Penalized Spline Regression Research Articles

Bayesian Inference of Finite Population Quantiles for Skewed Survey Data Using Skew-Normal Penalized Spline Regression.

Skewed data are common in sample surveys. In probability proportional to size sampling, we propose two Bayesian model-based predictive methods for estimating finite population quantiles with skewed sample survey data. We assume the survey outcome to follow a skew-normal distribution given the probability of selection and model the location and scale parameters of the skew-normal distribution as functions of the probability of selection. To allow a flexible association between the survey outcome and the probability of selection, the first method models the location parameter with a penalized spline and the scale parameter with a polynomial function, while the second method models both the location and scale parameters with penalized splines. Using a fully Bayesian approach, we obtain the posterior predictive distributions of the nonsampled units in the population and thus the posterior distributions of the finite population quantiles. We show through simulations that our proposed methods are more efficient and yield shorter credible intervals with better coverage rates than the conventional weighted method in estimating finite population quantiles. We demonstrate the application of our proposed methods using data from the 2013 National Drug Abuse Treatment System Survey.

Penalized I-spline monotone regression estimation

We propose a penalized regression spline estimator for monotone regression. To construct the estimator, we adopt the I-splines with the total variation penalty. The I-splines lend themselves to the monotonicity because of the simpler form of restrictions, and the total variation penalty induces a data-driven knot selection scheme. A coordinate descent algorithm is developed for the estimator. If the number of complexity parameter candidates sufficiently increases, the algorithm considers all possible monotone linear spline fits to the given data. The pruning process of the algorithm not only provides numerical stability, but also implements the data-driven knot selection. We also compute the maximum candidate of the complexity parameter to facilitate complexity parameter selection. Extensive numerical studies show that the proposed estimator captures spatially inhomogeneous behaviors of data, such as sudden jumps.

The sex-specific association between maternal paraben exposure and size at birth.

Parabens are a group of esters of parahydroxybenzoic acid and are utilized as antimicrobial preservatives in the majority of personal care products (PCPs). Epidemiological studies regarding the adverse effects of parabens on fetuses are still limited. The aim of this study was to determine the association between maternal paraben exposure and birth outcomes. One hundred and ninety-nine pregnant women were enrolled, and maternal urine was collected in the third trimester. The urine concentrations of four parabens (methyl (MP), ethyl (EP), propyl (PP), and butyl (BP)) were determined by ultra-performance liquid chromatography coupled with quadrupole time-of-flight mass spectrometry. Generalized additive model-penalized regression splines and a multivariable regression model were employed to determine the association between paraben exposure levels and birth outcomes. A causal mediation analysis was conducted to determine the mediation effect of oxidative stress on birth outcomes. The geometric means of urinary MP, EP, PP, and BP were 51.79, 1.26, 4.21, and 1.25 μg/g cre., respectively. In the penalized regression splines, sex-specific associations between maternal MP levels and birth outcomes were observed; a downward curvature was observed between the MP level and birth weight, length, head circumference, and thoracic circumference among female newborns. Pregnant women in the group with MP levels above the third quartile had neonates with significantly lower body weight (β = -215.98 g, p value = 0.02) compared to those in the group with MP levels lower than the third quartile. No significant mediation of oxidative stress was observed between maternal MP exposure and female birth weight. The estimated proportion mediated ranged from -6% to 15%. The negative association between maternal paraben exposure and female birth outcomes in relation to child development should be carefully considered.

Faster model matrix crossproducts for large generalized linear models with discretized covariates

