scholarly journals Smoothness and conditioning in generalised smoothing spline calculations

1988 ◽  
Vol 30 (1) ◽  
pp. 43-56 ◽  
Author(s):  
M. R. Osborne ◽  
Tania Prvan
Keyword(s):  
Conditional Expectation ◽  
Numerical Algorithms ◽  
Smoothing Splines ◽  
Smoothing Spline ◽  
Stochastic Formulation ◽  
Data Points ◽  
Usual Order

AbstractWe consider a generalisation of the stochastic formulation of smoothing splines, and discuss the smoothness properties of the resulting conditional expectation (generalised smoothing spline), and the sensitivity of the numerical algorithms. One application is to the calculation of smoothing splines with less than the usual order of continuity at the data points.

A smoothing spline-based regularization of the initial inverse problem in a two-dimensional heat equation

2008 ◽  
Vol 223 (2) ◽  
pp. 439-449 ◽  
Author(s):  
K Masood ◽  
M T Mustafa
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Smoothing Splines ◽  
Smoothing Spline ◽  
Hyperbolic Heat Conduction ◽  
Parabolic Model ◽  
Initial Profile ◽  
Conduction Model ◽  
Heat Conduction Model

A smoothing spline-based method and a hyperbolic heat conduction model is applied to regularize the recovery of the initial profile from a parabolic heat conduction model in two-dimensions. An ill-posed inverse problem involving recovery of the initial temperature distribution from measurements of the final temperature distribution is investigated. A hyperbolic heat conduction model is considered instead of a parabolic model and smoothing splines are applied to regularize the recovered initial profile. The comparison of the proposed procedure and parabolic model is presented graphically by examples.

scholarly journals Forecasting the Market Equity Premium: Does Nonlinearity Matter?

2021 ◽  
Vol 13 (5) ◽  
pp. 9
Author(s):  
Anwen Yin
Keyword(s):  
Computational Cost ◽  
Equity Premium ◽  
Smoothing Splines ◽  
Smoothing Spline ◽  
Nonlinear Relationship ◽  
Shrinkage Estimators ◽  
Forecast Combination ◽  
Nonlinear Method ◽  
Nonlinear Methods ◽  
Predictive Variables

We propose using the nonlinear method of smoothing splines in conjunction with forecast combination to predict the market equity premium. The smooth splines are flexible enough to capture the possible nonlinear relationship between the equity premium and predictive variables while controlling for complexity, overcoming the difficulties often attached to nonlinear methods such as computational cost, overfitting and interpretation. Our empirical results show that when used with forecast combination, the smoothing spline forecasts outperform many competing methods such as the adaptive combinations, shrinkage estimators and technical indicators, in delivering statistical and economic gains consistently.

scholarly journals Robust Identification of Temporal Biomarkers in Longitudinal Omics Studies

2021 ◽  
Author(s):  
Ahmed A. Metwally ◽  
Tom Zhang ◽  
Si Wu ◽  
Ryan Kellogg ◽  
Wenyu Zhou ◽  
...  
Keyword(s):  
Empirical Distribution ◽  
Temporal Patterns ◽  
R Package ◽  
Smoothing Splines ◽  
Omics Data ◽  
Differential Analysis ◽  
Time Intervals ◽  
Robust Identification ◽  
Data Points ◽  
Subject Dropout

Longitudinal studies increasingly collect rich 'omics' data sampled frequently over time and across large cohorts to capture dynamic health fluctuations and disease transitions. However, the generation of longitudinal omics data has preceded the development of analysis tools that can efficiently extract insights from such data. In particular, there is a need for statistical frameworks that can identify not only which omics features are differentially regulated between groups but also over what time intervals. Additionally, longitudinal omics data may have inconsistencies, including nonuniform sampling intervals, missing data points, subject dropout, and differing numbers of samples per subject. In this work, we developed a statistical method that provides robust identification of time intervals of temporal omics biomarkers. The proposed method is based on a semi-parametric approach, in which we use smoothing splines to model longitudinal data and infer significant time intervals of omics features based on an empirical distribution constructed through a permutation procedure. We benchmarked the proposed method on five simulated datasets with diverse temporal patterns, and the method showed specificity greater than 0.99 and sensitivity greater than 0.72. Applying the proposed method to the Integrative Personal Omics Profiling (iPOP) cohort revealed temporal patterns of amino acids, lipids, and hormone metabolites that are differentially regulated in male versus female subjects following a respiratory infection. In addition, we applied the longitudinal multi-omics dataset of pregnant women with and without preeclampsia, and the method identified potential lipid markers that are temporally significantly different between the two groups. We provide an open-source R package, OmicsLonDA (Omics Longitudinal Differential Analysis): https://bioconductor.org/packages/OmicsLonDA to enable widespread use.

