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Journal articles on the topic 'Linear estimation problems'

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Author: Grafiati

Published: 14 March 2023

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1

Florens, Jean-Pierre, and Anna Simoni. "REGULARIZING PRIORS FOR LINEAR INVERSE PROBLEMS." Econometric Theory 32, no. 1 (November 6, 2014): 71–121. http://dx.doi.org/10.1017/s0266466614000796.

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This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior we propose the posterior mean as an estimator and prove frequentist consistency of the posterior distribution. The latter provides the frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new data-driven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions such that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting. Under these conditions our data-driven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.

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del Álamo, Miguel, and Axel Munk. "Total variation multiscale estimators for linear inverse problems." Information and Inference: A Journal of the IMA 9, no. 4 (March 2, 2020): 961–86. http://dx.doi.org/10.1093/imaiai/iaaa001.

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Abstract Even though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on a $d$-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of $BV$ functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating $BV$ functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the $L^q$-risk, for any $q\in [1,\infty )$. This is to the best of our knowledge the first convergence result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101–126) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$ in which the minimax rate is slower than $n^{-1/(d+2\beta +2)}$, where $\beta$ is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of $BV$ functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

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3

Ross, G. J. S. "Estimation problems of non-linear functional relationships." Journal of Applied Statistics 17, no. 3 (January 1990): 299–306. http://dx.doi.org/10.1080/02664769000000002.

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4

Koo, Ja-Yong, and Han-Yeong Chung. "Log-density estimation in linear inverse problems." Annals of Statistics 26, no. 1 (February 1998): 335–62. http://dx.doi.org/10.1214/aos/1030563989.

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5

Volaufová, Júlia. "Some estimation problems in multistage linear models." Linear Algebra and its Applications 388 (September 2004): 389–97. http://dx.doi.org/10.1016/j.laa.2004.03.007.

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6

Adjali, M. H., and M. Laurent. "Thermal conductivity estimation in non-linear problems." International Journal of Heat and Mass Transfer 50, no. 23-24 (November 2007): 4623–28. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2007.03.005.

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7

Ran, Mengfei, and Yihe Yang. "Optimal Estimation of Large Functional and Longitudinal Data by Using Functional Linear Mixed Model." Mathematics 10, no. 22 (November 17, 2022): 4322. http://dx.doi.org/10.3390/math10224322.

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The estimation of large functional and longitudinal data, which refers to the estimation of mean function, estimation of covariance function, and prediction of individual trajectory, is one of the most challenging problems in the field of high-dimensional statistics. Functional Principal Components Analysis (FPCA) and Functional Linear Mixed Model (FLMM) are two major statistical tools used to address the estimation of large functional and longitudinal data; however, the former suffers from a dramatically increasing computational burden while the latter does not have clear asymptotic properties. In this paper, we propose a computationally effective estimator of large functional and longitudinal data within the framework of FLMM, in which all the parameters can be automatically estimated. Under certain regularity assumptions, we prove that the mean function estimation and individual trajectory prediction reach the minimax lower bounds of all nonparametric estimations. Through numerous simulations and real data analysis, we show that our new estimator outperforms the traditional FPCA in terms of mean function estimation, individual trajectory prediction, variance estimation, covariance function estimation, and computational effectiveness.

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ODEN, J. TINSLEY, SERGE PRUDHOMME, TIM WESTERMANN, JON BASS, and MARK E. BOTKIN. "ERROR ESTIMATION OF EIGENFREQUENCIES FOR ELASTICITY AND SHELL PROBLEMS." Mathematical Models and Methods in Applied Sciences 13, no. 03 (March 2003): 323–44. http://dx.doi.org/10.1142/s0218202503002520.

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In this paper, a method for deriving computable estimates of the approximation error in eigenvalues or eigenfrequencies of three-dimensional linear elasticity or shell problems is presented. The analysis for the error estimator follows the general approach of goal-oriented error estimation for which the error is estimated in so-called quantities of interest, here the eigenfrequencies, rather than global norms. A general theory is developed and is then applied to the linear elasticity equations. For the shell analysis, it is assumed that the shell model is not completely known and additional errors are introduced due to modeling approximations. The approach is then based on recovering three-dimensional approximations from the shell eigensolution and employing the error estimator developed for linear elasticity. The performance of the error estimator is demonstrated on several test problems.

