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This paper develops a nonparametric method for estimating the spectral density of multivariate stationary time series using basis expansion. A likelihood-based approach is used to fit the model through the minimization of a penalized Whittle negative log-likelihood. Then, a Newton-type algorithm is developed for the computation. In this method, we smooth the Cholesky factors of the multivariate spectral density matrix in a way that the reconstructed estimate based on the smoothed Cholesky components is consistent and positive-definite. In a simulation study, we have illustrated and compared our proposed method with other competitive approaches. Finally, we apply our approach to two real-world problems, Electroencephalogram signals analysis, E l N i n ~ o Cycle. [ABSTRACT FROM AUTHOR]

This research aims to explore and compare several nonparametric regression techniques, including smoothing splines, natural cubic splines, B-splines, and penalized spline methods. The focus is on estimating parameters and determining the optimal number of knots to forecast cyclic and nonlinear patterns, applying these methods to simulated and real-world datasets, such as Thailand's coal import data. Cross-validation techniques are used to control and specify the number of knots, ensuring the curve fits the data points accurately. The study applies nonparametric regression to forecast time series data with cyclic patterns and nonlinear forms in the dependent variable, treating the independent variable as sequential data. Simulated data featuring cyclical patterns resembling economic cycles and nonlinear data with complex equations to capture variable interactions are used for experimentation. These simulations include variations in standard deviations and sample sizes. The evaluation criterion for the simulated data is the minimum average mean square error (MSE), which indicates the most efficient parameter estimation. For the real data, monthly coal import data from Thailand is used to estimate the parameters of the nonparametric regression model, with the MSE as the evaluation metric. The performance of these techniques is also assessed in forecasting future values, where the mean absolute percentage error (MAPE) is calculated. Among the methods, the natural cubic spline consistently yields the lowest average mean square error across all standard deviations and sample sizes in the simulated data. While the natural cubic spline excels in parameter estimation, B-splines show strong performance in forecasting future values. [ABSTRACT FROM AUTHOR]

This research aims to explore and compare several nonparametric regression techniques, including smoothing splines, natural cubic splines, B-splines, and penalized spline methods. The focus is on estimating parameters and determining the optimal number of knots to forecast cyclic and nonlinear patterns, applying these methods to simulated and real-world datasets, such as Thailand’s coal import data. Cross-validation techniques are used to control and specify the number of knots, ensuring the curve fits the data points accurately. The study applies nonparametric regression to forecast time series data with cyclic patterns and nonlinear forms in the dependent variable, treating the independent variable as sequential data. Simulated data featuring cyclical patterns resembling economic cycles and nonlinear data with complex equations to capture variable interactions are used for experimentation. These simulations include variations in standard deviations and sample sizes. The evaluation criterion for the simulated data is the minimum average mean square error (MSE), which indicates the most efficient parameter estimation. For the real data, monthly coal import data from Thailand is used to estimate the parameters of the nonparametric regression model, with the MSE as the evaluation metric. The performance of these techniques is also assessed in forecasting future values, where the mean absolute percentage error (MAPE) is calculated. Among the methods, the natural cubic spline consistently yields the lowest average mean square error across all standard deviations and sample sizes in the simulated data. While the natural cubic spline excels in parameter estimation, B-splines show strong performance in forecasting future values.

To fit a set of quantile curves, Bayesian simultaneous quantile curve fitting methods face some challenges in properly specifying a feasible formulation and efficiently accommodating the non-crossing constraints. In this article, we propose a new minimization problem and develop its corresponding Bayesian analysis. The new minimization problem imposes two penalties to control not only the smoothness of fitted quantile curves but also the differences between quantile curves. This enables a direct inference on differences of quantile curves and facilitates improved information sharing among quantiles. After adopting B-spline approximation for the positive smoothing functions in the minimization problem, we specify the pseudo composite asymmetric Laplace likelihood and derive its priors. The computation algorithm, including partially collapsed Gibbs sampling for model parameters and Monte Carlo Expectation-Maximization algorithm for penalty parameters, are provided to carry out the proposed approach. The extensive simulation studies show that, compared with other candidate methods, the proposed approach yields more robust estimation. More advantages of the proposed approach are observed for the extreme quantiles, heavy-tailed random errors, and inference on the differences of quantiles. We also demonstrate the relative performances of the proposed approach and other competing methods through two real data analyses. [ABSTRACT FROM AUTHOR]

Partially linear models are useful tools to analyze data from economic, genetic, and other fields. Similar to other data analyses, the identification of influential observations that may be potential outliers is an important step beyond estimation in such models. The objective of this article is to develop some diagnostic measures for identifying influential observations in partially linear models when some of the covariates are measured with errors. Deletion measures are developed based on case deletion, mean shift outlier models, and the corrected likelihood of Nakamura (1990). The performance of the methods is illustrated by an artificial example and a real example. [ABSTRACT FROM AUTHOR]