Wood et al. (J Am Stat Assoc 112(519):1199–1210, 2017) developed methods for fitting penalized regression spline based generalized additive models, with of the order of 10^4 coefficients, to up to 10^8 data. The methods offered two to three orders of magnitude reduction in computational cost relative to the most efficient previous methods. Part of the gain resulted from the development of a set of methods for efficiently computing model matrix products when model covariates each take only a discrete set of values substantially smaller than the sample size [generalizing an idea first appearing in Lang et al. (Stat Comput 24(2):223–238, 2014)]. Covariates can always be rounded to achieve such discretization, and it should be noted that the covariate discretization is marginal. That is we do not rely on discretizing covariates jointly, which would typically require the use of very coarse discretization. The most expensive computation in model estimation is the formation of the matrix cross product mathbf{X}^{mathsf{T}}{mathbf{WX}} where mathbf{X} is a model matrix and {mathbf{W}} a diagonal or tri-diagonal matrix. The purpose of this paper is to present a simple, novel and substantially more efficient approach to the computation of this cross product. The new method offers, for example, a 30 fold reduction in cross product computation time for the Black Smoke model dataset motivating Wood et al. (2017). Given this reduction in computational cost, the subsequent Cholesky decomposition of mathbf{X}^{mathsf{T}}{mathbf{WX}} and follow on computation of (mathbf{X}^{mathsf{T}}{mathbf{WX}})^{-1} become a more significant part of the computational burden, and we also discuss the choice of methods for improving their speed.

Smoothing parameters of penalized spline nonparametric regression model using linear mixed model

Nonparametric regression is method of regression approach for data patterns of which regression curve is unknown. Meanwhile, spline is a type of segmented polynomial piecewise. Such segmented nature gives more flexibility than common polynomial models so that spline can adapt more effectively to local characteristics of function or data. The use of spline focuses on presence of behavior and patterns of data which in certain area has different characteristics from other areas. Data matching can be done by observing points on data in which an extreme change occurs in an area and consequently data patterns in each area differ. One approach using nonparametric regression is spline regression with penalized spline (P-spline). It is defined as a regression determined using method of least squares and roughness penalty. The article examines smoothing parameters of penalized spline nonparametric regression model. P-spline can be represented in linear mixed model with components of variance to control nonlinearity level of estimators of smooth functions. P-spline using linear mixed model approach can be estimated using either maximum likelihood or residual maximum likelihood. The results of research show that smooth function of best linear unbiased predictor in linear mixed model is equivalent to estimators of penalized spline regression.

Application of Bayesian penalized spline regression for internal modeling in life insurance

Solvency 2 requires insurance companies to compute a Best-Estimate of their Liabilities (BEL) as well as a Solvency Capital Requirement (SCR). Life insurance companies being in the business of selling participating contracts with financial guarantees have to rely on a Monte-Carlo approach to appropriately value their BEL, which is the source of a first Monte-Carlo statistical error. In addition, several insurance companies rely on a (partial) internal model to derive their SCR. In this context, insurance companies rely again on a Monte-Carlo approach to value their SCR, which is the source of a second Monte-Carlo statistical error. These later computations require evaluating the BEL several thousand times, which is not possible in practice, since one single Monte-Carlo BEL evaluation can be a computational burden. The BEL has therefore to be approximated by an analytic proxy function, which introduces an additional source of numerical approximation error. In this paper, we show how these three sources of error (statistical and numerical) are intrinsically related to one another. We show that to obtain the best possible SCR accuracy, the computing power invested in assessing the Monte-Carlo SCR should be directly related to that invested in computing the Monte-Carlo BEL. Interestingly, and to achieve these results, we introduce a novel proxy method, which is highly practical, modular, smooth and naturally relates the approximation errors to the Monte-Carlo statistical errors. Furthermore, our approach allows insurance companies to naturally and transparently start reporting confidence levels on their prudential reporting, which is not disclosed so far by insurance companies and would be a relevant information within solvency disclosures for the industry.

A Comparison of O’Sullivan Penalized Spline and Penalized Spline Based Truncated Power Basis Methods to Predict Ozone Concentrations

This paper aims to predict ozone concentrations based on the effects of ultraviolet light. Spline regression is piecewise polynomials that connect join points called knots. In spline regression, parameter estimation was fit by OLS (Ordinary Least Square) method. However, the OLS method leads to over parameterized and in the plot of estimated regression curve is fluctuated when using too much knots. Therefore, it needs an additional constraint which contains smoothing parameter, so that results an ideal fit. This parameter estimation method is known as PLS (Penalized Least Square) method. Spline regression using PLS method is called as penalized spline regression. O’Sullivan (1986) introduced a class of penalised splines based on B-spline basis functions. OPS (O’sullivan Penalized Spline) is a direct generalisation of smoothing splines that latter arises when the maximal number of B-spline basis functions included. One of the top performance measures the predicted regression curve that can be used is MSE (Mean Square Error). Results showed that the OPS method had a smaller MSE and GCV than the PSTPB (Penalized Spline Based Truncated Power Basis) method, so the use of the OPS method for predicting ozone against ultraviolet light was suitable to use.

