Linear regression is the task of modeling the relationship between a result variable and some explanatory variables by a linear rule. Linear regression is a standard tool of statistical analysis, with widespread applications spanning essentially all of the sciences. While the standard linear regression task seeks to model the majority of the data, we consider problems where a regression fit could exist for some subset or portion of the data, that does not necessarily model the majority of the data. We will consider cases in which the subset with a linear model is described by some simple condition; in other words, we desire to perform linear regression on this conditional distribution. Note that neither the condition nor the model is known in advance.

Despite the considerable success enjoyed by machine learning techniques in practice, numerous studies demonstrated that many approaches are vulnerable to attacks. An important class of such attacks involves adversaries changing features at test time to cause incorrect predictions. Previous investigations of this problem pit a single learner against an adversary. However, in many situations an adversary's decision is aimed at a collection of learners, rather than specifically targeted at each independently. We study the problem of adversarial linear regression with multiple learners. We approximate the resulting game by exhibiting an upper bound on learner loss functions, and show that the resulting game has a unique symmetric equilibrium. We present an algorithm for computing this equilibrium, and show through extensive experiments that equilibrium models are significantly more robust than conventional regularized linear regression.

This Ph.D. thesis deals with the optimization of several renewable energy resources development as well as the improvement of facilities management in oceanic engineering and airports, using computational hybrid methods belonging to AI to this end. Energy is essential to our society in order to ensure a good quality of life. This means that predictions over the characteristics on which renewable energies depend are necessary, in order to know the amount of energy that will be obtained at any time. The second topic tackled in this thesis is related to the basic parameters that influence in different marine activities and airports, whose knowledge is necessary to develop a proper facilities management in these environments. Within this work, a study of the state-of-the-art Machine Learning have been performed to solve the problems associated with the topics above-mentioned, and several contributions have been proposed: One of the pillars of this work is focused on the estimation of the most important parameters in the exploitation of renewable resources. The second contribution of this thesis is related to feature selection problems. The proposed methodologies are applied to multiple problems: the prediction of $H_s$, relevant for marine energy applications and marine activities, the estimation of WPREs, undesirable variations in the electric power produced by a wind farm, the prediction of global solar radiation in areas from Spain and Australia, really important in terms of solar energy, and the prediction of low-visibility events at airports. All of these practical issues are developed with the consequent previous data analysis, normally, in terms of meteorological variables.

We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the responses for a small subset of the points that are then used to construct linear predictions for all points in the dataset. The performance of the predictions is evaluated by the total square loss on all responses (the attained as well as the hidden ones). We show that a good approximate solution to this least squares problem can be obtained from just dimension $d$ many responses by using a joint sampling technique called volume sampling. Moreover, the least squares solution obtained for the volume sampled subproblem is an unbiased estimator of optimal solution based on all n responses. This unbiasedness is a desirable property that is not shared by other common subset selection techniques. Motivated by these basic properties, we develop a theoretical framework for studying volume sampling, resulting in a number of new matrix expectation equalities and statistical guarantees which are of importance not only to least squares regression but also to numerical linear algebra in general. Our methods also lead to a regularized variant of volume sampling, and we propose the first efficient algorithms for volume sampling which make this technique a practical tool in the machine learning toolbox. Finally, we provide experimental evidence which confirms our theoretical findings.

Fitness landscape analysis investigates features with a high influence on the performance of optimization algorithms, aiming to take advantage of the addressed problem characteristics. In this work, a fitness landscape analysis using problem features is performed for a Multi-objective Bayesian Optimization Algorithm (mBOA) on instances of MNK-landscape problem for 2, 3, 5 and 8 objectives. We also compare the results of mBOA with those provided by NSGA-III through the analysis of their estimated runtime necessary to identify an approximation of the Pareto front. Moreover, in order to scrutinize the probabilistic graphic model obtained by mBOA, the Pareto front is examined according to a probabilistic view. The fitness landscape study shows that mBOA is moderately or loosely influenced by some problem features, according to a simple and a multiple linear regression model, which is being proposed to predict the algorithms performance in terms of the estimated runtime. Besides, we conclude that the analysis of the probabilistic graphic model produced at the end of evolution can be useful to understand the convergence and diversity performances of the proposed approach.

Identifying the location of a disturbance and its magnitude is an important component for stable operation of power systems. We study the problem of localizing and estimating a disturbance in the interconnected power system. We take a model-free approach to this problem by using frequency data from generators. Specifically, we develop a logistic regression based method for localization and a linear regression based method for estimation of the magnitude of disturbance. Our model-free approach does not require the knowledge of system parameters such as inertia constants and topology, and is shown to achieve highly accurate localization and estimation performance even in the presence of measurement noise and missing data.

For statistical learning, categorical variables in a table are usually considered as discrete entities and encoded separately to feature vectors, e.g., with one-hot encoding. "Dirty" non-curated data gives rise to categorical variables with a very high cardinality but redundancy: several categories reflect the same entity. In databases, this issue is typically solved with a deduplication step. We show that a simple approach that exposes the redundancy to the learning algorithm brings significant gains. We study a generalization of one-hot encoding, similarity encoding, that builds feature vectors from similarities across categories. We perform a thorough empirical validation on non-curated tables, a problem seldom studied in machine learning. Results on seven real-world datasets show that similarity encoding brings significant gains in prediction in comparison with known encoding methods for categories or strings, notably one-hot encoding and bag of character n-grams. We draw practical recommendations for encoding dirty categories: 3-gram similarity appears to be a good choice to capture morphological resemblance. For very high-cardinality, dimensionality reduction significantly reduces the computational cost with little loss in performance: random projections or choosing a subset of prototype categories still outperforms classic encoding approaches.

Predictive models are extremely useful for forecasting future outcomes and estimating metrics that are impractical to measure. For example, data scientists could use predictive models to forecast crop yields based on rainfall and temperature, or to determine whether patients with certain traits are more likely to react badly to a new medication. Before we talk about linear regression specifically, let's remind ourselves what a typical data science workflow might look like. A lot of the time, we'll start with a question we want to answer, and do something like the following: Linear regression is one of the simplest and most common supervised machine learning algorithms that data scientists use for predictive modeling. In this post, we'll use linear regression to build a model that predicts cherry tree volume from metrics that are much easier for folks who study trees to measure. This post is part of our focus on nature data this month.

Valid causal inference in observational studies often requires controlling for confounders. However, in practice measurements of confounders may be noisy, and can lead to biased estimates of causal effects. We show that we can reduce the bias caused by measurement noise using a large number of noisy measurements of the underlying confounders. We propose the use of matrix factorization to infer the confounders from noisy covariates, a flexible and principled framework that adapts to missing values, accommodates a wide variety of data types, and can augment many causal inference methods. We bound the error for the induced average treatment effect estimator and show it is consistent in a linear regression setting, using Exponential Family Matrix Completion preprocessing. We demonstrate the effectiveness of the proposed procedure in numerical experiments with both synthetic data and real clinical data.

We extend the decomposition approach for learning Bayesian networks (BN) proposed by (Xie et al., 2006) to learning multivariate regression chain graphs (MVR CGs), which include BNs as a special case. The same advantages of this decomposition approach hold in the more general setting: reduces complexity and increased power of computational independence tests. Moreover, latent (hidden) variables can be represented in MVR CGs by using bidirected edges, and our algorithm correctly recovers any independence structure that is faithful to a MVR CG, thus greatly extending the range of applications of decomposition-based model selection techniques. While our new algorithm has the same complexity as the one in (Xie et al., 2006) for BNs, it requires larger components for general MVR CGs, to insure that sufficient data is present to estimate parameters.