Thin-Plate Spline Generalized Linear Models (TPS-GLMs) are an extension of Semiparametric Generalized Linear Models (SGLMs), because they allow a smoothing spline to be extended to two or more dimensions. This class of models allows modeling a set of data in which it is desired to incorporate the non-linear joint effects of some covariates to explain the variability of a certain variable of interest. In the spatial context, these models are quite useful, since they allow the effects of locations to be included, both in trend and dispersion, using a smooth surface. In this work, we extend the local influence technique for the TPS-GLM model in order to evaluate the sensitivity of the maximum penalized likelihood estimators against small perturbations in the model and data. We fit our model through a joint iterative process based on Fisher Scoring and weighted backfitting algorithms. In addition, we obtained the normal curvature for the case-weight perturbation and response variable additive perturbation schemes, in order to detect influential observations on the model fit. Finally, two data sets from different areas (agronomy and environment) were used to illustrate the methodology proposed here. [ABSTRACT FROM AUTHOR]

The seasonal cycle (SC) of surface air temperature is a substantial element in climatology. Some basic and important topics in climate change studies are based on accurate and reliable estimation of SCs, such as percentile‐based indices of extremes and probability density function (PDF) changes of daily temperatures. For both of them, most studies characterized SCs using averages of multi‐days windows, which is not smooth and accurate enough representing the extreme thresholds and climatological normal of SCs. It is necessary to construct smooth and reasonable SCs for more accurate estimation of temperature changes on extreme thresholds and PDFs. In this study, we propose a flexible method based on generalized additive models for location scale and shape and penalized b‐spline smoothing technique with respective distributions to construct smooth SCs and SC for extreme temperatures (SCETs). The accuracy of the constructed smooth SCETs is good with the estimation biases of percentiles tending to be zero. The constructed smooth SCET also exhibits good stability over time, such that the magnitude changes of temperatures on each calendar day are close to climatic changes of mean temperature when the concerned period shifts. Based on the constructed smooth SCs, climatic changes by seasons and PDFs between two periods, 1961–1990 and 1991–2020, are examined over China. The increase of thresholds for hot extremes in spring during the recent period is prominent, while the increase of thresholds for daytime cold extremes in summer over a part of central to southern China is also notable. The smooth SCs and SCETs based on our flexible statistical modelling framework can characterize daily extreme temperatures reasonably and accurately, and should be expected to have more applications for a better understanding of climate changes related to distributions and seasonal cycles. [ABSTRACT FROM AUTHOR]

The nonparametric approach is an appropriate approach for patterns of relationships between predictor variables and response variables that are not or have not been known in form. In other words, there is no complete information about the pattern of relationships between variables. Curve estimation is determined based on relationship patterns in existing data. The nonparametric approach has great flexibility for estimating regression curves. This study aims to form a model on investor interest factors in improving tourism investment decisions with a nonparametric approach. The nonparametric method used is the smoothing spline regression method. The smoothing spline method is used because the modeling results from the smoothing spline approach can follow the relationship model between variables contained in the data. Thus, this method really helps researchers to model relationships between variables that are not linear and whose linear form is unknown. The results of the analysis showed that the nonparametric smoothing spline regression analysis method could model data by 94.63%, indicates that data variance can be explained by 94.63% with models, while other variance outside the study explain the remaining 5.37%. That is, investment motivation is one of the most important factors to improve investment decisions.

In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties. [ABSTRACT FROM AUTHOR]

Tolerance intervals in regression allow the user to quantify, with a specified degree of confidence, bounds for a specified proportion of the sampled population when conditioned on a set of covariate values. While methods are available for tolerance intervals in fully-parametric regression settings, the construction of tolerance intervals for nonparametric regression models has been treated in a limited capacity. This paper fills this gap and develops likelihood-based approaches for the construction of pointwise one-sided and two-sided tolerance intervals for nonparametric regression models. A numerical approach is also presented for constructing simultaneous tolerance intervals. An appealing facet of this work is that the resulting methodology is consistent with what is done for fully-parametric regression tolerance intervals. Extensive coverage studies are presented, which demonstrate very good performance of the proposed methods. The proposed tolerance intervals are calculated and interpreted for analyses involving a fertility dataset and a triceps measurement dataset. [ABSTRACT FROM AUTHOR]

Fitting a smoothing spline model on a large‐scale dataset is daunting due to the high computational cost. In this study, we develop an efficient basis selection method for smoothing spline calculation. The key idea is to force a nonparametric function in an infinite‐dimensional functional space to reside in a relatively small and finite‐dimensional model space without the loss of too much prediction accuracy. Such an approximation naturally allows for much faster numerical calculation, especially for large datasets. Among various basis selection methods, space‐filling basis selection has been proven to be more efficient since its model space dimension is smaller than that of others. Despite algorithmic benefits, most of the space‐filling selection methods only take the overall space‐filling property into account. These methods may be less efficient when the underlying response surface is not isomorphic. To overcome this obstacle, we develop an efficient algorithm to improve projective uniformity for space‐filling basis selection. It has been proved that the proposed estimator has the same convergence rate as the full bases estimator. Compared with the standard approach, the proposed method significantly reduces the computational cost. Simulation and real data studies demonstrate the efficiency and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

