Linear Function Approximation Research Articles

Variable selection for recurrent event data with informative censoring

Recurrent events data with a terminal event (e.g., death) often arise in clinical and observational studies. Variable selection is an important issue in all regression analysis. In this paper, the authors first propose the estimation methods to select the significant variables, and then prove the asymptotic behavior of the proposed estimator. Furthermore, the authors discuss the computing algorithm to assess the proposed estimator via the linear function approximation and generalized cross validation method for determination of the tuning parameters. Finally, the finite sample estimation for the asymptotical covariance matrix is also proposed.

Confidence Bounds on Design Variables Using High-Dimensional Model Representation–Based Inverse Reliability Analysis

AbstractThis paper presents a novel solution procedure for inverse reliability problems with implicit response functions without requiring the derivatives of the response functions with respect to uncertain variables. This can be used to determine the unknown design parameters such that the prescribed reliability indexes are attained in the presence of mixed uncertain (both random and fuzzy) variables. The proposed computational procedure involves three steps: (1) probability of failure calculation using high-dimensional model representation (HDMR) for the limit state/performance function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral, and fast-Fourier transform for solving the convolution integral; (2) reliability index update; and (3) most probable point update. The limit state/performance function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. This...

Comparison of Linear and Nonlinear Programming Techniques for Animal Diet

Linear programming techniques have been extensively used for animal diet formulation for more than last fifty years. To overcome the drawback of linear approximation of objective function for diet formulation, a mathematical model based on nonlinear programming technique is proposed to measure animal performance in terms of milk yield and weight gain. At the second step, it compares the result of proposed program with that of linear programming model. Result of proposed model gives better results using nonlinear programming. Thus the study is an attempt to develop a nonlinear programming model for optimal planning and best use of nutrient ingredients.

Improved Approximation of Linear Threshold Functions

We prove two main results on how arbitrary linear threshold functions $${f(x) = {\rm sign}(w \cdot x - \theta)}$$ over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is $${\epsilon}$$ -close to a threshold function depending only on $${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$$ many variables, where $${{\rm Inf}(f)}$$ denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut's well-known theorem (Friedgut in Combinatorica 18(1):474---483, 1998), which states that every Boolean function f is $${\epsilon}$$ -close to a function depending only on $${2^{O({\rm Inf}(f)/\epsilon)}}$$ many variables, for the case of threshold functions. We complement this upper bound by showing that $${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$$ many variables are required for $${\epsilon}$$ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is $${\epsilon}$$ -close to a threshold function with integer weights at most $${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$$ This is an improvement, in the dependence on the error parameter $${\epsilon}$$ , on an earlier result of Servedio (Comput Complex 16(2):180---209, 2007) which gave a $${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180---209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.

A dual-kernel-based tracking approach for visual target

A dual-kernel-based tracking approach for visual target is proposed in this paper. The similarity between candidate and target model, and the contrast between candidate and its neighboring background are considered simultaneously when evaluating a target candidate. The similarity is measured by Bhattacharyya coefficient while the contrast is calculated with Jensen-Shannon divergence, and they are adaptively fused into a novel objective function. By maximizing the linear approximation of objective function, a dual-kernel target location-shift relation from current location to a new location is induced. According to the location-shift relation, the optimal target location can be recursively gained in the mean shift procedure. Experimental evaluations on several image sequences demonstrate that the proposed algorithm can gain more accurate target location and better identification power to false target, and it is also robust to deformation and partial occlusion.

Adaptive wavelet methods for the stochastic Poisson equation

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝ d . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.

Inverse structural reliability analysis under mixed uncertainties using high dimensional model representation and fast Fourier transform

This paper presents a novel solution procedure for inverse reliability problems with implicit response functions without requiring the derivatives of the response functions with respect to the uncertain variables, that can be used to determine the unknown design parameters such that prescribed reliability indices are attained in the presence of mixed uncertain (both random and fuzzy) variables. The proposed computational procedure involves three steps: (i) probability of failure calculation using High Dimensional Model Representation (HDMR) for the limit state/performance function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral, and fast Fourier transform for solving the convolution integral, (ii) reliability index update, and (iii) most probable point MPP update. The limit state function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. This is a versatile method that can solve even highly nonlinear problems or the problems with multiple parameters. The methodology developed is applicable for inverse reliability analysis involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through four examples involving explicit/implicit performance functions.

Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming

We consider the classical finite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm for finding the optimal state costs or Q-factors. The main difference is in the policy evaluation phase: instead of solving a linear system of equations, our algorithm requires solving an optimal stopping problem. The solution of this problem may be inexact, with a finite number of value iterations, in the spirit of modified policy iteration. The stopping problem structure is incorporated into the standard Q-learning algorithm to obtain a new method that is intermediate between policy iteration and Q-learning/value iteration. Thanks to its special contraction properties, our method overcomes some of the traditional convergence difficulties of modified policy iteration and admits asynchronous deterministic and stochastic iterative implementations, with lower overhead and/or more reliable convergence over existing Q-learning schemes. Furthermore, for large-scale problems, where linear basis function approximations and simulation-based temporal difference implementations are used, our algorithm addresses effectively the inherent difficulties of approximate policy iteration due to inadequate exploration of the state and control spaces.

Gardner&Lockman Model (2000) and its Application in Numerical Analysis of Composite Beams

The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. On the basis of the theory of the viscoelastic body of Arutyunian–Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time “t”, two independent Volterra integral equations of the second kind have been derived. The creep functions is suggested by the Gardner&Lockman prediction model for time dependent deformations of concrete and compared with FIP CEB MC90-99 model. The elastic modulus of concrete Ec(t) is assumed to be constant in time “t”. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated

Optimizing direct response in Internet display advertising

Internet display advertising has grown into a multi-billion dollar a year global industry and direct response campaigns account for about three-quarters of all Internet display advertising. In such campaigns, advertisers reach out to a target audience via some form of a visual advertisement (hereinafter also called “ad”) to maximize short-term sales revenue. In this study, we formulate an advertiser’s revenue maximization problem in direct response Internet display advertisement campaigns as a mixed integer program via piecewise linear approximation of the revenue function. A novelty of our approach is that ad location and content issues are explicitly incorporated in the optimization model. Computational experiments on a large-scale actual campaign indicate that adopting the optimal media schedule can significantly increase advertising revenues without any budget changes, and reasonably sized instances of the problem can be solved within short execution times.

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L 2 norm, and study their approximation properties over Hilbert subspaces of L 2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of M -estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.

Optimal Design of a Thomson-Coil Actuator Utilizing a Mixed-Integer-Discrete-Continuous Variables Global Optimization Algorithm

To improve the performance of a Thomson-coil actuator, a new global optimization algorithm is proposed with mixed integer-discrete-continuous variables. In the algorithm, an augmented objective function is constructed by introducing a penalty function and an extended function to treat the discrete quantities as continuous ones. For the discrete objective function, two kinds of continuous function approximation methods, step function and linear function approximations, are suggested. The coupling particles swarm optimization method is then applied to the augmented objective function to find all the global and local optimal points. Finally, the validity of the proposed method is proven by comparing obtained optimization results with those from the enumeration algorithm.

Adaptive multiresolution analysis based on anisotropic triangulations

A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between the approximated function and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the Lp norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of of the approximated function in case of C2 functions).

Value Function Approximation in Reinforcement Learning Using the Fourier Basis

We describe the Fourier basis, a linear value function approximation scheme based on the Fourier series. We empirically demonstrate that it performs well compared to radial basis functions and the polynomial basis, the two most popular fixed bases for linear value function approximation, and is competitive with learned proto-value functions.

Local Monte Carlo Simulation for the Reliability Sensitivity Analysis

An efficient numerical technique, namely the Local Monte Carlo Simulation method, is presented to assess the reliability sensitivity in this paper. Firstly some samples are obtained by the random sampling, then the local domain with a constant probability content corresponding to each sample point can be defined, finally the conditional reliability and reliability sensitivity corresponding to every local region can be calculated by using linear approximation of the limit state function. The reliability and reliability sensitivity can be estimated by the expectation of all the conditional reliability and reliability sensitivity. Three examples testify the applicability, validity and accuracy of the proposed method. The results computed by the Local Monte Carlo Simulation method and the Monte Carlo method are compared, which demonstrates that, without losing precision, the computational cost by the former method is much less than the later.

