Francisco J Prieto | Universidad Carlos III de Madrid (original) (raw)
Papers by Francisco J Prieto
Computers & Industrial Engineering, 2021
Support Vector Machines (SVMs), originally proposed for classifications of two classes, have beco... more Support Vector Machines (SVMs), originally proposed for classifications of two classes, have become a very popular technique in the machine learning field. For multi-class classifications, various single-objective models and multi-objective ones have been proposed. However, most of the singleobjective models consider neither the different costs of different misclassifications nor the users' preferences. Although multi-objective models have taken this drawback into account, they result in large and hard second-order cone programs (SOCPs), from which we get weakly Pareto-optimal solutions. In this paper, we propose a Projected Multi-objective SVM (PM), which is a multiobjective technique that works in a higher dimensional space than the object space. For PM, we can characterize the associated Pareto-optimal solutions. Additionally, it significantly alleviates the computational bottlenecks for classifications with large numbers of classes. From our experimental results, we can see PM outperforms the single-objective multi-class SVMs (based on an all-together method, one-against-all method and one-against-one method) and other multiobjective SVMs. Compared to the single-objective multi-class SVMs, PM provides a wider set of options designed for different misclassification, without sacrificing training time. Compared to other multi-objective methods, PM promises the out-of-sample quality of the approximation of the Pareto frontier, with a considerable reduction of the computational burden.
Networks and Spatial Economics, 2018
The ongoing transformations of power systems worldwide pose important challenges, both economic a... more The ongoing transformations of power systems worldwide pose important challenges, both economic and technical, for their appropriate planning and operation. A key approach to improve the efficiency of these systems is through demand-side management, i.e., to promote the active involvement of consumers in the system. In particular, the current trend is to conceive systems where electricity consumers can vary their load according to real-time price incentives, offered by retailing companies. Under this setting, retail competition plays an important role as inadequate prices or services may entail consumers switching to a rival retailer. In this work we consider a game theoretical model where asymmetric retailers compete in prices to increase their profits by accounting for the utility function of consumers. Consumer preferences for retailers are uncertain and distributed within a Hotelling line. We analytically characterize the equilibrium of a retailer duopoly, establishing its existence and uniqueness conditions for a wide class of utility functions. Furthermore, sensitivities of the equilibrium prices with respect to relevant model parameters C. Ruiz
Journal of Optimization Theory and Applications, 2017
In this work, we describe the efficient use of improved directions of negative curvature for the ... more In this work, we describe the efficient use of improved directions of negative curvature for the solution of bound-constrained nonconvex problems. We follow an interior-point framework, in which the key point is the inclusion of computational low-cost procedures to improve directions of negative curvature obtained from a factorisation of the KKT matrix. From a theoretical point of view, it is well known that these directions ensure convergence to second-order KKT points. As a novelty, we consider the convergence rate of the algorithm with exploitation of negative curvature information. Finally, we test the performance of our proposal on both CUTEr/st and simulated problems, showing empirically that the enhanced directions affect positively the practical performance of the procedure.
In this note we analyze the relationship between the direction obtained from the minimization of ... more In this note we analyze the relationship between the direction obtained from the minimization of the kurtosis coefficient of the projections of a mixture of multivariate normal distributions and the linear discriminant function. We show that both directions are closely related, and in particular that given two vector random variables having symmetric distributions with unknown means and the same covariance matrix, the direction which minimizes the kurtosis coefficient of the projection is the linear discriminant function. This result provides a way to compute the discriminant function between two normal populations in the case in which the means and the common covariance matrix are unknown.
