Mean Square Error Sense Research Articles

Multitask Diffusion Adaptation Over Networks

Adaptive networks are suitable for decentralized inference tasks. Recent works have intensively studied distributed optimization problems in the case where the nodes have to estimate a single optimum parameter vector collaboratively. However, there are many important applications that are multitask-oriented in the sense that there are multiple optimum parameter vectors to be inferred simultaneously, in a collaborative manner, over the area covered by the network. In this paper, we employ diffusion strategies to develop distributed algorithms that address multitask problems by minimizing an appropriate mean-square error criterion with l2-regularization. The stability and performance of the algorithm in the mean and mean-square error sense are analyzed. Simulations are conducted to verify the theoretical findings, and to illustrate how the distributed strategy can be used in several useful applications related to target localization and hyperspectral data unmixing.

A Class of Fast Exact Bayesian Filters in Dynamical Models With Jumps

We address the statistical filtering problem in dynamical models with jumps. When a particular application is adequately modeled by linear and Gaussian probability density functions with jumps, a usual method consists in approximating the optimal Bayesian estimate [in the sense of the minimum mean square error (MMSE)] in a linear and Gaussian jump Markov state space system (JMSS). Practical solutions include algorithms based on numerical approximations or on sequential Monte Carlo (SMC) methods. In this paper, we propose a class of alternative methods which consists in building statistical models which, locally, similarly model the problem of interest, but in which the computation of the MMSE estimate can be be computed exactly (without numerical nor SMC approximations) and at a computational cost which is linear in the number of observations.

Optimal periodic scheduling for remote state estimation under sensor energy constraint

The authors consider a sensor scheduling problem for a scalar system observed by two sensors. The measurements derived from the two local sensors are sent over a bandwidth-limited network to a remote estimator. The bandwidth constraint allows at most one packet can be sent at each time step. Upon receiving the data, the estimator computes the optimal estimate of the state in the minimum mean-squared error sense. In consideration of simplicity as well as practical and efficient implementation, the authors focus on periodic scheduling scheme. The authors assume that the sensor energy is so limited that in each period there exist open-loop predictions at some steps, which makes the problem more challenging. Using some novel tools, it was shown that the `as uniformly as possible' scheme, originally proposed in an early work for minimising the average estimation error covariance when sensor energy is just enough, still holds true in this limited energy case but the optimal steps of open-loop prediction depend on the system parameters. Such a relationship between scheduling rule and system parameters is novel and does not exist in early works where sensor energy is just enough. Numerical examples are provided to demonstrate the key ideas of the proposed work.

Estimation of the exponential distribution based on multiply Type I hybrid censored sample

The exponential distibution is one of the most popular distributions in analyzing the lifetime data. In this paper, we propose multiply Type I hybrid censoring. And this paper presents the statistical inference on the scale parameter for the exponential distribution when samples are multiply Type I hybrid censoring. The scale parameter is estimated by approximate maximum likelihood estimation methods using two different Taylor series expansion types (<TEX>$AMLE_I$</TEX>, <TEX>$AMLE_{II}$</TEX>). We also obtain the maximum likelihood estimator (MLE) of the scale parameter <TEX>${\sigma}$</TEX> under the proposed multiply Type I hybrid censored samples. We compare the estimators in the sense of the root mean square error (RMSE). The simulation procedure is repeated 10,000 times for the sample size n=20 and 40 and various censored schemes. The <TEX>$AMLE_{II}$</TEX> is better than <TEX>$AMLE_I$</TEX> in the sense of the RMSE.

A New Robust Optimization Model for the Vehicle Routing Problem with Stochastic Demands

A new robust model referred to as the conditional expectation-combinatorial semi-deviation robust model (CE-CSDRM) is proposed. The optimization objective adopted is minimization of the conditional expectation of the total transportation costs, subject to the constraint that the service capacity of the depot is maximized. This approach reduces the large deviations (in the sense of mean square error) encountered in the expectation-value semi-deviation robust model (E-SDRM). The robust measurement in E-SDRM only considers reducing the potential risk but ignores the loss of incidental benefits. Here, two adjustment factors are introduced. An ‘inner factor’ reduces both potential risk and lessens the loss of incidental benefits, thus the decision-maker can choose his or her own risk preferences. An ‘outer factor’ measures the infl uence on the optimization objective of the risk preference decided by the inner factor. A theoretical analysis shows that CE-CSDRM is a generalization of E-SDRM. Finally, the eff ectiveness of the new model is demonstrated using simulations.

