Varying Coefficient Regression Model Research Articles

Efficient Modeling of Distributed Energy Resources’ Impact on Electric Grid Technical Losses: A Dynamic Regression Approach

Technical losses in electrical grids are inherent inefficiencies induced by the transmission and distribution of electricity, resulting in energy losses that can reach up to 40% of the generated energy. These losses pose significant challenges to grid operators regarding energy sustainability, reliability, and economic viability. Distributed Energy Resources (DERs) offer promising solutions to lower technical losses by decentralizing energy generation and consumption, reducing the need for long-distance transmission and optimizing grid operation. Hence, estimating the impact of DERs on grid technical losses becomes paramount for grid operators and planners. In response, this article proposes the application of regression modeling and nonlinear curve fitting algorithms to provide a more nuanced understanding and better characterize the intricate interplay between DER deployment and technical losses. Through a comprehensive case study based on more than 1080 computer simulations, we demonstrate the effectiveness of our proposed dynamic polynomial varying coefficient regression model in estimating the impact of DERs on technical losses within electrical grids. The proposed model offers a simple and effective methodology that allows grid operators to gain insights into the nonlinear dynamics of DER integration and make quicker and more informed decisions regarding grid management strategies, infrastructure investments, and policy interventions. Also, this research contributes to advancing the field of grid optimization by offering a simple equation that enhances our ability and haste to assess and mitigate technical losses in the context of an evolving energy landscape characterized by increasing DER adoption.

Semi-supervised estimation for the varying coefficient regression model

<abstract><p>In many cases, the 'labeled' outcome is difficult to observe and may require a complicated or expensive procedure, and the predictor information is easy to be obtained. We propose a semi-supervised estimator for the one-dimensional varying coefficient regression model which improves the conventional supervised estimator by using the unlabeled data efficiently. The semi-supervised estimator is proposed by introducing the intercept model and its asymptotic properties are proven. The Monte Carlo simulation studies and a real data example are conducted to examine the finite sample performance of the proposed procedure.</p></abstract>

Efficient estimation in varying coefficient regression models

Varying coefficient models have been studied for many years, and local kernel estimation is an important method to estimate the coefficient functions. Since same bandwidth parameter is used for each coefficient function, local kernel estimation is not effective when the degree of smoothness is different for each coefficient function. In this paper, a new global estimation method based on kernel function and backfitting is proposed. In this method, different bandwidth parameters can be used for each coefficient function to improve the estimation accuracy, and asymptotic properties of the estimators are proved. To reduce the computational complexity, an adaptive global estimation method, which is based on an estimation of the bandwidth ratio and has only one bandwidth parameter to be selected, is proposed. Simulation results show that the proposed methods work well, and two real data examples further demonstrate the potential of the proposed methods.

Robust Variable Selection for Single-Index Varying-Coefficient Model with Missing Data in Covariates

As applied sciences grow by leaps and bounds, semiparametric regression analyses have broad applications in various fields, such as engineering, finance, medicine, and public health. Single-index varying-coefficient model is a common class of semiparametric models due to its flexibility and ease of interpretation. The standard single-index varying-coefficient regression models consist mainly of parametric regression and semiparametric regression, which assume that all covariates can be observed. The assumptions are relaxed by taking the models with missing covariates into consideration. To eliminate the possibility of bias due to missing data, we propose a probability weighted objective function. In this paper, we investigate the robust variable selection for a single-index varying-coefficient model with missing covariates. Using parametric and nonparametric estimates of the likelihood of observations with fully observed covariates, we examine the estimators for estimating the likelihood of observations. For variable selection, we use a weighted objective function penalized by a non-convex SCAD. Theoretical challenges include the treatment of missing data and a single-index varying-coefficient model that uses both the non-smooth loss function and the non-convex penalty function. We provide Monte Carlo simulations to evaluate the performance of our approach.

Varying-coefficient regression analysis for pooled biomonitoring.

