Gaussian Error Model Research Articles

Full-Duplex Bidirectional Secure Communications Under Perfect and Distributionally Ambiguous Eavesdropper's CSI

Consider a full-duplex (FD) bidirectional secure communication system, where two communication nodes, named Alice and Bob, simultaneously transmit and receive confidential information from each other, and an eavesdropper, named Eve, overhears the transmissions. Our goal is to maximize the sum secrecy rate (SSR) of the bidirectional transmissions by optimizing the transmit covariance matrices at Alice and Bob. To tackle this SSR maximization (SSRM) problem, we develop an alternating difference-of-concave (ADC) programming approach to alternately optimize the transmit covariance matrices at Alice and Bob. We show that the ADC iteration has a semi-closed-form beamforming solution, and is guaranteed to converge to a stationary solution of the SSRM problem. Besides the SSRM design, this paper also deals with a robust SSRM transmit design under a moment-based random channel state information (CSI) model, where only some roughly estimated first and second-order statistics of Eve's CSI are available, but the exact distribution or other high-order statistics is not known. This moment-based error model is new and different from the widely used bounded-sphere error model and the Gaussian random error model. Under the consider CSI error model, the robust SSRM is formulated as an outage probability-constrained SSRM problem. By leveraging the Lagrangian duality theory and DC programming, a tractable safe solution to the robust SSRM problem is derived. The effectiveness and the robustness of the proposed designs are demonstrated through simulations.

Texture edge-guided depth recovery for structured light-based depth sensor

The emergence of depth sensor facilitates the real-time and low-cost depth capture. However, the quality of its depth map is still inadequate for further applications due to holes, noises and artifacts existing within its depth information. In this paper, we propose an iterative depth boundary refinement framework to recover Kinect depth map. We extract depth edges and detect the incorrect regions, and then re-fill the incorrect regions until the depth edges are consistent with color edges. In the incorrect region detection procedure, we propose a RGB-D data edge detection method inspired by the recently developed deep learning. In the depth in-painting procedure, we propose a priority-determined fill order in which the high confidence pixels and strong edges are assigned to high priority. The actual depth values are computed by using a weighted cost filter, in which color, spatial similarity measures and Gaussian error model are considered. Experimental results demonstrate that the proposed method provides sharp and clear edges for the Kinect depth, and depth edges are aligned with the color edges.

An approximate model for pulsar navigation simulation

This paper presents an approximate model for the simulation of pulsar aided navigation systems. High fidelity simulations of these systems are computationally intensive and impractical for simulating periods of a day or more. Simulation of yearlong missions is done by abstracting navigation errors as periodic Gaussian noise injections. This paper presents an intermediary approximate model to simulate position errors for periods of several weeks, useful for building more accurate Gaussian error models. This is done by abstracting photon detection and binning, replacing it with a simple deterministic process. The approximate model enables faster computation of error injection models, allowing the error model to be inexpensively updated throughout a simulation. Testing of the approximate model revealed an optimistic performance prediction for non-millisecond pulsars with more accurate predictions for pulsars in the millisecond spectrum. This performance gap was attributed to noise which is not present in the approximate model but can be predicted and added to improve accuracy.

A Bayesian approach to improved calibration and prediction of groundwater models with structural error

AbstractNumerical groundwater flow and solute transport models are usually subject to model structural error due to simplification and/or misrepresentation of the real system, which raises questions regarding the suitability of conventional least squares regression‐based (LSR) calibration. We present a new framework that explicitly describes the model structural error statistically in an inductive, data‐driven way. We adopt a fully Bayesian approach that integrates Gaussian process error models into the calibration, prediction, and uncertainty analysis of groundwater flow models. We test the usefulness of the fully Bayesian approach with a synthetic case study of the impact of pumping on surface‐ground water interaction. We illustrate through this example that the Bayesian parameter posterior distributions differ significantly from parameters estimated by conventional LSR, which does not account for model structural error. For the latter method, parameter compensation for model structural error leads to biased, overconfident prediction under changing pumping condition. In contrast, integrating Gaussian process error models significantly reduces predictive bias and leads to prediction intervals that are more consistent with validation data. Finally, we carry out a generalized LSR recalibration step to assimilate the Bayesian prediction while preserving mass conservation and other physical constraints, using a full error covariance matrix obtained from Bayesian results. It is found that the recalibrated model achieved lower predictive bias compared to the model calibrated using conventional LSR. The results highlight the importance of explicit treatment of model structural error especially in circumstances where subsequent decision‐making and risk analysis require accurate prediction and uncertainty quantification.

