Penalized Spline Model Research Articles

Estimating the proportion of modern contraceptives supplied by the public and private sectors using a Bayesian hierarchical penalized spline model

Estimating the proportion of modern contraceptives supplied by the public and private sectors using a Bayesian hierarchical penalized spline model

A joint penalized spline smoothing model for the number of positive and negative COVID-19 tests.

One of the key tools to understand and reduce the spread of the SARS-CoV-2 virus is testing. The total number of tests, the number of positive tests, the number of negative tests, and the positivity rate are interconnected indicators and vary with time. To better understand the relationship between these indicators, against the background of an evolving pandemic, the association between the number of positive tests and the number of negative tests is studied using a joint modeling approach. All countries in the European Union, Switzerland, the United Kingdom, and Norway are included in the analysis. We propose a joint penalized spline model in which the penalized spline is reparameterized as a linear mixed model. The model allows for flexible trajectories by smoothing the country-specific deviations from the overall penalized spline and accounts for heteroscedasticity by allowing the autocorrelation parameters and residual variances to vary among countries. The association between the number of positive tests and the number of negative tests is derived from the joint distribution for the random intercepts and slopes. The correlation between the random intercepts and the correlation between the random slopes were both positive. This suggests that, when countries increase their testing capacity, both the number of positive tests and negative tests will increase. A significant correlation was found between the random intercepts, but the correlation between the random slopes was not significant due to a wide credible interval.

Tortoise or Hare? The Associations between Physical Activity Volume and Intensity Distribution and the Risk of All-Cause Mortality: A Large Prospective Analysis of the UK Biobank.

Analysis methods to determine the optimal combination of volume and intensity of objectively measured physical activity (PA) with prospective outcomes are limited. Participants in UK Biobank were recruited in the UK between 2006 and 2010. We linked the questionnaire and accelerometer with all-cause mortality data from the NHS Information Centre and NHS Central Register up to April 2021. We developed a novel method, extending the penalized spline model of Augustin et al. to a smooth additive Cox model for survival data, and estimated the prospective relationship between intensity distribution and all-cause mortality, adjusting for the overall volume of PA. We followed 84,166 men and women (aged 40-69) for an average of 6.4 years (range 5.3-7.9), with an observed mortality rate of 22.2 deaths per 1000. Survival rates differed by PA volume quartile, with poorer outcomes for the lowest PA volumes. Participants with more sedentary to light intensity PA (<100 milligravities (mg)) and/or less vigorous intensity PA (>250 mg) than average for a given volume of PA, had higher mortality rates than vice versa. Approximate hazard ratios were 0.83 (95% credible interval [CI]: 0.79, 0.88) for an average-risk profile compared to a high-risk profile and 0.80 (95% CI: 0.74, 0.87) for a low-risk profile compared to an average-risk profile. A high- versus low-risk profile has the equivalent of 15 min more slow walking, but 10 min less moderate walking. At low PA volumes, increasing overall volume suggests the most benefit in reducing all-cause mortality risk. However, at higher overall volumes, substituting lighter with more vigorous intensity activity suggests greater benefit.

토양 내 오염물질 농도 예측을 위한 베이지안 벌점 스플라인

For the diagnosis of soil contamination, it is necessary to accurately understand the distribution of pollutants in the soil of the survey site. However, since it is practically impossible to investigate all observable points in the survey site, the concentration of pollutants in the soil of the survey site is predicted through a spatial prediction model based on data obtained from some selected points in the survey site. However, due to the nature of the soil data, the size of the data is often insufficient, which can greatly reduce the accuracy of spatial prediction. Therefore, in this study, to solve this problem, we propose a Bayesian penalized spline model that predicts the concentration of contaminants in the soil of the survey site by using the soil quality monitoring network data provided by the Ministry of Environment(MOE) in Korea as prior information. In addition, in order to evaluate the performance of the proposed model, RMSE, MAE, MAPE were used to compare the prediction accuracy by data size with several comparative models. As a result of the performance evaluation, it was confirmed that the performance of the proposed model showed better performance than the comparative models. In particular, the smaller the data size, the better the performance of the proposed model compared to the comparative models. Therefore, the use of the proposed model can be considered when trying to understand the distribution of pollutants in the soil at the initial stage of soil survey, where soil data for the survey site is relatively scarce.