A noise-resistant linearization model that reveals the true nonlinearity of the sensor is essential for retrieving accurate physical displacement from the signals captured by sensing electronics. In this paper, we propose a novel information-driven smoothing spline linearization method, which innovatively integrates one new and three standard information criterions into a smoothing spline for the high-precision displacement sensors' linearization. Using theoretical analysis and Monte Carlo simulation, the proposed linearization method is demonstrated to outperform traditional polynomial and spline linearization methods for high-precision displacement sensors with a low noise to range ratio in the 10−5 level. Validation experiments were carried out on two different types of displacement sensors to benchmark the performance of the proposed method compared to the polynomial models and the the non-smoothing cubic spline. The results show that the proposed method with the new modified Akaike Information Criterion stands out compared to the other linearization methods and can improve the residual nonlinearity by over 50% compared to the standard polynomial model. After being linearized via the proposed method, the residual nonlinearities reach as low as ±0.0311% F.S. (Full Scale of Range), for the 1.5 mm range chromatic confocal displacement sensor, and ±0.0047% F.S., for the 100 mm range laser triangulation displacement sensor. [ABSTRACT FROM AUTHOR]

Single cell RNA sequencing (scRNA-seq) technologies provide researchers with an unprecedented opportunity to exploit cell heterogeneity. For example, the sequenced cells belong to various cell lineages, which may have different cell fates in stem and progenitor cells. Those cells may differentiate into various mature cell types in a cell differentiation process. To trace the behavior of cell differentiation, researchers reconstruct cell lineages and predict cell fates by ordering cells chronologically into a trajectory with a pseudo-time. However, in scRNA-seq experiments, there are no cell-to-cell correspondences along with the time to reconstruct the cell lineages, which creates a significant challenge for cell lineage tracing and cell fate prediction. Therefore, methods that can accurately reconstruct the dynamic cell lineages and predict cell fates are highly desirable.In this article, we develop an innovative machine-learning framework called Cell Smoothing Transformation (CellST) to elucidate the dynamic cell fate paths and construct gene networks in cell differentiation processes. Unlike the existing methods that construct one single bulk cell trajectory, CellST builds cell trajectories and tracks behaviors for each individual cell. Additionally, CellST can predict cell fates even for less frequent cell types. Based on the individual cell fate trajectories, CellST can further construct dynamic gene networks to model gene-gene relationships along the cell differentiation process and discover critical genes that potentially regulate cells into various mature cell types.

The limiting distributions of statistics used to test hypotheses about parameters on the boundary of their domains may provide very poor approximations to the finite‐sample behaviour of these statistics, even for very large samples. We review theoretical work on this problem, describe hard and soft boundaries and iceberg estimators, and give examples highlighting how the limiting results greatly underestimate the probability that the parameter lies on its boundary even in very large samples. We propose and evaluate some simple remedies for this difficulty based on normal approximation for the profile score function, and then outline how higher order approximations yield excellent results in a range of hard and soft boundary examples. We use the approach to develop an accurate test for the need for a spline component in a linear mixed model. [ABSTRACT FROM AUTHOR]

Many methods have been developed to study nonparametric function-on-function regression models. Nevertheless, there is a lack of model selection approach to the regression function as a functional function with functional covariate inputs. To study interaction effects among these functional covariates, in this article, we first construct a tensor product space of reproducing kernel Hilbert spaces and build an analysis of variance (ANOVA) decomposition of the tensor product space. We then use a model selection method with the L 1 criterion to estimate the functional function with functional covariate inputs and detect interaction effects among the functional covariates. The proposed method is evaluated using simulations and stroke rehabilitation data. [ABSTRACT FROM AUTHOR]

The extraordinary amounts of data generated nowadays pose heavy demands on computational resources and time, which hinders the implementation of various statistical methods. An efficient and popular strategy of downsizing data volumes and thus alleviating these challenges is subsampling. However, the existing methods either rely on specific assumptions for the underlying models or acquire partial information from the available data. For regression problems, we propose a novel approach, termed adaptive subsampling with the minimum energy criterion (ASMEC). The proposed method requires no explicit model assumptions and "smartly" incorporates information on covariates and responses. ASMEC subsamples possess two desirable properties: space-fillingness and spatial adaptiveness. We investigate the limiting distribution of ASMEC subsamples and their theoretical properties under the smoothing spline regression model. The effectiveness and robustness of the ASMEC approach are also supported by a variety of synthetic examples and two real-life examples. [ABSTRACT FROM AUTHOR]

The multiresponse semiparametric regression (MSR) model is a regression model with more than two response variables that are mutually correlated, and its regression function is composed of parametric and nonparametric components. The study objectives are propose a new method for estimating the MSR model using smoothing spline. Also, find the confidence interval (CI) of parameters and the distribution asymptotically of the model parameters estimator. Methods used in this study are reproducing kernel Hilbert space (RKHS) method and a developed penalized weighted least squares (PWLS), and apply pivotal quantity, central limit theorem, and theorems of Cramer-Wold and Slutsky. The results are an 100(1–α)% CI estimate and an asymptotic normal distribution for the parameters of the MSR model. In conclusion, the estimated MSR model is a combined components estimate of parametric and nonparametric which is linear to observation, and CIs of parameters depend on t distribution and estimator of parameters is asymptotically normally distributed. Future time, this study results can be used as theoretical bases to design standard growth charts of the toddlers which can then be used to assess the nutritional status of the toddlers.