Partially informative normal and partial spline models

2015 ◽  
Author(s):  
◽  
Sifan Liu
Keyword(s):  
Yield Curve ◽  
Smoothing Splines ◽  
Smoothing Spline ◽  
Linear Component ◽  
Function Estimation ◽  
Regression Problem ◽  
Adaptive Smoothing ◽  
University Of Missouri ◽  
Spline Models ◽  
Large P Small N

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] There is a well-known Bayesian interpretation of function estimation by spline smoothing using a limit of proper normal priors. This limiting prior has the same form with Partially Informative Normal (PIN), which was introduced in Sun et al. (1999). In this dissertation, we first discuss some properties of PIN. In terms of improper priors, we consider q-vague convergence as the convergence mode. Then, we apply the properties to several extensions of smoothing spline problems. Partial spline model, which contains a non-parametric part as regular smoothing spline together with a linear parametric part, is discussed. We perform simulation studies and applications on yield curves. Specifically, Nelson-Siege (NS) model is considered to construct the linear component. NS partial spline model is used for fitting single yield curve, while partial parallel and non-parallel spline models are used for multiple curves. Then, large p, small n regression problem associated with the generalized univariate smoothing spline, some studies on bin smoothing splines, adaptive smoothing splines and correlated smoothing splines are discussed.

On the equivalence of two stochastic approaches to spline smoothing

1986 ◽  
Vol 23 (A) ◽  
pp. 391-405 ◽  
Author(s):  
Craig F. Ansley ◽  
Robert Kohn
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Minimization Problem ◽  
Stochastic Models ◽  
Conditional Expectation ◽  
Smoothing Spline ◽  
Computational Algorithms ◽  
Spline Smoothing ◽  
Optimal Smoothing ◽  
Space Formulation

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

scholarly journals More efficient approximation of smoothing splines via space-filling basis selection

Biometrika ◽  
2020 ◽  
Vol 107 (3) ◽  
pp. 723-735
Author(s):  
Cheng Meng ◽  
Xinlian Zhang ◽  
Jingyi Zhang ◽  
Wenxuan Zhong ◽  
Ping Ma
Keyword(s):  
Computational Cost ◽  
Smoothing Splines ◽  
Basis Functions ◽  
Computational Time ◽  
Smoothing Spline ◽  
Selection Methods ◽  
Nonparametric Regression Model ◽  
Full Sample ◽  
Real World Datasets ◽  
Efficient Approximation

Summary We consider the problem of approximating smoothing spline estimators in a nonparametric regression model. When applied to a sample of size $n$, the smoothing spline estimator can be expressed as a linear combination of $n$ basis functions, requiring $O(n^3)$ computational time when the number $d$ of predictors is two or more. Such a sizeable computational cost hinders the broad applicability of smoothing splines. In practice, the full-sample smoothing spline estimator can be approximated by an estimator based on $q$ randomly selected basis functions, resulting in a computational cost of $O(nq^2)$. It is known that these two estimators converge at the same rate when $q$ is of order $O\{n^{2/(pr+1)}\}$, where $p\in [1,2]$ depends on the true function and $r > 1$ depends on the type of spline. Such a $q$ is called the essential number of basis functions. In this article, we develop a more efficient basis selection method. By selecting basis functions corresponding to approximately equally spaced observations, the proposed method chooses a set of basis functions with great diversity. The asymptotic analysis shows that the proposed smoothing spline estimator can decrease $q$ to around $O\{n^{1/(pr+1)}\}$ when $d\leq pr+1$. Applications to synthetic and real-world datasets show that the proposed method leads to a smaller prediction error than other basis selection methods.