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9

С. И., Носков,, and Базилевский, М. П. "Multiple Lv-estimation of Linear Regression Models." Успехи кибернетики / Russian Journal of Cybernetics, no. 4(12) (December 28, 2022): 32–40. http://dx.doi.org/10.51790/2712-9942-2022-3-4-04.

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для оценки моделей множественной линейной регрессии существует много различных математических методов: наименьших квадратов, модулей, антиробастного оценивания, Lv-оценивания, множественного оценивания. Целью данной работы является обобщение указанных методов оценивания единой функцией потерь. Сначала была сформулирована задача оценивания, в которой в качестве критериев минимизации выступают критерии для антиробастного и Lv-оценивания. Недостатком сформулированной задачи является то, что для ее численного решения затруднительно определять начальные значения параметров, поскольку переменные могут иметь разные масштабы. Кроме того, функция потерь для этой задачи является неоднородной, что также затрудняет процесс оценивания. Для решения этих проблем введен новый критерий, равный критерию антиробастного оценивания, возведенному в степень v. С помощью него и функции потерь для Lv-оценивания сформулирована задача множественного Lv-оценивания. Функционал этой задачи является однородным, поэтому для проведения множественного Lv-оценивания целесообразно нормировать исходные переменные и переходить к оценкам стандартизованной линейной регрессии. Предложен алгоритм, по которому рекомендуется проводить множественное Lv-оценивание. В результате проведения множественного Lv-оценивания формируется множество, содержащее оценки линейной регрессии, полученные как известными методами, так и новыми. Правильный выбор наилучших из полученного множества оценок пока остается открытой научной задачей. С помощью предложенного множественного Lv-оценивания успешно решена задача моделирования железнодорожных пассажирских перевозок Иркутской области. there are many methods for estimating multiple linear regression models: ordinary least squares, least absolute deviations, anti-robust estimation, Lv-estimation, and multiple estimations. The purpose of this work is to generalize these methods by a loss function. First, an estimation problem was formulated where the minimization criteria are the anti-robust and Lv-estimations. The disadvantage of this problem statement is that it is difficult to determine the initial values of the parameters for a numerical solution, since the variables may have different scales. Besides, the loss function is non-uniform, which also complicates the estimation. To solve these problems, we introduced a new criterion, equal to the anti-robust estimation criterion raised to the power v. We stated the problem of multiple Lv-estimation using the new criterion and the loss function. The functional of this problem is homogeneous, therefore, for multiple Lv-estimations, it is advisable to normalize the initial variables and then apply the standardized linear regression estimates. We also developed an algorithm for multiple Lv-estimations. A result of such estimations is a set containing linear regression estimates obtained both by the existing and new methods. The optimal choice of the best estimates from the set of estimates remains an open problem. We successfully simulated the passenger railway traffic in the Irkutsk region with the proposed multiple Lv-estimations.

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10

Endtmayer, Bernhard, Ulrich Langer, and Thomas Wick. "Multigoal-oriented error estimates for non-linear problems." Journal of Numerical Mathematics 27, no. 4 (December 18, 2019): 215–36. http://dx.doi.org/10.1515/jnma-2018-0038.

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Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

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11

Dimakopoulou, Maria, Zhengyuan Zhou, Susan Athey, and Guido Imbens. "Balanced Linear Contextual Bandits." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3445–53. http://dx.doi.org/10.1609/aaai.v33i01.33013445.

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Contextual bandit algorithms are sensitive to the estimation method of the outcome model as well as the exploration method used, particularly in the presence of rich heterogeneity or complex outcome models, which can lead to difficult estimation problems along the path of learning. We develop algorithms for contextual bandits with linear payoffs that integrate balancing methods from the causal inference literature in their estimation to make it less prone to problems of estimation bias. We provide the first regret bound analyses for linear contextual bandits with balancing and show that our algorithms match the state of the art theoretical guarantees. We demonstrate the strong practical advantage of balanced contextual bandits on a large number of supervised learning datasets and on a synthetic example that simulates model misspecification and prejudice in the initial training data.

