Robust Location Estimators Research Articles

Robust discriminant analysis

AbstractDiscriminant analysis (DA) is one of the most popular methods for classification due to its conceptual simplicity, low computational cost, and often solid performance. In its standard form, DA uses the arithmetic mean and sample covariance matrix to estimate the center and scatter of each class. We discuss and illustrate how this makes standard DA very sensitive to suspicious data points, such as outliers and mislabeled cases. We then present an overview of techniques for robust DA, which are more reliable in the presence of deviating cases. In particular, we review DA based on robust estimates of location and scatter, along with graphical diagnostic tools for visualizing the results of DA.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Robust Methods Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

Advancing the frontier of fish geolocation into the ocean’s midwaters

Tracking large-scale movements of fishes in the ocean’s midwaters, below the euphotic zone and above the seafloor, is extremely challenging. Archival satellite telemetry devices rely on light, sea surface temperature, or bottom depth data to estimate location. Consequently, geolocation of fishes inhabiting the twilight (mesopelagic: 200–1000 m) and midnight (bathypelagic: 1000–4000 m) zones has been restricted to hypothesized movement routes, thereby precluding a baseline ecological understanding against which to assess potential anthropogenic impacts. We assessed the viability of comparing depth-temperature profiles measured by animal-borne satellite tags against those from 3D ocean-resolving models and incorporated known locations from acoustic telemetry to enable a quantitative framework for deep-sea geolocation. Testing of alternative, data-driven likelihood scenarios on a deep-water shark species assemblage with marked variation in modal depth distributions confirmed that the methodological frontier of geolocation can be advanced into the twilight and midnight zones. We identify key limitations in deep-water geolocation, and ways to overcome them, identifying a viable path for robust location estimates that can help address the knowledge gap on fish movement ecology in the deep sea. Our findings suggest that leveraging state-of-the-art geolocation approaches, in combination with novel technologies, raises new opportunities for studying enigmatic deep-ocean ecosystems.

Co-seismic surface displacement of the June 21, 2022 MW6 Khōst, Afghanistan earthquake from InSAR observations

A robust estimation of the earthquake location, seismic moment, and fault geometry is essential for objective seismic hazard assessment. Seismic events in a remote location, specifically in the absence of seismic and GNSS networks, can be investigated effectively using the InSAR-based technique. This study adopts the Differential Interferometric SAR (DInSAR) technique to quantify the co-seismic surface displacement caused by the June 21, 2022, Khōst MW6 earthquake that occurred along the western plate boundary between the Indian and Eurasian plate. The interferograms show that the maximum surface deformation occurred on the northwest and southwest of the fault line. From coherence, the Line of Sight (LOS) displacement, and the co-seismic surface displacement analysis, it has been observed that surface deformation was most pronounced in the southwest region of the fault line, and the surface has moved to the opposite direction along the fault line, which indicates a sinistral slightly oblique strike-slip movement. This InSAR-based observation appears consistent with the seismic waveforms derived from co-seismic surface displacements. Further, it has been argued that the slip deficit accumulated during the period of the last about 48 years along the frontal region of the northward extension of the Suleiman range and associated fault zone is qualitatively estimated at about 1.5 m, which is consistent with the seismic waveforms derived finite slip model.

Localization of underground pipe jacking machinery: A reliable, real-time and robust INS/OD solution

Accurate positioning is important for guidance control in pipeline construction, but traditional approaches fail in GPS-denied environments. This paper addresses the underground localization problem for the pipe jacking machine by presenting an Inertial Navigation System (INS)/ODometry (OD) scheme. It is a loosely coupled navigation system that combines an INS and an OD to provide reliable, real-time, and robust location estimation in an enclosed and restricted space. The combination is achieved by a multi-sensor fusion algorithm to avoid the divergence of standalone INS. Compared with the traditional localization technologies, the advantage of the INS/OD scheme is that there is no performance degradation or service failure despite severe blockage of the thick soil layer due to the selection of dead reckoning-based sensors. Moreover, an Unbiased Finite Impulse Response (UFIR) filter serves as the fusion mechanism to improve the robustness against a prior knowledge of noise, considering unknown and time-varying noise statistics in practice. The INS/OD scheme is verified through a numerical simulation and employed on-site in a real underground sewage pipeline construction project. Despite the construction site being more than 10 meters deep underground, the INS/OD scheme continues to perform reliably and robustly in real-time, significantly benefiting the engineering guidance enterprise.

