data is used to illustrate this method; and results are compared with the standard Kohonen approach.

A robust self-tuning scheme for PI-and PD-type fuzzy controllers

Fuzzy Systems, IEEE Transactions on, Jan 1, 1999

... results for some typical second-order linear as well as nonlinear processes using both of ...... more

A possibilistic fuzzy c-means clustering algorithm

Fuzzy Systems, IEEE …, Jan 1, 2005

In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated bo... more In 1997, we proposed the fuzzy-possibilistic c-means (FPCM) model and algorithm that generated both membership and typicality values when clustering unlabeled data. FPCM constrains the typicality values so that the sum over all data points of typicalities to a cluster is one. The row sum constraint produces unrealistic typicality values for large data sets. In this paper, we propose a new model called possibilistic-fuzzy c-means (PFCM) model. PFCM produces memberships and possibilities simultaneously, along with the usual point prototypes or cluster centers for each cluster. PFCM is a hybridization of possibilistic c-means (PCM) and fuzzy c-means (FCM) that often avoids various problems of PCM, FCM and FPCM. PFCM solves the noise sensitivity defect of FCM, overcomes the coincident clusters problem of PCM and eliminates the row sum constraints of FPCM. We derive the first-order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima of the PFCM objective functional. Several numerical examples are given that compare FCM and PCM to PFCM. Our examples show that PFCM compares favorably to both of the previous models. Since PFCM prototypes are less sensitive to outliers and can avoid coincident clusters, PFCM is a strong candidate for fuzzy rule-based system identification.

Entropic thresholding

Signal processing, Jan 1, 1989

... 5(a) and 5(b) represent the input image of Abraham Lincoln and its gray level histogram, resp... more ... 5(a) and 5(b) represent the input image of Abraham Lincoln and its gray level histogram, respectively. ... [3] JN Kapur, PK Sahoo and AKC Wong, &amp;quot;A new method for grey-level picturethresholding using the entropy of the histogram&amp;quot;, Comp. Graphics, Vision and Image Proc. ...

Entropy: a new definition and its applications

… , Man and Cybernetics, IEEE Transactions on, Jan 1, 1991

Shannon's definition of entropy is critically examined and a new definition of classical entropy ... more Shannon's definition of entropy is critically examined and a new definition of classical entropy based on the exponential behavior of information-gain is proposed along with its justification. The concept is then extended to gray tone image for defining its global, local and conditional entropy. Based on these definitions four algorithms for object extraction are developed and implemented. One of these algorithms uses a Poisson distribution-based model of an ideal image. Finally, a concept of positional entropy giving an information regarding the location of an object in a scene is introduced.

A mixed c-means clustering model

Fuzzy Systems, 1997., …, Jan 1, 1997

A Mixed c-Means Clustering Model Nikhil R. Pal and Kuhu Pal Machine Intelligence Unlit Indian Sta... more