Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Bayesian variable selection with related predictors

In data sets with many predictors, algorithms for identifying a good subset of predictors are often used. Most such algorithms do not allow for any relationships between predictors. For example, stepwise regression might select a model containing an interaction AB but neither main effect A or B. This paper develops mathematical representations of this and other relations between predictors, which may then be incorporated in a model selection procedure. A Bayesian approach that goes beyond the standard independence prior for variable selection is adopted, and preference for certain models is interpreted as prior information. Priors relevant to arbitrary interactions and polynomials, dummy variables for categorical factors, competing predictors, and restrictions on the size of the models are developed. Since the relations developed are for priors, they may be incorporated in any Bayesian variable selection algorithm for any type of linear model. The application of the methods here is illustrated via the stochastic search variable selection algorithm of George and McCulloch (1993), which is modified to utilize the new priors. The performance of the approach is illustrated with two constructed examples and a computer performance dataset.     

Structured regularization for conditional Gaussian graphical models

Conditional Gaussian graphical models are a reparametrization of the multivariate linear regression model which explicitly exhibits (i) the partial covariances between the predictors and the responses, and (ii) the partial covariances between the responses themselves. Such models are particularly suitable for interpretability since partial covariances describe direct relationships between variables. In this framework, we propose a regularization scheme to enhance the learning strategy of the model by driving the selection of the relevant input features by prior structural information. It comes with an efficient alternating optimization procedure which is guaranteed to converge to the global minimum. On top of showing competitive performance on artificial and real datasets, our method demonstrates capabilities for fine interpretation, as illustrated on three high-dimensional datasets from spectroscopy, genetics, and genomics.     

Untangle the structural and random zeros in statistical modelings

Count data with structural zeros are common in public health applications. There are considerable researches focusing on zero-inflated models such as zero-inflated Poisson (ZIP) and zero-inflated Negative Binomial (ZINB) models for such zero-inflated count data when used as response variable. However, when such variables are used as predictors, the difference between structural and random zeros is often ignored and may result in biased estimates. One remedy is to include an indicator of the structural zero in the model as a predictor if observed. However, structural zeros are often not observed in practice, in which case no statistical method is available to address the bias issue. This paper is aimed to fill this methodological gap by developing parametric methods to model zero-inflated count data when used as predictors based on the maximum likelihood approach. The response variable can be any type of data including continuous, binary, count or even zero-inflated count responses. Simulation studies are performed to assess the numerical performance of this new approach when sample size is small to moderate. A real data example is also used to demonstrate the application of this method.     

Multivariate Latent Growth Models for Mixed Data with Covariate Effects

The paper presents an extension of a new class of multivariate latent growth models (Bianconcini and Cagnone, 2012) to allow for covariate effects on manifest, latent variables and random effects. The new class of models combines: (i) multivariate latent curves that describe the temporal behavior of the responses, and (ii) a factor model that specifies the relationship between manifest and latent variables. Based on the Generalized Linear and Latent Variable Model framework (Bartholomew and Knott, 1999), the response variables are assumed to follow different distributions of the exponential family, with item-specific linear predictors depending on both latent variables and measurement errors. A full maximum likelihood method is used to estimate all the model parameters simultaneously. Data coming from the Data WareHouse of the University of Bologna are used to illustrate the methodology.     

Generalized linear models with functional predictors

Summary. We present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function. The technique is particularly useful when only fragments of each curve have been observed. We demonstrate, on both simulated and real data sets, how this approach can be used to perform linear, logistic and censored regression with functional predictors. In addition, we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors. Finally, we extend the methodology to apply generalized linear models and principal components to standard missing data problems.     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

Structural vector autoregressive analysis for cointegrated variables

Summary Vector autoregressive (VAR) models are capable of capturing the dynamic structure of many time series variables. Impulse response functions are typically used to investigate the relationships between the variables included in such models. In this context the relevant impulses or innovations or shocks to be traced out in an impulse response analysis have to be specified by imposing appropriate identifying restrictions. Taking into account the cointegration structure of the variables offers interesting possibilities for imposing identifying restrictions. Therefore VAR models which explicitly take into account the cointegration structure of the variables, so-called vector error correction models, are considered. Specification, estimation and validation of reduced form vector error correction models is briefly outlined and imposing structural short- and long-run restrictions within these models is discussed. I thank an anonymous reader for comments on an earlier draft of this paper that helped me to improve the exposition.     