State estimation for discrete-time Markov jump linear systems with multiplicative noises and delayed mode measurements

In this paper, the state estimation problem for discrete-time Markov jump linear systems affected by multiplicative noises is considered. The available measurements for the system under consideration have two components: the first is the model measurement and the second is the output measurement, where the model measurement is affected by a fixed amount of delay. Using Bayes' rule and some results obtained in this paper, a novel suboptimal state estimation algorithm is proposed in the sense of minimum mean-square error under a lot of Gaussian hypotheses. The proposed algorithm is recursive and does not increase computational and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed algorithm.

Systematic DFT Frames: Principle, Eigenvalues Structure, and Applications

Motivated by a host of recent applications requiring some amount of redundancy, frames are becoming a standard tool in the signal processing toolbox. In this paper, we study a specific class of frames, known as discrete Fourier transform (DFT) codes, and introduce the notion of systematic frames for this class. This is encouraged by a new application of frames, namely, distributed source coding that uses DFT codes for compression. Studying their extreme eigenvalues, we show that, unlike DFT frames, systematic DFT frames are not necessarily tight. Then, we come up with conditions for which these frames can be tight. In either case, the best and worst systematic frames are established in the minimum mean-squared reconstruction error sense. Eigenvalues of DFT frames and their subframes play a pivotal role in this work. Particularly, we derive some bounds on the extreme eigenvalues DFT subframes which are used to prove most of the results; these bounds are valuable independently.

An Efficient Pilot Design Scheme for Sparse Channel Estimation in OFDM Systems

This paper investigates the pilot placement problem for sparse channel estimation in orthogonal frequency division multiplexing (OFDM) systems. Prompted by the success of the compressed sensing technique in recovering sparse signals from undersampled measurements, compressed sensing has been successfully applied for pilot-aided sparse channel estimation in OFDM systems to reduce the transmitted overhead. However, the selection of pilot tones significantly affects channel estimation performance. Seeking optimal pilot placement for sparse channel estimation, in the sense of minimum mean-square error of the channel estimation, through an exhaustive search of all possible pilot placements is extremely computationally intensive. To reduce the computational complexity and simultaneously maximize the accuracy of sparse channel estimation, cross-entropy optimization is introduced to determine the optimal pilot placement. Computer simulation results demonstrate that the pilot index sequences obtained using the proposed method performed better compared with those obtained using the conventional equispaced scheme and the random search method.

Implementation of the "Non-Local Bayes" (NL-Bayes) Image Denoising Algorithm

This article presents a detailed implementation of the Non-Local Bayes (NL-Bayes) image denoising algorithm. In a nutshell, NL-Bayes is an improved variant of NL-means. In the NL-means algorithm, each patch is replaced by a weighted mean of the most similar patches present in a neighborhood. Images being mostly self-similar, such instances of similar patches are generally found, and averaging them increases the SNR. The NL-Bayes strategy improves on NL-means by evaluating for each group of similar patches a Gaussian vector model. To each patch is therefore associated a mean (which would be the result of NL-means), but also a covariance matrix estimating the variability of the patch group. This permits to compute an optimal (in the sense of Bayesian minimal mean square error) estimate of each noisy patch in the group, by a simple matrix inversion. The implementation proceeds in two identical iterations, but the second iteration uses the denoised image of the first iteration to estimate better the mean and covariance of the patch Gaussian models. A discussion of the algorithm shows that it is close in spirit to several state of the art algorithms (TSID, BM3D, BM3D-SAPCA), and that its structure is actually close to BM3D. Thorough experimental comparison made in this paper also shows that the algorithm achieves the best state of the art on color images in terms of PSNR and image quality. On grey level images, it reaches a performance similar to the more complex BM3D-SAPCA (no color version is available for this last algorithm).

Covariance tapering for prediction of large spatial data sets in transformed random fields

The best linear unbiased predictor (BLUP) is called a kriging predictor and has been widely used to interpolate a spatially correlated random process in scientific areas such as geostatistics. However, if an underlying random field is not Gaussian, the optimality of the BLUP in the mean squared error (MSE) sense is unclear because it is not always identical with the conditional expectation. Moreover, in many cases, data sets in spatial problems are often so large that a kriging predictor is impractically time-consuming. To reduce the computational complexity, covariance tapering has been developed for large spatial data sets. In this paper, we consider covariance tapering in a class of transformed Gaussian models for random fields and show that the BLUP using covariance tapering, the BLUP and the optimal predictor are asymptotically equivalent in the MSE sense if the underlying Gaussian random field has the Matérn covariance function.