This study investigated the effects of various seasonal fitting techniques on the spatial distribution of the common mode errors taking the coordinate time series of the continuous GPS reference stations of the Crustal Movement Observation Network of China (CMONOC) as an example. First, the seasonal term of coordinate time series was calculated using constant amplitude harmonic fitting (CAF), continuous wavelet transform (CWT), and smoothing spline fitting (SPF). The seasonal term and linear trend were then removed to obtain the residual time series. Finally, to determine the common mode errors of residual time series, principal component analysis (PCA) was applied. The results indicate that 1) smoothing spline fitting is superior to constant amplitude harmonic fitting and continuous wavelet transform in its ability to fit short-term irregular seasonal signals. In comparison to constant amplitude harmonic fitting, N/E/U has root mean square error (RMSE) values of smoothing spline fitting that are lower by 25%, 20%, and 14.4%, respectively. Smoothing spline fitting also has a higher coefficient of determination than continuous wavelet transform and constant amplitude harmonic fitting. The coefficient of determination in the U direction is larger than that in the N and E directions. 2) Each order PC of the residual series fitted by smoothing spline fitting exhibits apparent spatial aggregation characteristics, with PC1 having a uniform spatial distribution and presenting a largely positive response. Nevertheless, the residual series obtained by constant amplitude harmonic fitting and continuous wavelet transform exhibits scattered spatial response distribution features in each order PC. Compared to N and E, U’s spatial response distribution is distinct. From north to south, the spatial response of PC1 in the U direction progressively diminishes. In addition to being much lower than that in other locations, the Sichuan–Yunnan region’s spatial response value of PC1 and PC3 also exhibits a clear negative reaction. The root mean square error value of the residual series after smoothing spline fitting is the least, and the filtering effect is the best when comparing the spatial filtering effect based on the three fitting methods. We also compared the root mean square error reduction ratio before and after spatial filtering, and the results showed that the root mean square error reduction ratio before and after the residual series obtained by smoothing spline fitting is slightly larger than that obtained by other methods.

Prediction modeling for clinical decision making is of great importance and needed to be updated frequently with the changes of patient population and clinical practice. Existing methods are either done in an ad hoc fashion, such as model recalibration or focus on studying the relationship between predictors and outcome and less so for the purpose of prediction. In this article, we propose a dynamic logistic state space model to continuously update the parameters whenever new information becomes available. The proposed model allows for both time‐varying and time‐invariant coefficients. The varying coefficients are modeled using smoothing splines to account for their smooth trends over time. The smoothing parameters are objectively chosen by maximum likelihood. The model is updated using batch data accumulated at prespecified time intervals, which allows for better approximation of the underlying binomial density function. In the simulation, we show that the new model has significantly higher prediction accuracy compared to existing methods. We apply the method to predict 1 year survival after lung transplantation using the United Network for Organ Sharing data. [ABSTRACT FROM AUTHOR]

This paper aims to restore some parameters of functionals using cubic splines to forecast rare events in finance and economics. The article considers the mathematical method for recovering an unknown function from many different functionals, such as the value of a function, the value of its first derivative, second derivative, as well as a definite integral over a certain interval. Moreover, all observations can occur with an error. Therefore, the author uses a method of recovering a function from different functionals observed with an error. The function is restored in the form of a cubic spline, which has a value-second derivative representation. The optimization problem consists in minimizing several sums of squares of the deviation at once, for ordinary values, for the first derivatives, for the second derivatives, for integrals, and for roughness penalty. For greater flexibility, weights have been introduced both for each group of observations and for each individual observation separately. The article shows in detail how the elements of each corresponding matrix are filled in. The appendix provides an implementation of the method as a FunctionalSmoothingSpline function in R language. Examples of using the method for the analysis and forecasting of rare (discrete) events in the economy are given. Formulas for calculating the cross-validation score CV (α) for the automatic procedure for determining the smoothing parameter α from the data in our problem of recovering a function by many functionals are shown. The paper concludes that the presented method makes it possible to analyze and predict rare events, which will allow you to prepare for such future events, get some benefit from this, or reduce possible risks or losses.

