Robust functional regression model for marginal mean and subject-specific inferences

Abstract

We introduce flexible robust functional regression models, using various heavy-tailed processes, including a Student t-process. We propose efficient algorithms in estimating parameters for the marginal mean inferences and in predicting conditional means as well interpolation and extrapolation for the subject-specific inferences. We develop bootstrap prediction intervals for conditional mean curves. Numerical studies show that the proposed model provides robust analysis against data contamination or distribution misspecification, and the proposed prediction intervals maintain the nominal confidence levels. A real data application is presented as an illustrative example.

Nonparametric random effects functional regression model using Gaussian process priors

“…We focused our discussion on error term with Gaussian distribution in this paper, but the estimation procedure can be extended to generalized models with functional data (Wang and Shi, 2014). The GP priors may also be replaced by other process priors, for example, the robust heavy tailed processes (Wang et al, 2017;Cao, Shi and Lee, 2017). However, such an extension may be not straightforward, since there are no close forms for parameter estimation and prediction of random effect when a heavy tailed process prior, e.g.…”

Robust Task Learning Based on Nonlinear Regression With Mixtures of Student-t Distributions

“…Similar to LS-SVM, we employ Lagrange multiplier method to solve the optimization problem (19). The Lagrangian function can be expressed as…”

“…In the objective functions (19) for MoT-STR and (31) for MoT-MTR, η are the weight of noise ε. From (14) and (30), η is inversely proportional to the noise under the t-distribution.…”

“…Also, the t-distribution can be viewed as an important member of the distribution family called scale mixtures of normal (SMN) distribution [15]. Recently, SMN distributions have been used in various regression models [16]- [19], which extending the theory of Gaussian models from the framework of distribution family.…”

Bayesian Estimation for the Extended t-process Regression Models with Independent Errors

“…However, it is well-known that the GPR model is susceptible to outliers. To overcome this problem, robust methods are developed based on t-process and other heavy tailed processes; for example, Shah et al (2014) used a simple t-process to replace a GP; Wang et al (2017) proposed an extended t-process regression model (eTPR); and Cao et al (2017) developed robust models based on other heavy-tailed processes such as Slash process and contaminated-normal process. Heavy-tailed processes, particularly t-process, have been used frequently in many different areas to build a robust model, for example, Yu et al (2007) and Zhang and Yeung (2010) used t-process to build a multi-task learning model, and Xu et al (2011) employed matrix-variate t-process and a variational approximation method to construct a sparse matrix-variate block model.…”