Orthogonal Matching Pursuit for Sparse Quantile Regression

Varying coefficient model (VCM) is extensively used in various scientific fields due to its capability of capturing the changing structure of predictors. Classical mean regression analysis is often complicated in the existence of skewed, heterogeneous and heavy-tailed data. For this purpose, this work employs the idea of model averaging and introduces a novel comprehensive approach by incorporating quantile-adaptive weights across different quantile levels to further improve both least square (LS) and quantile regression (QR) methods. The proposed procedure that adaptively takes advantage of the heterogeneous and sparse nature of input data can gain more efficiency and be well adapted to extreme event case and high-dimensional setting. Motivated by its nice properties, we develop several robust methods to reveal the dynamic close-to-truth structure for VCM and consistently uncover the zero and nonzero patterns in high-dimensional scientific discoveries. We provide a new iterative algorithm that is proven to be asymptotic consistent and can attain the optimal nonparametric convergence rate given regular conditions. These introduced procedures are highlighted with extensive simulation examples and several real data analyses to further show their stronger predictive power compared with LS, composite quantile regression (CQR) and QR methods.

This paper focuses on the Least Square (LS) regression using the mean and Quantile (M) regression analysis using median which is based on “well-Known” parametric estimation methodologies. Data from Oregon and California highway bridges were used for the comparison of the two methods. Relationships were developed to predict the unit cost of FRP repair work and FRP cost was found to have a high degree of correlation with FRP area for both Oregon and California. It was observed that the Cost Estimating Relationships (CERs) obtained by Quantile (M) regression method had the smaller Mean Absolute Deviation (MAD) values and lower Mean Absolute Percentage Error (MAPE) values than Least Square (LS) regression. The stuudy showed that Quantile Regression is much less sensitive to outliers than Least Squares Regression.

Objective: To assess various marginal effects of nutrient intakes, health behaviours and nutrition knowledge on the entire distribution of body mass index (BMI) across individuals. Design: Quantitative and distributional study. Setting: Taiwan Methods: This study applies Becker’s (1965) model of health production to construct an individual’s BMI determining function. Quantile regression method is used to analyse the entire distribution of BMI across a nationwide survey sample in Taiwan. Endogeneity associated with choice variables of nutrient intakes, health behaviours and nutrition knowledge in determining BMI is corrected using a two-stage least-squares method. Results: Empirical results indicate that energy, oleic acid and cholesterol increase BMI, while fibre, calcium and sodium reveal a reverse effect. For overweight and obese females, protein lowers BMI. For underweight and slightly to severely obese males, vitamin C reduces BMI. Jogging lowers BMI, but drinking raises non-obese individuals’ BMI. For males whose BMI is in the ideal weight to slightly overweight ranges, nutrition knowledge reduces BMI, but this effect is insignificant for females. Conclusions: Several important conclusions can be drawn. (1) Having a BMI which is too high or too low is a risk factor to health. (2) Quantile regression is a useful empirical method to evaluate crucial factors determining the entire distribution (high, medium and low) of BMI. (3) The econometric problem of endogeneity in regressors has to be considered and corrected when using a quantile regression approach. (4) BMI-relevant health education and policy can be evaluated and derived using quantile regression effects of an individual’s choices of nutrient intakes, health behaviours and nutrition knowledge.

This paper presents two probabilistic approaches based on bootstrap method and quantile regression (QR) method to estimate the uncertainty associated with solar photovoltaic (PV) power point forecasts. Solar PV output power forecasts are obtained using a hybrid intelligent model, which is composed of a data filtering technique based on wavelet transform (WT) and a soft computing model (SCM) based on radial basis function neural network (RBFNN) that is optimized by particle swarm optimization (PSO) algorithm. The point forecast capability of the proposed hybrid WT + RBFNN + PSO intelligent model is examined and compared with other hybrid models as well as individual SCM. The performance of the proposed bootstrap method in the form of probabilistic forecasts is compared with the QR method by generating different prediction intervals (PIs). Numerical tests using real data demonstrate that the point forecasts obtained from the proposed hybrid intelligent model can be effectively used to quantify PV power uncertainty. The performance of these two uncertainty quantification methods is assessed through reliability.

