Nonlinear Quantile Regression Research Articles

Day-ahead load probability density forecasting using monotone composite quantile regression neural network and kernel density estimation

Due to the aggregation of renewable resources and the fluctuation of demand side, exploring more advanced approaches to maintain the accuracy of short-term load forecasting has become a momentous task. The neural network-based quantile regression probability density forecasting can reveal the prediction uncertainty, which is critical in building energy-saving and reliable power grid. However, quantile curves estimated by previous quantile regression probability density forecasting model may cross each other, violating the basic properties of quantile and jeopardizing the flexibility of modeling. For counteracting this disadvantage, a probability density prediction method based on monotone composite quantile regression neural network and Gaussian kernel function (MCQRNNG) for day-ahead load is proposed. The proposed method guarantees non-crossing by simultaneously estimating multiple nonlinear quantile regression functions to obtain better estimation accuracy. Besides, three-dimensional grid search is adopted to adaptively determine the optimal network structure. Taking real load data carrying quantile crossing from Ottawa in Canada, Baden-Württemberg in Germany and a certain region in China, the performance of the MCQRNNG model is verified from three aspects of point, interval and probability prediction. Results show that the proposed model significantly outperforms the comparison model on the basis of effectively extricating the quantile crossing for power load forecasting.

Stand Volume Growth Modeling with Mixed-Effects Models and Quantile Regressions for Major Forest Types in the Eastern Daxing’an Mountains, Northeast China

The relative growth rate (RGRnv) is the standardized measurement of forest growth, whereby excluding the size differences between individuals allows their performance to be compared equally. The RGRnv model was developed using the National Forest Inventory (NFI) data on the Daxing’an Mountains, in Northeast China, which contain Dahurian larch (Larix gmelinii Rupr.), white birch (Betula platyphylla Suk.), and mixed coniferous–broadleaf forests. Four predictor variables—i.e., quadratic mean diameter (Dq), stand basal area (G), average tree height (Ha), and altitude (A)—and four different methods—i.e., the nonlinear mixed-effects models (NLME), three nonlinear quantile regression (NQR3), five nonlinear quantile regression (NQR5), and nine nonlinear quantile regression (NQR9) models—were used in this study. All the models were validated using the leave-one-out method. The results showed that (1) the mixed coniferous–broadleaf forest presented the highest RGRnv; (2) the RGRnv was negatively correlated with the four predictors, and the heteroscedasticity reduced significantly after the weighting function was integrated into the models; and (3) the quantile regression models performed better than NLME, and NQR9 outperformed both NQR3 and NQR5. To make more accurate predictions, parameters of the adjusted mixed-effects and quantile regression models should be recalculated and localized using sampled RGRnv in each region and then applied to predict all the other RGRnv of plots. MAPE% indicates the mean absolute percentage error. The values were stable when the sample numbers were greater than or equal to six across the three forest types, which showed relatively accurate and lowest-cost prediction results.

Quantile Regression: A Simplified Approach to a Goodness-of-fit Test

Recently, He and Zhu (2003) derived an omnibus goodness-of-fit test for linear or nonlinear quantile regression models based on a CUSUM process of the gradient vector, and they suggested using a particular sim ulation method for determining critical values for their test statistic. But despite the speed of modern computers, execution time can be high. One goal in this note is to suggest a slight modification of their method that eliminates the need for simulations among a collection of important and commonly occurring situations. For a broader range of situations, the modi fication can be used to determine a critical value as a function of the sample size (n), the number of predictors (q), and the quantile of interest (γ). This is in contrast to the He and Zhu approach where the critical value is also a function of the observed values of the q predictors. As a partial check on the suggested modification in terms of controlling the Type I error probability, simulations were performed for the same situations considered by He and Zhu, and some additional simulations are reported for a much wider range of situations.

Nonparametric inference for quantile cointegrations with stationary covariates

This paper considers the inference problems in nonlinear quantile regressions with both stationary and nonstationary covariates. The nonparametric local constant quantile estimator is proposed to estimate the unknown quantile regression function, whose asymptotic properties are established under quite general conditions. Specification testing of the quantile regression function is further considered through a statistic constructed based on the integrated squared distance between the parametric and the nonparametric estimators for the regression function. The test statistic is shown to converge to a random variable related to the local time of an Ornstein–Uhlenbeck process under the parametric null. The power of the test against local alternatives is also investigated. Additional asymptotic results on the null parametric quantile estimators and a bootstrap test are developed as well. Numerical results demonstrate that the proposed nonparametric estimator and the specification test enjoy attractive finite sample performance.

