Fixed-point Estimation Research Articles

A Bayesian multi-model inference methodology for imprecise moment-independent global sensitivity analysis of rock structures

Traditional global sensitivity analysis (GSA) neglects the epistemic uncertainties associated with the probabilistic characteristics (i.e. type of distribution type and its parameters) of input rock properties emanating due to the small size of datasets while mapping the relative importance of properties to the model response. This paper proposes an augmented Bayesian multi-model inference (BMMI) coupled with GSA methodology (BMMI-GSA) to address this issue by estimating the imprecision in the moment-independent sensitivity indices of rock structures arising from the small size of input data. The methodology employs BMMI to quantify the epistemic uncertainties associated with model type and parameters of input properties. The estimated uncertainties are propagated in estimating imprecision in moment-independent Borgonovo's indices by employing a reweighting approach on candidate probabilistic models. The proposed methodology is showcased for a rock slope prone to stress-controlled failure in the Himalayan region of India. The proposed methodology was superior to the conventional GSA (neglects all epistemic uncertainties) and Bayesian coupled GSA (B-GSA) (neglects model uncertainty) due to its capability to incorporate the uncertainties in both model type and parameters of properties. Imprecise Borgonovo's indices estimated via proposed methodology provide the confidence intervals of the sensitivity indices instead of their fixed-point estimates, which makes the user more informed in the data collection efforts. Analyses performed with the varying sample sizes suggested that the uncertainties in sensitivity indices reduce significantly with the increasing sample sizes. The accurate importance ranking of properties was only possible via samples of large sizes. Further, the impact of the prior knowledge in terms of prior ranges and distributions was significant; hence, any related assumption should be made carefully.

Fixed-Point Estimation by Iterative Strategies and Stability Analysis with Applications

In this study, we developed a new faster iterative scheme for approximate fixed points. This technique was applied to discuss some convergence and stability results for almost contraction mapping in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Moreover, some numerical experiments were investigated to illustrate the behavior and efficacy of our iterative scheme. The proposed method converges faster than symmetrical iterations of the S algorithm, Thakur algorithm and K* algorithm. Eventually, as an application, the nonlinear Volterra integral equation with delay was solved using the suggested method.

Quantitative evaluation of the role of Fangcang shelter hospitals in the control of Omicron transmission: A case study of the outbreak in Shanghai, China in 2022

Since Omicron began to spread in China, Shanghai has become one of the cities with more severe outbreaks. Under the comprehensive consideration of the vaccine coverage rate, the number of Fangcang shelter hospital beds and the number of designated hospital beds in Shanghai, this paper established a deterministic compartmental model and used the Nelder-Mead Simplex Direct Search Algorithm and chi-square values to estimate the model parameters. we calculate ℛ0 = 3.6429 when the number of beds in the Fangcang shelter hospital is relatively tight in the second stage and ℛ0 = 0.4974 in the fifth stage when there are enough beds in both Fangcang shelter hospital and designated hospital. Then we perform a sensitivity analysis on ℛ0 by using perturbation of fixed point estimation of model parameters in the fifth stage, and obtain three parameters that are more sensitive to ℛ0, which are transmission rate (β1d), proportion of the infectious (η) and the hospitalization rate of asymptomatic infected cases (δ1). Through simulation, we obtain that if the hospitalization rate of asymptomatic infections δ2 > 0.9373 or the transmission rate β1b < 0.0467, the second stage of Omicron transmission in Shanghai can be well controlled. Finally, we find the measure that converting the National Convention and Exhibition Center (NECC) into a Fangcang shelter hospital has played an important role in curbing the epidemic. Whether this temporary Fangcang shelter hospital is not built or delayed, the cumulative number of confirmed cases will both exceed 100,000, and the cumulative asymptomatic infections will both exceed 1 million. In addition, for a city of 10 million people, we obtain that if a permanent Fangcang shelter hospital with 17,784 beds is built ahead of epidemic, there will be no shortage of beds during the outbreak of Omicron. Our findings enrich the content of the impact of Fangcang shelter hospital beds on the spread of Omicron and confirm the correct policy adopted by the Chinese government.

