Approaches for Outlier Detection in Sparse High-Dimensional Regression Models

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

Measurement of the cosmic ray spectrum above 4×10184{\times}10^{18} eV using inclined events detected with the Pierre Auger Observatory

A measurement of the cosmic-ray spectrum for energies exceeding $4{\times}10^{18}$ eV is presented, which is based on the analysis of showers with zenith angles greater than $60^{\circ}$ detected with the Pierre Auger Observatory between 1 January 2004 and 31 December 2013. The measured spectrum confirms a flux suppression at the highest energies. Above $5.3{\times}10^{18}$ eV, the "ankle", the flux can be described by a power law $E^{-\gamma}$ with index $\gamma=2.70 \pm 0.02 \,\text{(stat)} \pm 0.1\,\text{(sys)}$ followed by a smooth suppression region. For the energy ($E_\text{s}$) at which the spectral flux has fallen to one-half of its extrapolated value in the absence of suppression, we find $E_\text{s}=(5.12\pm0.25\,\text{(stat)}^{+1.0}_{-1.2}\,\text{(sys)}){\times}10^{19}$ eV.Comment: Replaced with published version. Added journal reference and DO

Energy Estimation of Cosmic Rays with the Engineering Radio Array of the Pierre Auger Observatory

The Auger Engineering Radio Array (AERA) is part of the Pierre Auger Observatory and is used to detect the radio emission of cosmic-ray air showers. These observations are compared to the data of the surface detector stations of the Observatory, which provide well-calibrated information on the cosmic-ray energies and arrival directions. The response of the radio stations in the 30 to 80 MHz regime has been thoroughly calibrated to enable the reconstruction of the incoming electric field. For the latter, the energy deposit per area is determined from the radio pulses at each observer position and is interpolated using a two-dimensional function that takes into account signal asymmetries due to interference between the geomagnetic and charge-excess emission components. The spatial integral over the signal distribution gives a direct measurement of the energy transferred from the primary cosmic ray into radio emission in the AERA frequency range. We measure 15.8 MeV of radiation energy for a 1 EeV air shower arriving perpendicularly to the geomagnetic field. This radiation energy -- corrected for geometrical effects -- is used as a cosmic-ray energy estimator. Performing an absolute energy calibration against the surface-detector information, we observe that this radio-energy estimator scales quadratically with the cosmic-ray energy as expected for coherent emission. We find an energy resolution of the radio reconstruction of 22% for the data set and 17% for a high-quality subset containing only events with at least five radio stations with signal.Comment: Replaced with published version. Added journal reference and DO

Measurement of the Radiation Energy in the Radio Signal of Extensive Air Showers as a Universal Estimator of Cosmic-Ray Energy

We measure the energy emitted by extensive air showers in the form of radio emission in the frequency range from 30 to 80 MHz. Exploiting the accurate energy scale of the Pierre Auger Observatory, we obtain a radiation energy of 15.8 \pm 0.7 (stat) \pm 6.7 (sys) MeV for cosmic rays with an energy of 1 EeV arriving perpendicularly to a geomagnetic field of 0.24 G, scaling quadratically with the cosmic-ray energy. A comparison with predictions from state-of-the-art first-principle calculations shows agreement with our measurement. The radiation energy provides direct access to the calorimetric energy in the electromagnetic cascade of extensive air showers. Comparison with our result thus allows the direct calibration of any cosmic-ray radio detector against the well-established energy scale of the Pierre Auger Observatory.Comment: Replaced with published version. Added journal reference and DOI. Supplemental material in the ancillary file

Combined fit to the spectrum and composition data measured by the Pierre Auger Observatory including magnetic horizon effects

The measurements by the Pierre Auger Observatory of the energy spectrum and mass composition of cosmic rays can be interpreted assuming the presence of two extragalactic source populations, one dominating the flux at energies above a few EeV and the other below. To fit the data ignoring magnetic field effects, the high-energy population needs to accelerate a mixture of nuclei with very hard spectra, at odds with the approximate E$^{-2}$ shape expected from diffusive shock acceleration. The presence of turbulent extragalactic magnetic fields in the region between the closest sources and the Earth can significantly modify the observed CR spectrum with respect to that emitted by the sources, reducing the flux of low-rigidity particles that reach the Earth. We here take into account this magnetic horizon effect in the combined fit of the spectrum and shower depth distributions, exploring the possibility that a spectrum for the high-energy population sources with a shape closer to E$^{-2}$ be able to explain the observations