We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For univariate regression, we show that the solution of training a width-n shallow ReLU network is within n-1/2 of the function which fits the training data and whose difference from the initial function has the smallest 2-norm of the second derivative weighted by a curvature penalty that depends on the probability distribution that is used to initialize the network parameters. We compute the curvature penalty function explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and thence the solution function is the natural cubic spline interpolation of the training data. For stochastic gradient descent we obtain the same implicit bias result. We obtain a similar result for different activation functions. For multivariate regression we show an analogous result, whereby the second derivative is replaced by the Radon transform of a fractional Laplacian. For initialization schemes that yield a constant penalty function, the solutions are polyharmonic splines. Moreover, we show that the training trajectories are captured by trajectories of smoothing splines with decreasing regularization strength. [ABSTRACT FROM AUTHOR]

By clustering patients with the urologic chronic pelvic pain syndromes (UCPPS) into homogeneous subgroups and associating these subgroups with baseline covariates and other clinical outcomes, we provide opportunities to investigate different potential elements of pathogenesis, which may also guide us in selection of appropriate therapeutic targets. Motivated by the longitudinal urologic symptom data with extensive subject heterogeneity and differential variability of trajectories, we propose a functional clustering procedure where each subgroup is modeled by a functional mixed effects model, and the posterior probability is used to iteratively classify each subject into different subgroups. The classification takes into account both group-average trajectories and between-subject variabilities. We develop an equivalent state-space model for efficient computation. We also propose a cross-validation based Kullback–Leibler information criterion to choose the optimal number of subgroups. The performance of the proposed method is assessed through a simulation study. We apply our methods to longitudinal bi-weekly measures of a primary urological urinary symptoms score from a UCPPS longitudinal cohort study, and identify four subgroups ranging from moderate decline, mild decline, stable and mild increasing. The resulting clusters are also associated with the one-year changes in several clinically important outcomes, and are also related to several clinically relevant baseline predictors, such as sleep disturbance score, physical quality of life and painful urgency. for this article are available online. [ABSTRACT FROM AUTHOR]

Unmanned ground vehicles (UGVs) are technically complex machines to operate in difficult or dangerous environmental conditions. In recent years, there has been an increase in research on so called "following vehicles". The said concept introduces a guide—an object that sets the route the platform should follow. Afterwards, the role of the UGV is to reproduce the mentioned path. The article is based on the field test results of an outdoor localization subsystem using ultra-wideband technology. It focuses on determining the guide's route using a smoothing spline for constructing a UGV's path planning subsystem, which is one of the stages for implementing a "follow-me" system. It has been shown that the use of a smoothing spline, due to the implemented mathematical model, allows for recreating the guide's path in the event of data decay lasting up to a several seconds. The innovation of this article originates from influencing studies on the smoothing parameter of the estimation errors of the guide's location. [ABSTRACT FROM AUTHOR]

Influenza A viruses (IAV) are the only influenza viruses known to cause flu pandemics. Understanding the evolution of different sub‐types of IAV on their natural hosts is important for preventing and controlling the virus. We propose a mechanism‐based Bayesian multi‐level mixed‐effects model for characterising influenza viral dynamics, described by a set of ordinary differential equations (ODE). Both strain‐specific and subject‐specific random effects are included for the ODE parameters. Our models can characterise the common features in the population while taking into account the variations among individuals. The random effects selection is conducted at strain level through re‐parameterising the covariance parameters of the corresponding random effect distribution. Our method does not need to solve ODE directly. We demonstrate that the posterior computation can proceed via a simple and efficient Markov chain Monte Carlo algorithm. The methods are illustrated using simulated data and a real data from a study relating virus load estimates from influenza infections in ducks. [ABSTRACT FROM AUTHOR]

Pathway analysis is one way to determine whether there is a causal relationship between extrinsic and intrinsic factors. The linearity assumption is something that can change the model. The shape of the model is subject to linearity assumptions. Path analysis is parametric when the linearity assumption is true, whereas non-parametric path analysis is used when the non-linear shape is unknown and there is no knowledge of the data pattern. Non-linear path analysis is used when the non-linear shape and data pattern are unknown. This work aimed to combine the smoothing spline method and the Fourier series method to compute non-parametric path function and it is believed that they would be able to produce more flexible function estimations for data patterns since both have the benefit of being accurate or close to the real data pattern. As a result, we found that Fourier series and smoothing splines can be used in non-parametric path analysis only if the linearity assumption is violated. Non-parametric regression-based path analysis estimators were then obtained using the ordinary least squares (OLS) approach. It uses a non-parametric approach and therefore gives non-unique estimation results.

The goal of this retrospective study was to evaluate the effect of surgical timing on patient outcomes after spontaneous intracerebral hemorrhage (ICH). We also identified risk factors associated with poor prognosis. We reviewed all patients who underwent surgery for ICH between January 2014 and January 2021. The outcome was measured using the modified Rankin Scale (mRS) score at 6 months after the surgery. Patients with mRS 0–2 were considered having favorable outcomes, and those with mRS 3–5 were considered having unfavorable outcomes. The relationships of surgical timing with the risk of unfavorable outcomes were identified using the interaction and stratified analyses, and generalized additive and logistic regression models. A nomogram was established and evaluated using a receiver operating characteristic curve analysis, plotted decision curve, and calibration curve. We identified 53 patients with favorable outcomes and 144 with unfavorable outcomes. The number of cases who underwent surgery at >12 hours and <36 hours in the favorable outcome group was more than that in the unfavorable outcome group (P < 0.001). When the time to operating room (TOR) was less than 21 hours, a shorter TOR was associated with unfavorable outcomes, using the smoothing spline analysis (odds ratio = 0.8, P < 0.001). Finally, we developed a nomogram using systolic blood pressure, Glasgow Coma Scale, midline shift, hematoma volume, and TOR for predicting the unfavorable outcome. The area under the curve, accuracy, specificity, and sensitivity of nomogram were 0.90, 0.87, 0.72, and 0.93, respectively. Surgical timing between 12 and 26 hours after ICH was associated with favorable outcomes. The nomogram including systolic blood pressure, Glasgow Coma Scale, midline shift, hematoma volume, and TOR was reliable for predicting the ICH outcome. [ABSTRACT FROM AUTHOR]

