Unbiasedness Condition Research Articles

LRD spectral analysis of multifractional functional time series on manifolds

This paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (Fract Calc Appl Anal 25:1426–1458, 2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert–Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive–moving average (SPHARMA(p,q)) processes.

Over-the-Air Computation With Multiple Receivers: A Space-Time Approach

The emerging concept of Over-the-Air (OtA) computation has shown great potential for achieving resource-efficient data aggregation across large wireless networks. However, current research in this area has been limited to the standard many-to-one topology, where multiple nodes transmit data to a single receiver. In this study, we address the problem of applying OtA computation to scenarios with multiple receivers, and propose a novel communication design that exploits joint precoding and decoding over multiple time slots. To determine the optimal precoding and decoding vectors, we formulate an optimization problem that aims to minimize the mean squared error of the desired computations while satisfying the unbiasedness condition and power constraints. Our proposed multi-slot design is shown to be effective in saving communication resources (e.g., time slots) and achieving smaller estimation errors compared to the baseline approach of separating different receivers over time.

Doubly Robust Estimation for Correcting Position Bias in Click Feedback for Unbiased Learning to Rank

Clicks on rankings suffer from position bias: generally items on lower ranks are less likely to be examined—and thus clicked—by users, in spite of their actual preferences between items. The prevalent approach to unbiased click-based learning-to-rank (LTR) is based on counterfactual inverse-propensity-scoring (IPS) estimation. In contrast with general reinforcement learning, counterfactual doubly robust (DR) estimation has not been applied to click-based LTR in previous literature. In this article, we introduce a novel DR estimator that is the first DR approach specifically designed for position bias. The difficulty with position bias is that the treatment—user examination—is not directly observable in click data. As a solution, our estimator uses the expected treatment per rank, instead of the actual treatment that existing DR estimators use. Our novel DR estimator has more robust unbiasedness conditions than the existing IPS approach, and in addition, provides enormous decreases in variance: our experimental results indicate it requires several orders of magnitude fewer datapoints to converge at optimal performance. For the unbiased LTR field, our DR estimator contributes both increases in state-of-the-art performance and the most robust theoretical guarantees of all known LTR estimators.

Variable-sample method for the computation of stochastic Nash equilibrium

This article proposes a variable-sample method for the computation of stochastic stable Nash equilibrium, in which the objective functions are approximated, in each iteration, by the sample average approximation with different sample sizes. We start by investigating the contraction mapping properties under the variable-sample framework. Under some moderate conditions, it is shown that the accumulation points attained from the algorithm satisfy the first-order equilibrium conditions with probability one. Moreover, we use the asymptotic unbiasedness condition to prove the convergence of the accumulation points of the algorithm into the set of fixed points and prove the finite termination property of the algorithm. We also verify that the algorithm converges to the equilibrium even if the optimization problems in each iteration are solved inexactly. In the numerical tests, we comparatively analyze the accuracy error and the precision error of the estimators with different sample size schedules with respect to the sampling loads and the computational times. The results validate the effectiveness of the algorithm.

Bias-compensated augmented complex-valued NSAF algorithm and its low-complexity implementation

The performance of traditional adaptive filtering algorithms will be degraded when the input signal contains noise. To handle colored noisy input and output measurements in the complex domain, this paper proposes an augmented complex-valued estimation-input NSAF (ACEI-NSAF) algorithm, derived by using estimated noiseless input, which is obtained by the minimization of an instantaneous perturbation with both input and output data, to update the weight vectors. Benefiting from this, the proposed algorithm can achieve unbiased estimation of the unknown system in the presence of noisy inputs, thereby surpassing the augmented complex-valued NSAF (ACNSAF) algorithm. The mean stability, excess mean-square error, and computational complexity are investigated, and the unbiasedness condition of the proposed algorithm is presented. Subsequently, a low-complexity ACEI-NSAF (SACEI-NSAF) algorithm is introduced by using the subband selection approach. Experimental results in system identification demonstrate the effectiveness of the proposed algorithms and analysis.

