Unsupervised and supervised data classification via nonsmooth and global optimisation
- Bagirov, Adil, Rubinov, Alex, Sukhorukova, Nadezda, Yearwood, John
- Authors: Bagirov, Adil , Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Journal article
- Relation: Top Vol. 11, no. 1 (2003), p. 1-92
- Full Text:
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- Description: We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss results of numerical experiments.
- Description: C1
- Description: 2003000421
- Authors: Bagirov, Adil , Rubinov, Alex , Sukhorukova, Nadezda , Yearwood, John
- Date: 2003
- Type: Text , Journal article
- Relation: Top Vol. 11, no. 1 (2003), p. 1-92
- Full Text:
- Reviewed:
- Description: We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss results of numerical experiments.
- Description: C1
- Description: 2003000421
Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails
- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964
- Full Text:
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- Description: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964
- Full Text:
- Reviewed:
- Description: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.
Multivariate approximation by polynomial and generalized rational functions
- Díaz Millán, Reinier, Peiris, Vinesha, Sukhorukova, Nadezda, Ugon, Julien
- Authors: Díaz Millán, Reinier , Peiris, Vinesha , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2022
- Type: Text , Journal article
- Relation: Optimization Vol. 71, no. 4 (2022), p. 1171-1187
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
- Authors: Díaz Millán, Reinier , Peiris, Vinesha , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2022
- Type: Text , Journal article
- Relation: Optimization Vol. 71, no. 4 (2022), p. 1171-1187
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
Detecting K-complexes for sleep stage identification using nonsmooth optimization
- Moloney, David, Sukhorukova, Nadezda, Vamplew, Peter, Ugon, Julien, Li, Gang, Beliakov, Gleb, Philippe, Carole, Amiel, Hélène, Ugon, Adrien
- Authors: Moloney, David , Sukhorukova, Nadezda , Vamplew, Peter , Ugon, Julien , Li, Gang , Beliakov, Gleb , Philippe, Carole , Amiel, Hélène , Ugon, Adrien
- Date: 2012
- Type: Text , Journal article
- Relation: ANZIAM Journal Vol. 52, no. 4 (2012), p. 319-332
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- Description: The process of sleep stage identification is a labour-intensive task that involves the specialized interpretation of the polysomnographic signals captured from a patient's overnight sleep session. Automating this task has proven to be challenging for data mining algorithms because of noise, complexity and the extreme size of data. In this paper we apply nonsmooth optimization to extract key features that lead to better accuracy. We develop a specific procedure for identifying K-complexes, a special type of brain wave crucial for distinguishing sleep stages. The procedure contains two steps. We first extract "easily classified" K-complexes, and then apply nonsmooth optimization methods to extract features from the remaining data and refine the results from the first step. Numerical experiments show that this procedure is efficient for detecting K-complexes. It is also found that most classification methods perform significantly better on the extracted features. © 2012 Australian Mathematical Society.
- Authors: Moloney, David , Sukhorukova, Nadezda , Vamplew, Peter , Ugon, Julien , Li, Gang , Beliakov, Gleb , Philippe, Carole , Amiel, Hélène , Ugon, Adrien
- Date: 2012
- Type: Text , Journal article
- Relation: ANZIAM Journal Vol. 52, no. 4 (2012), p. 319-332
- Full Text:
- Reviewed:
- Description: The process of sleep stage identification is a labour-intensive task that involves the specialized interpretation of the polysomnographic signals captured from a patient's overnight sleep session. Automating this task has proven to be challenging for data mining algorithms because of noise, complexity and the extreme size of data. In this paper we apply nonsmooth optimization to extract key features that lead to better accuracy. We develop a specific procedure for identifying K-complexes, a special type of brain wave crucial for distinguishing sleep stages. The procedure contains two steps. We first extract "easily classified" K-complexes, and then apply nonsmooth optimization methods to extract features from the remaining data and refine the results from the first step. Numerical experiments show that this procedure is efficient for detecting K-complexes. It is also found that most classification methods perform significantly better on the extracted features. © 2012 Australian Mathematical Society.