Risk measurement of oil price based on Bayesian nonlinear quantile regression model

Oil price forecasting is one of the most challenging issues since it is noisy, non-stationary, and chaotic. In this paper, we design a Bayesian Nonlinear Quantile method consisting of a Threshold Improved model and an Adaptive MCMC model to improve predicting performance. Specifically, the threshold improve model is introduced to solve the problems caused by the completely asymmetric distribution, and the Adaptive MCMC model is used to get the optimal threshold. Besides, the two-stage framework is applied to improve traditional methods' performance, including the Indirect GARCH model and Asymmetric Slope model. The experimental results show that our approach provides a promising alternative to oil price prediction, and the framework also improve the performance of the traditional methods. The contribution of this paper is to improve the accuracy of the oil price forecasting model, and the framework applies to other energy prices as well.

Quantitative modeling of radioactive cesium concentrations in large omnivorous mammals after the Fukushima nuclear power plant accident

Large quantities of radionuclides released by the Fukushima nuclear power plant accident entered terrestrial and marine ecosystems. The resulting radioactive contamination of large omnivorous wild mammals such as wild boar (Sus scrofa) and Asian black bear (Ursus thibetanus) varied greatly depending on location, season, and time after the accident. Quantitative modeling of how such factors influence radionuclide burdens in these species is important for enhancing current knowledge of chronic radionuclide exposure consequences in mammalian populations, and for assessing potential human risks from consumption of contaminated animal meat. Here we modeled the time course of radioactive cesium (134Cs + 137Cs) concentrations in boar and black bears from Fukushima Prefecture over ~ 7 years after the accident, using nonlinear robust and quantile regressions and mixed-effects modeling. To estimate predictive performance, models fitted to the full data set were compared with those fitted only to the first 3.5 years of data, and tested on the last 3.5 years of data. Ecological half-lives for radioactive cesium, and magnitudes and phase shifts for sinusoidal seasonal oscillations in cesium burdens, were estimated by each analysis method for each species. These results can improve the understanding and prediction of radionuclide concentrations in large mammals that inhabit radioactively contaminated areas.

Estimating nitrogen, phosphorus, potassium, and sulfur uptake and requirement in soybean

Estimation of crop nutrient demand, seed nutrient removal, and nutrient use efficiency (yield to nutrient uptake ratio) are crucial for pursuing both balanced nutrition and more sustainable farming systems. However, the estimation of the nutrient requirements as the nutrient uptake per unit of seed yields is impaired in many situations due to the narrow variation of the dataset used to obtain these values or by the overgeneralization of considering a constant value for the nutrient demand at varying yield levels. Past studies focused on other crops and using linear models for estimation of the nutrient requirements, but not yet for soybeans (Glycine max L.). The aims of this research study were to: (i) quantify nitrogen (N), phosphorus (P), potassium (K), and sulfur (S) requirements in soybean and (ii) compare linear and non-linear (spherical) models in their relationship between plant and seed nutrient content all relative to seed yield at varying probabilities utilizing quantile regression. A large dataset from different studies conducted between 2009–2018 period, including data of seed yield, total biomass at physiological maturity, and N, P, K, and S uptake. Soybean seed yield ranged from 955 to 6525 kg ha−1, aboveground biomass from 1990 to 15,814 kg ha−1, and harvest index from 0.16 to 0.57. On average, nutrient uptake was 261 kg N ha−1, 25 kg P ha−1, 133 kg K ha−1, and 16 kg S ha−1 (N:P:K:S ratio = 17:1.6:8.5:1), while nutrient content in seeds averaged 191 kg N ha−1, 17 kg P ha−1, 54 kg K ha−1, and 9 kg S ha−1 (N:P:K:S ratio = 21:1.8:5.8:1). The spherical model described better than the linear model the relationship between plant nutrient uptake or nutrient content in seeds with seed yield in soybean, and thus, nutrient requirements per unit of yield decreased as seed yield increased. A relationship between nutrient internal efficiency and seed yield for the different percentiles as determined by the non-linear quantile regression offered probabilistic values for estimating nutrient uptake in soybean, providing useful information for obtaining more reliable estimates of nutrient balances at the system-level.

