Bound Of Variance Research Articles

Multiple scattering, radiative transfer, and parameter regimes

We have established the radiative transfer regime in a four-parameter space spanned by amplitude and scale of fluctuations, convergence rate, and spatial range. The derivation of radiative transfer equations from wave equations offers a better understanding of the underlying physics in radiative transfer theory. One approach is based on tools borrowed from quantum field theory (QFT) and employs the Dyson and Bethe–Salpeter equations as the foundation. This has allowed us to investigate the small scale fluctuation regime, kl = O ( ε ) . We also investigated this regime using the scale-separated asymptotics approach and found that the conditions on the parameters are the same as in the QFT approach if the convergence rate of the Dyson series is O ( ε 2 ) . Furthermore, we found that there is a multitude of regimes where radiative transfer manifests. We hence infer that for a fixed spatial range, increase in the scale size of fluctuations leads to decrease in the upper bound of variance of permittivity fluctuations along with diminished convergence rate of the Wigner series. Also, for a fixed scale size and convergence rate, the spatial range where radiative transport manifests decreases with increase in strength of fluctuations.

Estimation of small failure probabilities by partially Bayesian active learning line sampling: Theory and algorithm

Line sampling (LS) has proved to be a highly promising advanced simulation technique for assessing small failure probabilities. Despite the great interest in practical engineering applications, many efforts from the research community have been devoted to improving the standard LS. This paper aims at offering some new insights into the LS method, leading to an innovative method, termed ‘partially Bayesian active learning line sampling’ (PBAL-LS). The problem of evaluating the failure probability integral in the LS method is treated as a Bayesian, rather than frequentist, inference problem, which allows to incorporate our prior knowledge and model the discretization error. The Gaussian process model is used as the prior distribution for the distance function, and the posterior mean, and an upper bound of the posterior variance of the failure probability are derived. Based on the posterior statistics of the failure probability, we also put forward a learning function and a stopping criterion, which enable us to use active learning. Besides, an efficient algorithm is also designed to implement the PBAL-LS method, with the ability to automatically adjust the important direction and efficiently process the lines. Five numerical examples are studied to demonstrate the performance of the proposed PBAL-LS method against several existing methods.

Path Planning Method for Underwater Gravity-Aided Inertial Navigation Based on PCRB

Gravity-aided inertial navigation system (GAINS) is an important development in autonomous underwater vehicle (AUV) navigation. An effective path planning algorithm plays an important role in the performance of navigation in long-term underwater missions. By combining the gravity information obtained at each position with the error information from the INS, the posterior Cramér-Rao bound (PCRB) of GAINS is derived in this paper. The PCRB is the estimated lower bound of position variance for navigation along the planned trajectory. And the sum of PCRB is used as the minimum cost from the initial state to the current state in the state space, and the position error prediction variance of inertial navigation system (INS) is used as the minimum estimated cost of the path from the current state to the goal state in the A* algorithm. Thus, a path planning method with optimal navigation accuracy is proposed. According to simulation results, traveling along the path planned by the proposed method can rapidly improve the positioning accuracy while consuming just slightly more distance. Even when measuring noise changes, the planned path can still maintain optimal positioning accuracy, and high positioning accuracy is possible for any trajectory located within a certain range of the planned path.

Batching Adaptive Variance Reduction

Adaptive Monte Carlo variance reduction is an effective framework for running a Monte Carlo simulation along with a parameter search algorithm for variance reduction, whereas an initialization step is required for preparing problem parameters in some instances. In spite of the effectiveness of adaptive variance reduction in various fields of application, the length of the preliminary phase has often been left unspecified for the user to determine on a case-by-case basis, much like in typical sequential frameworks. This uncertain element may possibly be even fatal in realistic finite-budget situations, since the pilot run may take most of the budget, or possibly use up all of it. To unnecessitate such an ad hoc initialization step, we develop a batching procedure in adaptive variance reduction, and provide an implementable formula of the learning rate in the parameter search which minimizes an upper bound of the theoretical variance of the empirical batch mean. We analyze decay rates of the minimized upper bound towards the minimal estimator variance with respect to the predetermined computing budget, and provide convergence results as the computing budget increases progressively when the batch size is fixed. Numerical examples are provided to support theoretical findings and illustrate the effectiveness of the proposed batching procedure.