Human biomonitoring involves measuring the accumulation of contaminants in biological specimens (such as blood or urine) to assess individuals' exposure to environmental contamination. Due to the expensive cost of a single assay, the method of pooling has become increasingly common in environmental studies. The implementation of pooling starts by physically mixing specimens into pools, and then measures pooled specimens for the concentration of contaminants. An important task is to reconstruct individual-level statistical characteristics based on pooled measurements. In this article, we propose to use the varying-coefficient regression model for individual-level biomonitoring and provide methods to estimate the varying coefficients based on different types of pooled data. Asymptotic properties of the estimators are presented. We illustrate our methodology via simulation and with application to pooled biomonitoring of a brominated flame retardant provided by the National Health and Nutrition Examination Survey (NHANES).

Quantile Fuzzy Varying Coefficient Regression based on kernel function

The fuzzy varying coefficient regression model is a generalized version of fuzzy linear regression model. This kind of model is flexible and adaptable than fuzzy linear regression model. In this paper, we introduce a fuzzy varying coefficient regression model based on the quantile loss function and under the kernel function. Based on the presented goodness of fit indices, we show that the proposed approach is robust under the outlier data. Some applications of this approach are studied on some data sets and a simulated data set.

Measuring the symmetry of model errors for varying coefficient regression models based on correlation coefficient

In this paper, we propose a residuals based estimator of k-th correlation coefficient between the density function and distribution function for varying coefficient regression models, and further we use this k-th correlation coefficient to test whether the density function of the true model error is symmetric or not. First, we propose a moment based estimator of k-th correlation coefficient and present its asymptotic results. Second, we consider statistical inference of k-th correlation coefficient by using the empirical likelihood method, and the empirical likelihood statistic is shown to be asymptotically distributed as Chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the CEO dataset.

Robust functional coefficient selection for the single-index varying coefficients regression model

ABSTRACT A rank-based estimation and selection of the functional regression coefficients for the single-index varying coefficient regression model is considered. The estimation of the functional regression coefficients together with the selection procedure are carried out following a proposed back-fitting type computational algorithm by minimizing the rank-based objective function. Finite sample performance of the proposed estimator are evaluated via extensive Monte Carlo simulation studies. These demonstrate the robustness and efficiency of the proposed estimator compared to its least squares counterpart under different settings of the model error distribution. A real data example motivated by problems in deep-water fish ecology is given to illustrate the proposed methodology.

Clustered spatio-temporal varying coefficient regression model.

In regression analysis for spatio-temporal data, identifying clusters of spatial units over time in a regression coefficient could provide insight into the unique relationship between a response and covariates in certain subdomains of space and time windows relative to the background in other parts of the spatial domain and the time period of interest. In this article, we propose a varying coefficient regression method for spatial data repeatedly sampled over time, with heterogeneity in regression coefficients across both space and over time. In particular, we extend a varying coefficient regression model for spatial-only data to spatio-temporal data with flexible temporal patterns. We consider the detection of a potential cylindrical cluster of regression coefficients based on testing whether the regression coefficient is the same or not over the entire spatial domain for each time point. For multiple clusters, we develop a sequential identification approach. We assess the power and identification of known clusters via a simulation study. Our proposed methodology is illustrated by the analysis of a cancer mortality dataset in the Southeast of the U.S.

Empirical likelihood in varying-coefficient quantile regression with missing observations

In this paper, we focus on the partially linear varying-coefficient quantile regression model with observations missing at random (MAR), which include the responses or the responses and covariates MAR. Based on the local linear estimation of the varying-coefficient function in the model, we construct empirical log-likelihood ratio functions for unknown parameter in the linear part of the model, which are proved to be asymptotically weighted chi-squared distributions, further the adjusted empirical log-likelihood ratio functions are verified to converge to standard chi-squared distribution. The asymptotic normality of maximum empirical likelihood estimator for the parameter is also established. In order to do variable selection, we consider penalized empirical likelihood by using smoothly clipped absolute deviationv (SCAD) penalty, and the oracle property of the penalized likelihood estimator of the parameter is proved. Furthermore, Monte Carlo simulation and a real data analysis are undertaken to test the performance of the proposed methods.

