Dolores Romero Morales

Copenhagen Business School · Department of Economics

Counterfactual Analysis has shown to be a powerful tool in the burgeoning field of Explainable Artificial Intelligence. In Supervised Classification, this means associating with each record a so-called counterfactual explanation: an instance that is close to the record and whose probability of being classified in the opposite class by a given class...

In this paper, our goal is to enhance the interpretability of Generalized Linear Models by identifying the most relevant interactions between categorical predictors. Searching for interaction effects can quickly become a highly combinatorial, and thus computationally costly, problem when we have many categorical predictors or even a few of them but...

Data Envelopment Analysis (DEA) allows us to capture the complex relationship between multiple inputs and outputs in firms and organizations. Unfortunately, managers may find it hard to understand a DEA model and this may lead to mistrust in the analyses and to difficulties in deriving actionable information from the model. In this paper, we propos...

In this paper we introduce an approach to incorporate fairness constraints into Lasso regression. Assuming that a group of individuals need to be protected against discrimination, we address the problem of training the Lasso regression subject to a fairness constraint, enforcing that an equal proportion of individuals from the protected and non-pro...

Counterfactual explanations have become a very popular interpretability tool to understand and explain how complex machine learning models make decisions for individual instances. Most of the research on counterfactual explainability focuses on tabular and image data and much less on models dealing with functional data. In this paper, a counterfact...

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to have tools to understand how models arrive at decisions. Assuming an already trained Supervised Classification model, post-hoc explanations can be obtained via so-called counterfactual analysis: a counterfactual expl...

Tree ensembles are one of the most powerful methodologies in Machine Learning. In this paper, we investigate how to make tree ensembles more flexible to incorporate by design explainability and fairness. While explainability helps the user understand the key features that play a role in the classification task, with fairness we ensure that the ense...

In this note we show how to introduce fairness in a Support Vector Machine (SVM) classification model. Assuming that a group of individuals need to be protected against discrimination, we address the problem of training the classifier by jointly maximizing the classification performance (SVM margin) and equity (closeness between the distribution of...

Counterfactual Explanations are becoming a de-facto standard in post-hoc interpretable machine learning. For a given classifier and an instance classified in an undesired class, its counterfactual explanation corresponds to small perturbations of that instance that allows changing the classification outcome. This work aims to leverage Counterfactua...

In this paper, we tailor optimal randomized regression trees to handle multivariate functional data. A compromise between prediction accuracy and sparsity is sought. Whilst fitting the tree model, the detection of a reduced number of intervals that are critical for prediction, as well as the control of their length, is performed. Local and global s...

Counterfactual Analysis has shown to be a powerful tool in the burgeoning field of Explainable Artificial Intelligence. In Supervised Classification, this means associating with each record a so-called counterfactual explanation: an instance that is close to the record and whose probability of being classified in the positive class by a given class...

In this paper, we make Cluster Analysis more interpretable with a new approach that simultaneously allocates individuals to clusters and gives rule-based explanations to each cluster. The traditional homogeneity metric in clustering, namely the sum of the dissimilarities between individuals in the same cluster, is enriched by considering also, for...

Data Envelopment Analysis (DEA) allows us to capture the complex relationship between multiple inputs and outputs in firms and organizations. Unfortunately, managers may find it hard to understand a DEA model and this may lead to mistrust in the analyses and to difficulties in deriving actionable information from the model. In this paper, we propos...

In modelling the relative performance of a set of Decision Making Units (DMUs), a common challenge is to account for heterogeneity in the services they provide and the settings in which they operate. One solution is to include many features in the model and hereby to use a one-fits-all model that is sufficiently complex to account for this heteroge...

Many applications in data analysis study whether two categorical variables are independent using a function of the entries of their contingency table. Often, the categories of the variables, associated with the rows and columns of the table, are grouped, yielding a less granular representation of the categorical variables. The purpose of this is to...

In this paper, we tailor optimal randomized regression trees to handle multivariate functional data. A compromise between prediction accuracy and sparsity is sought. Whilst fitting the tree model, the detection of a reduced number of intervals that arecritical for prediction, as well as the control of their length, is performed. Local and global sp...

