Function Approximation Method Research Articles

Jackknife resample method for precision estimation of weighted total least squares

Few studies have been conducted on the precision estimation of weighted total least squares (WTLS) by using the approximate function probability distribution method. And the existing Monte Carlo method needs to simulate a lot, the amount of calculation is large and the results obtained are uncertain. In order to further improve the total least squares precision estimation theory, this paper introduces the Jackknife method into Geomatics data processing. Combining the Jackknife method and WTLS method, the delete-1 Jackknife method and delete-d Jackknife method are proposed. The biases and standard deviations or covariance of parameter estimations are calculated by these proposed methods. And the specific steps of the precision estimation of these two methods are given. Applying these methods to the linear regression model and the coordinate transformation model, and comparing with the approximate function method and the Monte Carlo method, we can see that the Jackknife methods for precision estimation can obtain more stable and reasonable precision results and are very adaptive. In order to get more reasonable precision results, the Jackknife method does spend much more time over total least squares when the amount of the observed data is large. But compared with Monte Carlo method, it can reduce the amount of calculation and improve the computational efficiency. The method in this paper could provide an idea for further study on the precision estimation for total least squares.

A Universal Empirical Dynamic Programming Algorithm for Continuous State MDPs

We propose universal randomized function approximation-based empirical value learning (EVL) algorithms for Markov decision processes. The “empirical” nature comes from each iteration being done empirically from samples available from simulations of the next state. This makes the Bellman operator a random operator. A parametric and a nonparametric method for function approximation using a parametric function space and a reproducing kernel Hilbert space respectively are then combined with EVL. Both function spaces have the universal function approximation property. Basis functions are picked randomly. Convergence analysis is performed using a random operator framework with techniques from the theory of stochastic dominance. Finite time sample complexity bounds are derived for both universal approximate dynamic programming algorithms. Numerical experiments support the versatility and computational tractability of this approach.

A meshless symplectic method for two-dimensional nonlinear Schrödinger equations based on radial basis function approximation

For two-dimensional nonlinear Schrödinger equations, we propose a meshless symplectic method based on radial basis function interpolation. With the method of lines, we first discretize the equation in spatial domain by using the radial basis function approximation method and obtain a finite-dimensional Hamiltonian system. Then appropriate time integrator is employed to derive the full-discrete symplectic scheme. Compared with the classical conservative methods that are only valid on uniform grids, our meshless method is conservative for both uniform grids and nonuniform nodes. The accuracy and conservation properties are analyzed in detail. Several numerical experiments are presented to demonstrate the accuracy and the conservation properties of our approach.

An Enhanced Analytical Calculation of the Phase Inductance of Switched Reluctance Machines

Accurate prediction of the phase inductance of switched reluctance machines (SRMs) is of crucial importance at the design stage because it is a key parameter to estimate the performance indices, such as the torque and loss. Instead of using the popular but time-consuming finite-element analysis (FEA) or the methods that require prior knowledge of magnetic fields from an FEA, such as function approximation and flux path methods, this paper proposes a novel fast and accurate analytical approach to determine the phase inductance of an SRM by solving the partial differential equations of magnetic potentials based on Maxwell's equations. In particular, an enhancement of the accuracy of the analytical model is accomplished by applying conformal mappings that deal with the non-radial/non-peripheral geometric structures when calculating the air-region permeance parameters. Moreover, a magnetic circuit network is applied to account for the steel saturation effects. The close agreement between the results of the proposed analytical method, FEAs and experimental measurements validate the analysis.