In recent years, the intersection of machine learning (ML) and climate science has yielded profound insights into understanding and predicting extreme climate events, particularly heatwaves and droughts. Various approaches have been suggested to define and model extreme events, including extreme value theory (Sura, 2011), random forests (e.g., (Weirich-Benet et al., 2023) and, more recently, deep learning (e.g., (Jacques-Dumas et al., 2022)). Within this context, quantile regression (QR) is valuable for modelling the relationship between variables by estimating the conditional quantiles of the response variable. This provides insights into the entire distribution rather than just the mean but also aids in unravelling the complex relationships among climate variables (Barbosa et al., 2011; Franzke, 2015). QR has been extended in many ways to address critical issues such as nonlinear relations, nonstationary processes, compound events, and the complexities of handling spatio-temporal data. This study presents a novel approach for predicting and better understanding heatwaves. We introduce an interpretable, nonlinear, non-parametric, and structured Spatio-Temporal Quantile Regression (STQR) method that incorporates the QR check function, commonly known as pinball loss, into machine learning models. We focus on analysing how the importance of predictors changes as the quantile being modelled increases. This allows us to circumvent arbitrary definitions of what constitutes a heatwave and instead observe if a natural definition of a heatwave emerges in predictor space. By analysing European heatwaves over recent decades using reanalysis and weather data, we demonstrate the advantages of our methodology over traditional extreme event modelling methods.ReferencesBarbosa, S.M., Scotto, M.G., Alonso, A.M., 2011. Summarising changes in air temperature over Central Europe by quantile regression and clustering. Nat. Hazards Earth Syst. Sci. 11, 3227–3233. https://doi.org/10.5194/nhess-11-3227-2011Franzke, C.L.E., 2015. Local trend disparities of European minimum and maximum temperature extremes. Geophys. Res. Lett. 42, 6479–6484. https://doi.org/10.1002/2015GL065011Jacques-Dumas, V., Ragone, F., Borgnat, P., Abry, P., Bouchet, F., 2022. Deep Learning-based Extreme Heatwave Forecast. Front. Clim. 4, 789641. https://doi.org/10.3389/fclim.2022.789641Sura, P., 2011. A general perspective of extreme events in weather and climate. Atmospheric Res. 101, 1–21. https://doi.org/10.1016/j.atmosres.2011.01.012Weirich-Benet, E., Pyrina, M., Jiménez-Esteve, B., Fraenkel, E., Cohen, J., Domeisen, D.I.V., 2023. Subseasonal Prediction of Central European Summer Heatwaves with Linear and Random Forest Machine Learning Models. Artif. Intell. Earth Syst. 2. https://doi.org/10.1175/AIES-D-22-0038.

Sparse quantile regression is a useful tool for variable selection, robust estimation and heteroscedasticity treatment in high-dimensional data analysis. Due to the non-smooth of quantile loss, the computation is heavier than the least square models. In the literature, there are various numerical methods for linear quantile regression, such as ADMM developed in Gu et al. [Technometrics. 2017;60(3):319–331]. However, the computation of multivariate quantile regression has not yet been fully resolved, especially when the dimension is high. Motivated by this, we will focus on the design of fast numerical algorithms for row-sparse multivariate quantile regression model. By virtue of proximal operator and Majorize–Minimization, four smoothed algorithms are designed. For all the obtained algorithms, we analyse their convergence and the parameter selection. We conduct plenty of simulations and four real data sets analysis. Finally, we conclude that the smoothed method is faster than the non-smooth method, especially when the number of predictors is large.

Quantile regression (QR) provides a flexible statistical framework for modeling the entire conditional distribution of the response variable, making it useful for analysis in various fields. Despite its advantages, existing methods for QR often encounter numerical challenges in high-dimensional settings, especially for those with ordinal responses. In this paper, we use a latent-response framework to construct a Bayesian hierarchical model to conduct parameter estimation and variable selection for ordinal QR. Using the asymmetric Laplace working likelihood and the horseshoe prior for the regression coefficients, we obtain the posterior samples to be screened by the sequential two-means clustering to identify significant predictors. Extensive numerical results via simulation studies and two real-data applications demonstrate the competitive performance of our approach over some other existing Bayesian methods for ordinal data analysis. The illustrative datasets on educational attainment and liver cancer methylation analyses highlight the practical utility of our proposed approach in both low- and high-dimensional scenarios.