A Study on the Effects of Sensor Position Error and the Placement of Calibration Emitter for Source Localization

Modern applications use mobile or randomly deployed sensors whose positions are not precise when locating a signal source. An estimator would require statistical knowledge of the sensor position errors to reach the optimum localization performance. Further accuracy improvement necessitates a calibration emitter whose position is known exactly to correct the sensor positions. Under Gaussian error model, this paper shows that when the covariance matrices of the sensor position errors and the measurement noise satisfy certain relation, taking the sensor position errors into account is not necessary and a simpler estimator that pretends the sensor position uncertainties are absent is sufficient to reach the optimum performance. The performance gain from a calibration emitter depends on where it is placed. We derive the optimum calibration position by improving the Fisher information matrix of the source location estimate. The optimum position is of theoretical interest and may not be practical. A suboptimum criterion for realistic calibration emitter placement is then proposed. We shall use TOA, TDOA, and AOA localizations to illustrate the derived results. Simulations support very much the theoretical developments and performance analysis.

A data-constrained large ensemble analysis of Antarctic evolution since the Eemian

Reconstructing the historical behaviour of the Antarctic Ice Sheet is essential for understanding and predicting climatic interactions, ice sheet sensitivities, and sea level contributions. Furthermore, to interpret such reconstructions with any degree of confidence, meaningful uncertainty estimates are required. Working toward this goal, this article presents results from a large-ensemble data-constrained study of Antarctic evolution over the Last Glacial cycle.Ice sheet chronologies have been generated using a 3-D glacial systems model that includes sheet/stream/shelf flow; a parametrized basal drag coefficient (accounting for sub-grid topographic roughness, sediment likelihood, and systematic model-to-observation thickness misfits); a sub-grid grounding-line flux parametrization; a visco-elastic bedrock response component; parametrized climate forcing; separate shelf and tidewater calving treatments; and a physically-motivated, empirically derived, sub-shelf melt component. There are 31 ensemble parameters to capture uncertainties in the glacial cycle climate, mass-balance processes, and ice dynamics. Once generated, the ensemble of model runs is constrained using a suite of observational data (including constraints on relative sea level, past ice thickness and grounding line retreat data-points, as well as present-day ice volumes and configuration) and an evaluation methodology that produces a misfit-to-observations score for each run. Assuming variants of a Gaussian error model, the scores are used to generate probability distributions for the past evolution of the ice sheet.In our ensemble-based analysis the Last Glacial Maximum (LGM) of Antarctic ice volume occurs at ∼25–24 ka. The LGM ice volume excess relative to present-day is likely between 5.6 and 14.3 m equivalent sea level (mESL), and with less confidence >10 mESL. There is little change in the grounded ice volume from 24 ka until 17–16 ka at which time widespread deglaciation commences. The Ross and Weddell Sea sectors are the largest sources of ice mass loss. The major period of grounding line retreat in the Weddell Sea sector, resulting in the present-day Ronne–Filchner shelf system, occurs after 10 ka; the grounding line in the Ross Sea sector start its major retreat phase after 12 ka. There is no significant contribution (<1.8 mESL) to meltwater pulse 1a.During the Eemian, the modelled Antarctic Ice Sheet has a minimal ice volume at 114 ka contributing ∼2 mESL relative to present-day. The mass loss is principally due to a major retreat of the grounding line in the Siple Coast sector of the Ross Sea.We present standard deviation plots of the ensemble results for present day and at the LGM. These plots highlight geographic areas that would most benefit from additional constraint and therefore offer priorities and guidance for future field campaigns.