Many real-life applications consider nominal categorical predictor variables that have a hierarchical structure, e.g. economic activity data in Official Statistics. In this paper, we focus on linear regression models built in the presence of this type of nominal categorical predictor variables, and study the consolidation of their categories to hav...

In recent years, supervised classification has been used to support or even replace human decisions in high stakes domains. The training of these algorithms uses historical data which might be biased against individuals with certain sensitive attributes. The increasing concern about potential biases has motivated anti-discrimination laws prohibitin...

Counterfactual explanations have become a very popular interpretability tool to understand and explain how complex machine learning models make decisions for individual instances. Most of the research on counterfactual explainability focuses on tabular and image data and much less on models dealing with functional data. In this paper, a counterfact...

In this paper, we model an optimal regression tree through a continuous optimization problem, where a compromise between prediction accuracy and both types of sparsity, namely local and global, is sought. Our approach can accommodate important desirable properties for the regression task, such as cost-sensitivity and fairness. Thanks to the smoothn...

We propose a method to reduce the complexity of Generalized Linear Models in the presence of categorical predictors. The traditional one-hot encoding, where each category is represented by a dummy variable, can be wasteful, difficult to interpret, and prone to overfitting, especially when dealing with high-cardinality categorical predictors. This p...

In this paper, we tackle the problem of enhancing the interpretability of the results of Cluster Analysis. Our goal is to find an explanation for each cluster, such that clusters are characterized as precisely and distinctively as possible, i.e., the explanation is fulfilled by as many as possible individuals of the corresponding cluster, true posi...

In this paper, we make Cluster Analysis more interpretable with a new approach that simultaneously allocates individuals to clusters and gives rule-based explanations to each cluster. The traditional homogeneity metric in clustering, namely the sum of the dissimilarities between individuals in the same cluster, is enriched by considering also, for...

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to be able to understand how models arrive at decisions. Assuming an already trained Supervised Classification model, an effective class of post-hoc explanations are counterfactual explanations, i.e., a set of actions t...

We propose a method to reduce the complexity of Generalized Linear Models in the presence of categorical predictors. The traditional one-hot encoding, where each category is represented by a dummy variable, can be wasteful, difficult to interpret, and prone to overfitting, especially when dealing with high-cardinality categorical predictors. This p...

In this paper, our goal is to enhance the interpretability of Generalized Linear Models by identifying the most relevant interactions between categorical predictors. In the presence of categorical predictors, searching for interaction effects can quickly become a highly combinatorial problem when we have many categorical predictors or even a few, b...

Since the seminal paper by Bates and Granger in 1969, a vast number of ensemble methods that combine different base regressors to generate a unique one have been proposed in the literature. The so-obtained regressor method may have better accuracy than its components, but at the same time it may overfit, it may be distorted by base regressors with...

Classification and regression trees, as well as their variants, are off-the-shelf methods in Machine Learning. In this paper, we review recent contributions within the Continuous Optimization and the Mixed-Integer Linear Optimization paradigms to develop novel formulations in this research area. We compare those in terms of the nature of the decisi...

Classification and Regression Trees (CARTs) are off-the-shelf techniques in modern Statistics and Machine Learning. CARTs are traditionally built by means of a greedy procedure, sequentially deciding the splitting predictor variable(s) and the associated threshold. This greedy approach trains trees very fast, but, by its nature, their classificatio...

In this paper, we tackle the problem of enhancing the interpretability of the results of Cluster Analysis. Our goal is to find an explanation for each cluster, such that clusters are characterized as precisely and distinctively as possible, i.e., the explanation is fulfilled by as many as possible individuals of the corresponding cluster, true posi...

We propose a method to reduce the complexity of Generalized Linear Models in the presence of categorical predictors. The traditional one-hot encoding, where each category is represented by a dummy variable, can be wasteful, difficult to interpret, and prone to overfitting, especially when dealing with high-cardinality categorical predictors. This p...