Classes and clusters in data analysis
- Rubinov, Alex, Sukhorukova, Nadezda, Ugon, Julien
- Authors: Rubinov, Alex , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2006
- Type: Text , Journal article
- Relation: European Journal of Operational Research Vol. 173, no. 3 (Sep 2006), p. 849-865
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- Description: We discuss the relation between classes and clusters in datasets with given classes. We examine the distribution of classes within obtained clusters, using different clustering methods which are based on different techniques. We also study the structure of the obtained clusters. One of the main conclusions, obtained in this research is that the notion purity cannot be always used for evaluation of accuracy of clustering techniques. (c) 2005 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003001593
- Authors: Rubinov, Alex , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2006
- Type: Text , Journal article
- Relation: European Journal of Operational Research Vol. 173, no. 3 (Sep 2006), p. 849-865
- Full Text:
- Reviewed:
- Description: We discuss the relation between classes and clusters in datasets with given classes. We examine the distribution of classes within obtained clusters, using different clustering methods which are based on different techniques. We also study the structure of the obtained clusters. One of the main conclusions, obtained in this research is that the notion purity cannot be always used for evaluation of accuracy of clustering techniques. (c) 2005 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003001593
The choice of a similarity measure with respect to its sensitivity to outliers
- Rubinov, Alex, Sukhorukova, Nadezda, Ugon, Julien
- Authors: Rubinov, Alex , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2010
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 709-721
- Full Text:
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- Description: This paper examines differences in the choice of similarity measures with respect to their sensitivity to outliers in clustering problems, formulated as mathematical programming problems. Namely, we are focusing on the study of norms (norm-based similarity measures) and convex functions of norms (function-norm-based similarity measures). The study consists of two parts: the study of theoretical models and numerical experiments. The main result of this study is a criterion for the outliers sensitivity with respect to the corresponding similarity measure. In particular, the obtained results show that the norm-based similarity measures are not sensitive to outliers whilst a very widely used square of the Euclidean norm similarity measure (least squares) is sensitive to outliers. Copyright © 2010 Watam Press.
- Authors: Rubinov, Alex , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2010
- Type: Text , Journal article
- Relation: Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms Vol. 17, no. 5 (2010), p. 709-721
- Full Text:
- Reviewed:
- Description: This paper examines differences in the choice of similarity measures with respect to their sensitivity to outliers in clustering problems, formulated as mathematical programming problems. Namely, we are focusing on the study of norms (norm-based similarity measures) and convex functions of norms (function-norm-based similarity measures). The study consists of two parts: the study of theoretical models and numerical experiments. The main result of this study is a criterion for the outliers sensitivity with respect to the corresponding similarity measure. In particular, the obtained results show that the norm-based similarity measures are not sensitive to outliers whilst a very widely used square of the Euclidean norm similarity measure (least squares) is sensitive to outliers. Copyright © 2010 Watam Press.
New algorithm to find a shape of a finite set of points
- Sukhorukova, Nadezda, Ugon, Julien
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2003
- Type: Text , Conference paper
- Relation: Paper presented at the Symposium on Industrial Optimisation and the 9th Australian Optimisation Day, Perth : 30th September, 2002
- Full Text:
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- Description: Very often in data classification problems we have to determine a shape of a finite set of points within datasets. One of the most common approaches to represent such sets is to determine them as collections of several groups of points. The goal of this project is to develop some algorithms to find a shape for each group. Numerical experiments using the Discrete Gradient method have been done. The results are presented.
- Description: E1
- Description: 2003000351
- Authors: Sukhorukova, Nadezda , Ugon, Julien
- Date: 2003
- Type: Text , Conference paper
- Relation: Paper presented at the Symposium on Industrial Optimisation and the 9th Australian Optimisation Day, Perth : 30th September, 2002
- Full Text:
- Reviewed:
- Description: Very often in data classification problems we have to determine a shape of a finite set of points within datasets. One of the most common approaches to represent such sets is to determine them as collections of several groups of points. The goal of this project is to develop some algorithms to find a shape for each group. Numerical experiments using the Discrete Gradient method have been done. The results are presented.
- Description: E1
- Description: 2003000351
Chebyshev multivariate polynomial approximation and point reduction procedure
- Sukhorukova, Nadezda, Ugon, Julien, Yost, David
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2021
- Type: Text , Journal article
- Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
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- Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2021
- Type: Text , Journal article
- Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Chebyshev multivariate polynomial approximation : alternance interpretation
- Sukhorukova, Nadezda, Ugon, Julien, Yost, David
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Book chapter
- Relation: 2016 Matrix Annals p. 177-182
- Full Text:
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- Description: In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Book chapter
- Relation: 2016 Matrix Annals p. 177-182
- Full Text:
- Reviewed:
- Description: In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.