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12

Aalto, Atte. "Iterative observer-based state and parameter estimation for linear systems." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 1 (January 2018): 265–88. http://dx.doi.org/10.1051/cocv/2017005.

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We propose an iterative method for joint state and parameter estimation using measurements on a time interval [0, T] for systems that are backward output stabilizable. Since this time interval is fixed, errors in initial state may have a big impact on the parameter estimate. We propose to use the back and forth nudging (BFN) method for estimating the system’s initial state and a Gauss–Newton step between BFN iterations for estimating the system parameters. Taking advantage of results on the optimality of the BFN method, we show that for systems with skew-adjoint generators, the initial state and parameter estimate minimizing an output error cost functional is an attractive fixed point for the proposed method. We treat both linear source estimation and bilinear parameter estimation problems.

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13

Yücalar, Fatih, Deniz Kilinc, Emin Borandag, and Akin Ozcift. "Regression Analysis Based Software Effort Estimation Method." International Journal of Software Engineering and Knowledge Engineering 26, no. 05 (June 2016): 807–26. http://dx.doi.org/10.1142/s0218194016500261.

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Estimating the development effort of a software project in the early stages of the software life cycle is a significant task. Accurate estimates help project managers to overcome the problems regarding budget and time overruns. This paper proposes a new multiple linear regression analysis based effort estimation method, which has brought a different perspective to the software effort estimation methods and increased the success of software effort estimation processes. The proposed method is compared with standard Use Case Point (UCP) method, which is a well-known method in this area, and simple linear regression based effort estimation method developed by Nassif et al. In order to evaluate and compare the proposed method, the data of 10 software projects developed by four well-established software companies in Turkey were collected and datasets were created. When effort estimations obtained from datasets and actual efforts spent to complete the projects are compared with each other, it has been observed that the proposed method has higher effort estimation accuracy compared to the other methods.

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14

Pupeikis, R., and A. Žebrauskaite. "Determination of Sufficient Sample Size for Linear Estimation Problems." IFAC Proceedings Volumes 19, no. 5 (May 1986): 77–80. http://dx.doi.org/10.1016/s1474-6670(17)59771-2.

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15

Navarro-Moreno, J., and J. C. Ruiz-Molina. "An efficient algorithm for continuous-discrete linear estimation problems." IEEE Signal Processing Letters 8, no. 12 (December 2001): 310–12. http://dx.doi.org/10.1109/97.975877.

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16

Fu, Zhengqing, and Lanlan Guo. "Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems." Complexity 2019 (November 25, 2019): 1–9. http://dx.doi.org/10.1155/2019/4861708.

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This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

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17

Sbarbaro, Daniel, Marko Vauhkonen, and Tor Arne Johansen. "Linear inverse problems and state estimation: Regularization, Observability and Convergence." IFAC Proceedings Volumes 45, no. 25 (2012): 212–16. http://dx.doi.org/10.3182/20120913-4-it-4027.00027.

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18

Eshagh, M. "Variance component estimation in linear ill-posed problems: TSVD issue." Acta Geodaetica et Geophysica Hungarica 45, no. 2 (June 2010): 184–94. http://dx.doi.org/10.1556/ageod.45.2010.2.4.

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19

Fernandez-Alcala, R. M., J. Navarro-Moreno, and J. C. Ruiz-Molina. "A Unified Approach to Linear Estimation Problems for Nonstationary Processes." IEEE Transactions on Information Theory 51, no. 10 (October 2005): 3594–601. http://dx.doi.org/10.1109/tit.2005.855595.

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20

Alessandri, A., M. Baglietto, and G. Battistelli. "On Quadratic Boundedness to Solve Estimation Problems for Linear Systems." IFAC Proceedings Volumes 36, no. 11 (June 2003): 109–14. http://dx.doi.org/10.1016/s1474-6670(17)35648-3.

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21

Rump, S. M. "Estimation of the Sensitivity of linear and nonlinear algebraic problems." Linear Algebra and its Applications 153 (July 1991): 1–34. http://dx.doi.org/10.1016/0024-3795(91)90207-d.