Classification and regression trees, as well as their variants, are off-the-shelf methods in Machine Learning. In this paper, we review recent contributions within the Continuous Optimization and the Mixed-Integer Linear Optimization paradigms to develop novel formulations in this research area. We compare those in terms of the nature of the decisi...

Since the seminal paper by Bates and Granger in 1969, a vast number of ensemble methods that combine different base regressors to generate a unique one have been proposed in the literature. The so-obtained regressor method may have better accuracy than its components , but at the same time it may overfit, it may be distorted by base regressors with...

In this paper, we model an optimal regression tree through a continuous optimization problem, where a compromise between prediction accuracy and both types of sparsity, namely local and global, is sought. Our approach can accommodate important desirableproperties for the regression task, such as cost-sensitivity and fairness. Thanks to thesmoothnes...

In this paper, we propose a mathematical optimization approach to cluster the rows and/or columns of contingency tables to detect possible statistical dependencies among the observed variables. With this, we obtain a clustered contingency table of smaller size, which is desirable when interpreting the statistical dependence results of the observed...

In this paper, we study linear regression models built on categorical predictor variables that have a hierarchical structure. For such variables, the categories are arranged as a directed tree, where the categories in the leaf nodes give the highest granularity in the representation of the variable. Instead of taking the fully detailed model, the u...

COVID-19 is an infectious disease that was first identified in China in December 2019. Subsequently COVID-19 started to spread broadly, to also arrive in Spain by the end of Jan-uary 2020. This pandemic triggered confinement measures, in order to reduce the expansion of the virus so as not to saturate the health care system. With the aim of providi...

Decision trees are popular Classification and Regression tools and, when small-sized, easy to interpret. Traditionally, a greedy approach has been used to build the trees, yielding a very fast training process; however, controlling sparsity (a proxy for interpretability) is challenging. In recent studies, optimal decision trees, where all decisions...

Decision trees are popular Classification and Regression tools and, when small-sized, easy to interpret. Traditionally, a greedy approach has been used to build the trees, yielding a very fast training process; however, controlling sparsity (a proxy for interpretability) is challenging. In recent studies, optimal decision trees, where all decisions...

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors' interpretability by building a sparse loading matrix. In this paper, we propose...

In this paper we propose an optimization model and a solution approach to visualize datasets which are made up of individuals observed along different time periods. These individuals have attached a time-dependent magnitude and a dissimilarity measure, which may vary over time. Difference of convex optimization techniques, namely, the so-called Dif...

This paper proposes an integrative approach to feature (input and output) selection in Data Envelopment Analysis (DEA). The DEA model is enriched with zero-one decision variables modelling the selection of features, yielding a Mixed Integer Linear Programming formulation. This single-model approach can handle different objective functions as well a...

This paper proposes an integrative approach to feature (input and output) selection in Data Envelopment Analysis (DEA). The DEA model is enriched with zero-one decision variables modelling the selection of features, yielding a Mixed Integer Linear Programming formulation. This single-model approach can handle different objective functions as well a...

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors' interpretability by building a sparse loading matrix. In this paper, we propose...

Decision trees are popular Regression and Classification tools, easy to interpret and with excellent performance. The training process is very fast, since a greedy approach is used to build the tree. In recent studies, optimal decision trees, where all decisions are optimized simultaneously, have shown a better learning performance. In this paper,...

Classification and Regression Trees (CARTs) are an off-the-shelf technique in modern Statistics and Machine Learning. CARTs are traditionally built by means of a greedy procedure, sequentially deciding the splitting predictor variable(s) and the associated threshold. This greedy approach trains trees very fast, but, by its nature, the classificatio...

In this article we develop a novel online framework to visualize news data over a time horizon. First, we perform a Natural Language Processing analysis, wherein the words are extracted, and their attributes, namely the importance and the relatedness, are calculated. Second, we present a Mathematical Optimization model for the visualization problem...

In this paper we develop a new framework to visualize datasets which are made up of individuals observed along different time periods. These individuals have attached a time-dependent magnitude and a dissimilarity measure, which may vary over time. A mathematical optimization model is proposed and solved by means of difference of convex optimizatio...