Flexible regression approach to propensity score analysis and its relationship with matching and weighting.

In propensity score analysis, the frequently used regression adjustment involves regressing the outcome on the estimated propensity score and treatment indicator. This approach can be highly efficient when model assumptions are valid, but can lead to biased results when the assumptions are violated. We extend the simple regression adjustment to a varying coefficient regression model that allows for nonlinear association between outcome and propensity score. We discuss its connection with some propensity score matching and weighting methods, and show that the proposed analytical framework can shed light on the intrinsic connection among some mainstream propensity score approaches (stratification, regression, kernel matching, and inverse probability weighting) and handle commonly used causal estimands. We derive analytic point and variance estimators that properly take into account the sampling variability in the estimated propensity score. Extensive simulations show that the proposed approach possesses desired finite sample properties and demonstrates competitive performance in comparison with other methods estimating the same causal estimand. The proposed methodology is illustrated with a study on right heart catheterization.

Smooth backfitting for errors-in-variables varying coefficient regression models

Varying coefficient models inherit the simplicity and easy interpretation of classical linear models while enjoying the flexibility of nonparametric models. They are very useful in analyzing the relation between a response and a set of predictors. There has been no study, however, on the estimation of varying coefficients when the predictors, on which the varying coefficients depend, are contaminated by measurement errors. A new kernel smoothing technique that is tailored to the structure of an underlying varying coefficient model as well as corrects for the bias due to the measurement errors is developed here. The estimators of the varying coefficients are given implicitly by solving a system of integral equations, whose implementation requires an iterative backfitting procedure. The existence of a unique solution and the convergence of the associated backfitting algorithm are established theoretically. Some numerical evidences that support the theory and demonstrate the success of the proposed methodology are presented.

Averaged and Integrated Estimations of Varying-Coefficient Regression Models with Dependent Observations

In practical applications, lots of data such as sequentially collected economic data often exhibit some evident dependence. This paper studies the varying-coefficient regression models with different smoothing variables when the data form a stationaryα-mixing sequence. Both the averaged and integrated estimators of coefficient functions are proposed. The asymptotic normalities of the proposed averaged and integrated estimators are also established.

Adjusting for baseline information in comparing the efficacy of treatments using bivariate varying-coefficient models

ABSTRACT In biomedical studies, patients' reaction to the treatment can be different depending on their health condition at baseline. In this paper, we develop a bivariate varying-coefficient regression model for longitudinal data with the baseline outcome. The proposed model enables the exploration of the dynamic trend of response variables over time and to provide an effective treatment based on an individual's baseline level of disease by allowing the coefficients to vary with time and baseline. The varying coefficients are modelled through basis function approximation and a set of basis functions is selected by the proposed criterion based on the empirical loglikelihood. After the proposed model is fitted to data, the hypothesis test is designed to evaluate the efficacy of treatments across baseline levels. Theoretical and empirical studies confirm that the proposed methods choose the most parsimonious model consistently and compare the treatment effects successfully across baseline levels. The entire procedure is illustrated with depression data analysis.

Parameter Estimation of Locally Compensated Ridge-Geographically Weighted Regression Model

Geographically weighted regression (GWR) is a spatial data analysis method where spatially varying relationships are explored between explanatory variables and a response variable. One unresolved problem with spatially varying coefficient regression models is local collinearity in weighted explanatory variables. The consequence of local collinearity is: estimation of GWR coefficients is possible but their standard errors tend to be large. As a result, the population values of the coefficients cannot be estimated with great precision or accuracy. In this paper, we propose a recently developed method to remediate the collinearity effects in GWR models using the Locally Compensated Ridge Geographically Weighted Regression (LCR-GWR). Our focus in this study was on reviewing the estimation parameters of LCR-GWR model. And also discussed an appropriate statistic for testing significance of parameters in the model. The result showed that Parameter estimation of LCR-GWR model using weighted least square method is , where the ridge parameter, λ, varies across space. The LCR-GWR is not necessarily calibrates the ridge regressions everywhere; only at locations where collinearity is likely to be an issue. And the parameter significance test using t-test, t = .