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22

Brauer, H., O. Kosch, U. Tenner, H. Wiechmann, and A. Arlt. "A modified linear estimation approach for solving biomagnetic inverse problems." IEEE Transactions on Magnetics 32, no. 3 (May 1996): 1298–301. http://dx.doi.org/10.1109/20.497483.

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23

Ananyev, B. I. "On some estimation problems for nonlinear dynamic systems." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 4 (December 2021): 562–77. http://dx.doi.org/10.35634/vm210403.

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Two problems of nonlinear guaranteed estimation for states of dynamical systems are considered. It is supposed that unknown measurable in $t$ disturbances are linearly included in the equation of motion and are additive in the measurement equations. These disturbances are constrained by nonlinear integral functionals, one of which is analog of functional of the generalized work. The studied problem consists in creation of the information sets according to measurement data containing the true position of the trajectory. The dynamic programming approach is used. If the first functional requires solving a nonlinear equation in partial derivatives of the first order which is not always possible, then for functional of the generalized work it is enough to find a solution of the linear Lyapunov equation of the first order that significantly simplifies the problem. Nevertheless, even in this case it is necessary to impose additional conditions on the system parameters in order for the system trajectory of the observed signal to exist. If the motion equation is linear in state variable, then many assumptions are carried out automatically. For this case the issue of mutual approximation of information sets via inclusion for different functionals is discussed. In conclusion, the most transparent linear quadratic case is considered. The statement is illustrated by examples.

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24

Pricop-Jeckstadt, Mihaela. "Optimal indirect estimation for linear inverse problems with discretely sampled functional data." Inverse Problems 37, no. 12 (October 25, 2021): 125001. http://dx.doi.org/10.1088/1361-6420/ac2d76.

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Abstract Optimal mean estimation from noisy independent pathes of a stochastic process that are indirectly observed is an issue of great interest in functional inverse problems. In this paper, minimax rates of convergence for a class of linear inverse problems with correlated noise, general source conditions and various degrees of ill-posedness are proven in a continuous setting, when the pathes are entirely observed, and in a projected framework. The phase transition phenomenon characteristic to the functional data analysis appears also here and the thresholds that separate the sparse and the dense data set scenarios are computed for different smoothness conditions. The common design proves to be a special case of the independent design in view of the interpretation of the sampling properties via s-numbers and the price to pay for the data correlation turns out to be high. Finally. numerical experiments involving Abel’s integral operator illustrate the goodness-of-fit of the Tikhonov estimator in various scenarios reflecting the common and independent design as well as sparse and dense sampling.

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Xia, Youshen, and Mohamed S. Kamel. "A Cooperative Recurrent Neural Network for Solving L1 Estimation Problems with General Linear Constraints." Neural Computation 20, no. 3 (March 2008): 844–72. http://dx.doi.org/10.1162/neco.2007.10-06-376.

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The constrained L1 estimation is an attractive alternative to both the unconstrained L1 estimation and the least square estimation. In this letter, we propose a cooperative recurrent neural network (CRNN) for solving L1 estimation problems with general linear constraints. The proposed CRNN model combines four individual neural network models automatically and is suitable for parallel implementation. As a special case, the proposed CRNN includes two existing neural networks for solving unconstrained and constrained L1 estimation problems, respectively. Unlike existing neural networks, with penalty parameters, for solving the constrained L1 estimation problem, the proposed CRNN is guaranteed to converge globally to the exact optimal solution without any additional condition. Compared with conventional numerical algorithms, the proposed CRNN has a low computational complexity and can deal with the L1 estimation problem with degeneracy. Several applied examples show that the proposed CRNN can obtain more accurate estimates than several existing algorithms.

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26

Grove, William M. "Maximizing Heritability of a Linear Combination of Traits." Psychological Reports 75, no. 1 (August 1994): 467–76. http://dx.doi.org/10.2466/pr0.1994.75.1.467.

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In 1971 Jones proposed an approximate procedure for finding that linear combination of scores which has maximum heritability in a twin sample. I give an exact small-sample procedure. I point out two problems: such procedures can over-optimize the heritability by capitalizing on chance, and confidence intervals and significance tests are needed. I give an approach using James-Stein shrinkage estimation and bootstrapped standard errors to address these problems. It appears that confidence intervals may be quite broad. To reduce the width of the confidence intervals, one can accept some small-sample bias in exchange for smaller sampling errors. The James-Stein approach to estimating coefficients is used to achieve reduced confidence interval width. I illustrate with a computational example using personality data.

