Varying Coefficient Models Research Articles

Variable screening for varying coefficient models with ultrahigh-dimensional survival data

In this article, we develop a variable screening method for varying coefficient hazards models of single-index form. The proposed method can be viewed as a natural survival extension of conditional correlation screening. An appealing feature of the proposed method is that it is applicable to many popularly used survival models, including the varying coefficient additive hazards model and the varying coefficient Cox model. The proposed method enjoys the sure screening property, and the number of the selected covariates can be bounded by a moderate order. Simulation studies demonstrate that our method performs well, and an empirical example is also presented.

Spatially varying coefficient models using reduced-rank thin-plate splines

Spatially varying coefficient (SVC) regression models are concerned about regression for spatial data, where regression coefficients may vary in space. This paper proposes a new approach for SVC modeling by representing regression coefficients using a class of multiresolution spline basis functions in a generalized-linear model framework. The proposed method provides flexible and parsimonious representations for regression coefficients. It enables commonly used (generalized) linear-regression packages for estimation, testing, and constructing confidence levels. We develop a fast estimation algorithm that simultaneously performs variable selection, detects spatial heterogeneity for each variable, and determines its complexity. We provide numerical examples and an application to a real estate dataset to demonstrate the proposed method’s effectiveness.

Local Walsh-average regression for single index varying coefficient models

The local Walsh-average regression has been shown to be a comparable method to the others in nonparametric regression. In this paper, we develop a new estimation procedure based on local Walsh-average regression for single index varying coefficient models. Under some general assumptions, we establish the asymptotic properties of the proposed estimators for both the parametric and nonparametric parts. We further demonstrate that the proposed estimates have great efficiency gains across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error compared with that of local polynomial regression estimates under the least squares loss. Even in the worst case scenarios, the asymptotic relative efficiency owns a lower bound equaling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric link functions respectively, versus the least squares estimators. A simulation study is conducted to assess the finite sample properties of the proposed procedure, and a real data analysis is followed to further illustrate the application of the proposed methodology.

A temporal construal theory explanation of the price-quality relationship in online dynamic pricing

According to construal theory, the desirability of a product influences consumers’ choices from a temporally, socially, spatially and hypothetically distant perspective. The purpose of this study is to test the temporal dimension of such theory with reference to hotel room prices, also distinguishing by firm size. Temporal distance is measured through the days of advance booking, in a varying-coefficient model reflecting both the dynamic pricing behaviour of the supply and the seasonal oscillation pattern of the demand. Results support the appropriateness of the temporal construal theory as an interpretative framework for the effectiveness of dynamic pricing, from the perspective of behavioural psychology. On average, the influence of quality on price in small hotels is lower than in large hotels; however, its variability is higher in small hotels. Large hotels charge a premium price during peak season dates, while small firms apply discounts, likely aiming to reach full occupation.

Nonlinear spatial and temporal decomposition provides insight for climate change effects on sub-Arctic herbivore populations

Global temperatures are increasing, affecting timing and availability of vegetation along with relationships between plants and their consumers. We examined the effect of population density, herd body condition in the previous year, elevation, plant productivity and phenology, snow, and winter onset on juvenile body mass in 63 semi-domesticated populations of Rangifer tarandus throughout Norway using spatiotemporal generalized additive models (GAMs) and varying coefficient models (VCMs). Optimal climate windows were calculated at both the regional and national level using a novel nonlinear climate window algorithm optimized for prediction. Spatial and temporal variation in effects of population and environmental predictors were considered using a model including covariates decomposed into spatial, temporal, and residual components. The performance of this decomposed model was compared to spatiotemporal GAMs and VCMs. The decomposed model provided the best fit and lowest prediction errors. A positive effect of herd body condition in the previous year explained most of the deviance in calf body mass, followed by a more complex effect of population density. A negative effect of timing of spring and positive effect of winter onset on juvenile body mass suggested that a snow free season was positive for juvenile body mass growth. Our findings suggest early spring onset and later winter permanent snow cover as reinforcers of early-life conditions which support more robust reindeer populations. Our methodological improvements for climate window analyses and effect size measures for decomposed variables provide important contributions to account for, measure, and interpret nonlinear relationships between climate and animal populations at large scales.

New solutions to the generalized (2+1)-D Boiti–Leon–Pempinelli equation

In this article, we will study the (2 + 1)-D Boiti–Leon–Pempinelli (BLP) model, which has an application in hydrodynamics. This model explains the evolution of water waves in hydrodynamics. We obtained new explicit solutions to the generalized (2 + 1)-D BLP equation using the improved tanh-coth method. This method proves to be a reliable and effective tool for solving nonlinear wave equations. Furthermore, different solitary wave solutions are constructed such as dark, bright, and kink solitons. The variable-coefficient models are more generic than their constant-coefficient counterparts are, the reported for the variable coefficient BLP equation are stable and different relative to those achieved using other techniques.