Strain engineering of transverse electric and transverse magnetic mode of material gain in GeSn/SiGeSn quantum wells

8-band k · p Hamiltonian together with envelope function approximation and planewave expansion method are applied to calculate the electronic band structure and material gain for Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw quantum wells (QWs) grown on virtual Ge1-zSnz substrates integrated with Si platform. It is clearly shown how both the emission wavelength in this material system can be controlled by the content of virtual substrate and the polarization of emitted light can be controlled via the built-in strain. In order to systematically demonstrate these possibilities, the transverse electric (TE) and transverse magnetic (TM) modes of material gain, and hence the polarization degree, are calculated for Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw (QWs) with the strain varying from tensile (ε = +1.5%) to compressive (ε = −0.9%). It has been predicted that the polarization can be changed from 100% TE to 80% TM. In addition, it has been shown that SiyGe1−x−ySnx barriers, lattice matched to the virtual Ge1-zSnz substrate (condition: y = 3.66(x-z)), may ensure a respectable quantum confinement for electrons and holes in this system. With such material features Ge1−wSnw/SiyGe1−x−ySnx/Ge1−wSnw QW structure unified with Ge1-zSnz/Si platform may be considered as a very prospective one for light polarization engineering.

A hybrid computational intelligence approach for predicting soil shear strength for urban housing construction: a case study at Vinhomes Imperia project, Hai Phong city (Vietnam)

This research proposes an alternative for estimating shear strength of soil based on a hybridization of Support Vector Regression (SVR) and Particle Swarm Optimization (PSO). SVR is used as a function approximation method for making prediction of the soil shear strength based on a set of twelve variables including sample depth, sand content, loam content clay content, moisture content, wet density, dry density, void ratio, liquid limit, plastic limit, plastic index, and liquid index. The hybrid framework, named as PSO–SVR, relies on PSO, as a metaheuristic, to optimize the training phase of the employed function approximator. A data set consisting of 443 soil samples associated with the experimental results of shear strength has been collected from a housing project in Vietnam. This data set is then used to train and verify the performance of the PSO–SVR model specifically constructed for shear strength estimation. The hybrid model has achieved a good modeling outcome with Root Mean Square Error (RMSE) = 0.038, Mean Absolute Percentage Error (MAPE) = 9.701%, and Coefficient of Determination (R2) = 0.888. Hence, the PSO–SVR model can be a potential alternative to be participated in the design phase of high-rise housing projects.

An Efficient SMO Algorithm for Solving Non-smooth Problem Arising in $$\varepsilon $$ ε -Insensitive Support Vector Regression

Classical support vector regression (C-SVR) is a powerful function approximation method, which is robust against noise and performs a good generalization, since it is formulated by a regularized error function employing the $$\varepsilon $$ -insensitiveness property. To exploit the kernel trick, C-SVR generally solves the Lagrangian dual problem. In this paper, an efficient sequential minimal optimization (SMO) algorithm with a novel easy to compute working set selection (WSS) based on the minimization of an upper bound on the difference between consecutive loss function values for solving a convex non-smooth dual optimization problem obtained by reformulating the dual problem of C-SVR with $$l_2$$ error loss function which is equivalent to the $$\varepsilon $$ -insensitive version of the LSSVR, is proposed. The asymptotic convergence to the optimum of the proposed SMO algorithm is also proved. This proposed SMO algorithm for solving non-smooth problem comprises both SMO algorithms for solving LSSVR and C-SVR. Indeed, it becomes equivalent to the SMO algorithm with second-order WSS for solving LSSVR when $$\varepsilon =0$$ . The proposed algorithm has the advantage of dealing with the optimization variables half the number of the ones in C-SVR, which results in lesser number of kernel related matrix evaluations than the standard SMO algorithm developed for C-SVR and improves the probability of the matrix outputs to have been precomputed and cached. Therefore, the proposed SMO algorithm results better training time than the standard SMO algorithm for solving C-SVR, especially with caching process. Moreover, the superiority of the proposed WSS over its first-order counterpart for solving the non-smooth optimization problem is presented.