Remodeling and Estimation for Sparse Partially Linear Regression Models

When the dimension of covariates in the regression model is high, one usually uses a submodel as a working model that contains significant variables. But it may be highly biased and the resulting estimator of the parameter of interest may be very poor when the coefficients of removed variables are not exactly zero. In this paper, based on the selected submodel, we introduce a two-stage remodeling method to get the consistent estimator for the parameter of interest. More precisely, in the first stage, by a multistep adjustment, we reconstruct an unbiased model based on the correlation information between the covariates; in the second stage, we further reduce the adjusted model by a semiparametric variable selection method and get a new estimator of the parameter of interest simultaneously. Its convergence rate and asymptotic normality are also obtained. The simulation results further illustrate that the new estimator outperforms those obtained by the submodel and the full model in the sense of mean square errors of point estimation and mean square prediction errors of model prediction.

Joint Cooperative-Transmit and Receive FDE for Single-Carrier Incremental Relaying

In this paper, we propose joint cooperative-transmit/receive frequency-domain equalization (cooperative-Tx/Rx FDE) with incremental relaying (IR) for broadband single-carrier (SC) transmission under the total and individual transmit power constraint at a source node (S) and a relay node (R). We derive the optimum cooperative-Tx/Rx FDE weights to minimize the mean square error (MSE) after packet combining at a destination node (\ssr D). We show that the optimum cooperative-Tx FDE weights allocate the transmit power in the minimum MSE (MMSE) sense in the frequency domain and in the maximal ratio transmission (MRT) sense in the spatial domain. To relax the condition that the complete channel state information (CSI) needs to be shared among all nodes, selection-based suboptimum cooperative-Tx FDE weights are derived. Computer simulation verifies the effectiveness of the proposed schemes and shows that the 5%-outage throughput of the system can be increased by around 30% over the conventional IR with only Rx FDE.

Automated landmark points detection by using a mixture of approaches: the vole-teeth case

This paper deals with the automated detection of a closed curve’s dominant points. We treat a curve as a 1-D function of the arc length. The problem of detecting dominant points is translated into seeking the extrema of the corresponding 1-D function. Three approaches for automated dominant points detection are presented: (1) an approach based on fitting polynomial, (2) an approach using 1-D computer registration and (3) an innovative approach based on a multi-resolution scheme, zero-crossing and hierarchical clustering. Afterwards, two methods are introduced based on the linearly and non-linearly mixing the results from the three approaches. We then mix the results in a mean-square error sense by using the linear and non-linear fittings, respectively. We experimentally demonstrate the problem of detecting 21 landmarks on 38 vole-teeth that by mixing, the detection accuracy is improved by up to 41.47 % with respect to the results for individual approaches, as applied within the mixture.

Tissue similarity maps (TSMs): A new means of mapping vascular behavior and calculating relative blood volume in perfusion weighted imaging

This study introduces a new processing means that uses the original signal (rather than contrast agent concentration) from dynamic susceptibility contrast (DSC) perfusion weighted imaging (PWI) to calculate a relative cerebral blood volume map and a tissue similarity map (TSM). Ten healthy volunteers and eight multiple sclerosis (MS) patients were studied using high resolution PWI. The TSM is found by choosing a reference region in one slice and comparing its signal in a mean squared error sense to the signal from every pixel in all images throughout the brain. The TSMs provide a means to determine which tissues have similar flow characteristics with high contrast and signal-to-noise ratios. The effective blood volume measured from this approach is nearly identical to that from conventional relative cerebral blood volume (rCBV) maps but with better signal-to-noise. Of interest is the fact that choosing one MS lesion as the reference tissue appears to be enough to find nearly all lesions throughout the brain. That is, these lesions all behave the same from a vascular point of view. The TSM results are robust within and across slices properly nulling the same type of tissue throughout the brain for a given reference region. TSM derived rCBV agrees well with the conventional derived rCBV using contrast agent concentration. TSM may provide a new means to study similarities between blood flow patterns in tissue in the brain and in better diagnosing vascular differences between tissues and lesions.

On the Multidimensional Signal Constellation Design for Few-Mode-Fiber-Based High-Speed Optical Transmission

In this paper, we propose an algorithm to determine the -dimensional optimum signal constellation design (DD-OSCD), in minimum mean-square error sense, for channel capacity achieving source distribution. We then describe a -dimensional coded modulation scheme, employing orbital angular momentum modes, two polarization states, and two in-phase/quadrature coordinates, suitable for use in future few-mode fiber (FMF) applications. Monte Carlo simulations indicate that the proposed low-density parity-check (LDPC)-coded polarization-division-multiplexed (PDM) quadrature-multiplexed (QM) DD-OSCD significantly outperforms conventional PDM-QAM (by 4.32 dB for 64-ary 3-D constellation at a BER of ). In addition, the DD-OSCD outperforms sphere packing constellations (by 0.33 dB for 8-ary 3-D constellation at a BER of ).