Evaluation of indicators for cyanobacterial risk in 108 temperate lakes using 23\xa0years of environmental monitoring data

BackgroundCyanobacterial blooms are of increasing concern for drinking water supply. In Sweden, a survey among drinking water producers showed that the sense of urgency was little. At 60% of the Swedish drinking water treatment plants, operators lacked monitoring strategies. To get a picture of the size of the problem the presence of cyanobacterial risk in 108 Swedish lakes was evaluated from 23 years of environmental monitoring data. The drivers and indicators for cyanobacterial growth were investigated by analyzing water quality in 9 lakes that have more frequent cyanobacteria bio-volume above the WHO drinking water alert level 1: 0.2 mm3L−1.ResultsThe study shows that the most common species in those lakes are Anabaena/Dolichospermum and Aphanizomenon followed by Microcystis, Woronichinia and Planktothrix, which can produce a variety of toxins such as anatoxins, cylindrospermopsins, microcystins and saxitoxins, supported by literature study. Our results show that cyanobacterial risk in those problematic 9 lakes are mainly nutrients driven with possibly contribution of increased organic matter. By applying non-linear quantile regression with total phosphorus (TP) as an example, we demonstrated that certain drivers such as TP can be useful for cyanobacterial risk assessment and provide control measures by setting nutrients targets. We also evidenced that cyanobacterial peaks presented at low TN:TP ratio while not necessarily vice versa. We also further evaluated that chlorophyll-a and transparency might be suitable as indicators for cyanobacterial blooms in certain lakes, while for most of the lakes, their connection is low.ConclusionNutrients are main drivers for higher cyanobacterial occurrence in the 9 lakes. We suggest TP concentrations should be investigated thoroughly to provide important knowledge which can be used to set nutrient targets to sustain safe drinking water supply and recreational services. The complexity of indicating cyanobacterial risk in a local condition was also highlighted in this study and future study is suggested. To classify different types of lake and identify their drivers and the similarities of species composition changes in those lakes will be future studied.

Time-varying comovement of stock and treasury bond markets in Europe: A quantile regression approach

Using the linear OLS and nonlinear quantile regressions, this study examines nature of dynamic comovement between stocks and treasury bonds in Europe and its main drivers of financial uncertainty and investors’ expectations about future economic state. Comovement of the EU asset markets is proxied by pair-wise realized correlations between stock and 10-year treasury bond returns in thirteen European countries from 1994Q1 to 2015Q4. From this study, a lower uncertainty for future financial markets increases comovement of EU stock and bond markets. Investors’ higher expectation for future economic growth has a positive effect on the market comovement. The empirical findings support that investors’ perceptions about the future state of the economy and financial (stock) market uncertainty are key factors for joint pricing of European stocks and treasury bonds. Importantly, the nonlinear quantile regressions in this study indicate evidence of nonlinear effects of the two economic drivers across the whole distribution of the dependent variable of comovment of the EU asset markets.

환태평양 지역의 주식-국채시장 간 시간가변적 동조화

Using the nonlinear quantile regression and realized moments techniques, this study finds evidence that financial uncertainty is negatively associated with a sustainable interdependence (i.e., comovement) of stock and government bond markets in the Pacific basin region. Besides, investors’ affirmative expectation for the future economic state is positively associated with comovement of the asset markets in this region. The former supports the validity of the flight-to-quality for the asset market comovement and the latter validates the wealth effectin this region. Our findings imply that interdependence of stock and bond markets is crucial for investors’ efficient diversification and sustainable economic growth via effective securities financing of companies in the country.

Confidence Limits of Word Identification Scores Derived Using Nonlinear Quantile Regression.

