The Application of Mixed Smoothing Spline and Fourier Series Model in Nonparametric Regression

In daily life, mixed data patterns are often found, namely, those that change at a certain sub-interval or that follow a repeating pattern in a certain trend. To handle this kind of data, a mixed estimator of a Smoothing Spline and a Fourier Series has been developed. This paper describes a simulation study of the estimator in nonparametric regression and its implementation in the case of poor households. The minimum Generalized Cross Validation (GCV) was used in order to select the best model. The simulation study used generation data with a Uniform distribution and a random error with a symmetrical Normal distribution. The result of the simulation study shows that the larger the sample size n, the better the mixed estimator as a model of nonparametric regression for all variances. The smaller the variance, the better the model for all combinations of samples n. Very poor households are characterized predominantly in their consumption of carbohydrates compared to that of fat and protein. The results of this study suggest that the distribution of assistance to poor households is not the same, because in certain groups there are poor households that consume higher carbohydrates, and some households may consume higher fats.

Download Full-text

Combination Estimation of Smoothing Spline and Fourier Series in Nonparametric Regression

Journal of Mathematics ◽

10.1155/2020/4712531 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Ni Putu Ayu Mirah Mariati ◽

I. Nyoman Budiantara ◽

Vita Ratnasari

Keyword(s):

Fourier Series ◽

Least Squares ◽

Nonparametric Regression ◽

Simulated Data ◽

Smoothing Spline ◽

Penalized Least Squares ◽

Second Stage ◽

Whole Process ◽

Mixed Estimator ◽

The Mean

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.

Download Full-text

Smoothing Parameter Selection Problem in Nonparametric Regression Based on Smoothing Spline: A Simulation Study

Journal of Applied Sciences ◽

10.3923/jas.2012.636.644 ◽

2012 ◽

Vol 12 (7) ◽

pp. 636-644 ◽

Cited By ~ 3

Author(s):

Dursun Aydin ◽

M. Seref Tuzemen

Keyword(s):

Nonparametric Regression ◽

Simulation Study ◽

Parameter Selection ◽

Smoothing Parameter ◽

Selection Problem ◽

Smoothing Spline ◽

Smoothing Parameter Selection

Download Full-text

Generalized Cross Validation (GCV) in Smoothing Spline Nonparametric Regression Models

Journal of Physics Conference Series ◽

10.1088/1742-6596/1808/1/012053 ◽

2021 ◽

Vol 1808 (1) ◽

pp. 012053

Author(s):

M Maharani ◽

D R S Saputro

Keyword(s):

Nonparametric Regression ◽

Regression Models ◽

Cross Validation ◽

Smoothing Spline ◽

Generalized Cross Validation

Download Full-text

A New Mixed Estimator in Nonparametric Regression for Longitudinal Data

Journal of Mathematics ◽

10.1155/2021/3909401 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Made Ayu Dwi Octavanny ◽

I Nyoman Budiantara ◽

Heri Kuswanto ◽

Dyah Putri Rahmawati

Keyword(s):

Longitudinal Data ◽

Least Squares ◽

Nonparametric Regression ◽

Cross Validation ◽

Weighted Least Squares ◽

Simulated Data ◽

Real Data ◽

New Method ◽

Mixed Estimator ◽

Data Yield

We introduce a new method for estimating the nonparametric regression curve for longitudinal data. This method combines two estimators: truncated spline and Fourier series. This estimation is completed by minimizing the penalized weighted least squares and weighted least squares. This paper also provides the properties of the new mixed estimator, which are biased and linear in the observations. The best model is selected using the smallest value of generalized cross-validation. The performance of the new method is demonstrated by a simulation study with a variety of time points. Then, the proposed approach is applied to a stroke patient dataset. The results show that simulated data and real data yield consistent findings.

Download Full-text

Selection of Optimal Smoothing Parameters in Mixed Estimator of Kernel and Fourier Series in Semiparametric Regression

Journal of Physics Conference Series ◽

10.1088/1742-6596/2123/1/012035 ◽

2021 ◽

Vol 2123 (1) ◽

pp. 012035

Author(s):

Andi Tenri Ampa ◽

I Nyoman Budiantara ◽

Ismaini Zain

Keyword(s):

Drinking Water ◽

Fourier Series ◽

Cross Validation ◽

Semiparametric Regression ◽

Gaussian Kernel ◽

Regression Estimation ◽

Mixed Estimator ◽

Optimal Smoothing ◽

Smoothing Parameters ◽

Selection Of

Abstract In this article, we propose a new method of selecting smoothing parameters in semiparametric regression. This method is used in semiparametric regression estimation where the nonparametric component is partially approximated by multivariable Fourier Series and partly approached by multivariable Kernel. Selection of smoothing parameters using the method with Generalized Cross-Validation (GCV). To see the performance of this method, it is then applied to the data drinking water quality sourced from Regional Drinking Water Company (PDAM) Surabaya by using Fourier Series with trend and Gaussian Kernel. The results showed that this method contributed a good performance in selecting the optimal smoothing parameters.