Iterative Channel Estimation with Frequency Replacement for SC-FDMA Systems

Single-carrier frequency division multiple access (SC-FDMA) is an attractive air interface scheme for broadband wireless communications. For the coherent demodulation of SC-FDMA systems, the receiver performs channel estimation to obtain the channel frequency response by using the reference signals multiplexed in the transmitted signal. In this paper, we propose an iterative frequency domain channel estimation scheme for SC-FDMA systems. To prevent severe noise enhancements due to the Gaussianity of the frequency domain data signal, we consider two low complexity algorithms, namely, hard and soft frequency replacements. We first analytically derive the mean square error (MSE) to characterize the effects of frequency replacement and post iterative filtering. By analyzing MSE performance, we present a design framework to develop the optimal frequency replacement. Based on the design framework, hard and soft frequency replacement algorithms are optimized in the MSE sense. Then, their low complexity adaptation methods are proposed for practical SC-FDMA systems, where the proposed frequency replacement is performed according to the reliability of the data estimates. Simulation results show that the proposed iterative channel estimation techniques effectively compensate for noise enhancements, and thus give good MSE and frame error rate performances.

OFDM Channel Estimation Based on Impulse Response Decimation: Analysis and Novel Algorithms

In this paper, OFDM data-aided channel estimation based on the decimation of the Channel Impulse Response (CIR) through the selection of the Most Significant Samples (MSS) is addressed. Our aim is to approach the Minimum Mean Square Error (MMSE) channel estimation performance, while avoiding the need for a-priori knowledge of channel statistics (KCS). The optimal set of samples is defined in the instantaneous and average senses. We derive lower bounds on the estimation mean-square error (MSE) performance for any MSS selection strategy. We show how MSS-based channel estimation can approach these MSE lower bounds. We introduce novel MSS strategies oriented towards instantaneous decimation (Instantaneous Energy Selection - IES), and windowed decimation (Average Energy Selection - AES). We also consider decimation via Threshold Crossing Selection (TCS), which we characterize analytically, to derive the optimum threshold in the minimum MSE sense. We also propose a sub-optimal method for threshold setting that does not require KCS. Finally, we provide numerical results in terms of both MSE estimation performance and Bit Error Rate (BER) of a coded OFDM system using the proposed channel estimators, to show that they indeed approach MMSE performance.

Steady State and Transient MSE Analysis of Convexly Constrained Mixture Methods

We investigate convexly constrained mixture methods to adaptively combine outputs of two adaptive filters running in parallel to model a desired unknown system. We compare several algorithms with respect to their mean-square error in the steady state, when the underlying unknown system is nonstationary with a random walk model. We demonstrate that these algorithms are universal such that they achieve the performance of the best constituent filter in the steady state if certain algorithmic parameters are chosen properly. We also demonstrate that certain mixtures converge to the optimal convex combination filter such that their steady-state performances can be better than the best constituent filter. We also perform the transient analysis of these updates in the mean and mean-square error sense. Furthermore, we show that the investigated convexly constrained algorithms update certain auxiliary variables through sigmoid nonlinearity, hence, in this sense, related.

A Weighted Inner Product Estimator in the Geometric Algebra of Euclidean 3-Space for Source Localization Using an EM Vector-sensor

In this paper, the source localization by utilizing the measurements of a single electromagnetic (EM) vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space. In order to describe the orthogonality among the electric and magnetic measurements, two multivectors of the geometric algebra of Euclidean 3-space (G3) are used to model the outputs of a spatially collocated EM vector-sensor. Two estimators for the wave propagation vector estimation are then formulated by the inner product between a vector and a bivector in the G3. Since the information used by the two estimators is different, a weighted inner product estimator is then proposed to fuse the two estimators together in the sense of the minimum mean square error (MMSE). Analytical results show that the statistical performances of the weighted inner product estimator are always better than its traditional cross product counterpart. The efficacy of the weighted inner product estimator and the correctness of the analytical predictions are demonstrated by simulation results.

Predetection Fusion With Doppler Measurements and Amplitude Information

In previous work, we discussed an efficient form of predetection fusion for use as a preprocessing step before tracking on data sets with large sensor networks of low-quality sensors, with particular eye to application in multistatic sonar. This 2-D version (position measurements only) was compared against an optimal (slow) technique. In this work, we present the 4-D (using position and Doppler measurements) and 5-D versions [using position, Doppler, and also aspect-dependent signal-to-noise ratio (SNR) measurements] of predetection fusion. We demonstrate that improved results, in the sense of root mean square error (RMSE) and number of declared targets-and consequently better tracking results-are possible when Doppler measurements and SNR information are incorporated into our algorithm.