PEMODELAN KURS DOLLAR AMERIKA SERIKAT TERHADAP RUPIAH MENGGUNAKAN REGRESI PENALIZED SPLINE DILENGKAPI GUI R

United States Dollar (USD) exchange rate movement against Rupiah is the main guideline for economic actors in making decisions. Exchange rate movement of USD against Rupiah is a time series data. One of the statistical methods that can be used for modelling time series data is ARIMA. ARIMA method data must be stationery and residuals must be normally distributed, independent, and constant variance, which means an alternative model is needed so that it is not bound by any assumptions, namely a nonparametric penalized spline regression model. Selling rate data of USD against Rupiah is modeled using nonparametric penalized spline regression because the assumptions in the ARIMA model are not fulfilled. Penalized spline regression modeling is using full search algorithm in determining knot points. Lambda values are tested from 0 to 100000 on order 2, 3, and 4. Optimal penalized spline model is a model with minimum GCV value. R GUI facilitate the process of selecting the best model. Data is divided into 2 parts, namely in sample data for model formation and out sample data for evaluating the best model performance based on MAPE value. Penalized spline regression modeling produces the best model, namely optimal penalized spline model with minimum GCV value achieved on 3rd order with 35 knot points and lambda value = 2007. 96,20% value of R Squared model indicates the model is a strong model. In the evaluation of the best model, the MAPE data out sample value is 0.65%. MAPE value indicates the model has very good forecasting ability.

Estimating the Interest Rate Term Structures of Treasury and Corporate Debt with Bayesian Penalized Splines

This paper provides a Bayesian approach to estimating the interest rate term structures of Treasury and corporate debt with a penalized spline model. Although the literature on term structure modeling is vast, to the best of our knowledge, all methods developed so far belong to the frequentist school. In this paper, we develop a two-step estimation procedure from a Bayesian perspective. The Treasury term structure is first estimated with a Bayesian penalized spline model. The smoothing parameter is naturally embedded in the model as a ratio of posterior variances and does not need to be selected as in the frequentist approach. The corporate term structure is then estimated by adding a credit spread to the estimated Treasury term structure, incorporating knowledge of the positive credit spread into the Bayesian model as an informative prior. In contrast to the frequentist method, the small sample size of the corporate debt poses no particular difficulty to the proposed Bayesian approach.

Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels

The present study uses penalized splines (p- spline) to estimate the functional relationship between the survey variable and the auxiliary variable in a complex survey design; where a population divided into clusters is in turn subdivided into strata. This study has considered a case of auxiliary information present at two levels; at both cluster and element levels. The study further applied model calibration technique by penalty function to estimate the population total. The calibration problems at both levels have been treated as optimization problems and solved using penalty functions to derive the estimators for this study. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, the study expects that the variable of interest’s fitted values meets such constraints. This study runs a Monte Carlo simulation to assess the finite sample performance of the penalized spline model calibrated estimator under complex survey data. Simulation studies were conducted to compare the efficiency of p-spline model calibrated estimator with Horvitz Thompson estimator (HT) by mean squared error (MSE) criterion. This study shows that the p-spline model-based estimator is generally more efficient than the HT in terms of the mean squared error. The results have also shown that the estimator obtained is unbiased, consistent and very robust because it does not fail if the model is misspecified for the data.

Joint penalized spline modeling of multivariate longitudinal data, with application to HIV-1 RNA load levels and CD4 cell counts.