The relation between degree of sensorineural hearing loss and maximum speech identification scores (PBmax) is commonly used in audiological diagnosis and rehabilitation. It is important to consider the relation between the degree of hearing loss and the lower boundary of PBmax, as the PBmax varies largely between subjects at a given degree of hearing loss. The present study determines the lower boundary by estimating the lower limit of the one-tailed 95% confidence limit (CL) for a Dantale I, word list, in a large group of young and older subjects with primarily sensorineural hearing loss. PBmax scores were measured using Dantale I, at 30 dB above the speech reception threshold or at the most comfortable level from 1,961 subjects with a wide range of pure-tone averages. A nonlinear quantile regression approach was applied to determine the lower boundary (95% CL) of PBmax scores. At a specific pure-tone average, if the measured PBmax is poorer than the lower boundary (95% CL) of PBmax, it may be considered disproportionately poor.

Probability Density Forecasting of Wind Speed Based on Quantile Regression and Kernel Density Estimation

Based on quantile regression (QR) and kernel density estimation (KDE), a framework for probability density forecasting of short-term wind speed is proposed in this study. The empirical mode decomposition (EMD) technique is implemented to reduce the noise of raw wind speed series. Both linear QR (LQR) and nonlinear QR (NQR, including quantile regression neural network (QRNN), quantile regression random forest (QRRF), and quantile regression support vector machine (QRSVM)) models are, respectively, utilized to study the de-noised wind speed series. An ensemble of conditional quantiles is obtained and then used for point and interval predictions of wind speed accordingly. After various experiments and comparisons on the real wind speed data at four wind observation stations of China, it is found that the EMD-LQR-KDE and EMD-QRNN-KDE generally have the best performance and robustness in both point and interval predictions. By taking conditional quantiles obtained by the EMD-QRNN-KDE model as the input, probability density functions (PDFs) of wind speed at different times are obtained by the KDE method, whose bandwidth is optimally determined according to the normal reference criterion. It is found that most actual wind speeds lie near the peak of predicted PDF curves, indicating that the probabilistic density prediction by EMD-QRNN-KDE is believable. Compared with the PDF curves of the 90% confidence level, the PDF curves of the 80% confidence level usually have narrower wind speed ranges and higher peak values. The PDF curves also vary with time. At some times, they might be biased, bimodal, or even multi-modal distributions. Based on the EMD-QRNN-KDE model, one can not only obtain the specific PDF curves of future wind speeds, but also understand the dynamic variation of density distributions with time. Compared with the traditional point and interval prediction models, the proposed QR-KDE models could acquire more information about the randomness and uncertainty of the actual wind speed, and thus provide more powerful support for the decision-making work.

The Changing Dynamics of Board Independence: A Copula Based Quantile Regression Approach

This paper examines the effect of board characteristics, especially board independence, on firm performance from a dynamic perspective through copula-based quantile regression approaches, which allow us to focus on changes at different points in the distribution of board characteristics. We find that the effect of board independence on Tobin’s Q, a proxy of firm value, is negatively associated with firm value, using ordinary least squares (OLS) regression. This negative effect using the conditional mean of the firm value does not hold across the conditional quantiles of the distribution of Tobin’s Q, and this finding is still held under both the linear and the nonlinear quantile regressions. We even lessen the assumption of distributions of multivariate board variables by employing parametric copula-based quantile regressions as well as nonparametric ones. The results support our findings. Our results suggest that estimating the quantile effect of board variables on firm value can provide more meaningful insight than just examining the conditional mean effect.

Improved convolutional neural network‐based quantile regression for regional photovoltaic generation probabilistic forecast

Nowadays, an increasing number of photovoltaic (PV) plants are becoming integrated into one regional power grid. Under this circumstance, the probabilistic forecast of regional PV power generation is of significance for the regional power system operation and control. This study presents a novel probabilistic forecast method for regional PV generation that integrates the convolutional neural network (CNN) with non-linear quantile regression (QR). In this method, the CNN structure is enhanced to extract the non-linear features of the input data and generate the non-linear QR function. As a result, the improved CNN can effectively process high-dimensional and complex input data and the non-linear QR model can provide quantile forecast information of regional PV power. The validity of the proposed method is verified by using it to forecast the regional PV generation from the clustered PV plants in the Weifang region of China.