Climate-sensitive radial increment model of Norway spruce in Tyrol based on a distributed lag model with penalized splines for year-ring time series

A novel methodological framework is presented for climate-sensitive modeling of annual radial stem increment using year-ring width time series. The approach is based on a generalized additive model with penalized regression splines together with a distributed time lag model taking into account smooth nonlinear effects of a series of monthly temperature and precipitation values, as well as their interactions. Climate effects are also assumed to vary smoothly with time lag. The model framework enables both the detrending of the individual time series and the regression modeling to be performed simultaneously in a single model step. The approach is applied to year-ring width time series of Norway spruce (Picea abies (L.) H. Karst.) trees in Tyrol, Austria. The marginal response curves show that tree growth is mainly promoted by high temperatures in late spring and early summer and by precipitation in fall and winter. Summer drought does not have a negative influence on the current year’s radial increment; however, when it is associated with high temperatures, it lowers the increment in the subsequent growth period. Higher winter precipitation in conjunction with lower temperatures has a positive effect. A significant non-climate related long-term growth trend is demonstrated, probably reflecting NOx and SO2 emission trends in Austria.

Night noise exposure and risk of death by suicide in adults living in metropolitan areas.

Noise is defined as "sound that is unwanted, unpleasant, or harmful to health." It may induce negative emotions and mental health problems and even may leads to increased suicide risk. Little attention has been paid so far to a potential link between noise and suicide. We investigated the association between nighttime environmental noise and suicide death in adults in the Republic of Korea. We analyzed the data from the National Health Insurance Service-National Sample Cohort, with a 4-year baseline (2002-2005) and an 8-year follow-up (2006-2013) assessment. A total of 155,492 adults constituted the study sample: younger adults (20-54 years, n=124,994), or older adults (≥55 years, n=30,498), and adults with mood and anxiety disorders (n=34,615). Suicide death was defined as per International Classification of Diseases-10 code X60-X84. Data on nighttime noise were obtained from the National Noise Information System. During the study period, 315 (0.2%) died of suicide. The incidence of suicide per 100,000 person-years was 25.71. With interquartile range increases in nighttime noise, the adjusted hazard ratio (HR) for suicide death was significantly increased: 1.32 (95% confidence interval [CI]: 1.02-1.70) for younger adults, 1.43 (95% CI: 1.01-2.02) for older adults, and 1.55 (95% CI: 1.10-2.19) for adults with mental illness. In the penalized regression spline models, the HRs for suicide death were gradually increased with environmental noise levels, and the risk was the highest for adults with mental illness. We found a significant association between exposure to nighttime noise and the risk of suicide death in adults in the Republic of Korea.

Advanced Modeling of Soil Biological Properties Using Visible Near Infrared Diffuse Reflectance Spectroscopy

Although visible near infrared diffuse reflectance spectroscopy (VisNIR DRS) is an emerging, rapid, non-destructive, and cost effective technology to predict a host of soil biological parameters, the traditional chemometric partial least squares regression (PLS) model often poses challenges during sensor development. In an effort to identify alternatives to the PLS model, three multivariate machine learning algorithms [PLS, penalized spline regression (PSR), and random forest (RF) regression]in conjunction with two spectral preprocessing methods [Savitzky-Golay first derivative and absorbance (ABS)] were compared with respect to 12 soil biological parameters of 123 soil samples. The RF model with ABS spectra successfully predicted all biological parameters with residual prediction deviation (RPD) ranging from 2.60 to 3.60 and outperformed PSR and PLS models. The best PSR model was obtained for total bacteria with an RPD of 2.70 and an r2 of 0.86 and among other variables, only Gram positive bacteria (RPD=2.63, r2=0.85), Gram negative bacteria (RPD=2.58, r2=0.85), and SOM (RPD=2.67, r2=0.86) were satisfactorily predicted, exhibiting r2>0.80 and RPD>2.5. Conversely, all variables except SOM (RPD=2.07) were poorly predicted by PLS models which had an RPD<2. Furthermore, linear discriminant analysis qualitatively clustered soils with different levels of microbial parameters. Summarily, the RF model with ABS spectra showed great promise in characterizing soil microbial communities with potential for such analysis in-situ.

