M-type smoothing spline ANOVA for correlated data

Abstract

a b s t r a c tThis paper concerns outlier robust non-parametric regression with smoothing splines for data that are possibly correlated. We define a robust smoother as the minimizer of a penalized robustified log likelihood. Our estimation algorithm uses iteratively reweighted least squares to estimate the regression function. We develop two types of robust methods for joint estimation of the smoothing parameters and the correlation parameters: indirect methods and direct methods, terms borrowed from the related generalized smoothing spline literature. The indirect methods choose those parameters by conveniently approximating the distribution of the working data at each iteration as Gaussian. The direct methods estimate those parameters to minimize an estimate of the loss between the truth and the final estimated regression. Indirect methods are computationally more efficient, but our empirical studies suggest that direct methods result in more accurate estimates. Finally, the methods are applied to a data set from a macaque Simian-Human Immunodeficiency Virus (SHIV) challenge study.

Estimation of Multiresponse Multipredictor Nonparametric Regression Model Using Mixed Estimator

“…These splines have been used and developed widely in several cases by many researchers. For example, Liu et al [14] and Gao and Shi [15] used M-type splines for analyzing the variance in correlated data, and for estimating regression functions of nonparametric and semiparametric regression models, respectively; Chamidah et al [16] used truncated splines to estimate mean arterial pressure for prediction purposes, Chamidah et al [17] and Lestari et al [18] developed truncated spline and smoothing spline estimators, respectively, for estimating semiparametric regression models and determining the asymptotic properties of the estimator; Tirosh et al [19], Irizarry [20], Adams et al [21,22], Lee [23], and Maharani and Saputro [24] discussed smoothing spline for problems of analyzing fractal-like signals, minimizing risk estimate, modeling ARMA observations and estimating smoothing parameter, selection smoothing parameter using simulation data, and determining GCV criterion, respectively; Wang [13], Wang and Ke [25], Gu [26], and Sun et al [27] discussed smoothing splines in ANOVA models; Wang et al [28] applied a bivariate smoothing spline to data of cortisol and ACTH hormones; Lu et al [29] used a penalized spline for analyzing current status data; Berry and Helwig [30] compared tuning methods for penalized splines; Islamiyati et al [31,32] developed a least square spline for estimating two responses of non-parametric regression models and discussed linear spline in the modeling of blood sugar; and Kirkby et al [33] estimated nonparametric density using B-Spline. Additionally, Osmani et al [34] estimated the coefficient of a rates model using kernel and spline estimators.…”

Reproducing Kernel Hilbert Space Approach to Multiresponse Smoothing Spline Regression Function

“…Due to the flexible nature of these splines, many researchers have been interested in using and developing them in several cases. For examples, M-type splines were used by [21] to analyze variance for correlated data, and by [22] for estimating both nonparametric and semiparametric regression functions; truncated splines have been discussed by [23] to estimate mean arterial pressure for prediction purpose and by [24] to estimate blood pressure for prediction and interpretation purposes. Additionally, Ref.…”

“…The solution to(21) can be obtained by taking the partial diferentiation Q(b, d) with respect to b and d. In this step, we obtain the estimations of b and d as follows: b = A T D −1 WA −Wy, and d = D −1 W I − A A T D −1 WA −T D −1 W y where D = WC + NΛI.…”