Modeling Wildland Firefighter Travel Rates by Terrain Slope: Results from GPS-Tracking of Type 1 Crew Movement

Escape routes keep firefighters safe by providing efficient evacuation pathways from the fire line to safety zones. Effectively utilizing escape routes requires a precise understanding of how much time it will take firefighters to traverse them. To improve this understanding, we collected GPS-tracked travel rate data from US Interagency Hotshot “Type 1” Crews during training in 2019. Firefighters were tracked while hiking, carrying standard loads (e.g., packs, tools, etc.) along trails with a precisely-measured terrain slope derived from airborne lidar. The effects of the slope on the instantaneous travel rate were assessed by three models generated using non-linear quantile regression, representing low (bottom third), moderate (middle third), and high (upper third) rates of travel, which were validated using k-fold cross-validation. The models peak at about a −3° (downhill) slope, similar to previous slope-dependent travel rate functions. The moderate firefighter travel rate model mostly predicts faster movement than previous slope-dependent travel rate functions, suggesting that firefighters generally move faster than non-firefighting personnel while hiking. Steepness was also found to have a smaller effect on firefighter travel rates than previously predicted. The travel rate functions produced by this study provide guidelines for firefighter escape route travel rates and allow for more accurate and flexible wildland firefighting safety planning.

Effect of strip shelterwood-cuts on the crown morphology plasticity of natural regenerated Pinus tabuliformis saplings in northeastern China.

The study analysed the effect of shelterwood-cut strips on the outermost crown profiles and crown characteristics of naturally regenerated Pinus tabuliformis saplings in northeastern China. A total of 49 regenerated saplings from shelterwood-cut strips and 30 from uncut strips were collected. Nonlinear quantile regression was used to develop the outermost crown profile model for the saplings from the shelterwood-cut and uncut strips. The quantile value suitable for describing the outermost crown profiles of the two types of strips was selected using nonparametric boundary regression. The difference in crown morphologies between the shelterwood-cut strips and uncut strips was compared. The results showed that with the same diameter at breast height, the crown radii of the uncut strip saplings were larger than those of the shelterwood-cut strip saplings within the range of 0.2–1.0 for the relative depth into the crown. The largest crown radius of the saplings from the uncut strips was larger than that of the saplings from the shelterwood-cut strips. The inflection points of the shelterwood-cut strip sapling crowns were larger than those of the uncut strip sapling crowns. The crown volume of the small uncut strip saplings was larger than that of the shelterwood-cut strip saplings, and the difference in crown volume decreased with increasing sapling size. The saplings in the early stage of the uncut strips showed a greater growth rate than those of the shelterwood-cut strips, but their growth rate slowed over the long term according to branch-length annual growth. The present study provides a reference for forest management strategy decision making in promoting natural regeneration.

Model-robust designs for nonlinear quantile regression.

We construct robust designs for nonlinear quantile regression, in the presence of both a possibly misspecified nonlinear quantile function and heteroscedasticity of an unknown form. The asymptotic mean-squared error of the quantile estimate is evaluated and maximized over a neighbourhood of the fitted quantile regression model. This maximum depends on the scale function and on the design. We entertain two methods to find designs that minimize the maximum loss. The first is local - we minimize for given values of the parameters and the scale function, using a sequential approach, whereby each new design point minimizes the subsequent loss, given the current design. The second is adaptive - at each stage, the maximized loss is evaluated at quantile estimates of the parameters, and a kernel estimate of scale, and then the next design point is obtained as in the sequential method. In the context of a Michaelis-Menten response model for an estrogen/hormone study, and a variety of scale functions, we demonstrate that the adaptive approach performs as well, in large study sizes, as if the parameter values and scale function were known beforehand and the sequential method applied. When the sequential method uses an incorrectly specified scale function, the adaptive method yields an, often substantial, improvement. The performance of the adaptive designs for smaller study sizes is assessed and seen to still be very favourable, especially so since the prior information required to design sequentially is rarely available.

Current use and performance of the different fetal growth charts in the Italian population