Jackknife based generalized resampling reliability approach for rock slopes and tunnels stability analyses with limited data: Theory and applications

An efficient resampling reliability approach was developed to consider the effect of statistical uncertainties in input properties arising due to insufficient data when estimating the reliability of rock slopes and tunnels. This approach considers the effect of uncertainties in both distribution parameters (mean and standard deviation) and types of input properties. Further, the approach was generalized to make it capable of analyzing complex problems with explicit/implicit performance functions (PFs), single/multiple PFs, and correlated/non-correlated input properties. It couples resampling statistical tool, i.e. jackknife, with advanced reliability tools like Latin hypercube sampling (LHS), Sobol's global sensitivity, moving least square-response surface method (MLS-RSM), and Nataf's transformation. The developed approach was demonstrated for four cases encompassing different types. Results were compared with a recently developed bootstrap-based resampling reliability approach. The results show that the approach is accurate and significantly efficient compared with the bootstrap-based approach. The proposed approach reflects the effect of statistical uncertainties of input properties by estimating distributions/confidence intervals of reliability index/probability of failure(s) instead of their fixed-point estimates. Further, sufficiently accurate results were obtained by considering uncertainties in distribution parameters only and ignoring those in distribution types. • A generalized and computationally efficient resampling reliability approach is proposed to consider the effect of statistical uncertainties in input rock properties arising due to limited data availability. • Approach can be used for problems with single/multiple performance functions, implicit/explicit performance functions and correlated/non-correlated input properties. • Approach was demonstrated for four real rock stability case studies – structurally controlled slope, stress-controlled slope, shallow circular tunnel, and deep underground nuclear repository in massive brittle rocks encompassing different problem types. • Approach was highly accurate and significantly computationally efficient for all types of problems when results were compared with recently proposed bootstrap based re-sampling reliability approach.

Supervised Multivariate Learning with Simultaneous Feature Auto-Grouping and Dimension Reduction

Abstract Modern high-dimensional methods often adopt the ‘bet on sparsity’ principle, while in supervised multivariate learning statisticians may face ‘dense’ problems with a large number of nonzero coefficients. This paper proposes a novel clustered reduced-rank learning (CRL) framework that imposes two joint matrix regularizations to automatically group the features in constructing predictive factors. CRL is more interpretable than low-rank modelling and relaxes the stringent sparsity assumption in variable selection. In this paper, new information-theoretical limits are presented to reveal the intrinsic cost of seeking for clusters, as well as the blessing from dimensionality in multivariate learning. Moreover, an efficient optimization algorithm is developed, which performs subspace learning and clustering with guaranteed convergence. The obtained fixed-point estimators, although not necessarily globally optimal, enjoy the desired statistical accuracy beyond the standard likelihood setup under some regularity conditions. Moreover, a new kind of information criterion, as well as its scale-free form, is proposed for cluster and rank selection, and has a rigorous theoretical support without assuming an infinite sample size. Extensive simulations and real-data experiments demonstrate the statistical accuracy and interpretability of the proposed method.

Adaptive Waveform Design with Multipath Exploitation Radar in Heterogeneous Environments

The problem of detecting point like targets over a glistening surface is investigated in this manuscript, and the design of an optimal waveform through a two-step process for a multipath exploitation radar is proposed. In the first step, a non-adaptive waveform is transmitted and a constrained Generalized Likelihood Ratio Test (GLRT) detector is deduced at reception which exploits multipath returns in the range cell under test by modelling the target echo as a superposition of the direct plus the multipath returns. Under the hypothesis of heterogeneous environments, thus by assuming a compound-Gaussian distribution for the clutter return, this latter is estimated in the range cell under test through the secondary data, which are collected from the out-of-bin cells. The Fixed Point Estimate (FPE) algorithm is applied in the clutter estimation, then used to design the adaptive waveform for transmission in the second step of the algorithm, in order to suppress the clutter coming from the adjacent cells. The proposed GLRT is also used at the end of the second transmission for the final decision. Extensive performance evaluation of the proposed detector and adaptive waveform for various multipath scenarios is presented. The performance analysis prove that the proposed method improves the Signal-to-Clutter Ratio (SCR) of the received signal, and the detection performance with multipath exploitation.

Propensities and Second Order Uncertainty: A Modified Taxi Cab Problem.

