Optimal Linear Estimation Research Articles

An efficient semi-analytical extreme value method for time-variant reliability analysis

Time-variant reliability analysis plays a vital role in improving the validity and practicability of product reliability evaluation over a specific time interval. Sampling-based extreme value method is the most direct way to implement accurate reliability assessment. Its adoption for time-variant reliability analysis, however, is limited due to the computational burden caused by repeatedly evaluating performance function. This paper proposes a semi-analytical extreme value method to improve the computational efficiency of extreme value method. The time-variant performance function is transformed into dependent instantaneous performance functions in which the stochastic processes are discretized by the expansion optimal linear estimation method to simulate the dependence among different time instants. Each instantaneous function is separately approximated by Taylor series expansion at the most probable point through instantaneous reliability analysis. Based on the approximated performance functions, the computational cost of sampling-based extreme value method is significantly reduced. Results of three numerical examples demonstrate the efficacy of the proposed method.

Means, quasi-scalar product and covariance of fuzzy numbers

In this paper, we discuss extreme properties of means of fuzzy numbers. The proposed new definition quasi-scalar product and covariance between fuzzy numbers, and their properties are considered. As an application of the introduced concepts, the problem of optimal linear approximation of a fuzzy number by a given system of fuzzy numbers is considered.

Computational aspects of the kNN local linear smoothing for some conditional models in high dimensional statistics

We consider the problem of nonparametric estimation of certain conditional models when the explanatory variable is functional. The models studied are the Conditional Expectation (CE), the Conditional Distribution Function (CDF) and the Conditional Probability Density (CPD). The estimators are constructed by combining the local linear method and the k-Nearest Neighbors (kNN) smoothing approach. For each of the three models, we define an optimal estimator associated with the best number of neighbors chosen using two bandwidth selection procedures which are the cross-validation criterion and the bootstrap smoothing rule. The asymptotic properties of these optimal kNN Local Linear Estimators ( NN- LLE) are established under some standard conditions. The performances of the finite samples of these estimators are then examined and compared through a Monte-carlo study. In addition, an application on the riboflavin content in the yogurt using the Near-infrared curves is carried out to demonstrate the usefulness of the models proposed in the point-wise prediction as well as for the predictive interval.

A Scalable and Energy-Efficient IoT System Supported by Cell-Free Massive MIMO

An Internet-of-Things (IoT) system supports a massive number of IoT devices wirelessly. We show how to use cell-free (CF) massive multiple input and multiple output (MIMO) to provide a scalable and energy-efficient IoT system. We employ optimal linear estimation with random pilots to acquire channel state information (CSI) for MIMO precoding and decoding. In the uplink (UL), we employ optimal linear decoder and utilize random matrix (RM) theory to obtain two accurate signal-to-interference plus noise ratio (SINR) approximations involving only large-scale fading coefficients. We derive several max–min type power control algorithms based on both exact SINR expression and RM approximations. Next we consider the power control problem for downlink (DL) transmission. To avoid solving a time-consuming quasiconcave problem that requires repeat tests for the feasibility of a second-order cone programming (SOCP) problem, we develop a neural network (NN) aided power control algorithm that results in 30 times reduction in computation time. This power control algorithm leads to scalable CF Massive MIMO networks in which the amount of computations conducted by each access point (AP) does not depend on the number of network APs. Both UL and DL power control algorithms allow visibly improve the system spectral efficiency (SE) and, more importantly, lead to multifold improvements in energy efficiency (EE), which is crucial for IoT networks.