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27

Liauh, C. T., and R. B. Roemer. "A Semilinear State and Parameter Estimation Algorithm for Inverse Hyperthermia Problems." Journal of Biomechanical Engineering 115, no. 3 (August 1, 1993): 257–61. http://dx.doi.org/10.1115/1.2895484.

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An improved state and parameter estimation algorithm has been developed which decreases the total computational time required to accurately reconstruct complete hyperthermia temperature fields. Within this improved iterative estimation algorithm, if the change in the unknown perfusion parameters is small a linear approximation scheme is implemented in which the old Jacobian matrix (the sensitivity matrix) is used, instead of recalculating the new Jacobian matrix for the next iteration. In the hyperthermia temperature estimation problem the relationship between the temperature and the blood perfusion based on the bioheat transfer equation is generally nonlinear. However, the temperature can be approximated as a linear function of the blood perfusion over a certain range thus allowing this improved approach to work. Results show that if the temperature is approximated as a linear (or quasi-linear) function of the blood perfusion, the linearizing approach considerably reduces the CPU time required to accurately reconstruct the temperature field. The limiting case of implementing this approach is to calculate the Jacobian matrix for each iteration, which is identical to the approach used in the original nonlinear algorithm. Critical values of determining whether or not there is a need to recalculate the new Jacobian matrix during the iterations are presented for several inverse hyperthermia temperature estimation problems.

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Martinsson, Per-Gunnar, and Joel A. Tropp. "Randomized numerical linear algebra: Foundations and algorithms." Acta Numerica 29 (May 2020): 403–572. http://dx.doi.org/10.1017/s0962492920000021.

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This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues.Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.

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Hung, Yu-Heng, Ping-Chun Hsieh, Xi Liu, and P. R. Kumar. "Reward-Biased Maximum Likelihood Estimation for Linear Stochastic Bandits." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 7874–82. http://dx.doi.org/10.1609/aaai.v35i9.16961.

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Modifying the reward-biased maximum likelihood method originally proposed in the adaptive control literature, we propose novel learning algorithms to handle the explore-exploit trade-off in linear bandits problems as well as generalized linear bandits problems. We develop novel index policies that we prove achieve order-optimality, and show that they achieve empirical performance competitive with the state-of-the-art benchmark methods in extensive experiments. The new policies achieve this with low computation time per pull for linear bandits, and thereby resulting in both favorable regret as well as computational efficiency.

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30

Mahmoud, Magdi S. "Robust stability and ℋ∞-estimation for uncertain discrete systems with state-delay." Mathematical Problems in Engineering 7, no. 5 (2001): 393–412. http://dx.doi.org/10.1155/s1024123x01001703.

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In this paper, we investigate the problems of robust stability and ℋ∞-estimation for a class of linear discrete-time systems with time-varying norm-bounded parameter uncertainty and unknown state-delay. We provide complete results for robust stability with prescribed performance measure and establish a version of the discrete Bounded Real Lemma. Then, we design a linear estimator such that the estimation error dynamics is robustly stable with a guaranteed ℋ∞-performance irrespective of the parameteric uncertainties and unknown state delays. A numerical example is worked out to illustrate the developed theory.

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Pánek, David, Pavel Karban, Tamás Orosz, and Ivo Doležel. "Comparison of simplified techniques for solving selected coupled electroheat problems." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 39, no. 1 (January 8, 2020): 220–30. http://dx.doi.org/10.1108/compel-06-2019-0244.