The study of people’s ability to engage in causal probabilistic reasoning has typically used fixed-point estimates for key figures. For example, in the classic taxi-cab problem, where a witness provides evidence on which of two cab companies (the more common ‘green’/less common ‘blue’) were responsible for a hit and run incident, solvers are told the witness’s ability to judge cab color is 80%. In reality, there is likely to be some uncertainty around this estimate (perhaps we tested the witness and they were correct 4/5 times), known as second-order uncertainty, producing a distribution rather than a fixed probability. While generally more closely matching real world reasoning, a further important ramification of this is that our best estimate of the witness’ accuracy can and should change when the witness makes the claim that the cab was blue. We present a Bayesian Network model of this problem, and show that, while the witness’s report does increase our probability of the cab being blue, it simultaneously decreases our estimate of their future accuracy (because blue cabs are less common). We presented this version of the problem to 131 participants, requiring them to update their estimates of both the probability the cab involved was blue, as well as the witness’s accuracy, after they claim it was blue. We also required participants to explain their reasoning process and provided follow up questions to probe various aspects of their reasoning. While some participants responded normatively, the majority self-reported ‘assuming’ one of the probabilities was a certainty. Around a quarter assumed the cab was green, and thus the witness was wrong, decreasing their estimate of their accuracy. Another quarter assumed the witness was correct and actually increased their estimate of their accuracy, showing a circular logic similar to that seen in the confirmation bias/belief polarization literature. Around half of participants refused to make any change, with convergent evidence suggesting that these participants do not see the relevance of the witness’s report to their accuracy before we know for certain whether they are correct or incorrect.

Persymmetric adaptive detection of distributed targets in compound-Gaussian sea clutter with Gamma texture

Abstract We consider the problem of detecting a distributed target in compound-Gaussian sea clutter with Gamma distributed texture components. We design the Rao, Wald, and generalized likelihood ratio test (GLRT) detectors according to the two-step method: the first step is to obtain the Rao, Wald tests and GLRT on the assumption that the texture or/and covariance matrix structure are known; then the maximum a posteriori probability estimate of the clutter texture and the fixed point estimate of covariance matrix exploiting persymmetry are employed to replace the known texture and covariance matrix in the tests derived in the first step. Remarkably these proposed detectors ensure constant false alarm rate with respect to the covariance matrix structure. The effectiveness of the proposed detectors is verified by using simulated and real sea clutter data.

A Generalization of the Fixed Point Estimate for Packet-Scaled Complex Covariance Matrix Estimation

In this paper, the problem of estimating complex-valued proportional covariance matrices is addressed. The obtained estimate generalizes the fixed point estimate to scaled packets of data, and is hence called the generalized fixed point estimate (GFPE). The statistical properties (bias, consistency, and asymptotical distributions) of the estimate are presented, and verified through simulations. As an example of application, a radar detection problem is considered. The GFPE is used in the well-known adaptive normalized matched filter (ANMF), and the obtained empirical probability of false alarms threshold curve is compared to the theoretical one. The ANMF is also used to compare the GFPE to the fixed point estimate.

Adaptive Coherency Matrix Estimation for Polarimetric SAR Imagery Based on Local Heterogeneity Coefficients

Polarimetric synthetic aperture radar (SAR) images usually contain a mixture of homogeneous and heterogeneous regions, which makes estimation of the coherency matrix a very challenging task. In this paper, we propose an adaptive coherency matrix estimation method that employs local heterogeneity coefficient and leverages the sample covariance matrix estimation to the homogeneous components and the fixed-point estimation to the heterogeneous components. Evaluations were conducted with synthetic polarimetric data and real-world SAR imagery, including UAVSAR, RADARSAT-2, and ESAR. Our experimental results demonstrated that the heterogeneity coefficient effectively characterizes the scattering property of ground objects, which enables adaptive estimation of the coherency matrix in high-resolution polarimetric SAR imagery. Our method was able to handle single- and multilook polarimetric SAR imagery gracefully. Compared with the sample covariance matrix estimator, the fixed-point estimator, and the Lee sigma filtering, our method achieved the best performance for retaining the spatial structure, suppressing speckles, and preserving polarimetric information of SAR imagery with different degrees of heterogeneity.