In Bayesian quantile smoothing spline [Thompson, P., Cai, Y., Moyeed, R., Reeve, D., & Stander, J. (2010). Bayesian nonparametric quantile regression using splines. Computational Statistics and Data Analysis, 54, 1138–1150.], a fixed-scale parameter in the asymmetric Laplace likelihood tends to result in misleading fitted curves. To solve this problem, we propose a new Bayesian quantile smoothing spline (NBQSS), which considers a random scale parameter. To begin with, we justify its objective prior options by establishing one sufficient and one necessary condition of the posterior propriety under two classes of general priors including the invariant prior for the scale component. We then develop partially collapsed Gibbs sampling to facilitate the computation. Out of a practical concern, we extend the theoretical results to NBQSS with unobserved knots. Finally, simulation studies and two real data analyses reveal three main findings. Firstly, NBQSS usually outperforms other competing curve fitting methods. Secondly, NBQSS considering unobserved knots behaves better than the NBQSS without unobserved knots in terms of estimation accuracy and precision. Thirdly, NBQSS is robust to possible outliers and could provide accurate estimation.

The article considers a problem of best smoothing in a strip, where the objective is to find a function f:[0,1]→ℝ$$ f:\kern0.3em \left[0,1\right]\to \mathbb{R} $$ that satisfies bilateral constraints on its values, d(t)≤f(t)≤e(t)$$ d(t)\le f(t)\le e(t) $$ for all 0≤t≤1$$ 0\le t\le 1 $$ and minimizes a weighted sum of the L2$$ {L}_2 $$‐norm of the second derivative and squared deviations from specified values, yi$$ {y}_i $$, at discrete points 0=t1

In this paper, a cubic integral smoothing spline with roughness penalty for restoring a function by integrals is described. A mathematical method for building such a spline is described in detail. The method is based on cubic integral spline with a penalty function, which minimizes the sum of squares of the difference between the observed integrals of the unknown function and the integrals of the spline being constructed, plus an additional penalty for the nonlinearity (roughness) of the spline. This method has a matrix form, and this paper shows in detail how to fill in each matrix. The parameter \(\alpha \) governs the desired smoothness of the restored function. Spline knots can be chosen independently of observations, and a weight can be defined for each observation for more control over the resulting spline shape. An implementation in the R language as function int_spline is given. The function int_spline is easy to use, with all arguments completely described and corresponding examples given. An example of the application of the method in rare event analysis and forecasting is given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the estimation problem in semiparametric linear mixed models when some of the covariates are measured with errors. The authors proposed the corrected score function estimators for the parametric and non parametric components. The resulting estimators are shown to be consistent and asymptotically normal. An iterative algorithm is proposed for estimating the parameters. Asymptotic normality of the estimators is also derived. Finite sample performance of the proposed estimators is assessed by Monte Carlo simulation studies. We further illustrate the proposed procedures by an application. [ABSTRACT FROM AUTHOR]

Objective: Modeling right-censored data becomes a challenging task in survival analysis, due to having an incomplete data structure. When the response variable is rightcensored, classical estimation methods cannot be used directly. Therefore, the censorship problem should be solved before the modeling process. The purpose of this study is to solve the censorship problem with synthetic data transformation and to make a comparison between the partial linear model (PLM) and feed forward neural networks (FFNN), two popular modeling procedures in recent years, in terms of model residuals. Thus, it is to study the behavior of methods. Material and Methods: This paper aims to estimate the effects of explanatory variables on a right-censored response variable whose distribution is unknown by two different methods, PLM and FFNN based on the spline smoothing method. The spline smoothing method is a mathematical approximation method used in PLM estimation. FFNN is a machine learning method that has become very popular recently and produces satisfactory models. To overcome the censorship problem, the right-censored response variable for the two mentioned methods has been replaced with synthetic data. Synthetic data transformation is a widely used censorship resolution method that allows censorship presence to be included in the prediction process. Results: To achieve the aim of the study, both simulation and real data studies were carried out and the results were presented. Conclusion: Kidney weakness data is used as an example of real data. When the results are examined, it is seen that FFNN is superior to PLM in both numerical studies. [ABSTRACT FROM AUTHOR]

Spectra are frequently used to depict the dependence features of a second-order stationary process. In this paper, the spatial log-spectral density is expressed by a new type of smoothing splines in the form of the summation of a linear expression of univariate bases and two quadratic forms of univariate bases. Based on this new type of smoothing splines, a Bayesian nonparametric method is proposed to estimate the spectral density of spatial data observed on a lattice. The proposed Bayesian approach uses a Hamiltonian Monte Carlo-within-Gibbs technique to fit smoothing splines to the spatial periodogram. Our technique produces an automatically smoothed spatial spectral estimate along with samples from the posterior distributions of the parameters to facilitate inference. [ABSTRACT FROM AUTHOR]

In this work, a hybrid-forecasting model is proposed. The model includes a recursive penalty P-spline with parameters adaptation based on evolutionary optimization algorithms. In short-term forecasting, especially in real-time systems, the urgent task is to increase the forecast speed without compromising its quality. High forecasting speed has been achieved by an economical computational scheme of a recurrent P-spline with a shallow depth of prehistory. When combined with the adaptation of some parameters of the P-spline, such an approach allows you to control the forecast accuracy.