Purpose This study aims to investigate the impact of green technological innovation, taxes and environmental pollution on the blue economy within OECD countries. Design/methodology/approach The Method of Moments Quantile Regression (MMQR) and Bootstrap Quantile Regression methods were used to investigate the relationship between Gross Domestic Product (GDP), Natural Resources Rents, carbon dioxide (CO2) emissions, green taxation, green innovation and blue economy. This study uses panel data covering the period from 1990 to 2022. Findings Green taxes demonstrate a stronger positive impact on the blue economy at lower quantiles, with this effect diminishing progressively across the distribution to higher quantiles. Green innovation exhibits a consistently robust positive relationship with the blue economy throughout the distribution, remaining statistically significant across all quantiles examined. Carbon emissions display a significant negative association with the blue economy across the entire conditional distribution, with this adverse effect intensifying at higher quantiles. The findings further reveal that both GDP and natural resources rents exert positive effects on the blue economy across all quantiles, suggesting their role as consistent drivers of blue economy development regardless of the level of blue economy performance. Originality/value By combining MMQR with Bootstrap Quantile Regression, this study conducts a detailed examination of how environmental and economic determinants influence blue economy performance across varying quantiles. The findings contribute to the existing body of knowledge on sustainable ocean economies and provide practical guidance for policymakers seeking to promote economic development while safeguarding environmental integrity.

Together with quantile regression methods, such a model would have the potential for clinical utility through automated quantitative comparison of individual cases relative to their age and gender-matched peer group. Our aim was to demonstrate the automated processing of digital clinical head CT data in the development of a clinically useful model of age-related changes of the brain in the first 2 decades of life. A total of 415 (209 female) consecutive, clinical head CTs with radiographically normal findings from patients from birth through 20 years of age were retrospectively selected and subjected to automated segmentation. Brain volume, brain parenchymal fraction, brain radiodensity, and brain radiomass were assessed as a function of patient age. Statistical modeling and quantile regression were performed. Brain volume increased from 400 cm3 at birth to 1350 cm3 at 20 years of age (>3-fold). Males had a slightly steeper growth trajectory than females, with approximately 8% difference in volume between the sexes established in the first few years of life. Brain parenchymal fraction was variable at younger than 2 years of age, stabilizing between 0.85 and 0.92 at 2-3 years of age. Brain mean radiodensity was lower at birth (24 HU) and increased through 3 years of age, after which it stabilized near 30 HU, an approximately 25% increase. The product of brain volume and mean brain radiodensity (radiomass), increased from 700 HU × mL at birth to 3900 HU × mL, a 5.6-fold increase, with approximately 5% difference between males and females at 20 years. Quantile regression enables a given metric to be interpreted relative to an age- and sex-matched peer group. Automated segmentation of clinical head CT images permitted the generation of a reference database for quantitative analysis of pediatric and adolescent brains. Quantile regression facilitates clinical application.

This study aims to construct the model for the length of hospital stay for patients with COVID-19 using quantile regression and Bayesian quantile approaches. The quantile regression models the relationship at any point of the conditional distribution of the dependent variable on several independent variables. The Bayesian quantile regression combines the concept of quantile analysis into the Bayesian approach. In the Bayesian approach, the Asymmetric Laplace Distribution (ALD) distribution is used to form the likelihood function as the basis for formulating the posterior distribution. All 688 patients with COVID-19 treated in M. Djamil Hospital and Universitas Andalas Hospital in Padang City between March-July 2020 were used in this study. This study found that the Bayesian quantile regression method results in a smaller 95% confidence interval and higher value than the quantile regression method. It is concluded that the Bayesian quantile regression method tends to yield a better model than the quantile method. Based on the Bayesian quantile regression method, it investigates that the length of hospital stay for patients with COVID-19 in West Sumatra is significantly influenced by Age, Diagnoses status, and Discharge status.