Robust confidence intervals for the process capability index Cpk with bootstrap improvement

The process capability index (PCI), C pk , one of the widely used tools for assessing the capability of a manufacturing process, expresses the deviation of the process mean from the midpoint of the specification limits. The C pk is known to perform well under the general assumption that the experimental data are normally distributed without contamination. Under this assumption, the sample mean and sample standard deviation are used for the estimation of the PCI. However, the sample mean and sample standard deviation are quite sensitive to data contamination and this will result in underperformance of C pk . Therefore, in this article, we propose alternatives to the conventional method by replacing the sample mean and sample standard deviation with robust location and scale estimators. We also propose a method for constructing a robust PCI C pk confidence interval which lends itself to robust statistical hypothesis testing. The robust hypothesis testing methods based on this confidence interval are shown to be quite efficient when the data are normally distributed yet also outperform the conventional method when data contamination exists.

On the use of robust estimators of multivariate location for heterogeneous data

The properties of orthogonal equivariance and componentwise increasingness are often discussed for estimators of multivariate location. The former property is linked to a coordinate-free nature of the data (as in spatial data), whereas the latter property is linked to an ordered nature of the data (typically formed by several univariate variables). Since both properties do not get along well with each other, if robustness in the presence of outliers is to be pursued, one must choose between them. Admittedly, most of the literature on multivariate estimation of location, in which affine equivariance is pursued, typically abandons the latter property. Unfortunately, in the more and more common presence of heterogeneous data (as in spatio-temporal data), both orthogonal equivariance (therefore, affine equivariance) and componentwise increasingness might bring very disappointing results. For this very reason, both properties should be forfeited and only required for some of the components.

Fast Robust Location and Scatter Estimation: A Depth-based Method

The minimum covariance determinant (MCD) estimator is ubiquitous in multivariate analysis, the critical step of which is to select a subset of a given size with the lowest sample covariance determinant. The concentration step (C-step) is a common tool for subset-seeking; however, it becomes computationally demanding for high-dimensional data. To alleviate the challenge, we propose a depth-based algorithm, termed as FDB, which replaces the optimal subset with the trimmed region induced by statistical depth. We show that the depth-based region is consistent with the MCD-based subset under a specific class of depth notions, for instance, the projection depth. With the two suggested depths, the FDB estimator is not only computationally more efficient but also reaches the same level of robustness as the MCD estimator. Extensive simulation studies are conducted to assess the empirical performance of our estimators. We also validate the computational efficiency and robustness of our estimators under several typical tasks such as principal component analysis, linear discriminant analysis, image denoise and outlier detection on real-life datasets. An R package FDB and additional results are available in the supplementary materials.

The effects of economic growth and fossil fuel consumption to climate change: Evidence from Mediterranean Europe by robust estimators

Climate change and global warming during the recent decades are posing formidable chal- lenges to ecosystems. Nevertheless, changing the climate system due to extreme weather events such as cold spells, high temperatures, droughts, and heat waves have been recorded all over the world. Particularly, it has become less accurate to predict the weather in some European regions using a short time series without considering the extreme values events in the estimated model. Thus, forecasting the behaviour of climate needs more accurate statisti- cal techniques to be used. The main objective of this experimental study is to detect the best robust scale or robust location estimator to model the relationship between CO2 emissions, fossil fuel consumption and gross domestic product by considering the influence of different types of extreme weather events in the panel data of Mediterranean Europe countries over the period 1960-2020. The findings show that the MM-estimator is the best robust estimator han- dling data with high efficiency and high breakdown point with the existence of different types of extreme weather events. In conclusion, the robust MM-estimator could be used to provide an innovative integrated climate-economic approach for the accurate prediction of carbon emissions.