BER analysis of EGC receiver over correlated Nakagami-m fading channels with phase error and asynchronous CCI

Performance analysis of an L-branch Equal Gain Combining (EGC) receiver with phase estimation error and asynchronous Co-Channel Interference (CCI) over correlated Nakagami-m fading channel is presented. Both the desired and interfering signals are assumed to be Nakagami-m distributed. The output Signal to Noise Ratio (SINR) of the EGC combiner is derived. Closed form expression for the n-th moment of the output SINR is derived. Using the n-th moment expression, Average Bit Error Probability (ABEP) for Binary Phase Shift Keying (BPSK) modulation is derived using Moment Generating Function (MGF) and Pade approximation method. Numerical and simulation results of ABEP for various desired and interfering signal parameters are obtained. ABEP results for various degrees of phase errors are also obtained. The results for special cases are verified with the available results.

Digital Predistorter for Short-Wave Power Amplifier With Improving Index Accuracy of Lookup Table Based on FPGA

The method of function approximation to improve index accuracy of the lookup table (LUT) of digital predistortion (DPD) is proposed in this paper. The algorithm utilizes Taylor Series method for LUT approximation. The power value of a forward signal is divided into an integer and a decimal to index corresponding LUT respectively, and the indexed predistortion value of the forward signal is approached by fitting according to Taylor series, which makes the LUT index precision obviously increased. Experiment shows that comparing with the traditional Memory Polynomial Model (MPM) DPD LUT method, this model can improve Adjacent Channels Power Ratio (ACPR) over 15 dB for high frequency (HF) power amplifier’s output signal, compensate for short-wave power amplifier nonlinear distortion effectively, and improve the short-wave radio communication quality greatly.

Deep Option Pricing - Term Structure Models

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options within the setting of interest rate term structure models. This aims to accelerate existing numerical methods which is important for applications like historical VaR or exposure calculation being used in financial institutions. With ANNs being a universal function approximation method, this method trains an ANN on synthetically generated data including term structures of yield and volatility. Then, within an VaR or exposure calculation instead of applying costly numerical methods for the financial model, the engine runs the trained ANN. This is faster and more efficient and allows (a) considering term structures of yields, (b) term structures of volatilities and (c) trade interpolation. We outline the generation of the training data, the neural net selection and propose further methods for optimization. In particular we consider a control variate method and the application of no-arbitrage conditions and regularization to the cost function used for learning and calibration. Finally, we test our approach on the Hull-White model with time-dependent term structure for volatility and the the Trolle-Schwartz model. The latter adds an un-spanned stochastic volatility to the rates dynamic. The numerical results show that the ANN solution, especially the one with the control variate, is accurate and reduces the computing time significantly.

Secrecy Energy Efficiency Optimization for Artificial Noise Aided Physical-Layer Security in OFDM-Based Cognitive Radio Networks

In this paper, we investigate the power allocation of primary base station (PBS) and cognitive base station (CBS) across different orthogonal frequency division multiplexing (OFDM) subcarriers for energy-efficient secure downlink communication in OFDM-based cognitive radio networks (CRNs) with the existence of an eavesdropper having multiple antennas. We propose a secrecy energy efficiency maximization (SEEM) scheme by exploiting the instantaneous channel state information (ICSI) of the eavesdropper, called ICSI based SEEM (ICSI-SEEM) scheme with a given total transmit power budget for different OFDM subcarriers of both PBS and CBS. As for the case when the eavesdropper's ICSI is unknown, we also propose an SEEM scheme through using the statistical CSI (SCSI) of the eavesdropper, namely SCSI based SEEM (SCSI-SEEM) scheme. Since the ICSI-SEEM and SCSI-SEEM problems are fractional and non-convex, we first transform them into equivalent subtractive problems, and then achieve approximate convex problems through employing the difference of two-convex functions approximation method. Finally, new two-tier power allocation algorithms are proposed to achieve $\varepsilon$-optimal solutions of our formulated ICSI-SEEM and SCSI-SEEM problems. Simulation results illustrate that the ICSI-SEEM has a better secrecy energy efficiency (SEE) performance than SCSI-SEEM, and moreover, the proposed ICSI-SEEM and SCSI-SEEM schemes outperform conventional SR maximization and EE maximization approaches in terms of their SEE performance.