Estimator for Multirate Sampling Systems With Multiple Random Measurement Time Delays

The optimal linear estimation problem is studied for multirate sampling systems with multiple random measurement time delays (TDs). The considered multirate sampling scheme is that the sensor uniformly samples at a slow rate and the state uniformly updates at a fast rate. Known stochastic variable sequences obeying Bernoulli distributions are adopted to depict random measurement TDs, including missing measurements as a special case. First, using a state iterating method, the original system with multirate sampling and delayed measurements is transformed into a state-space model with single-rate sampling and delay-free measurements at measurement sampling (MS) instants. Then, by utilizing projection theory, a nonaugmented recursive optimal linear state filter is presented based on the established model in the linear minimum variance sense, where the estimators for the process noise are involved. Furthermore, the state estimator at state update instants is achieved through filtering or prediction based on the filter at MS instants. Finally, the centralized fusion estimator and the distributed covariance intersection fusion estimator are proposed for the multisensor case. Simulation research on a vehicle suspension system verifies the effectiveness of the algorithms.

Statistical inference of earlier origins for the first flaked stone technologies

Identifying when hominins first produced Lomekwian, Oldowan, and Acheulean technologies is vital to multiple avenues of human origins research. Yet, like most archaeological endeavors, our understanding is currently only as accurate as the artifacts recovered and the sites identified. Here we use optimal linear estimation (OLE) modelling to identify the portion of the archaeological record not yet discovered, and statistically infer the date of origin of the earliest flaked stone technologies. These models provide the most accurate framework yet for understanding when hominins first produced these tool types. Our results estimate the Oldowan to have originated 2.617 to 2.644 Ma, 36,000 to 63,000 years earlier than current evidence. The Acheulean’s origin is pushed back further through OLE, by at least 55,000 years to 1.815 to 1.823 Ma. We were unable to infer the Lomekwian’s date of origin using OLE, but an upper bound of 5.1 million years can be inferred using alternative nonparametric techniques. These dates provide a new chronological foundation from which to understand the emergence of the first flaked stone technologies, alongside their behavioral and evolutionary implications. Moreover, they suggest there to be substantial portions of the artifact record yet to be discovered.

Recursive optimal linear attitude estimator based on star sensor and gyro

Recursive optimal linear attitude estimator based on star sensor and gyro

Impact of spatial variability of shear wave velocity on the lagged coherency of synthetic surface ground motions

The spatial incoherence of ground motion during an earthquake can have a significant effect on the dynamic response of engineering structures such as bridges, dams, nuclear power plants and lifeline facilities. The main objective of this paper is to study the effect of anisotropic heterogeneities in a soil layer overlying homogeneous bedrock on the lagged coherency of surface ground motion. A set of numerical experiments is performed based on 2D spatial variability of shear-wave velocities modeled as a homogeneous stationary random field and discretized by the EOLE method (Expansion Optimal Linear Estimation). Seismic ground motions were simulated using FLAC2D software in the 1–25 Hz band for a plane wave excitation with SV polarization. The soil is characterized by horizontal and vertical autocorrelation distances ranging between 5 and 20 m and 1 and 2 m, respectively, and a coefficient of variation of the shear-wave velocity varying between 5% and 40%. The synthetic seismograms calculated for 9 parameter sets (100 realizations each) clearly show seismic waves scattering and surface waves diffracted locally by the ground heterogeneities, generating large spatial variations in coherence mainly controlled by the coefficient of variation of shear-wave velocity. Consistently with existing models and experimental data, the numerical coherency curves decrease with frequency and receiver distance, however at a rate which is lower than that observed in the experimental data. This difference is probably due to intrinsic attenuation that is not accounted for in the simulations and/or to our 2D simulations that do not reproduce the complete wavefield. The numerical average coherency curves for each parameter set exhibit maxima within narrow frequency bands caused by the vertically trapped body waves and surface wave propagation properties within the average ground model. This interpretation is supported by experimental data recorded in the Koutavos-Argostoli valley (Greece).