This paper proposes a new smoothing technique based on rational function approximation using truncated total least squares ( P − T T L S) and compares it with the widely used smoothing spline method, which has become a very powerful smoothing technique in the semiparametric regression setting. Due to the nature of rational approximation, it generates a system of linear equations with multi-collinearities and errors in all its variables. The proposed method is mainly designed to deal with these problems, especially for solving error-contaminated systems and ill-conditioned issues. To indicate the ability of the proposed method, we perform simulation experiments under different conditions and employ a real-world data application. The outcomes from the studies show that the model parameters estimated by P − T T L S have lower variances than benchmarked the smoothing spline ( B − S S) technique. [ABSTRACT FROM AUTHOR]

In this research, smoothing spline was used as a suitable nonparametric method for estimating one of the most important time series models, which is the Nonlinear Additive Autoregressive Model (NAAR) due to the importance of this model in terms of its representation of many phenomena and trying to control these phenomena through using accurate statistical models and how to estimate them in an advanced numerical method. This method is one of the smoothing techniques in finding the best fit for the curve, as the main idea is to find a smoothing function to reduce the sum of the squares of the residuals of the model related to the research topic. This type of method was used widely in nonparametric regression and gave good results in matching the curve of functions, but the expansion here was by using this method in functions that are nonparametric and non-linear in time series. The simulation method was used with three non-linear models of different formulas using four types of sample sizes for identifying the performance of the estimation method and presenting it through drawing to ensure the operation of the method. As for the real data, the monthly dollar sales data of the Central Bank of Iraq were modeled for years 2015-2017 then estimating the model curve in the proposed method, as the results indicated matching the curve of the Central Bank of Iraq's sales of dollars for the period studied. The results showed that the proposed method for matching the NAAR model curve was the best when using non-linear functions of various formulas. [ABSTRACT FROM AUTHOR]

This paper focuses on nonparametric regression modeling of time-series observations with data irregularities, such as censoring due to a cutoff value. In general, researchers do not prefer to put up with censored cases in time-series analyses because their results are generally biased. In this paper, we present an imputation algorithm for handling auto-correlated censored data based on a class of autoregressive nonparametric time-series model. The algorithm provides an estimation of the parameters by imputing the censored values with the values from a truncated normal distribution, and it enables unobservable values of the response variable. In this sense, the censored time-series observations are analyzed by nonparametric smoothing techniques instead of the usual parametric methods to reduce modelling bias. Typically, the smoothing methods are updated for estimating the censored time-series observations. We use Monte Carlo simulations based on right-censored data to compare the performances and accuracy of the estimates from the smoothing methods. Finally, the smoothing methods are illustrated using a meteorological time- series and unemployment datasets, where the observations are subject to the detection limit of the recording tool. [ABSTRACT FROM AUTHOR]

Researchers in a variety of biomedical fields have utilized frequency domain properties of heart rate variability (HRV), or the elapsed time between consecutive heart beats. HRV is measured from the electrocardiograph signal through the interbeat interval series. Popular approaches for estimating power spectra from these interval data apply common spectral analysis methods that are designed for the analysis of evenly sampled time series. The application of these methods to the interbeat interval series, which is indexed over an uneven time grid, requires a bias-inducing transformation. The goal of this article is to explore the use of penalized sum of squares for nonparametric estimation of the spectrum of HRV directly from the interbeat intervals. A novel cross-validation procedure is introduced for smoothing parameter selection. Empirical properties of the proposed estimation procedure are explored and compared with popular methods in a simulation study. The proposed method is used in an analysis of data from an insomnia study, which seeks to illuminate the association between the power spectrum of HRV during different periods of sleep with response to behavioral therapy.

In recent years, most studies in the field of technology and trade have focused on the critical importance of technological change in explaining international trade patterns. Technological innovation is not taking place in all countries and it does not spread to all countries at the same time, therefore technological distance would exist among countries.This paper has explained the effect of technological distance on Iran’s trade relations, specifying a semi-parametric gravity model over 1995-2015. As expected, the estimation results of semi-parametric gravity model show that the effect of GDP on bilateral trade relations between Iran and each of the selected trading partners is positive and significant. The population of the exporter has negative impact while the importer’s population has positive effect on bilateral trade. Our empirical results indicate that the relationship between technological distance and bilateral trade is nonparametric, implying a general effect of technology on Iran’s bilateral trade relations. More clearly, in some ranges of the technological distance a negative effect appears on trade relations, while in other fractions of the relevant distance, the impact is positive, leading to an increase in bilateral trade relations between Iran and its trading partners.