Motivated by the need to jointly model the longitudinal trajectories of HIV viral load levels and CD4 counts during the primary infection stage, we propose a joint penalized spline modeling approach that can be used to model the repeated measurements from multiple biomarkers of various types (eg, continuous, binary) simultaneously. This approach allows for flexible trajectories for each marker, accounts for potentially time-varying correlation between markers, and is robust to misspecification of knots. Despite its advantages, the application of multivariate penalized spline models, especially when biomarkers may be of different data types, has been limited in part due to its seemingly complexity in implementation. To overcome this, we describe a procedure that transforms the multivariate setting to the univariate one, and then makes use of the generalized linear mixed effect model representation of a penalized spline model to facilitate its implementation with standard statistical software. We performed simulation studies to evaluate the validity and efficiency through joint modeling of correlated biomarkers measured longitudinally compared to the univariate modeling approach. We applied this modeling approach to longitudinal HIV-1 RNA load and CD4 count data from Southern African cohorts to estimate features of the joint distributions such as the correlation and the proportion of subjects with high viral load levels and high CD4 cell counts overtime.

Meta-analysis of individual patient data with semi-competing risks under the Weibull joint frailty–copula model

In meta-analysis of individual patient data with semi-competing risks, the joint frailty–copula model has been proposed, where frailty terms account for the between-study heterogeneity and copulas account for dependence between terminal and nonterminal event times. In the previous works, the baseline hazard functions in the joint frailty–copula model are estimated by the nonparametric model or the penalized spline model, which requires complex maximization schemes and resampling-based interval estimation. In this article, we propose the Weibull distribution for the baseline hazard functions under the joint frailty–copula model. We show that the Weibull model constitutes a conjugate model for the gamma frailty, leading to explicit expressions for the moments, survival functions, hazard functions, quantiles, and mean residual lifetimes. These results facilitate the parameter interpretation of prognostic inference. We propose a maximum likelihood estimation method and make our computer programs available in the R package, joint.Cox. We also show that the delta method is feasible to calculate interval estimates, which is a useful alternative to the resampling-based method. We conduct simulation studies to examine the accuracy of the proposed methods. Finally, we use the data on ovarian cancer patients to illustrate the proposed method.

Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels

Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.

Prediction of ventricular arrhythmias in patients with a spontaneous Brugada type 1 pattern: the key is in the electrocardiogram.

There is currently no reliable tool to quantify the risks of ventricular fibrillation or sudden cardiac arrest (VF/SCA) in patients with spontaneous Brugada type 1 pattern (BrT1). Previous studies showed that electrocardiographic (ECG) markers of depolarization or repolarization disorders might indicate elevated risk. We aimed to design a VF/SCA risk prediction model based on ECG analyses for adult patients with spontaneous BrT1. This retrospective multicentre international study analysed ECG data from 115 patients (mean age 45.1 ± 12.8 years, 105 males) with spontaneous BrT1. Of these, 45 patients had experienced VF/SCA and 70 patients did not experience VF/SCA. Among 10 ECG markers, a univariate analysis showed significant associations between VF/SCA and maximum corrected Tpeak-Tend intervals ≥100 ms in precordial leads (LMaxTpec) (P < 0.001), BrT1 in a peripheral lead (pT1) (P = 0.004), early repolarization in inferolateral leads (ER) (P < 0.001), and QRS duration ≥120 ms in lead V2 (P = 0.002). The Cox multivariate analysis revealed four predictors of VF/SCA: the LMaxTpec [hazard ratio (HR) 8.3, 95% confidence interval (CI) 2.4-28.5; P < 0.001], LMaxTpec + ER (HR 14.9, 95% CI 4.2-53.1; P < 0.001), LMaxTpec + pT1 (HR 17.2, 95% CI 4.1-72; P < 0.001), and LMaxTpec + pT1 + ER (HR 23.5, 95% CI 6-93; P < 0.001). Our multidimensional penalized spline model predicted the 1-year risk of VF/SCA, based on age and these markers. LMaxTpec and its association with pT1 and/or ER indicated elevated VF/SCA risk in adult patients with spontaneous BrT1. We successfully developed a simple risk prediction model based on age and these ECG markers.