Modelling the end of the Acheulean at global and continental levels suggests widespread persistence into the Middle Palaeolithic

The Acheulean is the longest cultural tradition ever practised by humans, lasting for over 1.5 million years. Yet, its end has never been accurately dated; only broad 300–150 thousand years ago (Kya) estimates exist. Here we use optimal linear estimation modelling to infer the extinction dates of the Acheulean at global and continental levels. In Africa and the Near East the Acheulean is demonstrated to end between 174 and 166 Kya. In Europe it is inferred to end between 141 and 130 Kya. The Acheulean’s extinction in Asia occurs later (57–53 Kya), while global models vary depending on how archaeological sites are selected (107–29 Kya). These models demonstrate the Acheulean to have remained a distinct cultural tradition long after the inception of Middle Palaeolithic technologies in multiple continental regions. The complexity of this scenario mirrors the increasingly dynamic nature of the Middle Pleistocene hominin fossil record, suggesting contemporaneous hominin populations to have practised distinct stone-tool traditions.

Confidence-Based Design Optimization for a More Conservative Optimum Under Surrogate Model Uncertainty Caused by Gaussian Process

Abstract Even though many efforts have been devoted to effective strategies to build accurate surrogate models, surrogate model uncertainty is inevitable due to a limited number of available simulation samples. Therefore, the surrogate model uncertainty, one of the epistemic uncertainties in reliability-based design optimization (RBDO), has to be considered during the design process to prevent unexpected failure of a system that stems from an inaccurate surrogate model. However, there have been limited attempts to obtain a reliable optimum taking into account the surrogate model uncertainty due to its complexity and computational burden. Thus, this paper proposes a confidence-based design optimization (CBDO) under surrogate model uncertainty to find a conservative optimum despite an insufficient number of simulation samples. To compensate the surrogate model uncertainty in reliability analysis, the confidence of reliability is brought to describe the uncertainty of reliability. The proposed method employs the Gaussian process modeling to explicitly quantify the uncertainty of a surrogate model. Thus, metamodel-based importance sampling and expansion optimal linear estimation are exploited to reduce the computational burden on confidence estimation. In addition, stochastic sensitivity analysis of the confidence is developed for CBDO, which is formulated to find a conservative optimum than an RBDO optimum at a specific confidence level. Numerical examples using mathematical functions and finite element analysis show that the proposed confidence analysis and CBDO can prevent overestimation of reliability caused by an inaccurate surrogate model.

OPTIMAL DESIGNS FOR SERIES ESTIMATION IN NONPARAMETRIC REGRESSION WITH CORRELATED DATA

In this paper we investigate the problem of designing experiments for series estimators in nonparametric regression models with correlated observations. We use projection based estimators to derive an explicit solution of the best linear oracle estimator in the continuous time model for all Markovian-type error processes. These solutions are then used to construct estimators, which can be calculated from the available data along with their corresponding optimal design points. Our results are illustrated by means of a simulation study, which demonstrates that the new series estimator has a better performance than the commonly used techniques based on the optimal linear unbiased estimators. Moreover, we show that the performance of the estimators proposed in this paper can be further improved by choosing the design points appropriately.

Linear Optimal Estimation for Discrete-time and Continuous-time Systems with Multiple Measurement Delays

In this paper, we investigate the linear optimal estimation problems of discrete-time and continuous-time systems with multiple state delays in measurements. For discrete-time systems, we obtain the linear optimal estimation of state by direct calculation of optimal gain in terms of the solution to a retarded Riccati-like difference equation instead of a group of Riccati difference equations. For continuous-time systems, we also obtain the analytical expression of linear optimal estimation without resorting to Riccati partial differential equations. All the Riccati equations are of the same dimension as the system to be estimated and the computational cost is much saved. Infinite horizon case is also studied by stability analysis. Kalman filter can be recovered from our result when delays disappear. A numerical example is provided to demonstrate the results.