Methods for solution of control problems of multicommodity stock with random demand

Control problems for multicommodity stock with random demand are considered. Special models and methods based on linear approximation of stock consumption functions are proposed for solution of these problems. Results of numerical experiments with proposed models and methods are presented.

Process Impact and Design Optimization on the Soft Yield of 25nm FinFET SRAM Cells

A simulation methodology to correlate process variations to the soft yield of 6-T FinFET SRAM cells is demonstrated. By using the TCAD tool which is calibrated to the recently published 25nm FinFET technology, process impacts on the device electrical characteristics are simulated and stored in behavioral models. Circuit simulations can then be carried out to calculate the read/write noise margin. Employing the linear function approximation of the noise margin with respect to the six individual process variation parameters in sigma, one can analytically solve the yield when noise margin drops to zero. The calculation results show that instead of Vt variation, parasitic resistance fluctuation dominates fails at high bias voltages. A simple design optimization guideline is also demonstrated to reduce Vmin of the given technology (to 0.5V in this work). This methodology can serve as useful guidelines for both process development and device design.

Sensitivity-based coordination in distributed model predictive control

Abstract A new distributed model-predictive control method is introduced, which is based on a novel distributed optimization algorithm, relying on a sensitivity-based coordination mechanism. Coordination and therefore overall optimality is achieved by means of a linear approximation of the objective functions of neighboring controllers within the objective function of each local controller. As for most of the distributed optimization methods, an iterative solution of the distributed optimal control problems is required. An analysis of the method with respect to its convergence properties is provided. For illustration, the sensitivity-driven distributed model-predictive control (S-DMPC) method is applied to a simulated alkylation process. An almost optimal control sequence can be achieved after only one iteration in this case.

A NOVEL ADAPTIVE NEURO-FUZZY UNSCENTED KALMAN FILTER FOR SLAM

Extended Kalman filter (EKF) has been used as a popular choice to solve simultaneous localization and mapping (SLAM) problem. However, SLAM algorithm based on EKF-SLAM has two serious drawbacks, namely the linear approximation of nonlinear functions and the calculation of Jacobin matrices. For solving these problems, SLAM algorithm based on unscented Kalman filter (UKF-SLAM) has been recently proposed. However, the performance of the UKF-SLAM and thus the quality of the estimation depends on the correct a priori knowledge of process and measurement noise covariance matrices respectively denoted by Qk and Rk. Imprecise knowledge of these statistics can cause significant degradation in performance. This article proposes the development of an adaptive neuro-fuzzy UKF (ANFUKF) for SLAM. The Adaptive neuro-fuzzy attempts to estimate the elements of Rk matrix in the UKF-SLAM algorithm at each sampling instant when measurement updating step is carried out. The adaptive neuro-fuzzy inference system (ANFIS) supervises the performance of the UKF-SLAM with the aim of reducing the mismatch between the theoretical and actual covariance of the innovation sequences. The free parameters of ANFIS are trained using the steepest gradient descent (GD) to minimize the differences of the actual value of the covariance of the residual with its theoretical value as much as possible. The simulation results show the effectiveness of the proposed algorithm.

Adaptive predistortion using threshold decomposition-based piecewise linear modeling

A new adaptive predistortion approach for nonlinear power amplifiers based on the piecewise linear (PWL) approximation of nonlinear functions is presented. The PWL model is obtained using threshold decomposition (TD) of the input-output characteristics. The TD based PWL model is used to develop a baseband predistorter using both the direct and the indirect learning architectures. The proposed predistorter is proved to provide a better performance in suppressing nonlinear distortion than conventional polynomial predistorters and a reduced hardware complexity as compared to lookup table (LUT) approach. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011. © 2011 Wiley Periodicals, Inc.