Data Fusion Methods for Indoor Positioning Systems Based on Channel State Information Fingerprinting.

Indoor signals are susceptible to NLOS propagation effects, multipath effects, and a dynamic environment, posing more challenges than outdoor signals despite decades of advancements in location services. In modern Wi-Fi networks that support both MIMO and OFDM techniques, Channel State Information (CSI) is now used as an enhanced wireless channel metric replacing the Wi-Fi received signal strength (RSS) fingerprinting method. The indoor multipath effects, however, make it less robust and stable. This study proposes a positive knowledge transfer-based heterogeneous data fusion method for representing the different scenarios of temporal variations in CSI-based fingerprint measurements generated in a complex indoor environment targeting indoor parking lots, while reducing the training calibration overhead. Extensive experiments were performed with real-world scenarios of the indoor parking phenomenon. Results revealed that the proposed algorithm proved to be an efficient algorithm with consistent positioning accuracy across all potential variations. In addition to improving indoor parking location accuracy, the proposed algorithm provides computationally robust and efficient location estimates in dynamic environments. A Cramer-Rao lower bound (CRLB) analysis was also used to estimate the lower bound of the parking lot location error variance under various temporal variation scenarios. Based on analytical derivations, we prove that the lower bound of the variance of the location estimator depends on the (i) angle of the base stations, (ii) number of base stations, (iii) distance between the target and the base station, djr (iv) correlation of the measurements, ρrjai and (v) signal propagation parameters σC and γ.

Multi‐agent consensus with stopping rules under bounded measurement noise

AbstractA multi‐agent consensus protocol is investigated over noisy undirected connected graphs. It is supposed that each agent can know her neighbors' states with bounded noise. A theoretical stopping rule having a probabilistic guarantee is established for the consensus protocol, which explicitly relates the number of iterations with the closeness of the agreement. The stopping rule is derived by utilizing Bernstein inequality that deals with a given noise bound explicitly. The sample complexity with respect to the number of agents is also investigated, where it is shown that the number of iterations required by the proposed stopping rule based on the bound of noise itself is generally smaller than that of the conventional stopping rules that utilize bound of noise variance. The results are demonstrated through numerical examples.

Magnetic Permeability Sensor Array Prototype to Evaluate Reservoir Phase Permeability in situ Downhole

A measurement system capable of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> measurements of formation permeability is described. The measurement method relies upon imaging the distribution of magnetic permeability fluid injected into a wellbore system through mutual inductance measurements between pairs of coils and relating the change in distribution of this fluid over time to a direct measurement of formation phase permeability. Analysis of the mutual inductance measurement method shows a unique response to distributions of magnetic permeability for various coil spacings and insensitivity to other confounding electrical parameters. The lower bound of measurement variance for this system is modeled for a continuum of coil spacings and radii of investigation for a perturbation in magnetic permeability.

SPCBPT

Bidirectional path tracing (BDPT) can be accelerated by selecting appropriate light sub-paths for connection. However, existing algorithms need to perform frequent distribution reconstruction and have expensive overhead. We present a novel approach, SPCBPT, for probabilistic connections that constructs the light selection distribution in sub-path space. Our approach bins the sub-paths into multiple subspaces and keeps the sub-paths in the same subspace of low discrepancy, wherein the light sub-paths can be selected by a subspace-based two-stage sampling method, i.e., first sampling the light subspace and then resampling the light sub-paths within this subspace. The subspace-based distribution is free of reconstruction and provides efficient light selection at a very low cost. We also propose a method that considers the Multiple Importance Sampling (MIS) term in the light selection and thus obtain an MIS-aware distribution that can minimize the upper bound of variance of the combined estimator. Prior methods typically omit this MIS weights term. We evaluate our algorithm using various benchmarks, and the results show that our approach has superior performance and can significantly reduce the noise compared with the state-of-the-art method.