Fault estimation for discrete time‐variant systems subject to actuator and sensor saturations

AbstractThis article studies the H∞ fault estimation (FE) problem for linear discrete time‐variant systems with actuator and sensor saturations. To handle the saturation nonlinearities for FE problem, a pair of an auxiliary linear model and an associated performance function augmenting from the conventional H∞ performance index are constructed. Based on this pair, the original FE problem is readdressed as a two‐step optimization issue with an indefinite quadratic cost function. For a feasible optimization solution, the partial equivalence between a stationary point for a quadratic optimization problem and a projection for vectors in indefinite inner‐product space is utilized. A Krein‐space based inner product interpretation for the indefinite cost function is established and optimal linear estimation technique is employed to derive the stationary point of the aforementioned indefinite quadratic cost function. The existence condition of the fault estimator is explicitly obtained, and the fault is reconstructed analytically via a forward recursion algorithm. Two examples are given to show the effectiveness of the proposed FE scheme.

Oops! I Shrunk the Sample Covariance Matrix Again: Blockbuster Meets Shrinkage

Abstract Existing shrinkage techniques struggle to model the covariance matrix of asset returns in the presence of multiple-asset classes. Therefore, we introduce a Blockbuster shrinkage estimator that clusters the covariance matrix accordingly. Besides the definition and derivation of a new asymptotically optimal linear shrinkage estimator, we propose an adaptive Blockbuster algorithm that clusters the covariance matrix even if the (number of) asset classes are unknown and change over time. It displays superior all-around performance on historical data against a variety of state-of-the-art linear shrinkage competitors. Additionally, we find that for small- and medium-sized investment universes the proposed estimator outperforms even recent nonlinear shrinkage techniques. Hence, this new estimator can be used to deliver more efficient portfolio selection and detection of anomalies in the cross-section of asset returns. Furthermore, due to the general structure of the proposed Blockbuster shrinkage estimator, the application is not restricted to financial problems.

On the Usage of Deterministic (Related-Key) Truncated Differentials and Multidimensional Linear Approximations for SPN Ciphers

Among the few works realising the search of truncated differentials (TD) and multidimensional linear approximations (MDLA) holding for sure, the optimality of the distinguisher should be confirmed via an exhaustive search over all possible input differences/masks, which cannot be afforded when the internal state of the primitive has a considerable number of words. The incomplete search is also a long-term problem in the search of optimal impossible differential (ID) and zerocorrelation linear approximation (ZCLA) since all available automatic tools operate under fixed input and output differences/masks, and testing all possible combinations of differences/masks is impracticable for now. In this paper, we start by introducing an automatic approach based on the constraint satisfaction problem for the exploration of deterministic TDs and MDLAs. Since we transform the exhaustive search into an inherent feature of the searching model, the issue of incomplete search is settled. This tool is applied to search for related-key (RK) TDs of AES-192, and a new related-key differential-linear (DL) distinguisher is identified with a TD with certainty. Due to the novel property of the distinguisher, the previous RK DL attack on AES-192 is improved. Also, the new distinguisher is explained from the viewpoint of differentiallinear connectivity table (DLCT) and thus can be regarded as the first application of DLCT in the related-key attack scenario. As the second application of the tool, we propose a method to construct (RK) IDs and ZCLAs automatically. Benefiting from the control of the nonzero fixed differential pattern and the inherent feature of exhaustive search, the new searching scheme can discover longer distinguishers and hence possesses some superiorities over the previous methods. This technique is implemented with several primitives, and the provable security bounds of SKINNY and Midori64 against impossible differential distinguishing attack are generalised.

Experimental characterization of spatial variability for random field modeling on struts of additively manufactured lattice structures

The accuracy of the numerical predictions of lattice structures relies on the quantification of the uncertainties introduced by additive manufacturing (AM). In this study, an efficient random field discretization framework is proposed to characterize the spatial variability of geometric uncertainties on the strut members of the lattice structures fabricated by a specific AM process called material extrusion. Individual struts are fabricated with various printing angles and diameters using PLA material. Diameter values of the fabricated samples are measured along the printing direction and the radial direction of the cross-section at each layer under an optical microscope. Spatial correlations are characterized based on the measurements and the correlation lengths are evaluated. Two random field discretization methods, namely Karhunen-Loève expansion (KLE) and Expansion Optimal Linear Estimation (EOLE) methods are investigated to reduce the dimensionality of the random field discretization. The KLE method was found to give better accuracy in terms of error means while EOLE can produce better local accuracy. An optimal mesh size determination approach is introduced for the random field discretization with the expansion methods based on the number of expansion terms. The voxel-based strut models are generated with the diameter parameters modeled by the proposed random fields discretization framework. The comparison of the generated models with the fabricated struts shows that the AM-introduced variability in the material extrusion process on the struts is well represented by the random field modeling technique.

