Quantile Curves Research Articles

The maximum growth rate of life on Earth

AbstractLife on Earth spans a range of temperatures and exhibits biological growth rates that are temperature dependent. While the observation that growth rates are temperature dependent is well known, we have recently shown that the statistical distribution of specific growth rates for life on Earth is a function of temperature (Corkreyet al., 2016). The maximum rates of growth of all life have a distinct limit, even when grown under optimal conditions, and which vary predictably with temperature. We term this distribution of growth rates the biokinetic spectrum for temperature (BKST). The BKST possibly arises from a trade-off between catalytic activity and stability of enzymes involved in a rate-limiting Master Reaction System (MRS) within the cell. We develop a method to extrapolate quantile curves for the BKST to obtain the posterior probability of the maximum rate of growth of any form of life on Earth. The maximum rate curve conforms to the observed data except below 0°C and above 100°C where the predicted value may be positively biased. The deviation below 0°C may arise from the bulk properties of water, while the degradation of biomolecules may be important above 100°C. The BKST has potential application in astrobiology by providing an estimate of the maximum possible growth rate attainable by terrestrial life and perhaps life elsewhere. We suggest that the area under the maximum growth rate curve and the peak rate may be useful characteristics in considerations of habitability. The BKST can serve as a diagnostic for unusual life, such as second biogenesis or non-terrestrial life. Since the MRS must have been heavily conserved the BKST may contain evolutionary relics. The BKST can serve as a signature summarizing the nature of life in environments beyond Earth, or to characterize species arising from a second biogenesis on Earth.

Local Quantile Regression

Conditional quantile curves provide a comprehensive picture of a response contingent on explanatory variables. Quantile regression is a technique to estimate such curves. In a flexible modeling framework, a specific form of the quantile is not a priori fixed. Indeed, the majority of applications do not per se require specific functional forms. This motivates a local parametric rather than a global fixed model fitting approach. A nonparametric smoothing estimate of the conditional quantile curve requires to consider a balance between local curvature and variance. In this paper, we analyze a method based on a local model selection technique that provides an adaptive estimate. Theoretical properties on mimicking the oracle choice are offered and applications to stock market and weather analysis are presented.

Quantile Regression in Risk Calibration

Financial risk control has always been challenging and becomes now an even harder problem as joint extreme events occur more frequently. For decision makers and government regulators, it is therefore important to obtain accurate information on the interdependency of risk factors. Given a stressful situation for one market participant, one likes to measure how this stress affects other factors. The CoVaR (Conditional VaR) framework has been developed for this purpose. The basic technical elements of CoVaR estimation are two levels of quantile regression: one on market risk factors; another on individual risk factor. Tests on the functional form of the two-level quantile regression reject the linearity. A flexible semiparametric modeling framework for CoVaR is proposed. A partial linear model (PLM) is analyzed. In applying the technology to stock data covering the crisis period, the PLM outperforms in the crisis time, with the justification of the backtesting procedures. Moreover, using the data on global stock markets indices, the analysis on marginal contribution of risk (MCR) defined as the local first order derivative of the quantile curve sheds some light on the source of the global market risk.

Composite Quantile Regression for the Single-Index Model

Quantile regression is in the focus of many estimation techniques and is an important tool in data analysis. When it comes to nonparametric specifications of the conditional quantile (or more generally tail) curve one faces, as in mean regression, a dimensionality problem. We propose a projection based single index model specifi- cation. For very high dimensional regressors X one faces yet another dimensionality problem and needs to balance precision vs. dimension. Such a balance may be achieved by combining semiparametric ideas with variable selection techniques.