Comparing the Effectiveness of Robust Statistical Estimators of Proficiency Testing Schemes in Outlier Detection

This study investigates the effectiveness of robust estimators of location and dispersion, used in proficiency testing and listed in ISO 13528:2015, in outlier detection. The models utilize (a) kernel density plots, (b) Z-factors, (c) Monte Carlo simulations, and (d) distributions derived from at most two contaminating distributions and one main Gaussian. The simulation parameters cover a wide range of those commonly encountered in proficiency testing (PT) schemes, so the results presented are of fairly general application. We chose a functional sub-optimal solution by grouping and classifying the model settings, resulting in five matrices readily usable for selecting the best robust estimator. Whenever at most half of the distribution of each contaminating population is outside the central distribution, there is only one optimal estimator. For all other cases, the five matrices provide the appropriate robust statistic. The proposed method applies to 95.1% of 144 results for an existing PT for cement. These actual datasets indicate that the Hampel estimator for the mean and the Q-method for the standard deviation provide the most appropriate performance statistic in 86.1% of the cases.

On weighted multivariate sign functions

Multivariate sign functions are often used for robust estimation and inference. We propose using data dependent weights in association with such functions. The proposed weighted sign functions retain desirable robustness properties, while significantly improving efficiency in estimation and inference compared to unweighted multivariate sign-based methods. Using weighted signs, we demonstrate methods of robust location estimation and robust principal component analysis. We extend the scope of using robust multivariate methods to include robust sufficient dimension reduction and functional outlier detection. Several numerical studies and real data applications demonstrate the efficacy of the proposed methodology.

On the measure of anchored Gaussian simplices, with applications to multivariate medians

We consider anchored Gaussian ℓ-simplices in the d-dimensional Euclidean space, that is, simplices with one fixed vertex y∈Rd and the remaining vertices X1,…,Xℓ randomly sampled from the d-variate standard normal distribution. We determine the distribution of the measure of such simplices for any d, any ℓ, and any anchor point y, which is of interest, e.g., when studying the asymptotic behaviour of U-statistics based on such simplex measures. We provide two proofs of the results. The first one is short but is not self-contained as it crucially relies on a technical result for non-central Wishart distributions. The second one is a simple and self-contained proof, that also provides some geometric insight on the results. Quite nicely, variations on this second argument reveal intriguing distributional identities on products of central and non-central chi-square distributions with Beta-distributed non-centrality parameters. We independently establish these distributional identities by making use of Mellin transforms. Beyond the aforementioned use to study the asymptotic behaviour of some U-statistics, our results do find natural applications in the context of robust location estimation, as we illustrate by considering a class of simplex-based multivariate medians that contains the celebrated spatial median and Oja median as special cases. Throughout, our results are confirmed by numerical experiments.

Robust optimal estimation of location from discretely sampled functional data

AbstractEstimating location is a central problem in functional data analysis, yet most current estimation procedures either unrealistically assume completely observed trajectories or lack robustness with respect to the many kinds of anomalies one can encounter in the functional setting. To remedy these deficiencies we introduce the first class of optimal robust location estimators based on discretely sampled functional data. The proposed method is based on M‐type smoothing spline estimation with repeated measurements and is suitable for both commonly and independently observed trajectories that are subject to measurement error. We show that under suitable assumptions the proposed family of estimators is minimax rate optimal both for commonly and independently observed trajectories and we illustrate its highly competitive performance and practical usefulness in a Monte‐Carlo study and a real‐data example involving recent Covid‐19 data.