Difference variance dispersion graphs for comparing response surface designs with applications in food technology

Variance dispersion graphs have become a popular tool in aiding the choice of a response surface design. Often differences in response from some particular point, such as the expected position of the optimum or standard operating conditions, are more important than the response itself. We describe two examples from food technology. In the first, an experiment was conducted to find the levels of three factors which optimized the yield of valuable products enzymatically synthesized from sugars and to discover how the yield changed as the levels of the factors were changed from the optimum. In the second example, an experiment was conducted on a mixing process for pastry dough to discover how three factors affected a number of properties of the pastry, with a view to using these factors to control the process. We introduce the difference variance dispersion graph (DVDG) to help in the choice of a design in these circumstances. The DVDG for blocked designs is developed and the examples are used to show how the DVDG can be used in practice. In both examples a design was chosen by using the DVDG, as well as other properties, and the experiments were conducted and produced results that were useful to the experimenters. In both cases the conclusions were drawn partly by comparing responses at different points on the response surface.     

Penalised spline estimation for generalised partially linear single-index models

Generalised linear models are frequently used in modeling the relationship of the response variable from the general exponential family with a set of predictor variables, where a linear combination of predictors is linked to the mean of the response variable. We propose a penalised spline (P-spline) estimation for generalised partially linear single-index models, which extend the generalised linear models to include nonlinear effect for some predictors. The proposed models can allow flexible dependence on some predictors while overcome the “curse of dimensionality”. We investigate the P-spline profile likelihood estimation using the readily available R package mgcv, leading to straightforward computation. Simulation studies are considered under various link functions. In addition, we examine different choices of smoothing parameters. Simulation results and real data applications show effectiveness of the proposed approach. Finally, some large sample properties are established.     

Algorithmic construction of nearly column-orthogonal designs

Computer models can describe complicated phenomena encountered in science and engineering fields. To use these models for scientific investigation, however, their generally long running time and mostly deterministic nature require a specially designed experiment. The column-orthogonal design (COD) is a popular choice for computer experiments. Because of the restriction on the orthogonality, however, only little CODs can be constructed. In this article, we propose two algorithms for constructing nearly CODs by rotating orthogonal arrays under two different criteria. Further, some obtained nearly CODs are nearly Latin hypercube designs. Some examples are provided to show the advantages of our algorithms. Some rotation matrices obtained via the algorithms are listed.     

Sparsistent and constansistent estimation of the varying-coefficient model with a diverging number of predictors

ABSTRACT

The varying-coefficient model is a flexible class of approaches that extends simple linear relationships between covariates and responses. Two related problems concerning these models are selecting relevant variables and determining non-varying coefficients among those relevant ones. In this paper we study the sparsistency and constansistency of the regularized estimation approach when the number of predictors diverges with the sample size. Here, constansistency refers to the desired property that the non-zero, non-varying coefficients are identified with probability tending to one.     

Bivariate symbolic regression models for interval-valued variables

Interval-valued variables have become very common in data analysis. Up until now, symbolic regression mostly approaches this type of data from an optimization point of view, considering neither the probabilistic aspects of the models nor the nonlinear relationships between the interval response and the interval predictors. In this article, we formulate interval-valued variables as bivariate random vectors and introduce the bivariate symbolic regression model based on the generalized linear models theory which provides much-needed exibility in practice. Important inferential aspects are investigated. Applications to synthetic and real data illustrate the usefulness of the proposed approach.     

A Discrete Variable Chain Graph for Applicants for Credit

A bank offering unsecured personal loans may be interested in several related outcome variables, including defaulting on the repayments, early repayment or failing to take up an offered loan. Current predictive models used by banks typically consider such variables individually. However, the fact that they are related to each other, and to many interrelated potential predictor variables, suggests that graphical models may provide an attractive alternative solution. We developed such a model for a data set of 15 variables measured on a set of 14 000 applications for unsecured personal loans. The resulting global model of behaviour enabled us to identify several previously unsuspected relationships of considerable interest to the bank. For example, we discovered important but obscure relationships between taking out insurance, prior delinquency with a credit card and delinquency with the loan.     