Residuals based Kolmogorov-Smirnov and Cramér-von Mises tests for varying coefficient models

In this article, the problem of estimating the model error distribution in a heteroscedastic varying coefficient regression model is considered. A residual-based estimator of the model error distribution is proposed, associated with its asymptotic results. Furthermore, the application of estimated error distribution for testing whether some varying coefficient components are constants or not is investigated. Test statistics based on the Kolmogorov-Smirnov and Cramér-von Mises type functionals of the estimated model error distribution are used to check the null hypothesis. A bootstrap procedure is further proposed to calculate the critical values. Simulation studies are conducted to demonstrate the performance of the proposed test statistics.

Interactions between ambient air pollutants and temperature on emergency department visits: Analysis of varying-coefficient model in Guangzhou, China

BackgroundAt present, there are few studies on the effect of short-term interactions between ambient air pollutants and temperature on cause-specific emergency department visits in China. This study aimed to explore their short-term interactions on cause-specific emergency department visits using data collected from a total of 65 public hospitals in Guangzhou city, south China. Material and methodsWe included a total of 226,443 emergency department visits which were diagnosed as neurological, respiratory and circulatory disease in Guangzhou from January 1, 2014 to December 31, 2017. Average daily concentrations of air pollutants including carbon monoxide (CO), particulate matter having a median diameter of 2.5 μm or less (PM2.5), sulfur dioxide (SO2), nitrogen dioxide (NO2) and ozone (O3) were collected from the Guangzhou Environmental Protection Bureau. We employed quasi-Poisson varying coefficient regression models to assess the interaction effects between air pollutants and daily temperature levels (DTLs) on emergency department visits for neurological, respiratory and circulatory diseases, respectively. ResultsAverage number of emergency department visits for neurological, respiratory and circulatory diseases were 92, 26 and 38, respectively. After controlling for other pollutants, meteorological factors and other time-varying confounders, we found the interactions between NO2 and the 1st DTL (3.4–17.1 °C), NO2 and the 2nd DTL (17.1–23.5 °C) for neurological emergency department visits were statistically significant, displaying a nonlinear relationship. Additionally, we found that the interactions between SO2 and the 4th DTL (27.4–31.1 °C) also had a significantly adverse effect on respiratory emergency department visits. ConclusionsOur findings provide novel evidence on SO2-by-temperature interactions, and NO2-by-temperature interactions for emergency department visits of cause-specific diseases.

General local rank estimation for single-index varying coefficient models

In this paper, we consider rank-based estimation of the index coefficient and the functional regression coefficients for the single-index varying coefficient regression model. The consistency and asymptotic normality of the proposed estimators are established under mild assumptions. An extensive Monte-Carlo simulation study demonstrates the robustness and efficiency of the proposed estimators compared to the least squares estimators. An application in ecology is provided and shows that the rank-based regression procedure effectively provides more accurate estimates compared to its least squares counterpart.

Estimation of the error distribution in a varying coefficient regression model

This paper deals with the estimation of the error distribution function in a varying coefficient regression model. We propose two estimators and study their asymptotic properties by obtaining uniform stochastic expansions. The first estimator is a residual-based empirical distribution function. We study this estimator when the varying coefficients are estimated by under-smoothed local quadratic smoothers. Our second estimator which exploits the fact that the error distribution has mean zero is a weighted residual-based empirical distribution whose weights are chosen to achieve the mean zero property using empirical likelihood methods. The second estimator improves on the first estimator. Bootstrap confidence bands based on the two estimators are also discussed.

Penalized spline estimation in varying coefficient models with censored data

We consider P-spline smoothing in a varying coefficient regression model when the response is subject to random right censoring. We introduce two data transformation approaches to construct a synthetic response vector that is used in a penalized least squares optimization problem. We prove the consistency and asymptotic normality of the P-spline estimators for a diverging number of knots and show by simulation studies and real data examples that the combination of a data transformation for censored observations with P-spline smoothing leads to good estimators of the varying coefficient functions.