Distributed ADMM With Synergetic Communication and Computation

In this article, we propose a novel distributed alternating direction method of multipliers (ADMM) algorithm with synergetic communication and computation, called SCCD-ADMM, to reduce the total communication and computation cost of the system. Explicitly, in the proposed algorithm, each node interacts with only part of its neighboring nodes, the number of which is progressively determined according to a heuristic searching procedure, which takes into account both the predicted convergence rate and the communication and computation costs at each iteration, resulting in a trade-off between communication and computation. Then the node chooses its neighboring nodes according to an importance sampling distribution derived theoretically to minimize the variance with the latest information it locally stores. Finally, the node updates its local information with a new update rule which adapts to the number of communication nodes. We prove the convergence of the proposed algorithm and provide an upper bound of the convergence variance brought by randomness. Extensive simulations validate the excellent performances of the proposed algorithm in terms of convergence rate and variance, the overall communication and computation cost, the impact of network topology as well as the time for evaluation, in comparison with the traditional counterparts.

Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance

We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis–Hastings and delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be competitive up to a factor depending on the supremum of the (marginal) IS weight. We elaborate upon the criterion in case of unbiased estimators as part of an auxiliary variable framework. We show how the criterion implies asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA corrections, essentially if the ratio of exact and approximate likelihoods is bounded. We also show that convergence of the IS chain can be less affected by unbounded high-variance unbiased estimators than PM and DA chains.

Lowering the Cramer-Rao lower bounds of variance in randomized response sampling

In this paper, we lower the Cramer-Rao Lower bound of variance due to Singh and Sedory (2011, 2012) for the Odumade and Singh (2009) model in the sense that we propose a randomized response model that is more efficient. We investigate the properties of the proposed model under various situations for protection and efficiency. The adjustment makes use of known proportion of unrelated characteristics. The situation where the proportions of the unrelated characteristics are unknown is also discussed. In addition, we wrote SAS codes to simulate various data sets in order to compute the simulated relative efficiency values.

Resampling-aware Weighting Functions for Bidirectional Path Tracing Using Multiple Light Sub-Paths

Bidirectional path tracing (BPT) with multiple importance sampling (MIS) is a popular technique for rendering realistic images. Recently, it has been shown that BPT can be improved by preparing multiple light sub-paths and by resampling a small number of light sub-paths from them to generate full paths with large contribution. Traditionally, for MIS weights, the balance heuristic has widely been used to minimize the upper bound of variance, where each full path is weighted in proportion to the probability of the path. Although the probability of the path can change due to the resampling process, the weighting functions used in the previous methods remain unaffected by the change in probability, resulting in less efficiency. To address this problem, we propose new weighting functions for BPT with multiple light sub-paths. Our main contribution is a precise formulation of the variance and the derivation of the weighting functions that can appropriately treat the change in probability. We demonstrate that our weighting functions significantly improve the image quality. We will release a simple version of our implementation as open source to ensure reproducibility.

Asymptotic algorithm for computing the sample variance of interval data

The problem of the sample variance computation for epistemic inter-val-valued data is, in general, NP-hard. Therefore, known efficient algorithms for computing variance require strong restrictions on admissible intervals like the no-subset property or heavy limitations on the number of possible intersections between intervals. A new asymptotic algorithm for computing the upper bound of the sample variance in a feasible time is proposed. Conditions required for its application with finite samples are discussed and some properties of the algorithm are also given. It appears that our new algorithm could be effectively applied in definitely more situations than methods used so far.

Improved Neymanian analysis for 2K factorial designs with binary outcomes

2K factorial designs are widely adopted by statisticians and the broader scientific community. In this short note, under the potential outcomes framework, we adopt the partial identification approach and derive the sharp lower bound of the sampling variance of the estimated factorial effects, which leads to an “improved” Neymanian variance estimator that mitigates the overestimation issue suffered by the classic Neymanian variance estimator.

