Fast Bayesian model assessment for nonparametric additive regression

This paper is about optimal estimation of the additive components of a nonparametric, additive isotone regression model. It is shown that asymp- totically up to first order, each additive component can be estimated as well as it could be by a least squares estimator if the other components were known. The algorithm for the calculation of the estimator uses backfitting. Conver- gence of the algorithm is shown. Finite sample properties are also compared through simulation experiments. In this paper we discuss nonparametric additive monotone regression models. We discuss a backfitting estimator that is based on iterative application of the pool adjacent violator algorithm to the additive components of the model. Our main result states the following oracle property. Asymptotically up to first order, each additive component is estimated as well as it would be (by a least squares estimator) if the other components were known. This goes beyond the classical finding that the estimator achieves the same rate of convergence, independently of the number of additive components. The result states that the asymptotic distribution of the estimator does not depend on the number of components. We have two motivations for considering this model. First of all we think that this is a useful model for some applications. For a discussion of isotonic additive regres- sion from a more applied point, see also Bacchetti (1), Morton-Jones et al. (32) and De Boer, Besten and Ter Braak (7). But our main motivation comes from statistical theory. We think that the study of nonparametric models with several nonparamet- ric components is not fully understood. The oracle property that is stated in this paper for additive isotone models has been shown for smoothing estimators in some other nonparametric models. This property is expected to hold if the estimation of the different nonparametric components is based on local smoothing where the lo- calization takes place in different scales. An example are additive models of smooth functions where each localization takes place with respect to another covariate. In Mammen, Linton and Nielsen (28) the oracle property has been verified for the lo- cal linear smooth backfitting estimator. As local linear estimators, also the isotonic least squares is a local smoother. The estimator is a local average of the response variable but in contrast to local linear estimators the local neighborhood is chosen � Research of this paper was supported by the Deutsche Forschungsgemeinschaft project MA

Fractional data that are restricted in the standard unit interval (0, 1) with a highly skewed distribution are commonly encountered. Such data arise in various areas, such as economics, finance, and medicine, among others. One natural idea to model such data is to use the beta family due to its flexibility to accommodate various density shapes. In this paper, we propose a nonparametric additive beta regression model along with a variable selection procedure, where the mean response is related to covariates through the combination of unknown functions of covariates, which can be approximated on a B-spline basis. By using this approximation method, we transform the problem of variable selection into the problem of selecting the groups of coefficients in the expansion. Based on the penalized likelihood method for group variable selection, we successfully select the significant covariates. Moreover, the estimation and selection consistencies and the properties of the penalized estimators are established. The simulation studies demonstrate that the performance of our proposed method is quite good. Finally, we apply the proposed method to body fat data, and we obtain several important findings with satisfactory selection and prediction performance.

Reducing CO2 emissions in China's manufacturing industry: Evidence from nonparametric additive regression models

We propose and investigate additive density regression, a novel additive functional regression model for situations where the responses are random distributions that can be viewed as random densities and the predictors are vectors. Data in the form of samples of densities or distributions are increasingly encountered in statistical analysis and there is a need for flexible regression models that accommodate random densities as responses. Such models are of special interest for multivariate continuous predictors, where unrestricted nonparametric regression approaches are subject to the curse of dimensionality. Additive models can be expected to maintain one-dimensional rates of convergence while permitting a substantial degree of flexibility. This motivates the development of additive regression models for situations where multivariate continuous predictors are coupled with densities as responses. To overcome the problem that distributions do not form a vector space, we utilize a class of transformations that map densities to unrestricted square integrable functions and then deploy an additive functional regression model to fit the responses in the unrestricted space, finally transforming back to density space. We implement the proposed additive model with an extended version of smooth backfitting and establish the consistency of this approach, including rates of convergence. The proposed method is illustrated with an application to the distributions of baby names in the United States.

Background: Nowadays, forecasting models based on nonparametric models have been developed in many branches of science such as mathematics and economics. However, the relatively complicated structure of these models has made them less practical in medical sciences. Objectives: In this study, we investigated the application of a nonparametric regression model to predict the psychological symptoms appearing six months after mild traumatic brain injury. We also made a comparison between the performance of nonparametric regression and artificial neural network models in predicting the psychological symptoms. Methods: In a six-month period during 2015 - 2017, information of 100 mild traumatic patients was included in a prospective cohort study. The data were then divided randomly into two groups of training (n = 50) and testing (n = 50) for 100 times. In the training group, the focus was on 100 artificial neural network and nonparametric regression models. However, in the testing group, a comparison was made between the values obtained using the two final models. To compare the models, the ROC curve and the accuracy rate were finally applied. Results: According to the obtained results, the nonparametric regression model showed an accuracy rate of 91.25% while the neural network model had an accuracy rate of 85.34%. In the experimental set for both neural network and nonparametric regression models, the area under the ROC curve appeared to be 81.51% and 85.73 %, respectively. Conclusions: The nonparametric regression model appeared to be more powerful than the neural network model in predicting psychological symptoms.