Journal of Statistical Physics, 2015
Support Vector Machines (SVMs) have become a very popular technique in the machinelearning field ... more Support Vector Machines (SVMs) have become a very popular technique in the machinelearning field for classification problems. It was originally proposed for classification of two classes. Various multiclass models with a single objective have been proposed mostly based on two families of methods: an all-together approach and a one-against-all approach. However, most of these single-objective models consider neither the different costs of misclassification nor the user's preferences. To overcome these drawbacks, multiobjective models have been proposed. In this paper we rewrite the different approaches that deal with the multiclass SVM using multiobjective techniques. These multiobjective techniques can give us weakly Pareto-optimal solutions. We propose a multiobjective technique called Projected Multiobjective All-Together (PMAT), which works in a higher-dimension space than the object space. With this technique, we can theoretically characterize the Pareto-optimal solution set. For these multiobjective techniques we get approximate sets of the Pareto-optimal solutions. For these sets, we use hypervolume and epsilon indicators to evaluate different multiobjective techniques. From the experimental results, we can see that (PMAT) outperfoms the other multiobjective techniques. When facing classification problems with very large numbers of classes, we suggest combining a tree method and multiobjective techniques.
2003 IEEE Power Engineering Society General Meeting (IEEE Cat. No.03CH37491)
This work describes a procedure that determines the optimal allocation for the yearly energy resu... more This work describes a procedure that determines the optimal allocation for the yearly energy resulting from random water inflows to the different subperiods of a year so that the expected benefits are maximized. Its main idea is to distribute the energy stored in reservoirs in each period into two parts: one is directly sold in the energy market, while the
Statistics & Probability Letters, 2000
In this note we analyze the relationship between the direction obtained from the minimization of ... more In this note we analyze the relationship between the direction obtained from the minimization of the kurtosis coe cient of the projections of a mixture of multivariate normal distributions and the linear discriminant function. We show that both directions are closely related and, in particular, that given two vector random variables having symmetric distributions with unknown means and the same covariance matrix, the direction which minimizes the kurtosis coe cient of the projection is the linear discriminant function. This result provides a way to compute the discriminant function between two normal populations in the case in which means and common covariance matrix are unknown.
Technometrics, 2001
In this article, we present a simple multivariate outlier-detection procedure and a robust estima... more In this article, we present a simple multivariate outlier-detection procedure and a robust estimator for the covariance matrix, based on the use of information obtained from projections onto the directions that maximize and minimize the kurtosis coef cient of the projected data. The properties of this estimator (computational cost, bias) are analyzed and compared with those of other robust estimators described in the literature through simulation studies. The performance of the outlier-detection procedure is analyzed by applying it to a set of well-known examples.
Journal of Multivariate Analysis, 2010
In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as intere... more In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher's linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.
Journal of Computational and Graphical Statistics, 2007
A powerful procedure for outlier detection and robust estimation of shape and location with multi... more A powerful procedure for outlier detection and robust estimation of shape and location with multivariate data in high dimension is proposed. The procedure searches for outliers in univariate projections on directions that are obtained both randomly, as in the Stahel-Donoho method, and by maximizing and minimizing the kurtosis coefficient of the projected data, as in the Peña and Prieto method. We propose modifications of both methods to improve their computational efficiency and combine them in a procedure which is affine equivariant, has a high breakdown point, is fast to compute and can be applied when the dimension is large. Its performance is illustrated with a Monte Carlo experiment and in a real dataset.
Siam J Optimization, 1995
The procedures for the identification of outlier observations that are most reliable are based on... more The procedures for the identification of outlier observations that are most reliable are based on the use of a robustified Mahalanobis distance, and have a very high computational cost even for small size problems. All these procedures present difficulties when applied to the identification of point-mass contaminations, where the outIiers are grouped into one or more clusters, separated from the sample. In this work a specific method for this contamination pattern is described, and shown to be able to handle successfully those cases where methods based on robust estimators (the Minimum Volume ElIipsiod estimator or the Stahel-Donoho estimator) fail. The method is simple, exploratory in nature, and straightforward to apply using any standard statistical software package.
In this report a new decomposition methodology for optimization problems is presented. The propos... more In this report a new decomposition methodology for optimization problems is presented. The proposed procedure is general, simple and efficient. It avoids most disadvantages of other common decomposition techniques, such as Lagrangian Relaxation or Augmented Lagrangian Relaxation. The new methodology is applied to a problem coming from interconnected power systems. The application of the new method to this problem allows the computation of an optimal coordinated but decentralized solution. Local and global convergence properties of the proposed decomposition algorithm are described. Numerical results show that the new decentralized methodology has a lower computational cost than other decomposition techniques, and in large-scale cases even lower than a centralized approach.