Weighting estimation under bipartite incidence graph sampling

Bipartite incidence graph sampling provides a unified representation of many sampling situations for the purpose of estimation, including the existing unconventional sampling methods, such as indirect, network or adaptive cluster sampling, which are not originally described as graph problems. We develop a large class of design-based linear estimators, defined for the sample edges and subjected to a general condition of design unbiasedness. The class contains as special cases the classic Horvitz-Thompson estimator, as well as the other unbiased estimators in the literature of unconventional sampling, which can be traced back to Birnbaum et al. (1965). Our generalisation allows one to devise other unbiased estimators in future, thereby providing a potential of efficiency gains. Illustrations are given for adaptive cluster sampling, line-intercept sampling and simulated graphs.

Optimal discrete-time unbiased filtering for systems with unknown inputs

This paper concerns the H ∞ functional filtering design problem for discrete-time systems with unknown inputs. Our interest is focused on the unbiased filtering satisfying the H ∞ performance. First the unbiasedness conditions of the estimation error are presented. The dynamic of this error is dependent on the present and future values of the noise. Different cases are presented, the first one is based on the introduction of an algebraic constraint leading to the dynamic error dependent only on the present noise, the second case is more general and can be treated by modifying the performance index or by using the descriptor systems formulation. The conditions of the existence of an unbiased filter are presented and the designs of the different filters, based on the bounded real lemma, are given in the form of the linear matrix inequalities (LMI). The presented approach unifies the design of functional, reduced-order and full-order filters for discrete-time systems. Three numerical examples are presented to illustrate the approach described in the paper and shows the performance of the proposed functional filter.

Error-disturbance uncertainty relations in Faraday measurements

We examine error-disturbance relations in the quantum measurement of spin systems using an atom-light interface scheme. We model a single spin-1/2 system that interacts with a polarized light meter via a Faraday interaction. We formulate the error and disturbance of the model and examine the uncertainty relations. We find that, for the coherent light meter in pure polarization, both the error and disturbance behave like the cyclic oscillations due to the Faraday rotation in both the light and spin polarizations. We also examine a class of polarization-squeezed light meter, where we apply the phase-space approximation and characterize the role of squeezing. We derive the error-disturbance relations for these cases and find that the Heisenberg-Arthurs-Kelly uncertainty is violated while the tight Branciard-Ozawa uncertainty always holds. We note that, in the limit of weak interaction strength, the error and disturbance come to obey the unbiasedness condition and hence the Heisenberg-Arthurs-Kelly relation holds. The work contributes to our understanding of the quantum measurement of spin systems under the atom-light interface framework and may hold potential applications in quantum metrology, quantum state estimation, and control.

Embedded Unbiasedness: Effect on Optimal FIR Filtering Estimates

In this paper, we give an analysis of the embedded unbiasedness (EU) on optimal finite impulse response (OFIR) estimates. By minimizing the mean square error (MSE) constrained by the unbiasedness condition, a new OFIR-EU filter is derived. We show that the OFIR-EU filter does not require the initial conditions, and occupies an intermediate place between the UFIR and OFIR filters. It is also shown that the MSEs of the OFIR-EU and OFIR filters diminish with the estimation horizon. A numerical example has demonstrated that the OFIR-UE filter has better robustness against temporary model uncertainties than the OFIR and Kalman filters.

Bias-Compensated Sign Algorithm for Noisy Inputs and Its Step-Size Optimization

Employing the traditional least-mean-square (LMS) algorithm to estimate the weight vector of an unknown system will result in an estimation bias when the input signal of the adaptive filter is corrupted by noise. This paper proposes a bias-compensated sign algorithm (BC-SA) to address this problem. Specifically, an unbiasedness condition is employed to develop a compensation term for the classical sign algorithm (SA) to reduce the estimation bias caused by noisy inputs. The proposed BC-SA can not only reduce the estimation bias but also exhibit robustness against impulsive noise. Then the mean and mean-square performance of the BC-SA is analyzed based on Price's theorem under some frequently used statistical assumptions. Moreover, the step-size of the BC-SA is optimized based on the developed theoretical mean-square deviation (MSD). Simulation results are finally provided to evaluate the convergence performance of the proposed algorithm and to examine the theoretical findings.