This chapter reviews the literature on variable selection in nonparametric and semiparametric regression models via shrinkage. We highlight recent developments on simultaneous variable selection and estimation through the methods of least absolute shrinkage and selection operator (Lasso), smoothly clipped absolute deviation (SCAD), or their variants, but restrict our attention to nonparametric and semiparametric regression models. In particular, we consider variable selection in additive models, partially linear models, functional/varying coefficient models, single index models, general nonparametric regression models, and semiparametric/nonparametric quantile regression models.

Two-stage efficient estimation of longitudinal nonparametric additive models

A non-parametric analysis of the driving factors of China's carbon prices

There are three nonparametric regression approaches, namely, parametric, nonparametric and semi-parametric regression. Nonparametric regression allows the response variable to follow a different curve from one predictor variable to another. In paired data, the components of predictor variable and the response variable are assumed to follow unknown data patterns, so that they can be approximated by a kernel based regression model and Fourier series. The base components are approximated by kernel functions and Fourier series functions. Error is assumed to be normally distributed with zero mean and constant variance. The aim of this research is to make a kernel-based nonparametric regression model and Fourier series on poverty data in the Province of Bali. The research phase method begins with the introduction of the kernel based nonparametric regression model and the Fourier series. The next step is to study the estimation of the regression curve. The function estimation results are highly dependent on the bandwidth, smoothing and oscillation parameters. The resulting model gives a value of R2=0.6278, means that the used variables can explain the model by 62.78 percent. From the obtained modeling re-sults, the Open Unemployment Rate has a positive effect on the percentage of poor people in Bali. Keywords: Nonparametric Regression; Modeling; Kernel; Fourier Series.

SUMMARY When there are covariate effects to be considered, multi-state survival analysis is dominated either by parametric Markov regression models or by semiparametric Markov regression models using Cox's (1972) proportional hazards models for transition intensities between the states. The purpose of this research work is to study alternatives to Cox's model in a general finite-state Markov process setting. We shall look at two alternative models, Aalen's (1989) nonparametric additive hazards model and Lin & Ying's (1994) semiparametric additive hazards model. The former allows the effects of covariates to vary freely over time, while the latter assumes that the regression coefficients are constant over time. With the basic tools of the product integral and the functional delta-method, we present an estimator of the transition probability matrix and develop the large-sample theory for the estimator under each of these two models. Data on 1459 HLA identical sibling transplants for acute leukaemia from the International Bone Marrow Transplant Registry serve as illustration.

Discussion: Nonparametric estimation of noisy integral equations of the second kind

In major domestic cities, the development of urban expressway is network oriented. The traffic flow forecasting system is the important prerequisite and foundation for realizing real-time traffic management and control. However, the traffic flow forecasting research is mainly based on highways. Research and application of short-term traffic forecasting for urban expressway is severely insufficient. Therefore, the study of urban expressway flow forecasting is discussed and a short-term traffic flow forecasting system for urban expressway based on k-NN nonparametric regression model is proposed in this study. First, the study analyzes the characteristics and needs of the urban expressway traffic flow, introduces the k-NN nonparametric regression model, and designs the short-term traffic flow forecasting system based on k-NN overall. Then, the short-term urban expressway flow forecasting system based on k-NN is established in three aspects: the historical database, the search mechanism and algorithm parameters, and the forecasting plan. Finally, a short-term traffic forecasting for urban expressway based on k-NN nonparametric regression model is developed in the VS2010 VC++ platform. Utilizing the Shanghai urban expressway section measured traffic flow data, the comparison of average and weighted k-NN nonparametric regression model is discussed and the reliability of the forecasting result is analyzed. Results show that the accuracy of the proposed method, under the five-minute interval, is over 90%, which best proves the reasonableness of the proposed forecasting model based on the k-NN nonparametric model.

Efficient Estimation of a Semiparametric Characteristic-Based Factor Model of Security Returns

Poverty is a complex problem that has an impact on various social problems such as education, unemployment, health and economic growth. Therefore, the problem of poverty is important to overcome in order to create population welfare. One of the analyses that can be used to model the percentage of poverty is regression analysis. Regression analysis is divided into two approaches, namely parametric and nonparametric. Parametric regression has several assumptions while, the only assumption nonparametric regression shape of the curve does not form a certain pattern. There are several approaches to nonparametric regression, one of which is the Fourier Series. The purpose of this study is to model the percentage of poverty in West Sumatra Province. The unclear shape of the curve in the data used is a consideration for using nonparametric regression. Then it is known that the data used in this study is data per region which tends to have a fluctuating nature. So it is suitable to use the Fourier series approach. In this research, nonparametric regression modeling with one, two, and three oscillation parameters was attempted. The best model was obtained which consisted of two oscillation parameters with a Generalized Cross Validation (GCV) value of 2.110 and R² of 92.44%.

Asymptotic normality of the additive regression components for continuous time processes