Robust, Scalable, and Fast Bootstrap Method for Analyzing Large Scale Data

In this paper we address the problem of performing statistical inference for large scale data sets i.e., Big Data. The volume and dimensionality of the data may be so high that it cannot be processed or stored in a single computing node. We propose a scalable, statistically robust and computationally efficient bootstrap method, compatible with distributed processing and storage systems. Bootstrap resamples are constructed with smaller number of distinct data points on multiple disjoint subsets of data, similarly to the bag of little bootstrap method (BLB) [1]. Then significant savings in computation is achieved by avoiding the re-computation of the estimator for each bootstrap sample. Instead, a computationally efficient fixed-point estimation equation is analytically solved via a smart approximation following the Fast and Robust Bootstrap method (FRB) [2]. Our proposed bootstrap method facilitates the use of highly robust statistical methods in analyzing large scale data sets. The favorable statistical properties of the method are established analytically. Numerical examples demonstrate scalability, low complexity and robust statistical performance of the method in analyzing large data sets.

Generalized Robust Shrinkage Estimator and Its Application to STAP Detection Problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data.

Distributed target detection with polarimetric MIMO radar in compound-Gaussian clutter

This paper mainly deals with distributed targets detection with a polarimetric MIMO radar against compound-Gaussian clutter dominated scenario with unknown covariance matrix. First, the general polarimetric detecting problem of the distributed targets is developed to the MIMO radar, and then, the fully adaptive Generalized Likelihood Ratio Test (GLRT) is devised according to the well known two-step design procedure. Three covariance matrix estimation strategies using the secondary data are introduced to make derived receiver fully adaptive. A thorough performance assessment is given, and via several numerical examples, the results highlight that the spatial and polarization diversities can be exploited to improve the detection performance of the distributed targets in compound-Gaussian background, and it outperforms the phased-array counterpart, the adaptive loss is completely acceptable in practical applications. Meanwhile, the fixed-point estimation strategy is more suitable to implement the adaptive detection algorithm.

Performance Analysis of Colocated MIMO Radars with Randomly Distributed Arrays in Compound-Gaussian Clutter

This paper focuses on the target detection problem for colocated MIMO radar with randomly distributed arrays in the presence of compound-Gaussian clutter with unknown Power Spectral Density (PSD). First, we derive the Generalized Likelihood Ratio Test (GLRT) assuming known covariance structure, and then, three different covariance estimation strategies, i.e., Sampled Covariance Matrix (SCM), Normalized Sampled Covariance Matrix (NSCM), and Fixed Point Estimation (FPE) Matrix, are introduced in place of the exact one to make the derived GLRT fully adaptive. Thorough numerical evaluations of the detection performance are provided and discussed.

GLRT-based Detection Algorithm for Polarimetric MIMO Radar Against SIRV Clutter

This paper mainly deals with target detecting problem using polarimetric Multiple Input Multiple Output (MIMO) radar against Spherically Invariant Random Vector (SIRV) clutter. First, we develop the MIMO signal model to two polarimetric channels and SIRV clutter-dominated scenario, and then the Generalized Likelihood Ratio Test (GLRT) is derived with known covariance structure. Meanwhile, three estimation strategies of covariance, such that Sampled Covariance Matrix (SCM), Normalized Sampled Covariance Matrix (NSCM) and Fixed Point Estimation (FPE) matrix, are introduced to make derived receiver fully adaptive. A thorough performance assessment is given by several numerical examples, and the results show that the polarimetric diversity and the spatial diversity can be exploited to improve the detection performance, and it outperforms the conventional polarimetric phased-array counterpart. Meanwhile, the FPE strategy is more suitable to implement the adaptive detection algorithm, the adaptive loss of which is completely acceptable in practical applications.

Rao and Wald tests design of polarimetric multiple-input multiple-output radar in compound-Gaussian clutter

In high–resolution radars or at low gazing angles, the clutter is more satisfied in the compound-Gaussian model. Meanwhile, the polarisation diversity can be exploited to enhance the detection performance. Motivated by extending the detection problem of multiple-input multiple-output radar to such cases, this study mainly addresses the adaptive detectors design with an unknown covariance matrix based on Rao and Wald criterions. The two-step design strategy is adopted. Three estimation strategies of covariance with secondary data, such as sampled covariance matrix (SCM), normalised sampled covariance matrix (NSCM) and fixed point estimation (FPE) matrix, are introduced to make derived receivers fully adaptive. A thorough performance assessment is given by several numerical examples, the results of which show that Rao and Wald tests can provide good detection performance in even spikier clutter, and the polarimetric diversity can also be exploited to improve the detection performance. Meanwhile, the FPE strategy is more suitable to implement the adaptive detection algorithms, and the adaptive loss is completely acceptable in practical applications.