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Purpose The purpose of this paper is to compare different reduced-order models for models of control of induction brazing process. In the presented application, the problem is to reconstruct temperature at the points of interests (hot spots) from information measured at accessible places. Design/methodology/approach The paper describes the process of induction brazing. It presents the full field model and evaluates the possibilities for obtaining reduced models for temperature estimation. The primary attention is paid to the model based on proper orthogonal decomposition (POD). Findings The paper shows that for the given application, it is possible to find low-order estimator. In the examined linear case, the best estimator was created using POD reduced model together with the linear Kalman filter. Research limitations/implications The authors are aware of two main limitations of the presented study: material properties are considered linear, which is not a completely realistic assumption. However, if strong coupling and nonlinear material parameters are considered, the model becomes unsolvable. The process and measurement uncertainties are not considered. Originality/value The paper deals with POD of multi-physics 3 D application of induction brazing. The paper compares 11 different methods for temperature estimator design.

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Navarro-Moreno, J., J. C. Ruiz-Molina, and M. J. Valderrama. "A solution to linear estimation problems using approximate Karhunen-Loeve expansions." IEEE Transactions on Information Theory 46, no. 4 (July 2000): 1677–82. http://dx.doi.org/10.1109/18.850715.

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33

Hoffmann, Marc, and Markus Reiss. "Nonlinear estimation for linear inverse problems with error in the operator." Annals of Statistics 36, no. 1 (February 2008): 310–36. http://dx.doi.org/10.1214/009053607000000721.

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34

Lai, P. Y., and Stephen M. S. Lee. "Estimation of central shapes of error distributions in linear regression problems." Annals of the Institute of Statistical Mathematics 65, no. 1 (April 17, 2012): 105–24. http://dx.doi.org/10.1007/s10463-012-0360-2.

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35

Nuzman, Carl J., and H. Vincent Poor. "Linear estimation of self-similar processes via Lamperti's transformation." Journal of Applied Probability 37, no. 2 (June 2000): 429–52. http://dx.doi.org/10.1239/jap/1014842548.

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Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

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36

Nuzman, Carl J., and H. Vincent Poor. "Linear estimation of self-similar processes via Lamperti's transformation." Journal of Applied Probability 37, no. 02 (June 2000): 429–52. http://dx.doi.org/10.1017/s0021900200015631.

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Abstract:

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

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37

Fritz, Marlon. "Improved Output Gap Estimates and Forecasts Using a Local Linear Regression." Engineering Proceedings 5, no. 1 (June 30, 2021): 32. http://dx.doi.org/10.3390/engproc2021005032.

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The output gap, the difference between potential and actual output, has a direct impact on policy decisions, e.g., monetary policy. Estimating this gap and its further analysis remain the subject of controversial debates due to methodological problems. We propose a local polynomial regression combined with a Self-Exciting Threshold AutoRegressive (SETAR) model and its forecasting extension for a systematic output gap estimation. Furthermore, local polynomial regression is proposed for the (multivariate) OECD production function approach and its reliability is demonstrated in forecasting output growth. A comparison of the proposed gap to the Hodrick–Prescott filter as well as to estimations by experts from the FED and OECD shows a higher correlation of our output gap with those from those economic institutions. Furthermore, sometimes gaps with different magnitude and different positions above or below the trend are observed between different methods. This may cause competing policy implications which can be improved with our results.

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38

Wu. "Solving Fuzzy Linear Programming Problems with Fuzzy Decision Variables." Mathematics 7, no. 7 (June 26, 2019): 569. http://dx.doi.org/10.3390/math7070569.

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The numerical method for solving the fuzzy linear programming problems with fuzzydecision variables is proposed in this paper. The difficulty for solving this kind of problem is thatthe decision variables are assumed to be nonnegative fuzzy numbers instead of nonnegative realnumbers. In other words, the decision variables are assumed to be membership functions. One of thepurposes of this paper is to derive the analytic formula of error estimation regarding the approximateoptimal solution. On the other hand, the existence of optimal solutions is also studied in this paper.Finally we present two numerical examples to demonstrate the usefulness of the numerical method.

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Johannes, Jan, Sébastien Van Bellegem, and Anne Vanhems. "CONVERGENCE RATES FOR ILL-POSED INVERSE PROBLEMS WITH AN UNKNOWN OPERATOR." Econometric Theory 27, no. 3 (October 11, 2010): 522–45. http://dx.doi.org/10.1017/s0266466610000393.