A Bayesian smoothing spline method for mortality modelling

2012 ◽  
Vol 6 (2) ◽  
pp. 284-306 ◽  
Author(s):  
Arto Luoma ◽  
Anne Puustelli ◽  
Lasse Koskinen
Keyword(s):  
Smoothing Splines ◽  
Smoothing Spline ◽  
Spline Method ◽  
Initial Model ◽  
Data Set ◽  
Death Rates ◽  
Final Model ◽  
Mortality Modelling ◽  
Improved Model ◽  
Bayesian Smoothing

AbstractWe propose a new method for two-dimensional mortality modelling. Our approach smoothes the data set in the dimensions of cohort and age using Bayesian smoothing splines. The method allows the data set to be imbalanced, since more recent cohorts have fewer observations. We suggest an initial model for observed death rates, and an improved model which deals with the numbers of deaths directly. Unobserved death rates are estimated by smoothing the data with a suitable prior distribution. To assess the fit and plausibility of our models we perform model checks by introducing appropriate test quantities. We show that our final model fulfils nearly all requirements set for a good mortality model.

scholarly journals Analyzing the effects of stand thinning on microclimates with semiparametric smoothing splines

2006 ◽  
Vol 36 (7) ◽  
pp. 1641-1648 ◽  
Author(s):  
Biing T Guan ◽  
Shih-Hao Weng ◽  
Shing-Rong Kuo ◽  
Tsung-Yi Chang ◽  
Hsin-Wu Hsu ◽  
...  
Keyword(s):  
Temporal Trends ◽  
Smoothing Splines ◽  
Semiparametric Model ◽  
Monitoring Data ◽  
Smoothing Spline ◽  
Temporal Response ◽  
Ecological Processes ◽  
Stand Thinning ◽  
Microclimate Monitoring ◽  
Better Than

Monitoring the effects of stand thinning on microclimates is an integral part of any thinning experiment. It is through its modifications of microclimates that thinning alters important ecological processes. An efficient analysis of microclimate-monitoring data should address both the effects of thinning regimes on, and the temporal response trends of, microclimates. Probably because of the difficulties in modeling temporal trends parametrically, an examination of the existing literature on thinning showed that only a few studies have attempted to address the second aspect. We propose the use of semiparametric smoothing splines to analyze monitoring data from thinning experiments. First, the concept of a smoothing spline is briefly described. We then provide an example in which semiparametric mixed-effects smoothing-spline models were used to analyze microclimate-monitoring data from a thinning experiment. The proposed approach not only successfully detected the effects of thinning, but also revealed interesting temporal trends. For each of the microclimatic variables, we also compared the performance of the fitted semiparametric model with that of a parametric model. In general, the semiparametric model performed better than its parametric counterpart. We also addresse some concerns in using the proposed approach.

On the equivalence of two stochastic approaches to spline smoothing

1986 ◽  
Vol 23 (A) ◽  
pp. 391-405 ◽  
Author(s):  
Craig F. Ansley ◽  
Robert Kohn
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Minimization Problem ◽  
Stochastic Models ◽  
Conditional Expectation ◽  
Smoothing Spline ◽  
Computational Algorithms ◽  
Spline Smoothing ◽  
Optimal Smoothing ◽  
Space Formulation

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

scholarly journals Analysis of Heart Transplant Survival Data Using Generalized Additive Models

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Masaaki Tsujitani ◽  
Yusuke Tanaka
Keyword(s):  
Heart Transplant ◽  
Survival Data ◽  
Proportional Hazards ◽  
Generalized Additive Models ◽  
Survival Function ◽  
Additive Model ◽  
Smoothing Splines ◽  
Additive Models ◽  
Smoothing Spline ◽  
Piecewise Exponential

The Stanford Heart Transplant data were collected to model survival in patients using penalized smoothing splines for covariates whose values change over the course of the study. The basic idea of the present study is to use a logistic regression model and a generalized additive model withB-splines to estimate the survival function. We model survival time as a function of patient covariates and transplant status and compare the results obtained using smoothing spline, partial logistic, Cox's proportional hazards, and piecewise exponential models.

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