In this paper we develop an online tool to visualize news data over a time horizon. First, we perform a Natural Language Processing analysis, where the words are extracted, and their attributes, namely the importance and the relatedness, are calculated. Second, we present a Mathematical Optimization model for the visualization problem and a numeric...

In this paper we address the problem of visualizing a frequency distribution and an ad-jacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and...

In this work a modeling framework and a solution approach have been presented for a multi-period stochastic mixed 0-1 problem arising in tactical supply chain planning (TSCP). A multistage scenario tree based scheme is used to represent the parameters’ uncertainty and develop the related Deterministic Equivalent Model. A cost risk reduction is perf...

In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the standard Multidimensional Scaling Analysis, is written as a global optimization problem whose objective is the difference of two convex f...

In this paper we address the problem of visualizing a set of individuals, which have attached a statistical value given as a proportion, and a dissimilarity measure. Each individual is represented as a region within the unit square, in such a way that the area of the regions represent the proportions and the distances between them represent the dis...

The support vector machine (SVM) is a state-of-the-art method in supervised classification. In this paper the Cluster Support Vector Machine (CLSVM) methodology is proposed with the aim to increase the sparsity of the SVM classifier in the presence of categorical features, leading to a gain in interpretability. The CLSVM methodology clusters catego...

In linear classifiers, such as the Support Vector Machine (SVM), a score is associated with each feature and objects are assigned to classes based on the linear combination of the scores and the values of the features. Inspired by discrete psychometric scales, which measure the extent to which a factor is in agreement with a statement, we propose t...

In this article, we study a biobjective economic lot-sizing problem with applications, among others, in green logistics. The first objective aims to minimize the total lot-sizing costs including production and inventory holding costs, whereas the second one minimizes the maximum production and inventory block expenditure. We derive (almost) tight c...

The default approach for tuning the parameters of a Support Vector Machine (SVM) is a grid search in the parameter space. Different metaheuristics have been recently proposed as a more efficient alternative, but they have only shown to be useful in models with a low number of parameters. Complex models, involving many parameters, can be seen as ext...

This paper studies a multistage stochastic programming (SP) model for large-scale network revenue management. We solve the model by means of the so-called expected future value (EFV) decomposition via scenario analysis, estimating the impact of the decisions made at a given stage on the objective function value related to the future stages. The EFV...

Recently, Support Vector Machines with the ramp loss (RLM) have attracted attention from the computational point of view. In this technical note, we propose two heuristics, the first one based on solving the continuous relaxation of a Mixed Integer Nonlinear formulation of the RLM and the second one based on the training of an SVM classifier on a r...

Data mining techniques often ask for the resolution of optimization problems. Supervised classification, and, in particular, support vector machines, can be seen as a paradigmatic instance. In this paper, some links between mathematical optimization methods and supervised classification are emphasized. It is shown that many different areas of mathe...

Consider a firm, called the buyer, that satisfies its demand over two periods by assigning both demands to a supplier via a second-price procurement auction; call this the Standard auction. In the hope of lowering its purchase cost, the firm is considering an alternative procedure in which it will also allow bids on each period individually, where...

The widely used Support Vector Machine (SVM) method has shown to yield good results in Supervised Classification problems. When the interpretability is an important issue, then classification methods such as Classification and Regression Trees (CART) might be more attractive, since they are designed to detect the important predictor variables and,...

Revenue management (RM) enhances the revenues of a company by means of demand-management decisions. An RM system must take into account the possibility that a booking may be canceled, or that a booked customer may fail to show up at the time of service (no-show). We review the Passenger Name Record data mining based cancellation rate forecasting mo...

The widely used support vector machine (SVM) method has shown to yield very good results in supervised classification problems. Other methods such as classification trees have become more popular among practitioners than SVM thanks to their interpretability, which is an important issue in data mining. In this work, we propose an SVM-based method th...

In this paper, we analyze equilibria in competitive environments under constraints across players’ strategies. This means that the action taken by one player limits the possible choices of the other players. In this context, the usual approach to show existence of equilibrium, Kakutani’s fixed point theorem, cannot be applied directly. In particula...