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This paper studies the estimation of a nonparametric functionϕfrom the inverse problemr=Tϕgiven estimates of the functionrand of the linear transformT. We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The two assumptions quantify the prior regularity ofϕand the prior link existing betweenTand the Hilbert scale. The approach provides a unified framework that allows us to compare various sets of structural assumptions found in the econometric literature. Moreover, general upper bounds are also derived for the risk of the estimator of the structural functionϕas well as that of its derivatives. It is shown that the bounds cover and extend known results given in the literature. Two important applications are also studied. The first is the blind nonparametric deconvolution on the real line, and the second is the estimation of the derivatives of the nonparametric instrumental regression function via an iterative Tikhonov regularization scheme.

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Ilyas, Muhammad, Agah D. Garnadi, and Sri Nurdiati. "Adaptive Mixed Finite Element Method for Elliptic Problems with Concentrated Source Terms." Indonesian Journal of Science and Technology 4, no. 2 (July 9, 2019): 263–69. http://dx.doi.org/10.17509/ijost.v4i2.18183.

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An adaptive mixed finite element method using the Lagrange multiplier technique is used to solve elliptic problems with delta Dirac source terms. The problem arises in the use of Chow-Anderssen linear functional methodology to recover coefficients locally in parameter estimation of an elliptic equation from a point-wise measurement. In this article, we used a posterior error estimator based on averaging technique as refinement indicators to produce a cycle of mesh adaptation, which is experimentally shown to capture singularity phenomena. Our numerical results showed that the adaptive refinement process successfully refines elements around the center of the source terms. The results also showed that the global error estimation is better than uniform refinement process in terms of computation time.

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Frumin, Leonid L. "Linear least squares method in nonlinear parametric inverse problems." Journal of Inverse and Ill-posed Problems 28, no. 2 (April 1, 2020): 307–12. http://dx.doi.org/10.1515/jiip-2019-0009.

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AbstractA generalization of the linear least squares method to a wide class of parametric nonlinear inverse problems is presented. The approach is based on the consideration of the operator equations, with the selected function of parameters as the solution. The generalization is based on the two mandatory conditions: the operator equations are linear for the estimated parameters and the operators have discrete approximations. Not requiring use of iterations, this approach is well suited for hardware implementation and also for constructing the first approximation for the nonlinear least squares method. The examples of parametric problems, including the problem of estimation of parameters of some higher transcendental functions, are presented.

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Ding, Litao, and Peter Mathé. "Minimax Rates for Statistical Inverse Problems Under General Source Conditions." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 603–8. http://dx.doi.org/10.1515/cmam-2017-0055.

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AbstractWe describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon [4]. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.

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Todorov, Emanuel. "Stochastic Optimal Control and Estimation Methods Adapted to the Noise Characteristics of the Sensorimotor System." Neural Computation 17, no. 5 (May 1, 2005): 1084–108. http://dx.doi.org/10.1162/0899766053491887.

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Optimality principles of biological movement are conceptually appealing and straightforward to formulate. Testing them empirically, however, requires the solution to stochastic optimal control and estimation problems for reasonably realistic models of the motor task and the sensorimotor periphery. Recent studies have highlighted the importance of incorporating biologically plausible noise into such models. Here we extend the linear-quadratic-gaussian framework—currently the only framework where such problems can be solved efficiently—to include control-dependent, state-dependent, and internal noise. Under this extended noise model, we derive a coordinate-descent algorithm guaranteed to converge to a feedback control law and a nonadaptive linear estimator optimal with respect to each other. Numerical simulations indicate that convergence is exponential, local minima do not exist, and the restriction to nonadaptive linear estimators has negligible effects in the control problems of interest. The application of the algorithm is illustrated in the context of reaching movements. A Matlab implementation is available at www.cogsci.ucsd.edu/∼todorov .

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Nakonechny, Alexander, Grigory Kudin, Petr Zinko, and Taras Zinko. "GUARANTEED ROOT-MEAN-SQUARE ESTIMATES OF LINEAR MATRIX TRANSFORMATIONS UNDER CONDITIONS OF STATISTICAL UNCERTAINTY." Journal of Automation and Information sciences 2 (March 1, 2021): 24–37. http://dx.doi.org/10.34229/1028-0979-2021-2-3.