Nonlinear growth curve modeling using penalized spline models: A gentle introduction.

This didactic article aims to provide a gentle introduction to penalized splines as a way of estimating nonlinear growth curves in which many observations are collected over time on a single or multiple individuals. We begin by presenting piecewise linear models in which the time domain of the data is divided into consecutive phases and a separate linear regression line is fitted in each phase. Linear splines add the feature that the regression lines fitted in adjacent phases are always joined at the boundary so there is no discontinuity in level between phases. Splines are highly flexible raising the fundamental tradeoff between model fit and smoothness of the curve. Penalized spline models address this tradeoff by introducing a penalty term to achieve balance between fit and smoothness. The linear mixed-effects model, familiar from multilevel analysis, is introduced as a method for estimating penalized spline models. Higher order spline models using quadratic or cubic functions which further enhance a smooth fit are introduced. Technical issues in estimation, hypothesis testing, and constructing confidence intervals for higher order penalized spline models are considered. We then use data from the Early Childhood Longitudinal Study to illustrate each step in fitting a higher order penalized spline model, and to illustrate hypothesis testing, the construction of confidence intervals, and the comparison of the functions in 2 groups (boys and girls). Extensive graphical illustrations are provided throughout. Annotated computer scripts using the R package nlme are provided in online supplemental materials. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Estimating Chinese Treasury yield curves with Bayesian smoothing splines

An improved Bayesian smoothing spline (BSS) model is developed to estimate the term structure of Chinese Treasury yield curves. The developed BSS model has a flexible function form which can model various yield curve shapes. As a nonparametric method different from Jarrow–Ruppert–Yu’s penalized splines, the BSS model does not need to choose the number of and locations for knots. Instead, this BSS model obtains the smoothing parameter as a by-product that does not need to be estimated. Furthermore, a dimension reduction procedure is developed to calculate an inverse matrix when implementing this BSS model. Finally, simulation results and an application illustrate the BSS model outperforms traditional parametric models and the penalized spline model.

Mixed Generalized Akaike Information Criterion for Small Area Models

Summary A mixed generalized Akaike information criterion xGAIC is introduced and validated. It is derived from a quasi-log-likelihood that focuses on the random effect and the variability between the areas, and from a generalized degree-of-freedom measure, as a model complexity penalty, which is calculated by the bootstrap. To study the performance of xGAIC, we consider three popular mixed models in small area inference: a Fay–Herriot model, a monotone model and a penalized spline model. A simulation study shows the good performance of xGAIC. Besides, we show its relevance in practice, with two real applications: the estimation of employed people by economic activity and the prevalence of smokers in Galician counties. In the second case, where it is unclear which explanatory variables should be included in the model, the problem of selection between these explanatory variables is solved simultaneously with the problem of the specification of the functional form between the linear, monotone or spline options.

PENALIZED SPLINE: A GENERAL ROBUST TRAJECTORY MODEL FOR ZIYUAN-3 SATELLITE

Abstract. Owing to the dynamic imaging system, the trajectory model plays a very important role in the geometric processing of high resolution satellite imagery. However, establishing a trajectory model is difficult when only discrete and noisy data are available. In this manuscript, we proposed a general robust trajectory model, the penalized spline model, which could fit trajectory data well and smooth noise. The penalized parameter λ controlling the smooth and fitting accuracy could be estimated by generalized cross-validation. Five other trajectory models, including third-order polynomials, Chebyshev polynomials, linear interpolation, Lagrange interpolation and cubic spline, are compared with the penalized spline model. Both the sophisticated ephemeris and on-board ephemeris are used to compare the orbit models. The penalized spline model could smooth part of noise, and accuracy would decrease as the orbit length increases. The band-to-band misregistration of ZiYuan-3 Dengfeng and Faizabad multispectral images is used to evaluate the proposed method. With the Dengfeng dataset, the third-order polynomials and Chebyshev approximation could not model the oscillation, and introduce misregistration of 0.57 pixels misregistration in across-track direction and 0.33 pixels in along-track direction. With the Faizabad dataset, the linear interpolation, Lagrange interpolation and cubic spline model suffer from noise, introducing larger misregistration than the approximation models. Experimental results suggest the penalized spline model could model the oscillation and smooth noise.

