Robust Estimation and Model Order Selection for Signal Processing

In this thesis, advanced robust estimation methodologies for signal processing are developed and analyzed. The developed methodologies solve problems concerning multi-sensor data, robust model selection as well as robustness for dependent data. The work has been applied to solve practical signal processing problems in different areas of biomedical and array signal processing. In particular, for univariate independent data, a robust criterion is presented to select the model order with an application to corneal-height data modeling. The proposed criterion overcomes some limitations of existing robust criteria. For real-world data, it selects the radial model order of the Zernike polynomial of the corneal topography map in accordance with clinical expectations, even if the measurement conditions for the videokeratoscopy, which is the state-of-the-art method to collect corneal-height data, are poor. For multi-sensor data, robust model order selection selection criteria are proposed and applied to the problem of estimating the number of sources impinging onto a sensor array. The developed criteria are based on a robust and efficient estimator of the covariance of the r-mode unfoldings of a complex valued data tensor. Both in the case of Gaussian noise and for a brief sensor failure, the proposed robust multi-dimensional schemes outperform their matrix computation based counterparts. In the context of robustness for multi-sensor data, we next investigate the problem of estimating the complex-valued amplitude of sinusoidal signals in a completely unknown heavy-tailed symmetric spatially and temporally independent and identically distributed (i.i.d.) sensor noise environment. A selection of non-robust and robust estimators are compared to a proposed semi-parametric robust estimator. A third research focus in the area of multi-sensor data is that of analyzing the robustness of spatial time-frequency distribution (STFD) estimators. We provide a robustness analysis framework that is based on the influence function. The influence function is a robustness measure that describes the bias impact of an infinitesimal contamination at an arbitrary point on the estimator, standardized by the fraction of contamination. In addition to the asymptotic analysis, we also give a definition of the finite sample counterpart of the influence function. Simulation results for the finite sample influence function confirm the analytical results and show the insensitivity to small departures in the distributional assumptions for some recently proposed robust STFD estimators. A large part of this thesis concerns the topic of obtaining and analyzing robust estimators in the dependent data setup. First, some practical issues concerning the detection, and robust estimation in presence of patient motion induced artifacts in biomedical measurements are addressed. In particular, we provide an artifact-cleaning algorithm for data collected with an electrocardiogram (ECG). This is especially important for the monitoring of patients with portable ECG recording devices, since these devices suffer severely from patient motion induced artifacts. A second real-world problem addressed in this doctoral project is that of forecasting the intracranial pressure (ICP) levels for patients who suffered a traumatic brain injury. This enables active and early interventions for more effective control of ICP levels. We propose a methodology which uses combined artifact detection and robust estimation after a data transformation into the empirical mode domain. Motivated by plethora of practical applications, we then focus on deriving and analyzing sophisticated robust estimation and model selection techniques for autoregressive moving-average (ARMA) models. A fast algorithm as well as a detailed statistical and robustness analysis of a novel robust and efficient estimator is given. For the proposed estimator, which is termed the bounded influence propagation (BIP) τ-estimator, we compute a complete statistical robustness analysis, which includes conditions for the consistency, as well as a proof of qualitative and quantitative robustness. The robustness is measured by means of the influence function, the maximum bias curve and the breakdown point. The fast algorithm of the proposed estimator is based on first computing a robust initial estimate of an autoregressive (AR) approximation from which the ARMA model parameters are derived. In this way, the ARMA model parameters are derived from the long AR approximation without further use of the outlier-contaminated observations. The estimator is very suitable and attractive for ARMA model selection purposes, since the computational cost of estimating all the candidate ARMA models approximately reduces to that of computing one long AR model. In the area of model selection for ARMA models, we propose and compare different robust model order selection criteria that are based on the BIP τ-estimator.