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Linear estimation of observations in conditions of various types of interference in order to obtain unbiased estimates is the subject of research in numerous scientific publications. The problem of linear regression analysis in conditions when the elements of vector observations are known matrices that allow small deviations from the calculated ones was studied in previous publications of the authors. Using the technology of pseudo inverse operators, as well as the perturbation method, the problem was solved under the condition that linearly independent matrices are subject to small perturbations. The parameters of the linear estimates were presented in the form of expansions in a small parameter. Over the past decades, solving linear estimation problems under uncertainty has been carried out within the framework of the well-known minimax estimation method. Formally, the problems that arise in this direction are solved in the presence of some spaces for unknown observation parameters, as well as spaces to which observation errors may belong. The coefficients of the linear estimates are determined in the process of optimizing the guaranteed mean-square error of the desired estimate. Thus, the subject of scientific research can be problems of linear estimation of unknown rectangular matrices based on observations from errors with unknown correlation matrices of errors: unknown matrices belong to some bounded set, correlation matrices of random perturbations of the observation vector are unknown, but it is possible to assume cases when they belong to one or another defined bounded set. Some formulations of problems of linear estimation of observations are investigated in the proposed publication. The problem of linear estimation for a vector of observations of a special form is considered, the components of which are known rectangular matrices that are subject to small perturbations. Variants of the problem statement are proposed, which allow obtaining an analytical solution in the first approximation of a small parameter. A test example is presented.

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Nobahari, Hadi, and Saeed Nasrollahi. "A non-linear estimation and model predictive control algorithm based on ant colony optimization." Transactions of the Institute of Measurement and Control 41, no. 4 (February 2019): 1123–38. http://dx.doi.org/10.1177/0142331218798680.

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A new heuristic controller, called the continuous ant colony controller, is proposed for non-linear stochastic Gaussian/non-Gaussian systems. The new controller formulates the state estimation and the model predictive control problems as a single stochastic dynamic optimization problem, and utilizes a colony of virtual ants to find and track the best estimated state and the best control signal. For this purpose, an augmented state space is defined. An integrated cost function is also defined to evaluate the points of the augmented state space, explored by the ants. This function minimizes simultaneously the state estimation error, tracking error, control effort and control smoothness. Ants search the augmented state space dynamically in a similar scheme to the optimization algorithm, known as the continuous ant colony system. Performance of the new model predictive controller is evaluated for three non-linear problems. The problems are a non-linear continuous stirred tank reactor, a non-linear cart and spring system, and the attitude control of a non-linear quadrotor. The results verify successful performance of the proposed algorithm from both estimation and control points of view.

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Becker, Roland, Michael Innerberger, and Dirk Praetorius. "Adaptive FEM for Parameter-Errors in Elliptic Linear-Quadratic Parameter Estimation Problems." SIAM Journal on Numerical Analysis 60, no. 3 (June 2022): 1450–71. http://dx.doi.org/10.1137/21m1458077.

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Schneider, René, and Gerd Wachsmuth. "A Posteriori Error Estimation for Control-Constrained, Linear-Quadratic Optimal Control Problems." SIAM Journal on Numerical Analysis 54, no. 2 (January 2016): 1169–92. http://dx.doi.org/10.1137/15m1020460.

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48

Broemelin, Lyle, Abdullah Mat Yusoff, and Joaquin Diaz. "Some bayesian solutions for problems of adaptive estimation in linear dynamic systems." Communications in Statistics - Theory and Methods 14, no. 2 (January 1985): 401–18. http://dx.doi.org/10.1080/03610928508828921.

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Fletcher, Alyson K., Parthe Pandit, Sundeep Rangan, Subrata Sarkar, and Philip Schniter. "Plug in estimation in high dimensional linear inverse problems a rigorous analysis." Journal of Statistical Mechanics: Theory and Experiment 2019, no. 12 (December 20, 2019): 124021. http://dx.doi.org/10.1088/1742-5468/ab321a.

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Suliman, Mohamed A., Tarig Ballal, and Tareq Y. Al-Naffouri. "Perturbation-based regularization for signal estimation in linear discrete ill-posed problems." Signal Processing 152 (November 2018): 35–46. http://dx.doi.org/10.1016/j.sigpro.2018.05.005.

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