Model diagnostics for smoothing spline ANOVA models

The author proposes some simple diagnostics for assessing the necessity of selected terms in smoothing spline ANOVA models. The elimination of practically insignificant terms generally enhances the interpretability of the estimates and sometimes may also have inferential implications. The diagnostics are derived from Kullback‐Leibler geometry and are illustrated in the settings of regression, probability density estimation, and hazard rate estimation.     

ADDITIVE MODELS WITH PREDICTORS SUBJECT TO MEASUREMENT ERROR

This paper develops a likelihood‐based method for fitting additive models in the presence of measurement error. It formulates the additive model using the linear mixed model representation of penalized splines. In the presence of a structural measurement error model, the resulting likelihood involves intractable integrals, and a Monte Carlo expectation maximization strategy is developed for obtaining estimates. The method's performance is illustrated with a simulation study.     

Improving logistic regression on the imbalanced data by a novel penalized log-likelihood function

Logistic regression is estimated by maximizing the log-likelihood objective function formulated under the assumption of maximizing the overall accuracy. That does not apply to the imbalanced data. The resulting models tend to be biased towards the majority class (i.e. non-event), which can bring great loss in practice. One strategy for mitigating such bias is to penalize the misclassification costs of observations differently in the log-likelihood function. Existing solutions require either hard hyperparameter estimating or high computational complexity. We propose a novel penalized log-likelihood function by including penalty weights as decision variables for observations in the minority class (i.e. event) and learning them from data along with model coefficients. In the experiments, the proposed logistic regression model is compared with the existing ones on the statistics of area under receiver operating characteristics (ROC) curve from 10 public datasets and 16 simulated datasets, as well as the training time. A detailed analysis is conducted on an imbalanced credit dataset to examine the estimated probability distributions, additional performance measurements (i.e. type I error and type II error) and model coefficients. The results demonstrate that both the discrimination ability and computation efficiency of logistic regression models are improved using the proposed log-likelihood function as the learning objective.     

Bayesian penalized B-spline estimation approach for epidemic models

Ordinary differential equations (ODEs) are normally used to model dynamic processes in applied sciences such as biology, engineering, physics, and many other areas. In these models, the parameters are usually unknown, and thus they are often specified artificially or empirically. Alternatively, a feasible method is to estimate the parameters based on observed data. In this study, we propose a Bayesian penalized B-spline approach to estimate the parameters and initial values for ODEs used in epidemiology. We evaluated the efficiency of the proposed method based on simulations using the Markov chain Monte Carlo algorithm for the Kermack–McKendrick model. The proposed approach is also illustrated based on a real application to the transmission dynamics of hepatitis C virus in mainland China.     

Identifying Latent Structures in Panel Data

This paper provides a novel mechanism for identifying and estimating latent group structures in panel data using penalized techniques. We consider both linear and nonlinear models where the regression coefficients are heterogeneous across groups but homogeneous within a group and the group membership is unknown. Two approaches are considered—penalized profile likelihood (PPL) estimation for the general nonlinear models without endogenous regressors, and penalized GMM (PGMM) estimation for linear models with endogeneity. In both cases, we develop a new variant of Lasso called classifier‐Lasso (C‐Lasso) that serves to shrink individual coefficients to the unknown group‐specific coefficients. C‐Lasso achieves simultaneous classification and consistent estimation in a single step and the classification exhibits the desirable property of uniform consistency. For PPL estimation, C‐Lasso also achieves the oracle property so that group‐specific parameter estimators are asymptotically equivalent to infeasible estimators that use individual group identity information. For PGMM estimation, the oracle property of C‐Lasso is preserved in some special cases. Simulations demonstrate good finite‐sample performance of the approach in both classification and estimation. Empirical applications to both linear and nonlinear models are presented.     

Outlier detection and robust variable selection via the penalized weighted LAD-LASSO method

This paper studies the outlier detection and robust variable selection problem in the linear regression model. The penalized weighted least absolute deviation (PWLAD) regression estimation method and the adaptive least absolute shrinkage and selection operator (LASSO) are combined to simultaneously achieve outlier detection, and robust variable selection. An iterative algorithm is proposed to solve the proposed optimization problem. Monte Carlo studies are evaluated the finite-sample performance of the proposed methods. The results indicate that the finite sample performance of the proposed methods performs better than that of the existing methods when there are leverage points or outliers in the response variable or explanatory variables. Finally, we apply the proposed methodology to analyze two real datasets.     

SEMIPARAMETRIC REGRESSION AND GRAPHICAL MODELS

Semiparametric regression models that use spline basis functions with penalization have graphical model representations. This link is more powerful than previously established mixed model representations of semiparametric regression, as a larger class of models can be accommodated. Complications such as missingness and measurement error are more naturally handled within the graphical model architecture. Directed acyclic graphs, also known as Bayesian networks, play a prominent role. Graphical model-based Bayesian 'inference engines', such as bugs and vibes , facilitate fitting and inference. Underlying these are Markov chain Monte Carlo schemes and recent developments in variational approximation theory and methodology.     

Tolerance limits under Poisson regression based on small-sample asymptotic methodology

This article explores the calculation of tolerance limits for the Poisson regression model based on the profile likelihood methodology and small-sample asymptotic corrections to improve the coverage probability performance. The data consist of n counts, where the mean or expected rate depends upon covariates via the log regression function. This article evaluated upper tolerance limits as a function of covariates. The upper tolerance limits are obtained from upper confidence limits of the mean. To compute upper confidence limits the following methodologies were considered: likelihood based asymptotic methods, small-sample asymptotic methods to improve the likelihood based methodology, and the delta method. Two applications are discussed: one application relating to defects in semiconductor wafers due to plasma etching and the other examining the number of surface faults in upper seams of coal mines. All three methodologies are illustrated for the two applications.     

Trend estimation of multivariate time series with controlled smoothness

This paper extends the univariate time series smoothing approach provided by penalized least squares to a multivariate setting, thus allowing for joint estimation of several time series trends. The theoretical results are valid for the general multivariate case, but particular emphasis is placed on the bivariate situation from an applied point of view. The proposal is based on a vector signal-plus-noise representation of the observed data that requires the first two sample moments and specifying only one smoothing constant. A measure of the amount of smoothness of an estimated trend is introduced so that an analyst can set in advance a desired percentage of smoothness to be achieved by the trend estimate. The required smoothing constant is determined by the chosen percentage of smoothness. Closed form expressions for the smoothed estimated vector and its variance-covariance matrix are derived from a straightforward application of generalized least squares, thus providing best linear unbiased estimates for the trends. A detailed algorithm applicable for estimating bivariate time series trends is also presented and justified. The theoretical results are supported by a simulation study and two real applications. One corresponds to Mexican and US macroeconomic data within the context of business cycle analysis, and the other one to environmental data pertaining to a monitored site in Scotland.