Comparison of Significant Approaches of Penalized Spline Regression (P-splines)

Over the last two decades P-Splines have become a popular modeling tool in a wide class of statistical contexts. Fundamentally, semiparametric regression methods combine the leads of parametric and nonparametric approaches to regression analysis, while in precise, penalized spline regression uses the knowledge of nonparametric spline smoothing as a generalization of smoothing splines that let more suppleness in a choice of model with respect to the basis functions and the penalty. The present article compares two significant approaches of penalized spline regression models named as p-splines based on different basis functions with numerous knot selections and various types of penalties. These model fits have been applied on Wood Strength data to compare by calculating nonlinear least square method; also approaches are compared on several aspects: numerical immovability, quality of fit, derivative estimation and smoothing. This comparison will help us to fit best suitable model for conforming best suitable conditions and scenarios.

Estimation of Covariance Matrix on Bi-Response Longitudinal Data Analysis with Penalized Spline Regression

The correlation assumption of the longitudinal data with bi-response occurs on the measurement between the subjects of observation and the response. It causes the auto-correlation of error, and this can be overcome by using a covariance matrix. In this article, we estimate the covariance matrix based on the penalized spline regression model. Penalized spline involves knot points and smoothing parameters simultaneously in controlling the smoothness of the curve. Based on our simulation study, the estimated regression model of the weighted penalized spline with covariance matrix gives a smaller error value compared to the error of the model without covariance matrix.

Serum folate levels and urinary arsenic methylation profiles in the US population: NHANES, 2003-2012.

Arsenic is a prevalent environmental contaminant, and its folate-dependent methylation is important for detoxification in the body. In this study, we investigated the association between serum folate levels and methylation using data from the US National Health and Nutrition Examination Survey (NHANES) (2003-2012) (N = 11,016). Multivariate linear regression and penalized spline regression models were used to examine the association and possible upper limit of folate level regarding its impact on methylation in children (≤18 years) and adults (>18 years), respectively. Serum folate levels, methylation metabolites including urinary monomethylarsonic acid (MMA(V)) and dimethylarsinic acid (DMA(V)), and demographic variables were extracted from NHANES. Results showed that urinary percentage of DMA(V) (%DMA(V)) was positively associated with log(serum folate levels) after adjustment in children (β = 1.93, p < 0.01); urinary percentage of MMA(V) (%MMA(V)) was positively associated with log (serum folate levels) after adjustment in adults (β = 0.40, p < 0.01). No upper limit of folate level regarding its impact on arsenic methylation was identified. More than 50% of Non-Hispanic black and smokers with high total urinary arsenic levels had low serum folate levels. Our results indicate that folate promotes arsenic methylation, but the patterns are different in children versus in adults. Future interventions may be needed for the population exposed to high level of arsenic but with low serum folate to protect against the potential adverse health effects of arsenic.

A Design-Based Approach to Small Area Estimation Using a Semiparametric Generalized Linear Mixed Model

Summary In small area estimation, non-parametric models with penalized spline regression have been demonstrated to be a useful tool in creating granular area estimates to provide supplemental information where samples are few or non-existent. This study further examines the ability of a semiparametric generalized linear mixed model to produce conforming estimates for multiple area levels. A mosaic analogy is used to describe this process. A design-based jackknife method is employed for variance calculation.

Impact of scale of aggregation on associations of cardiovascular hospitalization and socio-economic disadvantage.