Two curve Chebyshev approximation and its application to signal clustering
- Authors: Sukhorukova, Nadezda
- Date: 2019
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 356, no. (2019), p. 42-49
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved. In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved.
- Authors: Sukhorukova, Nadezda
- Date: 2019
- Type: Text , Journal article
- Relation: Applied Mathematics and Computation Vol. 356, no. (2019), p. 42-49
- Relation: http://purl.org/au-research/grants/arc/DP180100602
- Full Text:
- Reviewed:
- Description: In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved. In this paper, we extend a number of important results of the classical Chebyshev approximation theory to the case of simultaneous approximation of two or more functions. The need for this extension is application driven, since such kind of problems appears in the area of curve (signal) clustering. In this paper, we propose a new efficient algorithm for signal clustering and develop a procedure that allows one to reuse the results obtained at the previous iteration without recomputing the cluster centres from scratch. This approach is based on the extension of the classical de la Vallee-Poussin procedure originally developed for polynomial approximation. We also develop necessary and sufficient optimality conditions for two curve Chebyshev approximation, which is our core tool for curve clustering. These results are based on application of nonsmooth convex analysis. (C) 2019 Elsevier Inc. All rights reserved.
Automatic sleep stage identification: difficulties and possible solutions
- Sukhorukova, Nadezda, Stranieri, Andrew, Ofoghi, Bahadorreza, Vamplew, Peter, Saleem, Muhammad Saad, Ma, Liping, Ugon, Adrien, Ugon, Julien, Muecke, Nial, Amiel, Hélène, Philippe, Carole, Bani-Mustafa, Ahmed, Huda, Shamsul, Bertoli, Marcello, Levy, P, Ganascia, J.G
- Authors: Sukhorukova, Nadezda , Stranieri, Andrew , Ofoghi, Bahadorreza , Vamplew, Peter , Saleem, Muhammad Saad , Ma, Liping , Ugon, Adrien , Ugon, Julien , Muecke, Nial , Amiel, Hélène , Philippe, Carole , Bani-Mustafa, Ahmed , Huda, Shamsul , Bertoli, Marcello , Levy, P , Ganascia, J.G
- Date: 2010
- Type: Text , Conference proceedings
- Full Text:
- Description: The diagnosis of many sleep disorders is a labour intensive task that involves the specialised interpretation of numerous signals including brain wave, breath and heart rate captured in overnight polysomnogram sessions. The automation of diagnoses is challenging for data mining algorithms because the data sets are extremely large and noisy, the signals are complex and specialist's analyses vary. This work reports on the adaptation of approaches from four fields; neural networks, mathematical optimisation, financial forecasting and frequency domain analysis to the problem of automatically determing a patient's stage of sleep. Results, though preliminary, are promising and indicate that combined approaches may prove more fruitful than the reliance on a approach.
- Authors: Sukhorukova, Nadezda , Stranieri, Andrew , Ofoghi, Bahadorreza , Vamplew, Peter , Saleem, Muhammad Saad , Ma, Liping , Ugon, Adrien , Ugon, Julien , Muecke, Nial , Amiel, Hélène , Philippe, Carole , Bani-Mustafa, Ahmed , Huda, Shamsul , Bertoli, Marcello , Levy, P , Ganascia, J.G
- Date: 2010
- Type: Text , Conference proceedings
- Full Text:
- Description: The diagnosis of many sleep disorders is a labour intensive task that involves the specialised interpretation of numerous signals including brain wave, breath and heart rate captured in overnight polysomnogram sessions. The automation of diagnoses is challenging for data mining algorithms because the data sets are extremely large and noisy, the signals are complex and specialist's analyses vary. This work reports on the adaptation of approaches from four fields; neural networks, mathematical optimisation, financial forecasting and frequency domain analysis to the problem of automatically determing a patient's stage of sleep. Results, though preliminary, are promising and indicate that combined approaches may prove more fruitful than the reliance on a approach.
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