Coherency Matrix Estimation of Heterogeneous Clutter in High-Resolution Polarimetric SAR Images

This paper presents an application of the recent advances in the field of spherically invariant random vector (SIRV) modeling for coherency matrix estimation in heterogeneous clutter. The complete description of the polarimetric synthetic aperture radar (POLSAR) data set is achieved by estimating the span and the normalized coherency independently. The normalized coherency describes the polarimetric diversity, while the span indicates the total received power. The main advantages of the proposed fixed-point (FP) estimator are that it does not require any <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> information about the probability density function of the texture (or span) and that it can directly be applied on adaptive neighborhoods. Interesting results are obtained when coupling this FP estimator with an adaptive spatial support based on the scalar span information. Based on the SIRV model, a new maximum-likelihood distance measure is introduced for unsupervised POLSAR classification. The proposed method is tested with both simulated POLSAR data and airborne POLSAR images provided by the Radar Ae¿roporte¿ Multi-Spectral d'Etude des Signatures system. Results of entropy/alpha/anisotropy decomposition, followed by unsupervised classification, allow discussing the use of the normalized coherency and the span as two separate descriptors of POLSAR data sets.

Performance of the adaptive generalised matched subspace constant false alarm rate detector in non-Gaussian noise: an experimental analysis

The authors deal with the performance analysis of an adaptive version of the generalised matched subspace detector (GMSD) in compound-Gaussian clutter with unknown covariance matrix. The original GMSD was proposed to detect subspace signals in compound-Gaussian noise with known covariance matrix and ensures the constant false alarm rate (CFAR) property. In real situations, this assumption is unrealistic, which means that the covariance matrix must be estimated from training data. The authors use a robust estimate of the covariance matrix called the fixed-point estimate, recently proposed in the literature. The performance of the obtained adaptive detector, in terms of CFAR behaviour and probability of detection, is evaluated in the presence of real sea clutter data, collected by the McMaster IPIX radar.

Exact Maximum Likelihood Estimates for SIRV Covariance Matrix: Existence and Algorithm Analysis

In this paper, we investigate the existence and the algorithm analysis of an adaptive scheme that has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by spherically invariant random vector (SIRV), which is the product c of the square root of a positive unknown random variable tau and an independent Gaussian vector x,c=radic(tau) x. A similar line of work was undertaken in the context of compound Gaussian noise, and this paper extends the previous results in the case of SIRV modeled noise. More precisely, the fixed-point estimate to be studied verifies a nonlinear algebraic equation (E)x=f(x). The aim of this paper is twofold. First, we prove that (E) admits a unique solution x; secondly, we show that the corresponding iterative algorithm x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n+1</sub> =f(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) converges to x for every admissible initial condition.

Message Passing for Maximum Weight Independent Set

We investigate the use of message-passing algorithms for the problem of finding the max-weight independent set (MWIS) in a graph. First, we study the performance of the classical loopy max-product belief propagation. We show that each fixed point estimate of max-product can be mapped in a natural way to an extreme point of the LP polytope associated with the MWIS problem. However, this extreme point may not be the one that maximizes the value of node weights; the particular extreme point at final convergence depends on the initialization of max-product. We then show that if max-product is started from the natural initialization of uninformative messages, it always solves the correct LP -- if it converges. This result is obtained via a direct analysis of the iterative algorithm, and cannot be obtained by looking only at fixed points. The tightness of the LP relaxation is thus necessary for max-product optimality, but it is not sufficient. Motivated by this observation, we show that a simple modification of max-product becomes gradient descent on (a convexified version of) the dual of the LP, and converges to the dual optimum. We also develop a message-passing algorithm that recovers the primal MWIS solution from the output of the descent algorithm. We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. Finally, we show that any problem of MAP estimation for probability distributions over finite domains can be reduced to an MWIS problem. We believe this reduction will yield new insights and algorithms for MAP estimation.