Functional Data Analysis of Generalized Quantile Regressions

Generalized quantile regressions, including the conditional quantiles and expectiles as special cases, are useful alternatives to the conditional means for characterizing a conditional distribution, especially when the interest lies in the tails. We develop a functional data analysis approach to jointly estimate a family of generalized quantile regressions. Our approach assumes that the generalized quantile regressions share some common features that can be summarized by a small number of principal component functions. The principal component functions are modeled as splines and are estimated by minimizing a penalized asymmetric loss measure. An iterative least asymmetrically weighted squares algorithm is developed for computation. While separate estimation of individual generalized quantile regressions usually suffers from large variability due to lack of sufficient data, by borrowing strength across data sets, our joint estimation approach significantly improves the estimation efficiency, which is demonstrated in a simulation study. The proposed method is applied to data from 150 weather stations in China to obtain the generalized quantile curves of the volatility of the temperature at these stations. These curves are needed to adjust temperature risk factors so that gaussianity is achieved. The normal distribution of temperature variations is vital for pricing weather derivatives with tools from mathematical finance.

Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach

ABSTRACTElectricity load forecasts are an integral part of many decision-making processes in the electricity market. However, most literature on electricity load forecasting concentrates on deterministic forecasts, neglecting possibly important information about uncertainty. A more complete picture of future demand can be obtained by using distributional forecasts, allowing for more efficient decision-making. A predictive density can be fully characterized by tail measures such as quantiles and expectiles. Furthermore, interest often lies in the accurate estimation of tail events rather than in the mean or median. We propose a new methodology to obtain probabilistic forecasts of electricity load that is based on functional data analysis of generalized quantile curves. The core of the methodology is dimension reduction based on functional principal components of tail curves with dependence structure. The approach has several advantages, such as flexible inclusion of explanatory variables like meteorological forecasts and no distributional assumptions. The methodology is applied to load data from a transmission system operator (TSO) and a balancing unit in Germany. Our forecast method is evaluated against other models including the TSO forecast model. It outperforms them in terms of mean absolute percentage error and mean squared error. Supplementary materials for this article are available online.

Quantile processes for semi and nonparametric regression

A collection of quantile curves provides a complete picture of conditional distributions. A properly centered and scaled version of the estimated curves at various quantile levels gives rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. Applications of our general process convergence results include the construction of non-crossing quantile curves and the estimation of conditional distribution functions. As a result of independent interest, we obtain a series of Bahadur representations with exponential bounds for tail probabilities of all remainder terms. Bounds of this kind are potentially useful in analyzing statistical inference procedures under the divide-and-conquer setup.

Quantile regression in varying-coefficient models: non-crossing quantile curves and heteroscedasticity

Quantile regression is an important tool for describing the characteristics of conditional distributions. Population conditional quantile functions cannot cross for different quantile orders. Unfortunately estimated regression quantile curves often violate this and cross each other, which can be very annoying for interpretations and further analysis. In this paper we are concerned with flexible varying-coefficient modelling, and develop methods for quantile regression that ensure that the estimated quantile curves do not cross. A second aim of the paper is to allow for some heteroscedasticity in the error modelling, and to also estimate the associated variability function. We investigate the finite-sample performances of the discussed methods via simulation studies. Some applications to real data illustrate the use of the methods in practical settings.

Transmuted new generalized Weibull distribution for lifetime modeling

The Weibull family of lifetime distributions play a fundamental role in reliability engineering and life testing problems. This paper investigates the potential usefulness of transmuted new generalized Weibull (TNGW) distribution for modeling lifetime data. This distribution is an important competitive model that contains twenty-three lifetime distributions as special cases. We can obtain the TNGW distribution using the quadratic rank transmutation map (QRTM) technique. We derive the analytical shapes of the density and hazard functions for graphical illustrations. In addition, we explore some mathematical properties of the TNGW model including expressions for the quantile function, moments, entropies, mean deviation, Bonferroni and Lorenz curves and the moments of order statistics. The method of maximum likelihood is used to estimate the model parameters. Finally the applicability of the TNGW model is presented using nicotine in cigarettes data for illustration.

