A Study on Learning Parameters in Application of Radial Basis Function Neural Network Model to Rotor Blade Design Approximation

Meta-model sre generally applied to approximate multi-objective optimization, reliability analysis, reliability based design optimization, etc., not only in order to improve the efficiencies of numerical calculation and convergence, but also to facilitate the analysis of design sensitivity. The radial basis function neural network (RBFNN) is the meta-model employing hidden layer of radial units and output layer of linear units, and characterized by relatively fast training, generalization and compact type of networks. It is important to minimize some scattered noisy data to approximate the design space to prevent local minima in the gradient based optimization or the reliability analysis using the RBFNN. Since the noisy data must be smoothed out in order for the RBFNN to be applied as the meta-model to any actual structural design problem, the smoothing parameter must be properly determined. This study aims to identify the effect of various learning parameters including the spline smoothing parameter on the RBFNN performance regarding the design approximation. An actual rotor blade design problem was considered to investigate the characteristics of RBFNN approximation with respect to the range of spline smoothing parameter, the number of training data, and the number of hidden layers. In the RBFNN approximation of the rotor blade design, design sensitivity characteristics such as main effects were also evaluated including the performance analysis according to the variation of learning parameters. From the evaluation results of learning parameters in the rotor blade design, it was found that the number of training data had larger influence on the RBFNN meta-model accuracy than the spline smoothing parameter while the number of hidden layers had little effect on the performances of RBFNN meta-model.

A multigrid preconditioner for tensor product spline smoothing

Penalized spline smoothing is a well-established, nonparametric regression method that is efficient for one and two covariates. Its extension to more than two covariates is straightforward but suffers from exponentially increasing memory demands and computational complexity, which brings the method to its numerical limit. Penalized spline smoothing with multiple covariates requires solving a large-scale, regularized least-squares problem where the occurring matrices do not fit into storage of common computer systems. To overcome this restriction, we introduce a matrix-free implementation of the conjugate gradient method. We further present a matrix-free implementation of a simple diagonal as well as more advanced geometric multigrid preconditioner to significantly speed up convergence of the conjugate gradient method. All algorithms require a negligible amount of memory and therefore allow for penalized spline smoothing with multiple covariates. Moreover, for arbitrary but fixed covariate dimension, we show grid independent convergence of the multigrid preconditioner which is fundamental to achieve algorithmic scalability.

Mixed Spline Smoothing and Kernel Estimator in Biresponse Nonparametric Regression

Mixed estimators in nonparametric regression have been developed in models with one response. The biresponse cases with different patterns among predictor variables that tend to be mixed estimators are often encountered. Therefore, in this article, we propose a biresponse nonparametric regression model with mixed spline smoothing and kernel estimators. This mixed estimator is suitable for modeling biresponse data with several patterns (response vs. predictors) that tend to change at certain subintervals such as the spline smoothing pattern, and other patterns that tend to be random are commonly modeled using kernel regression. The mixed estimator is obtained through two-stage estimation, i.e., penalized weighted least square (PWLS) and weighted least square (WLS). Furthermore, the proposed biresponse modeling with mixed estimators is validated using simulation data. This estimator is also applied to the percentage of the poor population and human development index data. The results show that the proposed model can be appropriately implemented and gives satisfactory results.

PEMODELAN KASUS COVID-19 MENGGUNAKAN MODEL REGRESI NONPARAMETRIK

Pemodelan kasus positif COVID-19 perhari sangat sulit bahkan banyak gejala menunjukkan bahwa data yang diperoleh tidak menunjukkan suatu pola hubungan yang mudah untuk digambarkan. Untuk mengatasi kesulitan-kesulitan tersebut digunakan model regresi nonparametrik. Tujuan penelitian ini adalah medapatkan model terbaik dari pemodelan data kasus baru perhari COVID-19 di Jakarta menggunakan model regresi nonparametrik berupa regresi spline (cubic spline), smoothing spline dan MARS. Data kasus baru perhari COVID-19 di Jakarta yang digunakan adalah data kasus baru mulai tanggal 16 Maret 2020 sampai dengan 15 Agustus 2020. Data tersebut dibagi dalam dua kelompok yaitu data tanggal 16 Maret 2020 sampai dengan 6 Agustus 2020 sebagai data in sample yang digunakan sebagai pembentuk model regresi nonparametrik, dan data tanggal 7 Agustus 2020 sampai dengan 15 Agustus 2020 sebagai data out sample yang digunakan untuk memvalidasi model regresi nonparametrik. Hasil penelitian menunjukkan model regresi nonparametrik berupa MARS dengan BF=46, MI=1 dan MO=1 merupakan model terbaik dan sangat akurat dalam melakukan prediksi untuk kasus COVID-19 di Jakarta.