Although the Stahel-Donoho estimator with resampling seems to o¤er an excellent compro-mise betwe... more Although the Stahel-Donoho estimator with resampling seems to o¤er an excellent compro-mise between computational cost and bias and variability properties among high breakdown point scale estimators, it still presents signi…cantly high bias when the outliers are close to the uncontaminated sample. In this paper, we analyze di¤erent possible causes for these high values, and o¤er two pro-cedures, based on the use of alternative measures of scale for the projections de…ning the weights for the observations, that partially o¤set these undesirable e¤ects.
Studies in Classification, Data Analysis, and Knowledge Organization, 2006
ABSTRACT A projection method for robust estimation of shape and location in multivariate data and... more ABSTRACT A projection method for robust estimation of shape and location in multivariate data and cluster analysis is presented. The key idea of the procedure is to search for heterogeneity in univariate projections on directions that are obtained both randomly, using a modification of the Stahel-Donoho procedure, and by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto (2005). We show in a Monte Carlo study that the resulting procedure works well for robust estimation. Also, it preserves the good theoretical properties of the Stahel-Donoho method.
We present an e-cient implementation of an interior-point algorithm for non-convex bound constrai... more We present an e-cient implementation of an interior-point algorithm for non-convex bound constrained problems that uses good directions of negative curvature. These directions should improve the computational e-ciency of the procedure and ensure convergence to second-order KKT points. We analyze the practical behavior of the procedure and present two sets of numerical experiments to check the relevance of the algorithm.
Journal of Statistical Physics, 2014
We investigate a class of models related to the Bak-Sneppen model, initially proposed to study ev... more We investigate a class of models related to the Bak-Sneppen model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of "complex behavior" such as self-organized criticality that is often observed in physical and biological systems.
Computers & Industrial Engineering, 2021
Support Vector Machines (SVMs), originally proposed for classifications of two classes, have beco... more Support Vector Machines (SVMs), originally proposed for classifications of two classes, have become a very popular technique in the machine learning field. For multi-class classifications, various single-objective models and multi-objective ones have been proposed. However, most of the singleobjective models consider neither the different costs of different misclassifications nor the users' preferences. Although multi-objective models have taken this drawback into account, they result in large and hard second-order cone programs (SOCPs), from which we get weakly Pareto-optimal solutions. In this paper, we propose a Projected Multi-objective SVM (PM), which is a multiobjective technique that works in a higher dimensional space than the object space. For PM, we can characterize the associated Pareto-optimal solutions. Additionally, it significantly alleviates the computational bottlenecks for classifications with large numbers of classes. From our experimental results, we can see PM outperforms the single-objective multi-class SVMs (based on an all-together method, one-against-all method and one-against-one method) and other multiobjective SVMs. Compared to the single-objective multi-class SVMs, PM provides a wider set of options designed for different misclassification, without sacrificing training time. Compared to other multi-objective methods, PM promises the out-of-sample quality of the approximation of the Pareto frontier, with a considerable reduction of the computational burden.
Networks and Spatial Economics, 2018
The ongoing transformations of power systems worldwide pose important challenges, both economic a... more The ongoing transformations of power systems worldwide pose important challenges, both economic and technical, for their appropriate planning and operation. A key approach to improve the efficiency of these systems is through demand-side management, i.e., to promote the active involvement of consumers in the system. In particular, the current trend is to conceive systems where electricity consumers can vary their load according to real-time price incentives, offered by retailing companies. Under this setting, retail competition plays an important role as inadequate prices or services may entail consumers switching to a rival retailer. In this work we consider a game theoretical model where asymmetric retailers compete in prices to increase their profits by accounting for the utility function of consumers. Consumer preferences for retailers are uncertain and distributed within a Hotelling line. We analytically characterize the equilibrium of a retailer duopoly, establishing its existence and uniqueness conditions for a wide class of utility functions. Furthermore, sensitivities of the equilibrium prices with respect to relevant model parameters C. Ruiz
Journal of Optimization Theory and Applications, 2017
In this work, we describe the efficient use of improved directions of negative curvature for the ... more In this work, we describe the efficient use of improved directions of negative curvature for the solution of bound-constrained nonconvex problems. We follow an interior-point framework, in which the key point is the inclusion of computational low-cost procedures to improve directions of negative curvature obtained from a factorisation of the KKT matrix. From a theoretical point of view, it is well known that these directions ensure convergence to second-order KKT points. As a novelty, we consider the convergence rate of the algorithm with exploitation of negative curvature information. Finally, we test the performance of our proposal on both CUTEr/st and simulated problems, showing empirically that the enhanced directions affect positively the practical performance of the procedure.