Regularized Bayesian estimation for GEV-B-splines model

Large observed datasets are not stationary and/or depend on covariates, especially, in the case of extreme hydrometeorological variables. This causes the difficulty in estimation, using classical hydrological frequency analysis. A number of non-stationary models have been developed using linear or quadratic polynomial functions or B-splines functions to estimate the relationship between parameters and covariates. In this article, we propose regularised generalized extreme value model with B-splines (GEV-B-splines models) in a Bayesian framework to estimate quantiles. Regularisation is based on penalty and aims to favour parsimonious model especially in the case of large dimension space. Penalties are introduced in a Bayesian framework and the corresponding priors are detailed. Five penalties are considered and the corresponding priors are developed for comparison purpose as: Least absolute shrinkage and selection (Lasso and Ridge) and smoothing clipped absolute deviations (SCAD) methods (SCAD1, SCAD2 and SCAD3). Markov chain Monte Carlo (MCMC) algorithms have been developed for each model to estimate quantiles and their posterior distributions. Those approaches are tested and illustrated using simulated data with different sample sizes. A first simulation was made on polynomial B-splines functions in order to choose the most efficient model in terms of relative mean biais (RMB) and the relative mean-error (RME) criteria. A second simulation was performed with the SCAD1 penalty for sinusoidal dependence to illustrate the flexibility of the proposed approach. Results show clearly that the regularized approaches leads to a significant reduction of the bias and the mean square error, especially for small sample sizes (n < 100). A case study has been considered to model annual peak flows at Fort-Kent catchment with the total annual precipitations as covariates. The conditional quantile curves were given for the regularized and the maximum likelihood methods.

Depth-based nonparametric description of functional data, with emphasis on use of spatial depth

Statistical depth and related quantile functions, originally introduced for nonparametric description and analysis of multivariate data in a way sensitive to inherent geometry, are in active development for functional data and in this setting offer special options since the data may be visualized regardless of dimension. This paper provides depth-based methods for revealing the structure of a functional data set in terms of relevant sample quantile curves displayed at selected levels, and for constructing and displaying confidence bands for corresponding “population” versions. Also, the usual functional boxplot is enhanced, by adding inner fences to flag possible shape outliers, along with the outer fences that flag location outliers. This enables the boxplot to serve as a stand-alone tool for functional data, as with univariate and multivariate data. Further, the spatial depth approach, well-established for multivariate data, is investigated for nonparametric description of functional data along these lines. In comparison with four other commonly used depth approaches for functional data, over a range of actual and simulated data sets, the spatial depth approach is seen to offer a very competitive combination of robustness, efficiency, computational ease, simplicity, and versatility.

A complete procedure for multivariate index-flood model application

Multivariate frequency analyses are needed to study floods due to dependence existing among representative variables of the flood hydrograph. Particularly, multivariate analyses are essential when flood-routing processes significantly attenuate flood peaks, such as in dams and flood management in flood-prone areas. Besides, regional analyses improve at-site quantile estimates obtained at gauged sites, especially when short flow series exist, and provide estimates at ungauged sites where flow records are unavailable. However, very few studies deal simultaneously with both multivariate and regional aspects. This study seeks to introduce a complete procedure to conduct a multivariate regional hydrological frequency analysis (HFA), providing guidelines. The methodology joins recent developments achieved in multivariate and regional HFA, such as copulas, multivariate quantiles and the multivariate index-flood model. The proposed multivariate methodology, focused on the bivariate case, is applied to a case study located in Spain by using hydrograph volume and flood peak observed series. As a result, a set of volume-peak events under a bivariate quantile curve can be obtained for a given return period at a target site, providing flexibility to practitioners to check and decide what the design event for a given purpose should be. In addition, the multivariate regional approach can also be used for obtaining the multivariate distribution of the hydrological variables when the aim is to assess the structure failure for a given return period.

Forecasting SO2 pollution incidents by means of quantile curves based on additive models

More than 90% of the sulfur dioxide in the air comes from human sources. Because of the adverse health effects of high levels of sulfur dioxide, specific regulations have been adopted to manage and reduce the amount of sulfur dioxide produced. However, some SO2 emission incidents (i.e. emission exceeding the limits established by law) still occur. The aim of this paper is to predict time series of SO2 concentrations in order to estimate in advance high emission episodes and analyse the influence of previous series in the prediction. Previous studies aimed to forecast SO2 pollution incidents are based on estimating mean values. Instead, we propose the use of quantile curves obtained from additive models as they provide not only the mean but also the whole distribution of the pollution levels. A backfitting algorithm with local polynomial kernel smoothers was used to estimate the model, and critical values of the hypothesis test were obtained by means of bootstrapping. The performance of the method was evaluated using simulated data as well as real data drawn from an SO2 time series of a coal‐fired power station located in northern Spain. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Investigation of fatigue crack growth in a power plant steel under elevated temperatures

