So far, most of the researchers developed one type of estimator in nonparametric regression. But in reality, in daily life, data with mixed patterns were often encountered, especially data patterns which partly changed at certain subintervals, and some others followed a recurring pattern in a certain trend. The estimator method used for the data pattern was a mixed estimator method of smoothing spline and Fourier series. This regression model was approached by the component smoothing spline and Fourier series. From this process, the mixed estimator was completed using two estimation stages. The first stage was the estimation with penalized least squares (PLS), and the second stage was the estimation with least squares (LS). Those estimators were then implemented using simulated data. The simulated data were gained by generating two different functions, namely, polynomial and trigonometric functions with the size of the sample being 100. The whole process was then repeated 50 times. The experiment of the two functions was modeled using a mixture of the smoothing spline and Fourier series estimators with various smoothing and oscillation parameters. The generalized cross validation (GCV) minimum was selected as the best model. The simulation results showed that the mixed estimators gave a minimum (GCV) value of 11.98. From the minimum GCV results, it was obtained that the mean square error (MSE) was 0.71 and R² was 99.48%. So, the results obtained indicated that the model was good for a mixture estimator of smoothing spline and Fourier series.

中文翻译：

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非参数回归中平滑样条和傅里叶级数的组合估计

到目前为止，大多数研究人员在非参数回归中开发了一种估计器。但是实际上，在日常生活中，经常会遇到混合模式的数据，特别是在某些子间隔部分改变的数据模式，而其他一些则以一定的趋势遵循重复的模式。用于数据模式的估计器方法是平滑样条和傅里叶级数的混合估计器方法。该回归模型通过分量平滑样条和傅里叶级数逼近。通过此过程，使用两个估算阶段完成了混合估算器。第一阶段是采用最小二乘法（PLS）的估计，第二阶段是采用最小二乘（LS）的估计。然后使用模拟数据实施这些估计量。通过生成两个不同的函数（即样本大小为100的多项式函数和三角函数）获得模拟数据。然后将整个过程重复50次。使用平滑样条和具有各种平滑和振荡参数的傅里叶级数估计器的混合模型来模拟这两个函数的实验。最小化的通用交叉验证（GCV）被选为最佳模型。仿真结果表明，混合估计量的最小值（GCV）值为11.98。从最小的GCV结果可以得出，均方误差（MSE）为0.71，而 使用平滑样条和具有各种平滑和振荡参数的傅里叶级数估计器的混合模型来模拟这两个函数的实验。最小化的通用交叉验证（GCV）被选为最佳模型。仿真结果表明，混合估计量的最小值（GCV）值为11.98。从最小的GCV结果可以得出，均方误差（MSE）为0.71，而 使用平滑样条和具有各种平滑和振荡参数的傅里叶级数估计器的混合模型来模拟这两个函数的实验。选择最小化的通用交叉验证（GCV）作为最佳模型。仿真结果表明，混合估计量的最小值（GCV）值为11.98。从最小的GCV结果可以得出，均方误差（MSE）为0.71，而R ²为99.48％。因此，获得的结果表明该模型对于平滑样条和傅里叶级数的混合估计是好的。