In this note we analyze the relationship between the direction obtained from the minimization of ... more In this note we analyze the relationship between the direction obtained from the minimization of the kurtosis coefficient of the projections of a mixture of multivariate normal distributions and the linear discriminant function. We show that both directions are closely related, and in particular that given two vector random variables having symmetric distributions with unknown means and the same covariance matrix, the direction which minimizes the kurtosis coefficient of the projection is the linear discriminant function. This result provides a way to compute the discriminant function between two normal populations in the case in which the means and the common covariance matrix are unknown.
Journal of Statistical Physics, 2015
Support Vector Machines (SVMs) have become a very popular technique in the machinelearning field ... more Support Vector Machines (SVMs) have become a very popular technique in the machinelearning field for classification problems. It was originally proposed for classification of two classes. Various multiclass models with a single objective have been proposed mostly based on two families of methods: an all-together approach and a one-against-all approach. However, most of these single-objective models consider neither the different costs of misclassification nor the user's preferences. To overcome these drawbacks, multiobjective models have been proposed. In this paper we rewrite the different approaches that deal with the multiclass SVM using multiobjective techniques. These multiobjective techniques can give us weakly Pareto-optimal solutions. We propose a multiobjective technique called Projected Multiobjective All-Together (PMAT), which works in a higher-dimension space than the object space. With this technique, we can theoretically characterize the Pareto-optimal solution set. For these multiobjective techniques we get approximate sets of the Pareto-optimal solutions. For these sets, we use hypervolume and epsilon indicators to evaluate different multiobjective techniques. From the experimental results, we can see that (PMAT) outperfoms the other multiobjective techniques. When facing classification problems with very large numbers of classes, we suggest combining a tree method and multiobjective techniques.
2003 IEEE Power Engineering Society General Meeting (IEEE Cat. No.03CH37491)
This work describes a procedure that determines the optimal allocation for the yearly energy resu... more This work describes a procedure that determines the optimal allocation for the yearly energy resulting from random water inflows to the different subperiods of a year so that the expected benefits are maximized. Its main idea is to distribute the energy stored in reservoirs in each period into two parts: one is directly sold in the energy market, while the
Statistics & Probability Letters, 2000
In this note we analyze the relationship between the direction obtained from the minimization of ... more In this note we analyze the relationship between the direction obtained from the minimization of the kurtosis coe cient of the projections of a mixture of multivariate normal distributions and the linear discriminant function. We show that both directions are closely related and, in particular, that given two vector random variables having symmetric distributions with unknown means and the same covariance matrix, the direction which minimizes the kurtosis coe cient of the projection is the linear discriminant function. This result provides a way to compute the discriminant function between two normal populations in the case in which means and common covariance matrix are unknown.
Technometrics, 2001
In this article, we present a simple multivariate outlier-detection procedure and a robust estima... more In this article, we present a simple multivariate outlier-detection procedure and a robust estimator for the covariance matrix, based on the use of information obtained from projections onto the directions that maximize and minimize the kurtosis coef cient of the projected data. The properties of this estimator (computational cost, bias) are analyzed and compared with those of other robust estimators described in the literature through simulation studies. The performance of the outlier-detection procedure is analyzed by applying it to a set of well-known examples.
Journal of Multivariate Analysis, 2010
In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as intere... more In this paper we study the properties of a kurtosis matrix and propose its eigenvectors as interesting directions to reveal the possible cluster structure of a data set. Under a mixture of elliptical distributions with proportional scatter matrix, it is shown that a subset of the eigenvectors of the fourth-order moment matrix corresponds to Fisher's linear discriminant subspace. The eigenvectors of the estimated kurtosis matrix are consistent estimators of this subspace and its calculation is easy to implement and computationally efficient, which is particularly favourable when the ratio n/p is large.