The definition of inspection intervals on the basis of fracture mechanics requires beside the knowledge of the load e.g. in terms of stress intensity solutions also the knowledge of the material parameters e.g. in terms of crack growth data. For this reason, fatigue crack growth curves were determined for a ferritic-martensitic power plant steel at elevated temperatures. Therefore, an experimental setup was developed. The fatigue crack growth tests are performed on C(T)-specimens, which are manufactured with varied orientations from a high pressure bypass valve (HP-Bypass) which was exchanged at a revision.The isothermal test for determining the threshold values as well as the fatigue crack growth data in the Paris-regime have been performed at room temperature and at temperatures between T = 300 °C and T = 600 °C. Beside the temperature, the R-ratio, the specimen orientation, the frequency and the normalized K-gradient are varied. The influences of all investigated parameters are evaluated and the main factors are identified. For statistical evaluation of the crack growth data quantile curves are created from the crack growth data as a function of the main factors.

Outcome-Based Quality Control by a Dental Reference Profile of a Population-Based Study (SHIP-0).

Objectives. The aim was to develop an instrument for quality control in dental practices. We compared the number of teeth of subjects of the Study of Health in Pomerania (SHIP-0) with those from patients of dental practices. Methods. Patients from seven dental practices (n = 1,497) were randomly sampled by age strata and gender for a period of two years. Dental status derived from patient files was transformed into practice profiles using age-specific number of teeth as a parameter. Practice profiles were compared with a nomogram, which was based on the age-specific number of teeth of 3,990 SHIP-0 participants regularly visiting the dentist. Further, negative binomial regression models were evaluated to model associations between the number of teeth with age and dental practices, including interactions. Results. The practice profiles ranged between the 45th and 95th quantile curves of the reference population SHIP-0. The rate ratios (RR) for the number of missing teeth ranged from 0.37 to 0.67 (p < 0.001) between the different dental practices, indicating lower risk for higher numbers of missing teeth in comparison to SHIP-0. Conclusions. This study showed considerable differences between dental practices and the reference population of SHIP-0 regarding the pattern of tooth loss and confirms the value of nomograms to compare age-specific numbers of teeth between patients of dental practices and a population-based-study as a tool for quality control. For further analyses, the socioeconomic status of patients and relevant risk factors will be used to adjust for structural differences in order to improve the validity of the comparisons.

Simultaneous estimation of linear conditional quantiles with penalized splines

We consider smooth estimation of the conditional quantile process in linear models using penalized splines. For linear quantile regression problems, usually separate models are fitted at a finite number of quantile levels and then information from different quantiles is combined in interpreting the results. We propose a smoothing method based on penalized splines that computes the conditional quantiles all at the same time. We consider both fixed-knots and increasing-knots asymptotics of the estimator and show that it converges to a multivariate Gaussian process. Simulations show that smoothing can result in more accurate estimation of the conditional quantiles. The method is further illustrated on a real data set. Empirically (although not theoretically) we observe that the crossing quantile curves problem can often disappear using the smoothed estimator.

Spatio-temporal characteristics of the extreme precipitation by L-moment-based index-flood method in the Yangtze River Delta region, China