Non-Bayesian Estimation Framework for Signal Recovery on Graphs

Graph signals arise from physical networks, such as power and communication systems, or as a result of a convenient representation of data with complex structure, such as social networks. We consider the problem of general graph signal recovery from noisy, corrupted, or incomplete measurements and under structural parametric constraints, such as smoothness in the graph frequency domain. In this paper, we formulate the graph signal recovery as a non-Bayesian estimation problem under a weighted mean-squared-error (WMSE) criterion, which is based on a quadratic form of the Laplacian matrix of the graph and its trace WMSE is the Dirichlet energy of the estimation error w.r.t. the graph. The Laplacian-based WMSE penalizes estimation errors according to their graph spectral content and is a difference-based cost function which accounts for the fact that in many cases signal recovery on graphs can only be achieved up to a constant addend. We develop a new Cram\'er-Rao bound (CRB) on the Laplacian-based WMSE and present the associated Lehmann unbiasedness condition w.r.t. the graph. We discuss the graph CRB and estimation methods for the fundamental problems of 1) A linear Gaussian model with relative measurements; and 2) Bandlimited graph signal recovery. We develop sampling allocation policies that optimize sensor locations in a network for these problems based on the proposed graph CRB. Numerical simulations on random graphs and on electrical network data are used to validate the performance of the graph CRB and sampling policies.

New H∞ dynamic observer design for time-delay systems subject to disturbances

This paper presents a new approach to design an H dynamic observer (DO) for multi-delayed linear systems subject to -norm disturbances. This observer generalises the existing results on the proportional observer (PO), the proportional integral observer (PIO), and the DO. The proposed design approach is derived from the solution of LMIs based on the parametrization results of algebraic constraints. These algebraic constraints can be easily obtained from the unbiasedness conditions of the estimation error. The obtained results are illustrated by a numerical example to show the performances of the proposed observer.

Cramér-Rao Bound Under Norm Constraint

The constrained Cramer-Rao bound (CCRB) is a benchmark for constrained parameter estimation. However, the CCRB unbiasedness conditions are too strict and thus, the CCRB may not be a lower bound for estimators under constraints. The recently developed Lehmann-unbiased-CCRB (LU-CCRB) was shown to be a lower bound for the commonly used constrained maximum likelihood (CML) estimator performance in cases where the CCRB is not. In constrained parameter estimation, the estimator is usually required to satisfy the constraints. However, the LU-CCRB is a lower bound for Lehmann-unbiased estimators that do not necessarily satisfy the constraints. In this letter, we consider the norm constraint and derive a novel bound, called norm-constrained CCRB (NC-CCRB), which is a lower bound on the mean-squared-error matrix trace of Lehmann-unbiased estimators that satisfy the norm constraint. The NC-CCRB is shown to be tighter than the LU-CCRB. In the simulations, we consider a linear estimation problem under norm constraint in which the proposed NC-CCRB better predicts the performance of the CML estimator than the CCRB trace and the LU-CCRB.

Optimal biased Kriging: Homeogram tapering and applications to geoid undulations in Korea

AbstractThis article studies the Optimal Biased Kriging (OBK) approach which is an alternative geostatistical method that gives up the unbiasedness condition of Ordinary Kriging (OK) to gain an improved Mean Squared Prediction Error (MSPE). The system of equations for the optimal linear biased Kriging predictor is derived and itsMSPE is compared with that of Ordinary Kriging. A major impediment in implementing this system of equations and performing Kriging interpolation with massive datasets is the inversion of the spatial coherency matrix. This problem is investigated and a novel method, called “homeogram tapering”, which exploits spatial sorting techniques to create sparse matrices for efficient matrix inversion, is described. Finally, as an application, results from experiments performed on a geoid undulation dataset from Korea are presented. A precise geoid is usually the indispensable basis for meaningful hydrological studies over wide areas. These experiments use the theory presented here along with a relatively new spatial coherency measure, called the homeogram, also known as the non-centered covariance function.

Cram$\acute{\text{e}}$r–Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

The constrained Cramer-Rao bound (CCRB) is a lower bound on the mean-squared-error (MSE) of estimators that satisfy some unbiasedness conditions. Although the CCRB unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood (CML) estimator, in the non-asymptotic region these conditions are usually too strict and the commonly used estimators, such as the CML estimator, do not satisfy them. Therefore, the CCRB may not be a lower bound on the MSE matrix of such estimators. In this paper, we propose a new definition for unbiasedness under constraints, denoted by C-unbiasedness, which is based on using Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account the parametric constraints. In addition to C-unbiasedness, a Cramer-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. This bound is a scalar bound that depends on the chosen weighted combination of estimation errors. It is shown that C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the set of estimators that satisfy the CCRB unbiasedness conditions is a subset of the set of C-unbiased estimators and the proposed LU-CCRB may be an informative lower bound in cases where the corresponding CCRB is not. In the simulations, we examine linear and nonlinear estimation problems under nonlinear constraints in which the CML estimator is shown to be C-unbiased and the LU-CCRB is an informative lower bound on the WMSE, while the corresponding CCRB on the WMSE is not a lower bound and is not informative in the non-asymptotic region.