A statistical framework for recovering pseudo-dynamic networks from static data.

The collection of temporal or perturbed data is often a prerequisite for reconstructing dynamic networks in most cases. However, these types of data are seldom available for genomic studies in medicine, thus significantly limiting the use of dynamic networks to characterize the biological principles underlying human health and diseases. We proposed a statistical framework to recover disease risk-associated pseudo-dynamic networks (DRDNet) from steady-state data. We incorporated a varying coefficient model with multiple ordinary differential equations to learn a series of networks. We analyzed the publicly available Genotype-Tissue Expression data to construct networks associated with hypertension risk, and biological findings showed that key genes constituting these networks had pivotal and biologically relevant roles associated with the vascular system. We also provided the selection consistency of the proposed learning procedure and evaluated its utility through extensive simulations. DRDNet is implemented in the R language, and the source codes are available at https://github.com/chencxxy28/DRDnet/. Supplementary data are available at Bioinformatics online.

A flexible movement model for partially migrating species

We propose a flexible model for a partially migrating species, which we demonstrate using yearly paths for golden eagles (Aquila chrysaetos). Our model relies on a smoothly time-varying potential surface defined by a number of attractors. We compare our proposed approach using varying coefficients to a latent-state model, which we define differently for migrating, dispersing, and local individuals. While latent-state models are more common in the existing animal movement literature, varying coefficient models have various benefits including the ability to fit a wide range of movement strategies without the need for major model adjustments. We compare simulations from the models for three individuals to illustrate the ability of our model to better describe movement behavior for specific movement strategies. We also demonstrate the flexibility of our model by fitting several individuals whose movement behavior is less stereotypical.

A zero-inflated mixture spatially varying coefficient modeling of cholera incidences

Spatial disease modeling remains an important public health tool. For cholera, the presence of zero counts is common. The Poisson model is inadequate to (1) capture over-dispersion, and (2) distinguish between excess zeros arising from non-susceptible and susceptible populations. In this study, we develop zero-inflated (ZI) mixture spatially varying coefficient (SVC) models to (1) distinguish between the sources of the excess zeros and (2) uncover the spatially varying effects of precipitation and temperature (LST) on cholera. We demonstrate the potential of the models using cholera data from Ghana. A striking observation is that the Poisson model outperformed the ZI mixture models in terms of fit. The ZI Negative Binomial (ZINB) outperformed the ZI Poisson (ZIP) model. Subject to our objectives, we make inferences using the ZINB model. The proportion of zeros estimated with the ZINB model is 0.41 and exceeded what would have been estimated using a Poisson model which is 0.35. We observed the spatial trends of the effects of precipitation and LST to have both increasing and decreasing gradients; an observation implying that the use of only the global coefficients would lead to wrong inferences. We conclude that (1) the use of ZI mixture models has epidemiological significance. Therefore, its choice over the Poisson model should be based on an epidemiological concept rather than model fit and, (2) the extension of ZI mixture models to accommodate spatially varying coefficients uncovered remarkable varying effects of the covariates. These findings have significant implications for public health monitoring of cholera.

Sparse regression for low-dimensional time-dynamic varying coefficient models with application to air quality data

Time dynamic varying coefficient models play an important role in applications of biology, medicine, environment, finance, etc. Traditional methods, such as kernel smoothing and spline smoothing, are popular. But explicit expressions are unavailable using these methods, and the convergence rate of coefficient function estimators is slow. To address these problems, we expand the varying component with appropriate basis functions. And then we solve a sparse regression problem via a sequential thresholded least-squares estimator. The “parameterization” leads to explicit expressions and fast computation speed. Convergence of the sequential thresholded least squares algorithm is guaranteed. The asymptotic distribution of the coefficient function estimator is derived under certain assumptions. Our simulation shows the proposed method has higher precision and computing speed. Finally, our proposed method is applied to the study of PM concentration in Beijing. We analyze the relationship between PM and other impact factors.

Postnatal exposure to PM2.5 and weight trajectories in early childhood.