The regionalization methods, which “trade space for time” by pooling information from different locations in the frequency analysis, are efficient tools to enhance the reliability of extreme quantile estimates. This paper aims at improving the understanding of the regional frequency of extreme precipitation by using regionalization methods, and providing scientific background and practical assistance in formulating the regional development strategies for water resources management in one of the most developed and flood-prone regions in China, the Yangtze River Delta (YRD) region. To achieve the main goals, L-moment-based index-flood (LMIF) method, one of the most popular regionalization methods, is used in the regional frequency analysis of extreme precipitation with special attention paid to inter-site dependence and its influence on the accuracy of quantile estimates, which has not been considered by most of the studies using LMIF method. Extensive data screening of stationarity, serial dependence, and inter-site dependence was carried out first. The entire YRD region was then categorized into four homogeneous regions through cluster analysis and homogenous analysis. Based on goodness-of-fit statistic and L-moment ratio diagrams, generalized extreme-value (GEV) and generalized normal (GNO) distributions were identified as the best fitted distributions for most of the sub-regions, and estimated quantiles for each region were obtained. Monte Carlo simulation was used to evaluate the accuracy of the quantile estimates taking inter-site dependence into consideration. The results showed that the root-mean-square errors (RMSEs) were bigger and the 90 % error bounds were wider with inter-site dependence than those without inter-site dependence for both the regional growth curve and quantile curve. The spatial patterns of extreme precipitation with a return period of 100 years were finally obtained which indicated that there are two regions with highest precipitation extremes and a large region with low precipitation extremes. However, the regions with low precipitation extremes are the most developed and densely populated regions of the country, and floods will cause great loss of human life and property damage due to the high vulnerability. The study methods and procedure demonstrated in this paper will provide useful reference for frequency analysis of precipitation extremes in large regions, and the findings of the paper will be beneficial in flood control and management in the study area.

Estimation of Non‐Crossing Quantile Regression Curves

Quantile regression methods have been widely used in many research areas in recent years. However conventional estimation methods for quantile regression models do not guarantee that the estimated quantile curves will be non‐crossing. While there are various methods in the literature to deal with this problem, many of these methods force the model parameters to lie within a subset of the parameter space in order for the required monotonicity to be satisfied. Note that different methods may use different subspaces of the space of model parameters. This paper establishes a relationship between the monotonicity of the estimated conditional quantiles and the comonotonicity of the model parameters. We develope a novel quasi‐Bayesian method for parameter estimation which can be used to deal with both time series and independent statistical data. Simulation studies and an application to real financial returns show that the proposed method has the potential to be very useful in practice.

Reference Curves Estimation Using Conditional Quantile and Radial Basis Function Network with Mass Constraint

This paper focuses on the improvement of reference curves building \(Y=q(X)\) using a fast algorithm, robust against outliers. Our method consists in plugging a radial basis function neural network in the local linear quantile regression estimation proposed by Yu and Jones (QYJ). This neural network (QRBFc) is designed with a constructive algorithm, introducing a constraint on its integral over the input space. After explaining the different models and algorithms, we compare the QYJ and QRBFc estimators with the quantile regression neural network (QRNN) implemented by A. J. Cannon through simulations with a known underlying model using the R software. We observe that the QRBFc estimator reduces the mean absolute deviation error obtained with other estimators by about 16 %, the introduction of the constraint allowing to lower the number of neurons and therefore the computation time. Finally, using a database of 416 electroencephalograms recorded on preterm infants, we compare the QYJ, QRBFc and QRNN models for the building of brain maturation curves which are based on the dependence of the mean duration of interburst intervals (called IBIs—periods of quiescence between periods of normal electrical activity) with the age. The pathological infants represent 12 % of the total population. Denoting by \(S_A\) the set of individuals whose coordinates (age, mean IBI length) are above the \(90~\%\)-quantile curve, the QRBFc network improves by \(16.5~\%\) the number of pathological infants in \(S_A\) compared to QYJ, when QRNN proved to be too unstable.

An Effective Nonparametric Quantile Regression Method for Solving the Crossing Problem in Data Fitting Process

Nonparametric quantile regression method can be used as the first choice for some biostatistical data. Since the nonparametric quantile regression curves are estimated individually, the quantile curves can cross, leading to an invalid distribution estimation for the response. A simple nonparametric quantile regression method is proposed to avoid the crossing problem. The method uses nonparametric conditional density function estimate instead of the conditional distribution estimate to assure quantile regression function monotonous. Both a simulation study and an analysis of real salmon lustrousness data show the significant improvement of the method in solving the quantile crossing problem for some kind of biological data.