Modeling high temperature deformation characteristics of AA7020 aluminum alloy using substructure-based constitutive equations and mesh-free approximation method

This research was aimed to assess the potential of a radial basis function (RBF) approximation method against the dislocation substructure-based constitutive model in predicting high-temperature deformation behavior of the AA7020 aluminum alloy. Hot compression tests were performed over a range of strain rate of 0.1–100 s−1 and a range of temperature of 350–500 °C up to a strain of 0.6. The hot deformation behavior of the alloy was first described by a substructure kinetic-based constitutive equation, with the effects of strain, strain rate and temperature together with dynamic recovery parameters taken into consideration. A RBF approximation method was then developed to model the flow behavior of the material. The RBF model, as a kind of novel mesh-free function estimation approach, was trained and tested with the obtained datasets from the hot compression tests. The performance of the developed analytical and neural computational models was evaluated using statistical criteria. The results showed that the RBF model was more proficient and accurate in predicting the hot deformation behavior of this aluminum alloy than the substructure-based constitutive model.

Optimal Size and Placement of Water Hammer Protective Devices in Water Conveyance Pipelines

Positive and negative pressure waves caused by water hammer possibly may lead to high damages to the water conveyance pipelines. To decrease the negative effects of pressure waves, costly equipment are implemented in pipelines. An economic design of these devices that also provides the safety of the pipeline against water hammer pressure waves and cavitation can be achieved by simulation and optimization tools. In this paper, the simulation task was carried out by meta modeling. The accuracy of three meta models: artificial neural network (ANN), support vector regression (SVR) and adaptive neuro-fussy inference system (ANFIS) was evaluated. According to the results, SVR was identified as the inferior method due to low capability of generalization, ANFIS as the median, and ANN as the superior method for function approximation. Then, ANN was coupled with an evolutionary algorithm (EA), Differential Evolution (DE) to find the optimal size and location of water hammer control devices in a water pipeline. Optimization was carried out on two single- and multi-objective approaches. The results showed that multi-objective optimization approach presents better designs than the single objective approach and optimal designs obtained by both approaches outperform the current setup of the water hammer facilities in terms of both costs and functionality. The single objective-based design could decrease the costs up to 12.5% whereas multi-objective approach was able to reach nearly 30% cost saving with higher level of the safety against cavitation. Results also showed that air chamber is the most effective device and air-valves have little effect for pipeline protection against water hammer.

Reduction of Aliasing Artifacts by Sign Function Approximation in Light Field Depth Estimation Based on Foreground–Background Separation

A sign function approximation method for depth from light field (DFLF) based on the foreground–background separation (FBS) is proposed. From a signal processing viewpoint, the FBS-based method can be considered as a bridge between the cost-based DFLF methods and depth model-based ones. The proposed sign function approximation corresponds to the winner-takes-all (WTA) approaches as the cost-based methods do. Experimental results on the synthetic images show that the proposed method reasonably performs in terms of the mean squared error as the state-of-the-art methods do. Especially, by using a suitable WTA method in the framework of our previous FBS-based DFLF work, the proposed method effectively reduces the angular aliasing artifacts in the resulting disparity maps of both the synthetic and real images.

The State Following Approximation Method.

