'Scuola Normale Superiore - Edizioni della Normale'
Abstract
Modern regression studies often encompass a very large number of potential predictors,
possibly larger than the sample size, and sometimes growing with the sample
size itself. This increases the chances that a substantial portion of the predictors
is redundant, as well as the risk of data contamination. Tackling these problems is
of utmost importance to facilitate scientific discoveries, since model estimates are
highly sensitive both to the choice of predictors and to the presence of outliers. In
this thesis, we contribute to this area considering the problem of robust model selection
in a variety of settings, where outliers may arise both in the response and
the predictors. Our proposals simplify model interpretation, guarantee predictive
performance, and allow us to study and control the influence of outlying cases on
the fit.
First, we consider the co-occurrence of multiple mean-shift and variance-inflation
outliers in low-dimensional linear models. We rely on robust estimation techniques
to identify outliers of each type, exclude mean-shift outliers, and use restricted
maximum likelihood estimation to down-weight and accommodate variance-inflation
outliers into the model fit. Second, we extend our setting to high-dimensional linear
models. We show that mean-shift and variance-inflation outliers can be modeled as
additional fixed and random components, respectively, and evaluated independently.
Specifically, we perform feature selection and mean-shift outlier detection through
a robust class of nonconcave penalization methods, and variance-inflation outlier
detection through the penalization of the restricted posterior mode. The resulting
approach satisfies a robust oracle property for feature selection in the presence of
data contamination – which allows the number of features to exponentially increase
with the sample size – and detects truly outlying cases of each type with asymptotic
probability one. This provides an optimal trade-off between a high breakdown point
and efficiency. Third, focusing on high-dimensional linear models affected by meanshift
outliers, we develop a general framework in which L0-constraints coupled with
mixed-integer programming techniques are used to perform simultaneous feature
selection and outlier detection with provably optimal guarantees. In particular,
we provide necessary and sufficient conditions for a robustly strong oracle property,
where again the number of features can increase exponentially with the sample size,
and prove optimality for parameter estimation and the resulting breakdown point.
Finally, we consider generalized linear models and rely on logistic slippage to perform
outlier detection and removal in binary classification. Here we use L0-constraints
and mixed-integer conic programming techniques to solve the underlying double
combinatorial problem of feature selection and outlier detection, and the framework
allows us again to pursue optimality guarantees.
For all the proposed approaches, we also provide computationally lean heuristic
algorithms, tuning procedures, and diagnostic tools which help to guide the analysis.
We consider several real-world applications, including the study of the relationships
between childhood obesity and the human microbiome, and of the main drivers of
honey bee loss. All methods developed and data used, as well as the source code to
replicate our analyses, are publicly available