Simulation optimization using stochastic kriging with robust statistics

Metamodels are widely used as fast surrogates to facilitate the optimization of simulation models. Stochastic kriging (SK) is an effective metamodeling tool for a mean response surface implied by stochastic simulation. In SK, it is usually assumed that the experimental data are normally distributed and uncontaminated. However, these assumptions can be easily violated in many practical applications. This paper proposes a new type of SK for simulation models that may have non-Gaussian responses; this new SK uses robust estimators of location (or central tendency) and scale (or variability) that are well-known in the literature on robust statistics. Statistical properties of the robust estimators used in this paper are briefly analyzed and the performances of the proposed methods are compared through numerical examples of different features. The comparison results show that the proposed robust SK with the robust estimators is quite efficient, no matter whether the standard assumptions hold or not.

A new robust multivariate mode estimator for eye-tracking calibration.

We propose in this work a new method for estimating the main mode of multivariate distributions, with application to eye-tracking calibration. When performing eye-tracking experiments with poorly cooperative subjects, such as infants or monkeys, the calibration data generally suffer from high contamination. Outliers are typically organized in clusters, corresponding to fixations in the time intervals when subjects were not looking at the calibration points. In this type of multimodal distributions, most central tendency measures fail at estimating the principal fixation coordinates (the first mode), resulting in errors and inaccuracies when mapping the gaze to the screen coordinates. Here, we developed a new algorithm to identify the first mode of multivariate distributions, named BRIL, which relies on recursive depth-based filtering. This novel approach was tested on artificial mixtures of Gaussian and Uniform distributions, and compared to existing methods (conventional depth medians, robust estimators of location and scatter, and clustering-based approaches). We obtained outstanding performances, even for distributions containing very high proportions of outliers, both grouped in clusters and randomly distributed. Finally, we demonstrate the strength of our method in a real-world scenario using experimental data from eye-tracking calibrations with Capuchin monkeys, especially for highly contaminated distributions where other algorithms typically lack accuracy.

Measurement bias invariant moving horizon estimation in the presence of outliers for data reconciliation of nonlinear dynamic systems

Some empirical findings in the literature regarding the two closely related areas of data reconciliation and state estimation for nonlinear dynamic systems suggest that the use of robust regression M-estimators is sufficient to implicitly cope with biased measurements. We present a counterexample to that belief on a dynamic model of two CSTRs in the presence of both measurement bias and outliers where such an approach gives rise to wrong transient behaviour of the reconciled values of the process variables. In the light of the above, we introduce and examine a method to deal simultaneously with outliers and explicitly with constant measurement bias. It is based on the location invariance for constant biased measurements assured by a robust estimator of scale, whereas for unbiased measurements the same robust estimator of scale is combined with a robust location estimator as a robustified version of the root mean square error. These two statistics are then combined together using the desirability function. Biased measurements are detected as a sequence of consecutive differences between measured and reconciled values of the same sign. Its performance is assessed numerically in the counterexample and compared with that of the correntropy M-estimator to show the superiority of the proposed data reconciliation method.

A robust Hotelling test statistic for one sample case in high dimensional data

The Hotelling statistic is used to test the hypothesis about the location parameter of multivariate Gaussian distribution, and it is significantly sensitive to outliers. Also, we cannot calculate it when the sample size is less than the number of variables because this statistic needs the inverse of the covariance matrix, and the sample covariance matrix is singular in high dimensional data. Although a new approach, based on shrinkage estimation, was proposed to solve this singularity problem, this estimator is still sensitive to outliers. On the other hand, a robust one sample Hotelling statistic was proposed by using the minimum covariance determinant (MCD) estimates instead of classical ones. Since the MCD estimates cannot be calculated when this statistic cannot be used in high-dimensional data. This study proposes to use the minimum regularized covariance determinant (MRCD) estimator instead of classical or MCD. The MRCD estimator is a robust location and scatter estimator, which can be calculated in high-dimensional data. We obtain the asymptotic distribution of the proposed test statistic using Monte Carlo simulations and examine the power and robustness properties of the test statistic with simulated datasets. As a result, we show that the approximate distribution of the test statistic is proper, and the proposed robust test statistic can be used to test the hypothesis about the location parameter of contaminated high dimensional data. Finally, we construct an R function in the MVTests package to perform our proposed test statistic.