Optimal Linear Instrumental Variables Approximations

This paper studies the identification and estimation of the optimal linear approximation of a structural regression function. The parameter in the linear approximation is called the Optimal Linear Instrumental Variables Approximation (OLIVA). This paper shows that a necessary condition for standard inference on the OLIVA is also sufficient for the existence of an IV estimand in a linear model. The instrument in the IV estimand is unknown and may not be identified. A Two-Step IV (TSIV) estimator based on Tikhonov regularization is proposed, which can be implemented by standard regression routines. We establish the asymptotic normality of the TSIV estimator assuming neither completeness nor identification of the instrument. As an important application of our analysis, we robustify the classical Hausman test for exogeneity against misspecification of the linear structural model. We also discuss extensions to weighted least squares criteria. Monte Carlo simulations suggest an excellent finite sample performance for the proposed inferences. Finally, in an empirical application estimating the elasticity of intertemporal substitution (EIS) with US data, we obtain TSIV estimates that are much larger than their standard IV counterparts, with our robust Hausman test failing to reject the null hypothesis of exogeneity of real interest rates.

A New Method for Determining the Degree of Controllability of State Variables for the LQR Problem Using the Duality Theorem

The control performance of a dynamic system can be checked by the degree of controllability. In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. We carried out the calculation of the degree of controllability for the linear quadratic optimal control (LQR) problem using a duality theorem. Compared with the traditional measures of controllability such as determinant, trace, and maximal eigenvalue of the inverse controllability Gramian, the proposed degree of controllability was developed for each state variable and takes into account both the controllability Gramian and the cost function. The new method is convenient to apply to LQR problem. In the numerical simulation, we determined the influence of the model parameters on the degree of controllability. Besides that, we analyzed the degree of controllability, which gives an insight into the relationship between the system model design and the control performance.

Linear Regression Estimation Methods for Inferring Standard Values of Snow Load in Small Sample Situations

The aim of this paper is to establish a new method for inferring standard values of snow load in small sample situations. Due to the incomplete meteorological data in some areas, it is often necessary to infer the standard values of snow load in the conditions of small samples in engineering, but the point estimation methods of classical statistics adopted till now do not take into account the influences of statistical uncertainty, and the inference results are always aggressive. In order to overcome the above shortcomings, according to the basic principle of optimal linear unbiased estimation and invariant estimation of the minimum type I distribution parameters and the tantile, using the least square method, the linear regression estimation methods for inferring standard values of snow load in small sample situations are proposed, which can take into account two cases such as parameter-free and known coefficient of variation, and the predicted formulas of snow load standard values are given, respectively. Through numerical integration and Monte Carlo numerical simulation, the numerical table of correlation coefficients is established, which is more convenient for the direct application of inferential formulas. According to the results of theoretical analysis and examples, when using the indirect point estimation methods to infer the standard values of snow load in the conditions of small samples, the inference results are always small. The linear regression estimation method is suitable for inferring standard values of snow load in the conditions of small samples, which can give more reasonable results. When using the linear regression estimation to infer standard values of snow load in practical application, even if the coefficient of variation is unknown, it can set the upper limit value of the coefficient of variation according to the experience; meanwhile, according to the parameter-free and known coefficient of variation, the estimation is carried out, respectively, and the smaller value of the two is taken as the final estimate. The method can be extended to the statistical inference of variable load standard values such as wind load and floor load.

Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

We introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.