ObjectivesThe choice of growth charts impacts on screening, diagnosis and clinical management of fetal growth abnormalities. The objectives of the study were to evaluate: 1) the clinical practice at a national level among tertiary referral centers in the use of fetal biometric growth charts; and 2) the impact on fetal growth screening of existing national and international growth charts. Study designA questionnaire was sent to 14 Italian tertiary referral centers to explore biometric reference growth charts used in clinical practice. National and international (Intergrowth-21st and World Health Organization) fetal growth charts were tested on a large national cohort of low risk women with singleton uneventful pregnancy derived from a retrospective cross-sectional multicenter study (21 centers). The percentage of fetuses with biometric measurements below and above the 10th and 90th percentile for each biometric parameter and gestational week were calculated for each growth chart. The percentile curves of the study population were calculated by non-linear quantile regressions. ResultsTwelve Italian centers (86 %) answered to the questionnaire showing a wide discrepancy in the use of growth charts for fetal biometry. The cohort included 7347 pregnant women. By applying Intergrowth-21st growth charts the percentage of fetuses with head circumference, abdominal circumference and femur length below the 10th centile was 3.9 %, 3.6 % and 2.3 %, and above the 90th centile 29.9 %, 32.5 % and 46 %, respectively. The percentages for the World Health Organization growth charts for head and abdominal circumferences and femur length were: below the 10th centile 6.3 %, 7.2 % and 5.3 %, and above 90th centile 22.8 %, 21.3 % and 31.9 %, respectively. ConclusionsThe wide discrepancy in clinical use of fetal growth charts in Italian centers warrants the adoption of an uniform set of charts. Our data suggest that immediate application into clinical practice of international growth charts might result into an under-diagnosis of small for gestational age fetuses and, especially, in an over-diagnosis of large for gestational age fetuses with major consequences for clinical practice. On these grounds, there is an urgent need for a nationwide study for the prospective evaluation of international growth charts and, if needed, the construction and adoption of methodologically robust national growth charts.

Non-Stationary Flood Frequency Analysis Using Cubic B-Spline-Based GAMLSS Model

Under changing environments, the most widely used non-stationary flood frequency analysis (NFFA) method is the generalized additive models for location, scale and shape (GAMLSS) model. However, the model structure of the GAMLSS model is relatively complex due to the large number of statistical parameters, and the relationship between statistical parameters and covariates is assumed to be unchanged in future, which may be unreasonable. In recent years, nonparametric methods have received increasing attention in the field of NFFA. Among them, the linear quantile regression (QR-L) model and the non-linear quantile regression model of cubic B-spline (QR-CB) have been introduced into NFFA studies because they do not need to determine statistical parameters and consider the relationship between statistical parameters and covariates. However, these two quantile regression models have difficulties in estimating non-stationary design flood, since the trend of the established model must be extrapolated infinitely to estimate design flood. Besides, the number of available observations becomes scarcer when estimating design values corresponding to higher return periods, leading to unreasonable and inaccurate design values. In this study, we attempt to propose a cubic B-spline-based GAMLSS model (GAMLSS-CB) for NFFA. In the GAMLSS-CB model, the relationship between statistical parameters and covariates is fitted by the cubic B-spline under the GAMLSS model framework. We also compare the performance of different non-stationary models, namely the QR-L, QR-CB, and GAMLSS-CB models. Finally, based on the optimal non-stationary model, the non-stationary design flood values are estimated using the average design life level method (ADLL). The annual maximum flood series of four stations in the Weihe River basin and the Pearl River basin are taken as examples. The results show that the GAMLSS-CB model displays the best model performance compared with the QR-L and QR-CB models. Moreover, it is feasible to estimate design flood values based on the GAMLSS-CB model using the ADLL method, while the estimation of design flood based on the quantile regression model requires further studies.

Effects of stand variables on stemflow and surface runoff in pine-oak forests in northern Mexico

The flow of water in temperate forests depends on the amount of precipitation, type of soil, topographic features, and forest cover, among other factors. Unlike the first three, forest cover can be modified by silvicultural treatments, the effects of which manifest in the quality and quantity of water, as well as in the transport of sediments and soil nutrients. The objective of this study was to analyze the effect of some stand variables on surface runoff and stemflow in pine-oak forests of northern Mexico. The stand variables included tree diameter at breast height, basal area, canopy cover, and volume. They were collected in eight 0.1-ha circular plots, measured in 2016 and re-measured in 2018. Nonlinear quantile regression was used to determine the best-fit relationships between the variables. Results indicated that surface runoff was most closely and inversely related to basal area. Stemflow was related to diameter at breast height, while showing no statistical significance. A stemflow funneling ratio did show an inverse, statistically-significant relationship with diameter at breast height. These results can help determine best forest management regimes compatible with the quantity and quality of water fluxes in this type of ecosystem.