Two-stage estimation for a normal mean having a known lower bound of variance with final sample size defined via Gini’s mean difference and mean absolute deviation

Revisiting Stein (1945, 1949) as well as Mukhopadhyay and Duggan (1997), we have proposed new two-stage procedures under both minimum risk point estimation and fixed-width confidence interval configurations for a normal mean μ when a lower bound of variance is known to us. New unbiased estimators based on sample standard deviation, Gini’s mean difference (GMD), and mean absolute deviance (MAD) are constructed to define the final sample sizes. The new procedures enjoy both asymptotic first-order and second-order properties, followed by simulated performances. Real data illustrations of the marigold data are also included.

Lower Bound of variance for Multipartite Structure of Hidden Markov Model

Lower Bound of variance for Multipartite Structure of Hidden Markov Model

Lower Bound of Variance for Multipartite Structure of Hidden Markov Model

Objective: This study derives an expression for obtaining lower bound of variance for estimates of hidden Markov model. Also, this study provides a parametric procedure for lower bound of variance. Methods/Statistical Analysis: In study a multipartite form of hidden Markov model considered. The states of model observed at time “2t-1” and “2t” respectively. The lower bound expression obtained by Louis methodology. The secondary data is used to study the validity of proposed lower bound. This dataset is a time series and named as “tree ring width”. Study also defines a parametric procedure for computing lower bound of variance. Findings: The study obtained the lower bound of variance for the maximum likelihood estimates of model using real-world data. The study compares the results of variance obtained by two procedures for various combinations of the states. It is found that for some combination of states, the lower bound of variance by two approaches found almost same. While for many combinations of states the lower bound of variance from proposed procedure found smaller than parametric procedure. The overall comparison lower bound of variance for maximum likelihood by proposed method explaining less variation in dataset then conventional i.e. parametric procedure. Conclusion: The study concluded that results for variance by derived expression found sharpen than those of parametric procedure. Keywords: Hidden Markov Model, Lower Bound Variance, Maximum Likelihood, Multipartite Structure, Parametric Procedure

Optimizing Adaptive Importance Sampling by Stochastic Approximation

We investigate the optimality of adaptive importance sampling by stochastic approximation. To address the well-known difficulty in the implementation of stochastic approximation, that is, the extreme performance sensitivity to the choice of decreasing learning rates, we focus on constant learning rates for a finite computing budget, derived via minimization of an upper bound of the theoretical variance of the empirical mean, not via accelerated convergence of the objective function as in the existing stochastic gradient framework. Strong convexity of the objective function lowers the upper bound of the theoretical variance, while the convexity parameter is not required for implementation. In contrast to existing schemes based on sample average approximation, the proposed approach based on stochastic approximation has the potential to run reasonably robust adaptive importance sampling with significantly less computing cost. We provide numerical results to support the theoretical findings and to illustrate ...

Potential Accuracy of Estimation of the Information Parameter of a Homodyne Laser Doppler Vibrometer Signal

This paper describes the analysis of the potential accuracy of estimation of the signal modulation index of a homodyne laser Doppler vibrometer with frequency modulation and known and unknown non-information parameters of the signal. The Rao — Cramer inequality is used to obtain expressions for calculating the lower bound of variance of the modulation index estimate and study its relationship with a signal/noise mixture. Recommendations on the choice of conditions for measuring the amplitude of vibro-displacement by the homodyne Doppler laser vibrometer are given.

Bayesian Cramer-Rao Lower Bound of Variances under Ranked Set Sampling*

Ranked set sampling (RSS) is regarded as a substitute of simple random sampling (SRS). Under SRS, obtained samples are onlyindependent, whereas in cases of balanced RSS, observations are independent order statistics. This paper presents a method to calculate the Bayesian Cramer Rao (BCR) bound when the statistical model is based on independent order statistics. This bound is also compared with independent simple random samples. In context of order samples where this classical Cramer Rao (CR) bound is hard to evaluate, this study presents a closed-form expression of a BCR bound. The efficiency in BCR bound due to independent order observations as compared to independent samples, is procure on the basis of theoretical and simulation results. Through this one can evaluate the significance of using independent order statistics as compare to independent samples for the gains in accuracy to estimate the parameters and reduction in required sample size. The procured results demonstrate that BCR bound based on independent order statistics is more compact, efficient observational economic than SRS based bound ofestimator.