Robust Estimation of Contact Force and Location for Magnetic-Field-Based Soft Tactile Sensor Considering Magnetic Source Inconsistency.

Flexible magnetic-field-based tactile sensors (FMFTS) have numerous advantages including low cost, ease of manufacture, simple wiring, high sensitivity, and so on. Flexible magnetic-field-based tactile sensors need to be calibrated before use to build accurate mapping between contact force and magnetic field intensity measured by magnetic sensors; however, when considering remanence inconsistency of magnetic source, each FMFTS needs to be calibrated independently to enhance accuracy, and the complex preparation prevents FMFTS from being used conveniently. A robust estimation method of contact force and location that can tolerate remanence inconsistency of magnetic source in FMFTS is proposed. Firstly, the position and orientation of magnetic source were tracked using the Levenberg–Marquart algorithm, and the tracking results were insensitive to the remanence of magnetic source with appropriate cost function. Secondly, the mapping between magnitude and location of contact force and position and orientation of magnetic source was built with calibration of one sensor; the mapping only depends on the structural response of flexible substrate, and thus can be extended to estimate external force and location for other sensors with the same structure. The proposed method was evaluated in both simulations and experiments, and the results confirm that the estimation of magnitude and location of external force for FMFTS with the same structure and different remanence could reach acceptable accuracy, depending on single calibration. The proposed method can be used to simplify the calibration procedure and remove the barrier for large-scale application of FMFTS and replacement of damaged FMFTS.

FUZZY ROBUST ESTIMATES OF LOCATION AND SCALE PARAMETERS OF A FUZZY RANDOM VARIABLE

A random variable is a variable whose components are random values. To characterise a random variable, the arithmetic mean is widely used as an estimate of the location parameter, and variation as an estimate of the scale parameter. The disadvantage of the arithmetic mean is that it is sensitive to extreme values, outliers in the data. Due to that, to characterise random variables, robust estimates of the location and scale parameters are widely used: the median and median absolute deviation from the median. In real situations, the components of a random variable cannot always be estimated in a deterministic way. One way to model the initial data uncertainty is to use fuzzy estimates of the components of a random variable. Such variables are called fuzzy random variables. In this paper, we examine fuzzy robust estimates of location and scale parameters of a fuzzy random variable: fuzzy median and fuzzy median of the deviations of fuzzy component values from the fuzzy median.

Real-time discriminant analysis in the presence of label and measurement noise

Quadratic discriminant analysis (QDA) is a widely used classification technique. Based on a training dataset, each class in the data is characterized by an estimate of its center and shape, which can then be used to assign unseen observations to one of the classes. The traditional QDA rule relies on the empirical mean and covariance matrix. Unfortunately, these estimators are sensitive to label and measurement noise which often impairs the model’s predictive ability. Robust estimators of location and scatter are resistant to this type of contamination. However, they have a prohibitive computational cost for large scale industrial experiments. We present a novel QDA method based on a recent real-time robust algorithm. We additionally integrate an anomaly detection step to classify the most atypical observations into a separate class of outliers. Finally, we introduce the label bias plot, a graphical display to identify label and measurement noise in the training data. The performance of the proposed approach is illustrated in a simulation study with huge datasets, and on real datasets about diabetes and fruit.

Robust Location Estimation based on TDOA and FDOA using Outlier Detection Algorithm

Robust Location Estimation based on TDOA and FDOA using Outlier Detection Algorithm