Journal of Computational and Graphical Statistics, 2007
A powerful procedure for outlier detection and robust estimation of shape and location with multi... more A powerful procedure for outlier detection and robust estimation of shape and location with multivariate data in high dimension is proposed. The procedure searches for outliers in univariate projections on directions that are obtained both randomly, as in the Stahel-Donoho method, and by maximizing and minimizing the kurtosis coefficient of the projected data, as in the Peña and Prieto method. We propose modifications of both methods to improve their computational efficiency and combine them in a procedure which is affine equivariant, has a high breakdown point, is fast to compute and can be applied when the dimension is large. Its performance is illustrated with a Monte Carlo experiment and in a real dataset.
Siam J Optimization, 1995
The procedures for the identification of outlier observations that are most reliable are based on... more The procedures for the identification of outlier observations that are most reliable are based on the use of a robustified Mahalanobis distance, and have a very high computational cost even for small size problems. All these procedures present difficulties when applied to the identification of point-mass contaminations, where the outIiers are grouped into one or more clusters, separated from the sample. In this work a specific method for this contamination pattern is described, and shown to be able to handle successfully those cases where methods based on robust estimators (the Minimum Volume ElIipsiod estimator or the Stahel-Donoho estimator) fail. The method is simple, exploratory in nature, and straightforward to apply using any standard statistical software package.
In this report a new decomposition methodology for optimization problems is presented. The propos... more In this report a new decomposition methodology for optimization problems is presented. The proposed procedure is general, simple and efficient. It avoids most disadvantages of other common decomposition techniques, such as Lagrangian Relaxation or Augmented Lagrangian Relaxation. The new methodology is applied to a problem coming from interconnected power systems. The application of the new method to this problem allows the computation of an optimal coordinated but decentralized solution. Local and global convergence properties of the proposed decomposition algorithm are described. Numerical results show that the new decentralized methodology has a lower computational cost than other decomposition techniques, and in large-scale cases even lower than a centralized approach.
Although the Stahel-Donoho estimator with resampling seems to o¤er an excellent compro-mise betwe... more Although the Stahel-Donoho estimator with resampling seems to o¤er an excellent compro-mise between computational cost and bias and variability properties among high breakdown point scale estimators, it still presents signi…cantly high bias when the outliers are close to the uncontaminated sample. In this paper, we analyze di¤erent possible causes for these high values, and o¤er two pro-cedures, based on the use of alternative measures of scale for the projections de…ning the weights for the observations, that partially o¤set these undesirable e¤ects.
Studies in Classification, Data Analysis, and Knowledge Organization, 2006
ABSTRACT A projection method for robust estimation of shape and location in multivariate data and... more ABSTRACT A projection method for robust estimation of shape and location in multivariate data and cluster analysis is presented. The key idea of the procedure is to search for heterogeneity in univariate projections on directions that are obtained both randomly, using a modification of the Stahel-Donoho procedure, and by maximizing and minimizing the kurtosis coefficient of the projected data, as proposed by Peña and Prieto (2005). We show in a Monte Carlo study that the resulting procedure works well for robust estimation. Also, it preserves the good theoretical properties of the Stahel-Donoho method.
We present an e-cient implementation of an interior-point algorithm for non-convex bound constrai... more We present an e-cient implementation of an interior-point algorithm for non-convex bound constrained problems that uses good directions of negative curvature. These directions should improve the computational e-ciency of the procedure and ensure convergence to second-order KKT points. We analyze the practical behavior of the procedure and present two sets of numerical experiments to check the relevance of the algorithm.
Journal of Statistical Physics, 2014
We investigate a class of models related to the Bak-Sneppen model, initially proposed to study ev... more We investigate a class of models related to the Bak-Sneppen model, initially proposed to study evolution. The BS model is extremely simple and yet captures some forms of "complex behavior" such as self-organized criticality that is often observed in physical and biological systems.