Discretization Error Estimation in Multidisciplinary Simulations

This paper proposesmethods to estimate the discretization error in the system output of coupledmultidisciplinary simulations. In such systems, the governing equations for each discipline are numerically solved by a different computational code, and each discipline has different mesh size parameters. A classic example of multidisciplinary analysis involves fluid–structure interaction, where the element sizes in fluid and structure meshes are typically different. The general case of three-dimensional steady-state problems is considered in the current paper, where mesh refinement is possible in all three spatial directions for each discipline. Two aspects of discretization error, which are of interest in multidisciplinary analysis, are considered: disciplinary mesh sizes and the mismatch of disciplinary meshes at the interface at which boundary conditions are exchanged. Two alternate representations for the discretization error for the previously specified generic case are presented: 1) ignoring mesh mismatch at the interface and 2) considering mesh mismatch at the interface. Polynomial, rational function, and Gaussian process error models are used to represent the discretization error. The proposed error models are illustrated using a threedimensional fluid–structure interaction problem of an aircraft wing.

Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. Lévy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the Lévy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the Lévy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.

A set-membership approach for high integrity height-aided satellite positioning

A Robust Set-Inversion via Interval Analysis method in a bounded-error framework is used to compute three-dimensional location zones in real time, at a given confidence level. This approach differs significantly from the usual Gaussian error model paradigm, since the satellite positions and the pseudorange measurements are represented by intervals encompassing the true value with a particular level of confidence. The method computes a location zone recursively, using contractions and bisections of an arbitrarily large initial location box. Such an approach can also handle an arbitrary number of erroneous measurements using a q-relaxed solver and allows the integration of geographic and cartographic information such as digital elevation models or three-dimensional maps. With enough data redundancy, inconsistent measurements can be detected and even rejected. The integrity risk of the location zone comes only from the measurement bounds settings, since the solver is guaranteed. A method for setting these bounds for a particular location zone confidence level is proposed. An experimental validation using real L1 code measurements and a digital elevation model is also reported in order to illustrate the performance of the method on real data.

ExPlanes: Exploring Planes in Triplet Data

Summary Many methods for the analysis of gene expression-, protein- or metabolite-data focus on the investigation of binary relationships, while the underlying biological processes creating this data may generate relations of higher than bivariate complexity. We give a novel method ExPlanes that helps to explore certain types of ternary relationships in a statistically robust, Bayesian framework. To arrive at an characterization of the data structure contained in triplet data we investigate 2-dimensional planes being the only linear structures that cannot be inferred from projections of the data. The key part of our methodology is the definition of a robust, Bayesian plane posterior under the assumption of an invariant prior and a Gaussian error model. A numerical representation of the plane posterior can be explored interactively. Beyond this purely Bayesian approach we can use the plane posterior to construct a family of posterior-based test statistics that allow testing the data for different plane related hypotheses. To demonstrate practicability we queried triplets of metabolic data from a plant crossing experiment for the presence of plane-, line- and point-structures by using posterior-based test statistics and were able to show their distinctiveness.