Boosting techniques for nonlinear time series models

Many of the popular nonlinear time series models require a priori the choice of parametric functions which are assumed to be appropriate in specific applications. This approach is mainly used in financial applications, when sufficient knowledge is available about the nonlinear structure between the covariates and the response. One principal strategy to investigate a broader class on nonlinear time series is the Nonlinear Additive AutoRegressive (NAAR) model. The NAAR model estimates the lags of a time series as flexible functions in order to detect non-monotone relationships between current and past observations. We consider linear and additive models for identifying nonlinear relationships. A componentwise boosting algorithm is applied for simultaneous model fitting, variable selection, and model choice. Thus, with the application of boosting for fitting potentially nonlinear models we address the major issues in time series modelling: lag selection and nonlinearity. By means of simulation we compare boosting to alternative nonparametric methods. Boosting shows a strong overall performance in terms of precise estimations of highly nonlinear lag functions. The forecasting potential of boosting is examined on the German industrial production (IP); to improve the model’s forecasting quality we include additional exogenous variables. Thus we address the second major aspect in this paper which concerns the issue of high dimensionality in models. Allowing additional inputs in the model extends the NAAR model to a broader class of models, namely the NAARX model. We show that boosting can cope with large models which have many covariates compared to the number of observations.     

Bootstrap estimation of the variance of the error term in monotonic regression models

The variance of the error term in ordinary regression models and linear smoothers is usually estimated by adjusting the average squared residual for the trace of the smoothing matrix (the degrees of freedom of the predicted response). However, other types of variance estimators are needed when using monotonic regression (MR) models, which are particularly suitable for estimating response functions with pronounced thresholds. Here, we propose a simple bootstrap estimator to compensate for the over-fitting that occurs when MR models are estimated from empirical data. Furthermore, we show that, in the case of one or two predictors, the performance of this estimator can be enhanced by introducing adjustment factors that take into account the slope of the response function and characteristics of the distribution of the explanatory variables. Extensive simulations show that our estimators perform satisfactorily for a great variety of monotonic functions and error distributions.     

TREE-BASED REGRESSION FOR A CIRCULAR RESPONSE

ABSTRACT

The application of conventional statistical methods to directional data generally produces erroneous results. Various regression models for a circular response have been presented in the literature, however these are unsatisfactory either in the limited relationships that can be modeled, or the limitations on the number or type of covariates admissible. One difficulty with circular regression is devising a meaningful regression function. This problem is exacerbated when trying to incorporate both linear and circular variables as covariates. Due to these complexities, circular regression is ripe for exploration via tree-based methods, in which a formal regression function is not needed, but where insight into the general structure and relationship between predictors and the response may be obtained. A basic framework for regression trees, predicting a circular response from a combination of circular and linear predictors, will be presented.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

A spatial model for the needle losses of pine-trees in the forests of Baden-Württemberg: an application of Bayesian structured additive regression

Summary.  The data that are analysed are from a monitoring survey which was carried out in 1994 in the forests of Baden-Württemberg, a federal state in the south-western region of Germany. The survey is part of a large monitoring scheme that has been carried out since the 1980s at different spatial and temporal resolutions to observe the increase in forest damage. One indicator for tree vitality is tree defoliation, which is mainly caused by intrinsic factors, age and stand conditions, but also by biotic (e.g. insects) and abiotic stresses (e.g. industrial emissions). In the survey, needle loss of pine-trees and many potential covariates are recorded at about 580 grid points of a 4 km × 4 km grid. The aim is to identify a set of predictors for needle loss and to investigate the relationships between the needle loss and the predictors. The response variable needle loss is recorded as a percentage in 5% steps estimated by eye using binoculars and categorized into healthy trees (10% or less), intermediate trees (10–25%) and damaged trees (25% or more). We use a Bayesian cumulative threshold model with non-linear functions of continuous variables and a random effect for spatial heterogeneity. For both the non-linear functions and the spatial random effect we use Bayesian versions of P -splines as priors. Our method is novel in that it deals with several non-standard data requirements: the ordinal response variable (the categorized version of needle loss), non-linear effects of covariates, spatial heterogeneity and prediction with missing covariates. The model is a special case of models with a geoadditive or more generally structured additive predictor. Inference can be based on Markov chain Monte Carlo techniques or mixed model technology.