BackgroundThere are numerous studies that show an increased incidence of cardiovascular disease with increasing levels of socio-economic disadvantage. Exposures that might influence the relationship include elements of the built environment and social systems that shape lifestyle risk behaviors. In Canberra (the Australian capital city) there has been a particular housing policy to create ‘mixed-tenure’ neighborhoods so that small pockets of disadvantage are surrounded by more affluent residences (known as a ‘salt-and-pepper’ pattern). This may contribute to a scatter of higher incidence rates in very small areas in this population that may be obscured if aggregated data are used. This study explored the effect of changing the scale of the spatial units used in small area disease modelling, aiming to understand the impact of this issue and the implications for local public health surveillance.MethodsThe residence location of hospitalized individuals were aggregated to two differently scaled area units. First, the Australian Bureau of Statistics Statistical Area 2 (SA2) which is normally used as the basis for deidentification and release of health data. Second, these data were aggregated to a smaller level: the Statistical Area 1 (SA1). Generalized Additive Models with penalized regression splines were used to assess the association of age-sex-standardized rates for cardiovascular disease hospital admissions with disadvantage.ResultsThe relationships observed were different between the two types of spatial units. The SA1 level exposure-response curve for rates against the disadvantage index extended in a linear fashion above the midrange level, while that found at SA2-level suggested a curvilinear form with no evidence that rates increased with higher disadvantage beyond the midrange.ConclusionOur result supports findings of other work that has found disadvantage increases risk of cardiovascular disease. The shape of the curves suggest a difference in associations of cardiovascular disease rates with disadvantage scores between SA1 versus SA2. From these results it can be concluded that scale of analysis does influence the understanding of geographical patterns of socio-economic disadvantage and cardiovascular disease morbidity. Health surveillance and interventions in Canberra should take into account the impact of the scale of aggregation on the association between disadvantage and cardiovascular disease observed.

Time to Amyloid Positivity and Preclinical Changes in Brain Metabolism, Atrophy, and Cognition: Evidence for Emerging Amyloid Pathology in Alzheimer's Disease.

Background: Aβ pathology is associated with longitudinal changes of brain metabolism, atrophy, and cognition, in cognitively healthy elders. However, Aβ information is usually measured cross-sectionally and dichotomized to classify subjects as Aβ-positive or Aβ-negative, making it difficult to evaluate when brain and cognitive changes occur with respect to emerging Aβ pathology. In this study, we use longitudinal Aβ information to combine the level and rate of change of Aβ to estimate the time to Aβ-positivity for each subject and test this temporal proximity to significant Aβ pathology for associations with brain structure, metabolism, and cognition.Methods: In 89 cognitively healthy elders with up to 10 years of follow-up, we estimated the points at which rates of fluorodeoxyglucose (FDG) PET, MRI, and cognitive and functional decline begin to accelerate with respect to the time to Aβ-positivity. Points of initial acceleration in rates of decline were estimated using mixed-effects models with penalized regression splines.Results: Acceleration of rates of FDG PET were observed to occur 20+ years before the conventional threshold for Aβ-positivity. Subtle signs of cognitive dysfunction were observed 10+ years before Aβ-positivity.Conclusions: Aβ may have subtle associations with other hallmarks of Alzheimer's disease before Aβ biomarkers reach conventional thresholds for Aβ-positivity. Therefore, we propose that emerging Aβ pathology occurs many years before cognitively healthy elders reach the current threshold for Aβ positivity (preclinical AD). To allow prevention in the earliest disease stages, AD clinical trials may be designed to also include subjects with Aβ biomarkers in the sub-threshold range.