A function approximation method is developed which aims to approximate a function in a small neighborhood of a state that travels within a compact set. The method provides a novel approximation strategy for the efficient approximation of nonlinear functions for real-time simulations and experiments. The development is based on the theory of universal reproducing kernel Hilbert spaces over the n -dimensional Euclidean space. Several theorems are introduced which support the development of this state following (StaF) method. In particular, it is shown that there is a bound on the number of kernel functions required for the maintenance of an accurate function approximation as a state moves through a compact set. In addition, a weight update law, based on gradient descent, is introduced where arbitrarily close accuracy can be achieved provided the weight update law is iterated at a sufficient frequency, as detailed in Theorem 4. An experience-based approximation method is presented which utilizes the samples of the estimations of the ideal weights to generate a global approximation of a function. The experience-based approximation interpolates the samples of the weight estimates using radial basis functions. To illustrate the StaF method, the method is utilized for derivative estimation, function approximation, and is applied to an adaptive dynamic programming problem where it is demonstrated that the stability is maintained with a reduced number of basis functions.

Inflation with Gauss-Bonnet coupling

We consider inflationary models with the inflaton coupled to the Gauss-Bonnet term assuming a special relation $\delta_1=2\lambda\epsilon_1$ between the two slow-roll parameters $\delta_1$ and $\epsilon_1$. For the slow-roll inflation, the assumed relation leads to the reciprocal relation between the Gauss-Bonnet coupling function $\xi(\phi)$ and the potential $V(\phi)$, and it leads to the relation $r=16(1-\lambda)\epsilon_1$ that reduces the tensor-to-scalar ratio $r$ by a factor of $1-\lambda$. For the constant-roll inflation, we derive the analytical expressions for the scalar and tensor power spectra, the scalar and tensor spectral tilts, and the tensor-to-scalar ratio to the first order of $\epsilon_1$ by using the method of Bessel function approximation. The tensor-to-scalar ratio is reduced by a factor of $1-\lambda+\lambda\tilde \eta$. Comparing the derived $n_s$-$r$ with the observations, we obtain the constraints on the model parameters $\tilde\eta$ and $\lambda$.

Second-Order Approximation Function Method for Precision Estimation of Total Least Squares

Second-Order Approximation Function Method for Precision Estimation of Total Least Squares

Dynamic pricing for managed lanes with multiple entrances and exits

Priced managed lanes are increasingly being used to better utilize the existing capacity of the roadway to relieve congestion and offer reliable travel time to road users. In this paper, we investigate the optimization problem for pricing managed lanes with multiple entrances and exits which seeks to maximize the revenue and minimize the total system travel time (TSTT) over a finite horizon. We propose a lane choice model where travelers make online decisions at each diverge point considering all routes on a managed lane network. We formulate the problem as a deterministic Markov decision process and solve it using the value function approximation (VFA) method for different initializations. We compare the performance of the toll policies predicted by the VFA method against the myopic revenue policy which maximizes the revenue only at the current timestep and two heuristic policies based on the measured densities on the managed and general purpose lanes (GPLs). We test the results on four different test networks. The primary findings from our research suggest the usefulness of the VFA method for determining dynamic tolls. The best-found objective value from the method at its termination is better than other heuristics for all test networks with average improvements in the objective ranging between 10% and 90% for revenue maximization and 0–27% for TSTT minimization. Certain VFA initializations obtain best-found toll profiles within first 5–50 iterations which warrants computational time savings. Our findings also indicate that the revenue-maximizing optimal policies follow the “jam-and-harvest” behavior where the GPLs are pushed towards congestion in the earlier time steps to generate higher revenue in the later time steps, a characteristic not observed for the policies minimizing TSTT.

Approximation methods for lognormal characteristic functions

ABSTRACTThe characteristic function of the lognormal distribution is of interest in a number of scientific fields yet an analytic solution remains elusive, making reliable and efficient approximations necessary. In this article, we build on the results of N. C. Beaulieu and A. Saberali in ‘New approximations to the lognormal characteristic function’, by introducing a Taylor- and Bessel function-based partial expansion of the integrand and a Chebyshev quadrature approach. Through computer simulations we show that the Taylor expansion remains accurate and efficient for all commonly computed values, and specify the range of values for which the other two approaches show a significantly stronger performance.

Comparative Analysis of Harmonic Function Approximation Methods

Comparative Analysis of Harmonic Function Approximation Methods