Background:Inconsistent evidence has assessed the impact of air pollution exposure on children’s growth trajectories. We investigated the role of 90-day average postnatal fine particulate matter (PM2.5) exposures by estimating the magnitude of effects at different ages, and the change in child weight trajectory by categories of exposure.Methods:We obtained weight values from electronic health records at each hospital visit (males = 1859, females = 1601) from birth to 6 years old children recruited into the Boston-based Children’s HealthWatch cohort (2009–2014). We applied mixed models, adjusting for individual and maternal confounders using (1) varying-coefficient models allowing for smooth non-linear interaction between age and PM2.5, (2) factor-smooth interaction between age and PM2.5 quartiles. Additionally, we stratified by sex and low birthweight (LBW) status (≤2500 g).Results:Using varying-coefficient models, we found that PM2.5 significantly modified the association between age and weight in males, with a positive association in children younger than 3 years and a negative association afterwards. In boys, for each 10 µg/m3 increase in PM2.5 we found a 2.6% increase (95% confidence interval = 0.8, 4.6) in weight at 1 year of age and a −0.6% (95% confidence interval = −3.9, 2.9) at 5 years. We found similar but smaller changes in females, and no differences comparing growth trajectories across quartiles of PM2.5. Most of the effects were in LBW children and null for normal birthweight children.Conclusions:This study suggests that medium-term postnatal PM2.5 may modify weight trajectories nonlinearly in young children, and that LBW babies are more susceptible than normal-weight infants.

Using an Eigenvector Spatial Filtering-Based Spatially Varying Coefficient Model to Analyze the Spatial Heterogeneity of COVID-19 and Its Influencing Factors in Mainland China

The COVID-19 pandemic has led to many deaths and economic disruptions across the world. Several studies have examined the effect of corresponding health risk factors in different places, but the problem of spatial heterogeneity has not been adequately addressed. The purpose of this paper was to explore how selected health risk factors are related to the pandemic infection rate within different study extents and to reveal the spatial varying characteristics of certain health risk factors. An eigenvector spatial filtering-based spatially varying coefficient model (ESF-SVC) was developed to find out how the influence of selected health risk factors varies across space and time. The ESF-SVC was able to take good control of over-fitting problems compared with ordinary least square (OLS), eigenvector spatial filtering (ESF) and geographically weighted regression (GWR) models, with a higher adjusted R2 and lower cross validation RMSE. The impact of health risk factors varied as the study extent changed: In Hubei province, only population density and wind speed showed significant spatially constant impact; while in mainland China, other factors including migration score, building density, temperature and altitude showed significant spatially varying impact. The influence of migration score was less contributive and less significant in cities around Wuhan than cities further away, while altitude showed a stronger contribution to the decrease of infection rates in high altitude cities. The temperature showed mixed correlation as time passed, with positive and negative coefficients at 2.42 °C and 8.17 °C, respectively. This study could provide a feasible path to improve the model fit by considering the problem of spatial autocorrelation and heterogeneity that exists in COVID-19 modeling. The yielding ESF-SVC coefficients could also provide an intuitive method for discovering the different impacts of influencing factors across space in large study areas. It is hoped that these findings improve public and governmental awareness of potential health risks and therefore influence epidemic control strategies.

Examining the vintage effect in hedonic pricing using spatially varying coefficients models: a case study of single-family houses in the Canton of Zurich

This article examines the spatially varying effect of age on single-family house (SFH) prices. Age has been shown to be a key driver for house depreciation and is usually associated with a negative price effect. In practice, however, there exist deviations from this behavior which are referred to as vintage effects. We estimate a spatially varying coefficients (SVC) model to investigate the spatial structures of vintage effects on SFH pricing. For SFHs in the Canton of Zurich, Switzerland, we find substantial spatial variation in the age effect. In particular, we find a local, strong vintage effect primarily in urban areas compared to pure depreciative age effects in rural locations. Using cross validation, we assess the potential improvement in predictive performance by incorporating spatially varying vintage effects in hedonic models. We find a substantial improvement in out-of-sample predictive performance of SVC models over classical spatial hedonic models.

B-spline estimation in varying coefficient models with correlated errors

<abstract><p>The varying coefficient model assumes that the regression function depends linearly on some regressors, and that the regression coefficients are smooth functions of other predictor variables. It provides an appreciable flexibility in capturing the underlying dynamics in data and avoids the so-called "curse of dimensionality" in analyzing complex and multivariate nonlinear structures. Existing estimation methods usually assume that the errors for the model are independent; however, they may not be satisfied in practice. In this study, we investigated the estimation for the varying coefficient model with correlated errors via B-spline. The B-spline approach, as a global smoothing method, is computationally efficient. Under suitable conditions, the convergence rates of the proposed estimators were obtained. Furthermore, two simulation examples were employed to demonstrate the performance of the proposed approach and the necessity of considering correlated errors.</p></abstract>