Download Full-text

Regresi Nonparametrik dengan Pendekatan Deret Fourier pada Data Debit Air Sungai Citarum

Jurnal Matematika MANTIK ◽

10.15642/mantik.2018.4.2.75-82 ◽

2018 ◽

Vol 4 (2) ◽

pp. 75-82

Author(s):

Intaniah Ratna Nur Wisisono ◽

Ade Irma Nurwahidah ◽

Yudhie Andriyana

Keyword(s):

Fourier Series ◽

Nonparametric Regression ◽

River Discharge ◽

Cross Validation ◽

Least Square ◽

Nonparametric Regression Model ◽

Optimal Bandwidth ◽

Ordinary Least Square ◽

Calculation Results ◽

Series Technique

River discharge is one of the factors that affect the occurrence of floods. It varies over time and hence we need to predict the flood risk. Since the plot of the data changes periodically showing a sines and cosines pattern, a nonparametric technique using Fourier series approach may be interesting to be applied. Fourier series can be estimated using OLS (Ordinary Least Square). In a Fourier series, nonparametric regression the level of subtlety of its function is determined by their bandwidth (K). Optimal bandwidth determined using the GCV (Generalized Cross Validation) method. From the calculation results, we have optimal bandwidth which is equal to 16 with R2 is 0.7295 which means that 72.95% of the total variance in the river discharge variable can be explained by the Fourier series nonparametric regression model. Comparing to a classical time series technique, ARIMA Box Jenkins, we obtained ARIMA (1,0,0) with RMSE 83.10 while using Fourier series approach generate a smaller RMSE 50.51.

Download Full-text

Penerapan Generalized Cross Validation dalam Model Regresi Smoothing Spline pada Produksi Ubi Jalar di Jawa Tengah

Indonesian Journal of Applied Statistics ◽

10.13057/ijas.v1i2.26250 ◽

2019 ◽

Vol 1 (2) ◽

pp. 117

Author(s):

Trionika Dian Wahyuningsih ◽

Sri Sulistijowati Handajani ◽

Diari Indriati

Keyword(s):

Sweet Potato ◽

Nonparametric Regression ◽

Cross Validation ◽

Animal Feed ◽

Organic Fertilizer ◽

Specific Pattern ◽

Production Model ◽

Smoothing Spline ◽

Urea Fertilizer ◽

Generalized Cross Validation

Sweet Potato is a useful plant as a source carbohydrates, proteins, and is used as an animal feed and ingredient industry. Based on data from the Badan Pusat Statistik (BPS), the production fluctuations of the sweet potato in Central Java from year to year are caused by many factor. The production of sweet potato and the factors that affected it if they are described into a pattern of relationships then they do not have a specific pattern and do not follow a particular distribution, such as harvest area, the allocation of subsidized urea fertilizer, and the allocation of subsidized organic fertilizer. Therefore, the production model of sweet potato could be applied into nonparametric regression model. The approach used for nonparametric regression in this study is smoothing spline regression. The method used in regression smoothing spline is generalized cross validation (GCV). The value of the smoothing parameter (λ) is chosen from the minimum GCV value. The results of the study show that the optimum λ value for the factors of harvest area, urea fertilizer and organic fertilizer are 5.57905e-14, 2.51426e-06, and 3.227217e-13 that they result a minimum GCV i.e 2.29272e-21, 1.38391e-16, and 3.46813e-24. Keywords: Sweet potato; nonparametric; smoothing spline; generalized cross validation.

Download Full-text

Visualizing Profiles of Large Datasets of Weighted and Mixed Data

Mathematics ◽

10.3390/math9080891 ◽

2021 ◽

Vol 9 (8) ◽

pp. 891

Author(s):

Aurea Grané ◽

Alpha A. Sow-Barry

Keyword(s):

Multidimensional Scaling ◽

Random Sample ◽

Simulation Study ◽

Clustering Algorithm ◽

Computational Cost ◽

Interpolation Formula ◽

Large Datasets ◽

Mixed Data ◽

Multivariate Techniques ◽

High Computational Cost

This work provides a procedure with which to construct and visualize profiles, i.e., groups of individuals with similar characteristics, for weighted and mixed data by combining two classical multivariate techniques, multidimensional scaling (MDS) and the k-prototypes clustering algorithm. The well-known drawback of classical MDS in large datasets is circumvented by selecting a small random sample of the dataset, whose individuals are clustered by means of an adapted version of the k-prototypes algorithm and mapped via classical MDS. Gower’s interpolation formula is used to project remaining individuals onto the previous configuration. In all the process, Gower’s distance is used to measure the proximity between individuals. The methodology is illustrated on a real dataset, obtained from the Survey of Health, Ageing and Retirement in Europe (SHARE), which was carried out in 19 countries and represents over 124 million aged individuals in Europe. The performance of the method was evaluated through a simulation study, whose results point out that the new proposal solves the high computational cost of the classical MDS with low error.

Download Full-text