Array processing of teleseismic body waves with the USArray

We introduce a novel method of array processing for measuring arrival times and relative amplitudes of teleseismic body waves recorded on large aperture seismic arrays. The algorithm uses a robust stacking algorithm with three features: (1) an initial ‘reference’ signal is required for initial alignment by cross-correlation; (2) a robust stacking method is used that penalizes signals that are not well matched to the stack; and (3) an iterative procedure alternates between cross-correlation with the current stack and the robust stacking algorithm. This procedure always converges in a few iterations making it well suited for interactive processing. We describe concepts behind a graphical interface developed to utilize this algorithm for processing body waves. We found it was important to compute several data quality metrics and allow the analyst to sort on these metrics. This is combined with a ‘pick cutoff’ function that simplifies data editing. Application of the algorithm to data from the USArray show four features of this method. (1) The program can produce superior results to that produced by a skilled analyst in approximately 1/5 of the time required for conventional interactive picking. (2) We show an illustrative example comparing residuals from S and SS for an event from northern Chile. The SS data show a remarkable ±10 s residual pattern across the USArray that we argue is caused by propagation approximately parallel to the subduction zones in Central and South America. (3) Quality metrics were found to be useful in identifying data problems. (4) We analyzed 50 events from the Tonga-Fiji region to compare residuals produced by this new algorithm with those measured by interactive picking. Both sets of residuals are approximately normally distributed, but corrupted by about 5% outliers. The scatter of the data estimated by waveform correlation was found to be approximately 1/2 that of the hand picked data. The outlier populations of both data sets are likely produced by cycle skips, but the distribution of hand picked data show a more diffuse departure from a Gaussian error model.

ExPlanes: exploring planes in triplet data.

Many methods for the analysis of gene expression-, protein- or metabolite-data focus on the investigation of binary relationships, while the underlying biological processes creating this data may generate relations of higher than bivariate complexity. We give a novel method ExPlanes that helps to explore certain types of ternary relationships in a statistically robust, Bayesian framework. To arrive at an characterization of the data structure contained in triplet data we investigate 2-dimensional planes being the only linear structures that cannot be inferred from projections of the data. The key part of our methodology is the definition of a robust, Bayesian plane posterior under the assumption of an invariant prior and a Gaussian error model. A numerical representation of the plane posterior can be explored interactively. Beyond this purely Bayesian approach we can use the plane posterior to construct a family of posterior-based test statistics that allow testing the data for different plane related hypotheses. To demonstrate practicability we queried triplets of metabolic data from a plant crossing experiment for the presence of plane-, line- and point-structures by using posterior-based test statistics and were able to show their distinctiveness.

Measurement Delay Estimation for Kalman Filter in Networked Control Systems

This work presents a method to estimate an unknown varying delay of measurements and subsequent Kalman filtering in the networked control system (NCS) framework. The delay estimation algorithm is based on the Gaussian error model and a network delay model, either a probability distribution or a Markov-chain. This method is used to tackle the problem with varying delays in a NCS. The undelayed output of the plant is estimated with a Kalman filter and used for control. The probability of a wrong delay estimate is derived. The estimation and control performance is evaluated with simulations. From the control perspective, it has comparable performance to the case with known delays.

Performance of the Zero Forcing precoding MIMO broadcast systems with channel estimation errors

In this paper, the effect of channel estimation errors upon the Zero Forcing (ZF) precoding Multiple Input Multiple Output Broadcast (MIMO BC) systems was studied. Based on the two kinds of Gaussian estimation error models, the performance analysis is conducted under different power allocation strategies. Analysis and simulation show that if the covariance of channel estimation errors is independent of the received Signal to Noise Ratio (SNR), imperfect channel knowledge deteriorates the sum capacity and the Bit Error Rate (BER) performance severely. However, under the situation of orthogonal training and the Minimum Mean Square Error (MMSE) channel estimation, the sum capacity and BER performance are consistent with those of the perfect Channel State Information (CSI) with only a performance degradation.