In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard L 2 space using the centered log-ratio transformation. The resulting functions satisfy the zero integral constraint. This is a key to propose a new spline basis, holding the same property, and consequently to build a new class of spline functions, called compositional splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing compositional splines and possible orthonormalization of the spline basis which might be useful in some applications. Finally, statistical processing of densities using the new approximation tool is demonstrated in case of simplicial functional principal component analysis with anthropometric data. [ABSTRACT FROM AUTHOR]

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Generalised additive models (GAMs) provide flexible models for a wide array of data sources. In the past, improvements of GAM estimation have focused on the smoothers used in the local scoring algorithm used for estimation, but poor prediction for non-Gaussian data motivates the need for robust estimation of GAMs. In this paper, rank-based estimation, as a robust and efficient alternative to the likelihood-based estimation of GAMs, is proposed. It is shown that rank GAM estimators can be obtained through iteratively reweighted likelihood-based GAM estimation which we call the iterated regularised rank quasi-likelihood (IRRQL). Simulation experiments support the use of rank-based GAM estimation for heavy-tailed or contaminated sources of data. [ABSTRACT FROM AUTHOR]

Some aspects of the brain change volume at a fairly constant rate across the life span, i.e., linear maturation rate with respect to age, whereas other aspects of the brain change volume at different rates for younger vs. older age ranges, i.e., non-linear changes. Various forms of non-linear maturation likely reflect different biological mechanisms such that theoretical distinctions between maturation patterns ought to be considered. Simulated data with known maturation patterns and a single critical age characterizing a qualitative change in maturation were used to establish the validity of a non-parametric fitting method, the smoothing spline, combined with processing steps for determining the form of the pattern and the associated critical age. Multiple classes of models were assessed, including model-free, bottom up approaches. Three categories of maturation patterns were explored: U-shaped, with a change in direction; sigmoidal, with an isolated period of change; accelerating, with changes in amplitude but not direction. As noise is a limiting factor in curve fitting, smoothing splines were fit to data with idealized low and realistic noise levels. The smoothing spline was shown to contain the relevant information to extract the critical ages of all maturation patterns in the form of derivative zero points, but the previously proposed method of using third derivative zero points worked only for the accelerating category. Therefore, an additional classification step was included to first determine the category of maturation pattern. Classification accuracy and identification of the calculated critical age within 5 years of the actual critical age was found to be perfect for low noise and high for realistic noise levels. To demonstrate the applicability of the method, a reevaluation of published biological data previously analyzed using third derivative zero points to determine critical ages was carried out for 17 aspects of MRI scans from 1,100 subjects. For a majority of non-linear areas, new critical ages were identified. Further modifications to the analysis procedure could include a wider set of maturation patterns and the inclusion of multiple critical ages to help determine distinctions between brain areas in the timing of developmental or degenerative events that influence their volume.

In this paper the possibility of development of the recursive digital filter using a P-spline is considered. The frequency and time response of the spline filter for real-time data are analytically obtained and investigated. The influence of the P-spline parameters on effectiveness of interpretation is explored with input metrical data. The patterns obtained during spline filter frequency analysis are confirmed by an example of Doppler function restoration.

Many brain structures show a complex, non-linear pattern of maturation and age-related change. Often, quadratic models (beta(0) + beta(1)age + beta(2)age(2) + epsilon) are used to describe such relationships. Here, we demonstrate that the fitting of quadratic models is substantially affected by seemingly irrelevant factors, such as the age-range sampled. Hippocampal volume was measured in 434 healthy participants between 8 and 85 years of age, and quadratic models were fit to subsets of the sample with different age-ranges. It was found that as the bottom of the age-range increased, the age at which volumes appeared to peak was moved upwards and the estimated decline in the last part of the age-span became larger. Thus, whether children were included or not affected the estimated decline between 60 and 85 years. We conclude that caution should be exerted in inferring age-trajectories from global fit models, e.g. the quadratic model. A nonparametric local smoothing technique (the smoothing spline) was found to be more robust to the effects of different starting ages. The results were replicated in an independent sample of 309 participants.

Semiparametric mixed-effects models (SMM) have received increasing attention in recent years because of the greater flexibility in analysing longitudinal trajectories. However, the normality assumption of SMM may be unrealistic when outliers occur in the data. This paper presents a semiparametric extension of the multivariate t linear mixed-effects model (MtLMM), called the multivariate t semiparametric mixed model (MtSMM). To be specific, the MtSMM incorporates a parametric linear function related to the fixed covariate effects and random effects which have a joint multivariate t distribution together with an arbitrary nonparametric smooth function to capture the unexpected patterns. A computationally analytical EM-based algorithm is developed for carrying out maximum likelihood estimation of the MtSMM. Simulation studies and a real example concerning the analysis of PBCseq data are used to investigate the empirical behaviour of the proposed methodology. [ABSTRACT FROM AUTHOR]