Variable projection algorithms with sparse constraint for separable nonlinear models

Separable nonlinear models are widely used in various fields such as time series analysis, system modeling, and machine learning, due to their flexible structures and ability to capture nonlinear behavior of data. However, identifying the parameters of these models is challenging, especially when sparse models with better interpretability are desired by practitioners. Previous theoretical and practical studies have shown that variable projection (VP) is an efficient method for identifying separable nonlinear models, but these are based on \(L_2\) penalty of model parameters, which cannot be directly extended to deal with sparse constraint. Based on the exploration of the structural characteristics of separable models, this paper proposes gradient-based and trust-region-based variable projection algorithms, which mainly solve two key problems: how to eliminate linear parameters under sparse constraint;and how to deal with the coupling relationship between linear and nonlinear parameters in the model. Finally, numerical experiments on synthetic data and real time series data are conducted to verify the effectiveness of the proposed algorithms.

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.

Nonparametric VSS-APA based on precise background noise power estimate

The adaptive algorithm used for echo cancellation(EC) system needs to provide 1) low misadjustment and 2) high convergence rate. The affine projection algorithm(APA) is a better alternative than normalized least mean square(NLMS) algorithm in EC applications where the input signal is highly correlated. Since the APA with a constant step-size has to make compromise between the performance criteria 1) and 2), a variable step-size APA(VSS-APA) provides a more reliable solution. A nonparametric VSS-APA(NPVSS-APA) is proposed by recovering the background noise within the error signal instead of cancelling the a posteriori errors. The most problematic term of its variable step-size formula is the value of background noise power(BNP). The power difference between the desired signal and output signal, which equals the power of error signal statistically, has been considered the BNP estimate in a rough manner. Considering that the error signal consists of background noise and misalignment noise, a precise BNP estimate is achieved by multiplying the rough estimate with a corrective factor. After the analysis on the power ratio of misalignment noise to background noise of APA, the corrective factor is formulated depending on the projection order and the latest value of variable step-size. The new algorithm which does not require any a priori knowledge of EC environment has the advantage of easier controllability in practical application. The simulation results in the EC context indicate the accuracy of the proposed BNP estimate and the more effective behavior of the proposed algorithm compared with other versions of APA class.

A two-stage variable selection strategy for supersaturated designs with multiple responses

A supersaturated design （SSD）, whose run size is not enough for estimating all the main effects, is commonly used in screening experiments. It offers a potential useful tool to investigate a large number of factors with only a few experimental runs. The associated analysis methods have been proposed by many authors to identify active effects in situations where only one response is considered. However, there are often situations where two or more responses are observed simultaneously in one screening experiment, and the analysis of SSDs with multiple responses is thus needed. In this paper, we propose a two-stage variable selection strategy, called the multivariate partial least squares-stepwise regression （MPLS-SR） method, which uses the multivariate partial least squares regression in conjunction with the stepwise regression procedure to select true active effects in SSDs with multiple responses. Simulation studies show that the MPLS-SR method performs pretty good and is easy to understand and implement.

Optimization design of anti-seismic engineering measures for intake tower based on non-dominated sorting genetic algorithm-Ⅱ

High-rise intake towers in high-intensity seismic areas are prone to structural safety problems under vibration.Therefore,effective and low-cost anti-seismic engineering measures must be designed for protection.An intake tower in northwest China was considered the research object,and its natural vibration characteristics and dynamic response were first analyzed using the mode decomposition response spectrum method based on a three-dimensional finite element model.The non-dominated sorting genetic algorithm-II(NSGA-II)was adopted to optimize the anti-seismic scheme combination by comprehensively considering the dynamic tower response and variable project cost.Finally,the rationality of the original intake tower antiseismic design scheme was evaluated according to the obtained optimal solution set,and recommendations for improvement were proposed.The method adopted in this study may provide significant references for designing anti-seismic measures for high-rise structures such as intake towers located in high-intensity earthquake areas.

ON THE SEPARABLE NONLINEAR LEAST SQUARES PROBLEMS

Separable nonlinear least squares problems are a special class of nonlinear least squares problems, where the objective functions are linear and nonlinear on different parts of variables. Such problems have broad applications in practice. Most existing algorithms for this kind of problems are derived from the variable projection method proposed by Golub and Pereyra, which utilizes the separability under a separate framework. However, the methods based on variable projection strategy would be invalid if there exist some constraints to the variables, as the real problems always do, even if the constraint is simply the ball constraint. We present a new algorithm which is based on a special approximation to the Hessian by noticing the fact that certain terms of the Hessian can be derived from the gradient. Our method maintains all the advantages of variable projection based methods, and moreover it can be combined with trust region methods easily and can be applied to general constrained separable nonlinear problems. Convergence analysis of our method is presented and numerical results are also reported.

Estimation of Sunflower Seed Yield Using Partial Least Squares Regression and Artificial Neural Network Models

Statistical models can efficiently establish the relationships between crop growth and environmental conditions while explicitly quantifying uncertainties. This study aimed to test the efficiency of statistical models established using partial least squares regression(PLSR) and artificial neural network(ANN) in predicting seed yields of sunflower(Helianthus annuus). Two-year field trial data on sunflower growth under different salinity levels and nitrogen(N) application rates in the Yichang Experimental Station in Hetao Irrigation District, Inner Mongolia, China, were used to calibrate and validate the statistical models. The variable importance in projection score was calculated in order to select the sensitive crop indices for seed yield prediction. We found that when the most sensitive indices were used as inputs for seed yield estimation, the PLSR could attain a comparable accuracy(root mean square error(RMSE) = 0.93 t ha-1, coefficient of determination(R^2) = 0.69) to that when using all measured indices(RMSE = 0.81 t ha-1,R^2= 0.77). The ANN model outperformed the PLSR for yield prediction with different combinations of inputs of both microplots and field data. The results indicated that sunflower seed yield could be reasonably estimated by using a small number of crop characteristic indices under complex environmental conditions and management options(e.g., saline soils and N application). Since leaf area index and plant height were found to be the most sensitive crop indices for sunflower seed yield prediction, remotely sensed data and the ANN model may be joined for regional crop yield simulation.