PENALIZED SPLINE: A GENERAL ROBUST TRAJECTORY MODEL FOR ZIYUAN-3 SATELLITE

Owing to the dynamic imaging system, the trajectory model plays a very important role in the geometric processing of high resolution satellite imagery. However, establishing a trajectory model is difficult when only discrete and noisy data are available. In this manuscript, we proposed a general robust trajectory model, the penalized spline model, which could fit trajectory data well and smooth noise. The penalized parameter λ controlling the smooth and fitting accuracy could be estimated by generalized cross-validation. Five other trajectory models, including third-order polynomials, Chebyshev polynomials, linear interpolation, Lagrange interpolation and cubic spline, are compared with the penalized spline model. Both the sophisticated ephemeris and on-board ephemeris are used to compare the orbit models. The penalized spline model could smooth part of noise, and accuracy would decrease as the orbit length increases. The band-to-band misregistration of ZiYuan-3 Dengfeng and Faizabad multispectral images is used to evaluate the proposed method. With the Dengfeng dataset, the third-order polynomials and Chebyshev approximation could not model the oscillation, and introduce misregistration of 0.57 pixels misregistration in across-track direction and 0.33 pixels in along-track direction. With the Faizabad dataset, the linear interpolation, Lagrange interpolation and cubic spline model suffer from noise, introducing larger misregistration than the approximation models. Experimental results suggest the penalized spline model could model the oscillation and smooth noise.

A Penalized Spline-Based Attitude Model for High-Resolution Satellite Imagery

Attitude models play a prominent role in the geometric processing of high-resolution satellite imagery (HRSI). Because of the high accuracy of the matching algorithm, attitude oscillations can occur in HRSI. Various methods for correcting this attitude oscillation with parallax observations have been proposed. However, few researchers have attempted to model the oscillation from the attitude records or have taken noise into consideration. In this paper, a penalized spline-based attitude model is proposed, which can model the oscillation with piecewise and continuously differentiable polynomials and smooth out the attitude noise with a penalty function. The balance between the fitting accuracy and noise smoothing is controlled by a penalty parameter, which is estimated by generalized cross-validation. Given that the attitude error introduces distortions into sensor-corrected images, the band-to-band registration of multispectral images is used to validate the attitude model. Five multispectral data sets captured by ZiYuan-3 are used to demonstrate the effectiveness of the proposed method. Compared with third-degree polynomials and cubic spline interpolation, the penalized spline model delivers the best performance by limiting the misregistration caused by the attitude model to within 0.1 pixels.

Assessing spatial variability of soil petroleum contamination using visible near-infrared diffuse reflectance spectroscopy

Visible near-infrared (VisNIR) diffuse reflectance spectroscopy (DRS) is a rapid, non-destructive method for sensing the presence and amount of total petroleum hydrocarbon (TPH) contamination in soil. This study demonstrates the feasibility of VisNIR DRS to be used in the field to proximally sense and then map the areal extent of TPH contamination in soil. More specifically, we evaluated whether a combination of two methods, penalized spline regression and geostatistics could provide an efficient approach to assess spatial variability of soil TPH using VisNIR DRS data from soils collected from an 80 ha crude oil spill in central Louisiana, USA. Initially, a penalized spline model was calibrated to predict TPH contamination in soil by combining lab TPH values of 46 contaminated and uncontaminated soil samples and the first-derivative of VisNIR reflectance spectra of these samples. The r(2), RMSE, and bias of the calibrated penalized spline model were 0.81, 0.289 log(10) mg kg(-1), and 0.010 log(10) mg kg(-1), respectively. Subsequently, the penalized spline model was used to predict soil TPH content for 128 soil samples collected over the 80 ha study site. When assessed with a randomly chosen validation subset (n = 10) from the 128 samples, the penalized spline model performed satisfactorily (r(2) = 0.70; residual prediction deviation = 2.0). The same validation subset was used to assess point kriging interpolation after the remaining 118 predictions were used to produce an experimental semivariogram and map. The experimental semivariogram was fitted with an exponential model which revealed strong spatial dependence among soil TPH [r(2) = 0.76, nugget = 0.001 (log(10) mg kg(-1))(2), and sill 1.044 (log(10) mg kg(-1))(2)]. Kriging interpolation adequately interpolated TPH with r(2) and RMSE values of 0.88 and 0.312 log(10) mg kg(-1), respectively. Furthermore, in the kriged map, TPH distribution matched with the expected TPH variability of the study site. Since the combined use of VisNIR prediction and geostatistics was promising to identify the spatial patterns of TPH contamination in soils, future research is warranted to evaluate the approach for mapping spatial variability of petroleum contaminated soils.