Grid-search event location with non-Gaussian error models

This study employs an event location algorithm based on grid search to investigate the possibility of improving seismic event location accuracy by using non-Gaussian error models. The primary departure from the Gaussian error model that is considered is the explicit use of non-Gaussian probability distributions in defining optimal estimates of location parameters. Specifically, the class of generalized Gaussian distributions is considered, which leads to the minimization of L p norms of arrival time residuals for arbitrary p ≥ 1 . The generalized Gaussian error models are implemented both with fixed standard errors assigned to the data and with an iterative reweighting of the data on a station/phase basis. An implicit departure from a Gaussian model is also considered, namely, the use of a simple outlier rejection criterion for disassociating arrivals during the grid-search process. These various mechanisms were applied to the ISC phase picks for the IWREF reference events, and the resulting grid-search solutions were compared to the GT locations of the events as well as the ISC solutions. The results indicate that event mislocations resulting from the minimization of L p residual norms, with p near 1, are generally better than those resulting from the conventional L 2 norm minimization (Gaussian error assumption). However, this result did not always hold for mislocations in event depth. Further, outlier rejection and iterative reweighting, applied with L 2 minimization, performed nearly as well as L 1 minimization in some cases. The results of this study suggest that ISC can potentially improve its location capability with the use of global search methods and non-Gaussian error models. However, given the limitations of this study, further research, including the investigation of other statistical and optimization techniques not addressed here, is needed to assess this potential more completely.

Uncertainty analysis in matched-field geoacoustic inversions

Quantifying uncertainty for parameter estimates obtained from matched-field geoacoustic inversions using a Bayesian approach requires estimation of the uncertainties in the data due to ambient noise as well as modeling errors. In this study, the variance parameter of the Gaussian error model, hereafter called error variance, is assumed to describe the data uncertainty. In practice, this parameter is not known a priori, and choosing a particular value is often difficult. Hence, to account for the uncertainty in error variance, several methods are introduced for implementing both the full and empirical Bayesian approaches. A full Bayesian approach that permits uncertainty of the error variance to propagate through the parameter estimation processes is a natural way of incorporating the uncertainty of error variance. Due to the large number of unknown parameters in the full Bayesian uncertainty analysis, an alternative, the empirical Bayesian approach, is developed, in which the posterior distributions of model parameters are conditioned on a point estimate of the error variance. Comparisons between the full and empirical Bayesian inferences of model parameters are presented using both synthetic and experimental data.

A probabilistic signal-strength-based evaluation methodology for sensor network deployment

The deployment of sensor networks has attracted a lot of attention recently. In essence this issue is concerned with how well a sensing field is monitored by sensors to achieve a particular coverage. In this paper, we propose a signal-strength-based approach to evaluate how well a sensing field is covered/monitored. We first formulate object tracking by a single sensor as a Gaussian-error model. Then we establish an error model on location estimation, given that the location of an object is known. This leads to a model to evaluate a sensor network with the given locations of sensors. We then apply the result to several applications, such as adding more sensor nodes for error reduction, and scheduling power modes (awake or sleep) of sensors, and demonstrate our simulation results.

Implementation techniques for geometric branch-and-bound matching methods

Algorithms for geometric matching and feature extraction that work by recursively subdividing transformation space and bounding the quality of match have been proposed in a number of different contexts and become increasingly popular over the last few years. This paper describes matchlist-based branch-and-bound techniques and presents a number of new applications of branch-and-bound methods, among them, a method for globally optimal partial line segment matching under bounded or Gaussian error, point matching under a Gaussian error model with subpixel accuracy and precise orientation models, and a simple and robust technique for finding multiple distinct object instances. It also contains extensive reference information for the implementation of such matching methods under a wide variety of error bounds and transformations. In addition, the paper contains a number of benchmarks and evaluations that provide new information about the runtime behavior of branch-and-bound matching algorithms in general, and that help choose among different implementation strategies, such as the use of point location data structures and space/time tradeoffs involving depth-first search.

Analysis of RAKE reception with MRC and imperfect weight estimation for binary coherent orthogonal signaling

We consider a RAKE receiver for coherent binary orthogonal signaling employing predetection maximal-ratio combining (MRC), in which the tap weights are estimated by adding two matched filtered outputs using the two reference signals. The weight estimation errors here are not independent of the additive channel noise, and therefore do not fit into the Gaussian error model for MRC. Instead of analyzing the distribution of the weight estimation errors, we find the characteristic function of the decision variable, and from it we obtain a formula for the symbol error probability.

Distributions of projective invariants and model-based machine vision

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.