Generalized Additive Models for Gigadata: Modeling the U.K. Black Smoke Network Daily Data

We develop scalable methods for fitting penalized regression spline based generalized additive models with of the order of 104 coefficients to up to 108 data. Computational feasibility rests on: (i) a new iteration scheme for estimation of model coefficients and smoothing parameters, avoiding poorly scaling matrix operations; (ii) parallelization of the iteration’s pivoted block Cholesky and basic matrix operations; (iii) the marginal discretization of model covariates to reduce memory footprint, with efficient scalable methods for computing required crossproducts directly from the discrete representation. Marginal discretization enables much finer discretization than joint discretization would permit. We were motivated by the need to model four decades worth of daily particulate data from the U.K. Black Smoke and Sulphur Dioxide Monitoring Network. Although reduced in size recently, over 2000 stations have at some time been part of the network, resulting in some 10 million measurements. Modeling at a daily scale is desirable for accurate trend estimation and mapping, and to provide daily exposure estimates for epidemiological cohort studies. Because of the dataset size, previous work has focused on modeling time or space averaged pollution levels, but this is unsatisfactory from a health perspective, since it is often acute exposure locally and on the time scale of days that is of most importance in driving adverse health outcomes. If computed by conventional means our black smoke model would require a half terabyte of storage just for the model matrix, whereas we are able to compute with it on a desktop workstation. The best previously available reduced memory footprint method would have required three orders of magnitude more computing time than our new method. Supplementary materials for this article are available online.

Concentration-response of short-term ozone exposure and hospital admissions for asthma in Texas

BackgroundShort-term exposure to ozone has been associated with asthma hospital admissions (HA) and emergency department (ED) visits, but the shape of the concentration-response (C-R) curve is unclear. MethodsWe conducted a time series analysis of asthma HAs and ambient ozone concentrations in six metropolitan areas in Texas from 2001 to 2013. Using generalized linear regression models, we estimated the effect of daily 8-hour maximum ozone concentrations on asthma HAs for all ages combined, and for those aged 5–14, 15–64, and 65+years. We fit penalized regression splines to evaluate the shape of the C-R curves. ResultsUsing a log-linear model, estimated risk per 10ppb increase in average daily 8-hour maximum ozone concentrations was highest for children (relative risk [RR]=1.047, 95% confidence interval [CI]: 1.025–1.069), lower for younger adults (RR=1.018, 95% CI: 1.005–1.032), and null for older adults (RR=1.002, 95% CI: 0.981–1.023). However, penalized spline models demonstrated significant nonlinear C-R relationships for all ages combined, children, and younger adults, indicating the existence of thresholds. We did not observe an increased risk of asthma HAs until average daily 8-hour maximum ozone concentrations exceeded approximately 40ppb. ConclusionOzone and asthma HAs are significantly associated with each other; susceptibility to ozone is age-dependent, with children at highest risk. C-R relationships between average daily 8-hour maximum ozone concentrations and asthma HAs are significantly curvilinear for all ages combined, children, and younger adults. These nonlinear relationships, as well as the lack of relationship between average daily 8-hour maximum and peak ozone concentrations, have important implications for assessing risks to human health in regulatory settings.

CDBTools - Evaluate Positron Annihilation Coincidence Doppler Broadening Spectrum

CDBTools is a lightweight and easy to use application designed to provide you with an analysis tool for coincidence Doppler broadening (CDB) of the positron annihilation energy spectrum files. This application enables you to analyze the CDB and ratio curves, plots as well as graphs representing the evolution of the orbital electron momentum spectrum. The CDB extraction is provided by selectable filters applied at the diagonal line of the input matrix. To achieve CDB ratio curves with minimal error caused by 511keV peak shift, spectrum curves are recalculated by penalized regression spline.

Univariate Generalized Additive Models for Simulated Stationary and Non-Stationary Generalized Pareto Distribution

Generalized additive models as a predictor in regression approaches, are made up over cubic spline basis and penalized regression splines. Despite of linear predictor in GLM, generalized additive models use a sum of smooth functions of covariates as a predictor. The data which are used in this study have generalized Pareto distribution and have been simulated by inversion method. The data are generated in two types, the stationary case and the non-stationary case. The method of root mean square of errors as a method of measurement is used for comparison between power of predictions which are based on penalized regression splines as a method in univariate generalized additive models and linear regression based on maximum likelihood estimation. The finding of this research illustrates that the amount of accuracy of estimation of parameter of location in UGAM approach as an alternative promising of modelling through each specialized GPD's models, has less RMSE in compare with MLE.