Efficient estimation and computation in generalized varying coefficient models with unknown link and variance functions for large-scale data

Efficient estimation and computation in generalized varying coefficient models with unknown link and variance functions for large-scale data

Situating Politics: Spatial Heterogeneity and the Study of Political History

AbstractWhile quantitative methods are routinely used to examine historical materials, critics take issue with the use of global regression models that attach a single parameter to each predictor, thereby ignoring the effects of time and space, which together define the context in which historical events unfold. This problem can be addressed by allowing for parameter heterogeneity, as highlighted by the proliferation of work on the use of time-varying parameter models. In this article, I show how this approach can be extended to the case of spatial data using spatially varying coefficient models, with an eye toward the study of electoral politics, where the use of spatial data is especially common in historical settings. Toward this end, I revisit a critical case in the field of quantitative history: the rise of electoral Populism in the American West in the period between 1890 and 1896. Upending popular narratives about the correlates of third-party support in the late nineteenth century, I show that the association between third-party vote share and traditional predictors such as economic hardship and ethnic composition varied considerably from one place to the next, giving rise to distinct varieties of electoral Populism—a finding that is missed by global models, which mistake the mathematically particular for the historically general. These findings have important theoretical and empirical implications for the study of political action in a world where parameter heterogeneity is increasingly recognized as a standard feature of modern social science.

Optimum Tuning Parameter Selection in Generalized lasso for Clustering with Spatially Varying Coefficient Models

Spatial clustering with spatially varying coefficient models is useful for determining the region with common effects of variables in spatial data. This study focuses on selecting the optimum tuning parameter of the generalized lasso for clustering with the spatially varying coefficient model. The k-fold cross-validation (CV) may fail to split spatial data into a training set and a testing set, if a region contains only a few observations. Moreover, the k-fold CV is known to give a biased estimate of the out-of-sample prediction error. Therefore, we investigated the performance of approximate leave-one-out cross-validation (ALOCV) in comparison with k-fold CV for selecting the tuning parameter in a simulation study on 2-dimensional grid. The ALOCV yielded smaller error than k-fold CV and could detect edges with differences shrunk by generalized lasso appropriately. Then, the ALOCV for selecting the optimum tuning parameter of the generalized lasso in fitting the spatially varying coefficient model is applied to the Chicago crime data. The result of selection by ALOCV was in accordance with the conclusion suggested in the preceding literature. Clustering into regions in advance for making k-fold CV feasible may lead to a wrong result of clustering with a spatially varying coefficient model.

Estimation of Hilbertian varying coefficient models

Estimation of Hilbertian varying coefficient models

Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model

Lakes on the Tibet Plateau (TP) have a significant impact on the water cycle and water balance, and it is important to monitor changes in lake area and identify the influencing factors. Existing research has failed to quantitatively identify the changes and influencing factors of lakes in different regions of the TP. Thus, an eigenvector spatial filtering based spatially varying coefficient (ESF-SVC) model was used to analyze the relationship between lake area and climatic and terrain factors in the inner watershed of the TP from 2000 to 2015. A comparison with ordinary regression and spatial models showed that the ESF-SVC model eliminates spatial autocorrelation and has the best model fit and complexity. The experiments demonstrated that precipitation, snow melt, and permafrost moisture release, as well as the area of vegetation and elevation difference in the watershed, can significantly promote the expansion of lakes, while evapotranspiration and days of mean daily temperature above zero have an inhibitory effect on lake area expansion. The degree of influence of each factor also differs significantly over time and across regions. Spatially quantitative modeling of lake area in the TP using the ESF-SVC method is a new attempt to provide novel ideas for lake research.

Interpretable Modeling for Short- and Medium-Term Electricity Demand Forecasting

We consider the problem of short- and medium-term electricity demand forecasting by using past demand and daily weather forecast information. Conventionally, many researchers have directly applied regression analysis. However, interpreting the effect of weather on the demand is difficult with the existing methods. In this study, we build a statistical model that resolves this interpretation issue. A varying coefficient model with basis expansion is used to capture the nonlinear structure of the weather effect. This approach results in an interpretable model when the regression coefficients are nonnegative. To estimate the nonnegative regression coefficients, we employ nonnegative least squares. Three real data analyses show the practicality of our proposed statistical modeling. Two of them demonstrate good forecast accuracy and interpretability of our proposed method. In the third example, we investigate the effect of COVID-19 on electricity demand. The interpretation would help make strategies for energy-saving interventions and demand response.