Multivariate functional response regression, with application to fluorescence spectroscopy in a cervical pre-cancer study

Abstract

Many scientific studies measure different types of high-dimensional signals or images from the same subject, producing multivariate functional data. These functional measurements carry different types of information about the scientific process, and a joint analysis that integrates information across them may provide new insights into the underlying mechanism for the phenomenon under study. Motivated by fluorescence spectroscopy data in a cervical pre-cancer study, a multivariate functional response regression model is proposed, which treats multivariate functional observations as responses and a common set of covariates as predictors. This novel modeling framework simultaneously accounts for correlations between functional variables and potential multi-level structures in data that are induced by experimental design. The model is fitted by performing a two-stage linear transformation—a basis expansion to each functional variable followed by principal component analysis for the concatenated basis coefficients. This transformation effectively reduces the intra-and inter-function correlations and facilitates fast and convenient calculation. A fully Bayesian approach is adopted to sample the model parameters in the transformed space, and posterior inference is performed after inverse-transforming the regression coefficients back to the original data domain. The proposed approach produces functional tests that flag local regions on the functional effects, while controlling the overall experiment-wise error rate or false discovery rate. It also enables functional discriminant analysis through posterior predictive calculation. Analysis of the fluorescence spectroscopy data reveals local regions with differential expressions across the pre-cancer and normal samples. These regions may serve as biomarkers for prognosis and disease assessment.

Fast covariance estimation for multivariate sparse functional data

“…For regression with multivariate functional responses, see Zhu et al. (), Luo and Qi (), Li, Huang, and Zhu, (), Wong, Li, and Zhu (), Zhu, Morris, Wei, and Cox (), Kowal, Matteson, and Ruppert (), and Qi and Luo (). Graphical models for multivariate functional data are studied in (Zhu, Strawn, & Dunson, ) and Qiao, Guo, and James ().…”

Comparison and contrast of two general functional regression modelling frameworks

“…AR( p ), Matern, CAR) indexed by correlation parameters ρ k . For multivariate functional data, ${\mathrm{bold-italicR}}_k={\mathrm{bold-italicP}}_k^{-1}$, where P k is a basis coefficient-specific precision matrix that can be fit nonparametrically using principles of Gaussian graphical models (Zhang, et al 2016b), or alternatively principal components can be defined across the functions for each basis k (Zhu, et al 2016b). When the data consists of repeated spatially/temporally correlated or multivariate functions for each subject, R k is block diagonal.…”

“…For simplicity we first describe the Gaussian FMM with conditionally independent random effect and residual error functions, for which the random effect functions Uhmfalse(false) are iid mean zero Gaussian Processes with covariance Qhfalse(,false) and the residual error functions Eifalse(false) are iid mean zero Gaussian Processes with covariance Sfalse(,false). As described below, this framework has been generalized to allow parametrically specified correlation structures across the N functional residuals to accommodate correlated functional data, including AR(p), Matern (Zhu et al, 2016c), and conditional autoregressive (CAR) (Zhang et al, 2016a) processes between functions, or to handle multivariate functional data via functional graphical models (Zhang et al, 2016b). These processes could be assumed to be stationary, with covariance parameters common across t , or nonstationary, allowing the covariance parameters to themselves vary across t .…”

“…Lee, et al (2016) demonstrated how to perform semiparametric functional regression with terms nonparametric in both x and t using this framework, and how it can accommodate longitudinally correlated functional data. Zhu, et al (2016a) developed unified methods for functional regression and discrimination for sonar-terrain data, and Zhu, et al (2016b) developed methods for multivariate functional response regression for fluorescence spectroscopy data. Zhang, et al (2016b) introduced functional graphical models that can be used to model multivariate functional data.…”

Concentration of FAD as a marker for cervical precancer detection

“…Intrinsic fluorescence [19][20][21] has also been extracted to understand the biochemical changes with disease progression through different experimental 22 and simulation-based techniques. [23][24][25][26] The use of different algorithms, such as machine learning, 27 wavelet analysis, 9 and other multivariate algorithms 10,20,28,29 have made detection more effective and robust.…”