A battery capacity estimation method based on the equivalent circuit model and quantile regression using vehicle real-world operation data

BackgroundThere is a well-documented social gradient in obesity in most developed countries. Many previous studies have conventionally categorised individuals according to their body mass index (BMI), focusing on those above a certain threshold and thus ignoring a large amount of the BMI distribution. Others have used linear BMI models, relying on mean effects that may mask substantial heterogeneity in the effects of socioeconomic variables across the population.MethodIn this study, we measure the social gradient of the BMI distribution of the adult population in Spain over the past two decades (1993–2014), using unconditional quantile regressions. We use three socioeconomic variables (education, income and social class) and evaluate differences in the corresponding effects on different percentiles of the log-transformed BMI distribution. Quantile regression methods have the advantage of estimating the socioeconomic effect across the whole BMI distribution allowing for this potential heterogeneity.ResultsThe results showed a large and increasing social gradient in obesity in Spain, especially among females. There is, however, a large degree of heterogeneity in the socioeconomic effect across the BMI distribution, with patterns that vary according to the socioeconomic indicator under study. While the income and educational gradient is greater at the end of the BMI distribution, the main impact of social class is around the median BMI values. A steeper social gradient is observed with respect to educational level rather than household income or social class.ConclusionThe findings of this study emphasise the heterogeneous nature of the relationship between social factors and obesity across the BMI distribution as a whole. Quantile regression methods might provide a more suitable framework for exploring the complex socioeconomic gradient of obesity.Electronic supplementary materialThe online version of this article (doi:10.1186/s12939-016-0454-1) contains supplementary material, which is available to authorized users.

Purpose – This paper aims to investigate whether there is heterogeneity in the relationship between the bank loan interest rate and its determinants using the quantile regression method and to reconcile some conflicting findings in prior literature. Design/methodology/approach – First, the effects of 18 determinants were examined on the bank loan interest rate using the ordinary least squares method (OLS). Second, it was investigated whether the relationship between the loan rate and its determinants is heterogeneous across quantiles of loan rates using the quantile regression method. Findings – Considerable heterogeneity was found in the relationship between the loan rate and its determinants. Specifically, a determinant that is beneficial for the bank loan rate, on average, as revealed by the OLS method may become unimportant or even detrimental for firms located at extremely high or low loan rate quantiles. By revealing extreme heterogeneity in the relationship between the loan rate and some of its determinants, the authors potentially explain two conflicting findings in prior literature. Originality/value – The conventional OLS method masks the heterogeneity in the relationship between the bank loan interest rate and its determinants. Quantile regression can be used to supplement the OLS estimates to gain a more detailed and complete picture of the relationship between the dependent variable and explanatory variables.

In this study, a stepwise cluster forecasting (SCF) framework is proposed for probabilistic prediction for monthly streamflow through integrating stepwise cluster analysis and quantile regression methods. The developed SCF method can capture discrete and nonlinear relationships between explanatory and response variables. A cluster tree was generated through the SCF method for reflecting complex relationships between independent (i.e. explanatory) and dependent (i.e. response) variables in the hydrologic system. Quantile regression approach was employed to construct probabilistic relationships between inputs and outputs in each leaf of the cluster tree. The developed SCF method was applied for monthly streamflow prediction in Xiangxi River based on the gauged records at Xingshan gauging station and related meteorological data. The performance of the SCF method was evaluated through indices of percent bias (PBIAS), RMSE-observations standard deviation ratio (RSR), and Nash-Sutcliffe efficiency coefficient (NSE). Two new indices, the relative distance to the bounds (RDB) and the degree of uncertainty (DOU) were proposed to reflect the uncertainty of the predictions from SCF model. The results showed that the uncertainty of the predictions was acceptable and would not change significantly in calibration and validation periods. Quantile regression was integrated into prediction process of the SCF approach to provide probabilistic forecasts. The 90 % confidence intervals could well bracket the observations in both calibration and validation periods. Comparison among different forecasting techniques showed the effectiveness of the proposed method.

Body mass index is a measure of nutritional status of an individual. Malnutrition is a leading public health problem in developing countries like Nigeria, it is also a major cause of morbidity and mortality. In this study, Body mass index is modeled using ordinary least squares method and quantile regression method. Data is collected from Antiretroviral therapy Clinic in Federal Medical Centre, Jalingo. Variables in the data collected are the Body mass index, age, weight, height, sex and occupation of the patients. Results showed that the ordinary least square regression and quantile regression at 25th percentile, median percentile, 75th percentile and 95th percentile fit the data. Weight, age, sex and height of patients are significant in determining the BMI of the patients when OLS method is applied. While weight, sex and height of patients are significant in determining the BMI of the patients. It is also discovered that OLS method fits the data more than quantile regression method using AIC and MSE.