The Use of Calibration Weighting for Variance Estimation Under Systematic Sampling: Applications to Forest Cover Assessment

The purpose of this note is to propose a variance estimator under non-measurable designs that exploits the existence of an auxiliary variable well correlated with the survey variable of interest. Under non-measurable designs, the Sen–Yates–Grundy variance estimator generates a downward bias that can be reduced using a calibration weighting based on the auxiliary variable. Conditions of approximate unbiasedness for the resulting calibration estimator are given. The application to systematic sampling is considered. The proposal proves to be effective for estimating the variance of the forest cover estimator in remote sensing-based surveys, owing to the strong correlation between the reference data, available from a systematic sample, and the satellite map data, available for the whole population and hence exploited as an auxiliary variable. Supplementary materials accompanying this paper appear online.

H∞ dynamic observer design for linear discrete-time systems

The objective of this paper is to propose a new form of H∞ dynamic observer for linear discrete-time systems. The widely used proportional observer (PO) and proportional integral observer (PIO) can be considered as particular cases of the proposed observer. The observer design is derived from the solution of linear matrix inequalities (LMIs), based on the solutions of algebraic constraints obtained from the unbiasedness conditions of estimation error. A numerical example is provided to show the performance of the proposed observer, compared with PO and PIO.

Linear Optimal Unbiased Filter for Time-Variant Systems Without Apriori Information on Initial Conditions

In this technical note, an optimal unbiased filter (OUF) is derived for time-variant systems to relax the initial condition assumption in Kalman filter (KF). By minimizing the mean square errors subject to the unbiasedness condition a solution is derived in a batch computation form first. To facilitate the on-line application, a recursive realization is further developed. The effect of removing the initial condition assumption on the estimation performance is analysed, and we show that the proposed algorithm converges to the KF asymptotically. Two-state harmonic model and four-state moving target tracking model are employed to demonstrate that the OUF can improve transient estimation performance significantly and can be used in place of the KF when the apriori information about the initial state values is not available.

Great expectations? evidence from Colombia’s exchange rate survey

In this paper, we use the largest exchange rate survey in Colombia to test for the rational expectations hypothesis, the presence of a time-varying risk premium and the accuracy of exchange rate forecasts. Our findings indicate that episodes of exchange rate appreciation preceded expectations of further appreciation in the short run, but were marked by depreciations in the long run. This reversal largely explains the stabilizing pattern of expectations. Additionally, we find that the forward discount differed from future exchange rate changes due to the rejection of the unbiasedness condition and to the presence of a time-varying risk premium. Finally, we find that only short run expectations were able to outperform a random walk process as well as models of extrapolative, adaptive, and regressive expectations. Long-run expectations, on the other hand, behaved poorly in terms of forecasting accuracy.

Unbiased, optimal, and in‐betweens: the trade‐off in discrete finite impulse response filtering

In this survey, the authors examine the trade-off between the unbiased, optimal, and in-between solutions in finite impulse response (FIR) filtering. Specifically, they refer to linear discrete real-time invariant state-space models with zero mean noise sources having arbitrary covariances (not obligatorily delta shaped) and distributions (not obligatorily Gaussian). They systematically analyse the following batch filtering algorithms: unbiased FIR (UFIR) subject to the unbiasedness condition, optimal FIR (OFIR) which minimises the mean square error (MSE), OFIR with embedded unbiasedness (EU) which minimises the MSE subject to the unbiasedness constraint, and optimal UFIR (OUFIR) which minimises the MSE in the UFIR estimate. Based on extensive investigations of the polynomial and harmonic models, the authors show that the OFIR-EU and OUFIR filters have higher immunity against errors in the noise statistics and better robustness against temporary model uncertainties than the OFIR and Kalman filters.