PARTICULATE AIR POLLUTION AND SURVIVAL: PROPORTIONALITY ASSUMPTION, TIMING OF DOSE, AND DOSE-RESPONSE RELATIONSHIP IN THE HARVARD SIX CITIES STUDY

Background and Aims: The association of fine particulate air pollution (PM2.5) with mortality is well established, however questions have been raised about the sensitivity of results to model specification, including whether covariates are proportional and whether the dose-response is linear. We addressed these issues using an extended follow-up of the Harvard Six Cities Study through 2006, incorporating more recent lower exposures. Methods: 8111 white adults were included. Personal characteristics were collected at enrollment (1974). Vital status has been updated using the National Death Index. Annual PM2.5 was measured by study or US EPA monitors. We replicated the original results for PM2.5 and all-cause mortality using a Cox regression and repeated with a Poisson survival analysis with separate hazard rates for each year of follow-up. To relax the proportionality assumption, we included interactions between year of follow-up and smoking, sex, and education. We examined the shape of the dose response relationship using penalized splines and the fit of models with different lags. Results: Poisson and Cox models produced similar results, which did not change with time-varying covariate effects. Akaike’s information criterion was lower for models using current year’s PM2.5 than for the models including earlier years. Each 10µg/m3 increase in PM2.5 exposure in the year of death was associated with an adjusted RR=1.13 (95% CI, 1.06-1.20). The penalized spline model chose one degree of freedom, indicating a linear dose-response relationship. The results did not show any rationale for threshold in the association, and the effects of PM2.5 did not change by time period. Conclusions: Including more recent observations with exposures below the US standard (15µg/m3), we found a strong linear association without any threshold between PM2.5 and all-cause mortality. Results showed that current year was the best exposure window and that the toxicity of PM2.5 has not changed over time.

The Hodrick-Prescott Filter: A special case of penalized spline smoothing

We prove that the Hodrick-Prescott Filter (HPF), a commonly used method for smoothing econometric time series, is a special case of a linear penalized spline model with knots placed at all observed time points (except the first and last) and uncorrelated residuals. This equivalence then furnishes a rich variety of existing data-driven parameter estimation methods, particularly restricted maximum likelihood (REML) and generalized cross-validation (GCV). This has profound implications for users of HPF who have hitherto typically relied on subjective choice, rather than estimation, for the smoothing parameter. By viewing estimates as roots of an appropriate quadratic estimating equation, we also present a new approach for constructing confidence intervals for the smoothing parameter. The method is akin to a parametric bootstrap where Monte Carlo simulation is replaced by saddlepoint approximation, and provides a fast and accurate alternative to exact methods, when they exist, e.g. REML. More importantly, it is also the only computationally feasible method when no other methods, exact or otherwise, exist, e.g. GCV. The methodology is demonstrated on the Gross National Product (GNP) series originally analyzed by Hodrick and Prescott (1997). With proper attention paid to residual correlation structure, we show that REML-based estimation delivers an appropriate smooth for both the GNP series and its returns.