R-语言程序

response functions
Yi (t ) X ia Ba (t ) Z il U l (t ) Ei (t ) a 1 l 1
fixed effect function random effect functions
p
m
residual error functions
Conclusions
Introduction
Functional Data:
– Ideal units of observation: curves – Observed data: curves sampled on fine grid
– – In recent years, these type of data are being increasingly encountered in practice with new technologies taking automated measurements Examples: accelerometer data, EEG data, ECG data, time series, various genomics and proteomics data (MALDI-MS) Scanned images w/ intensities representing measurement of interest Data set involves multiple images; from multiple individuals, and often multiple images per individual Interested in combining information across images to make inferences about populations or interventions Examples: fMRI data, 2D gel electrophoresis proteomics data, PET data, Time-space data
Types of Functional Regression
Functional regression: Linear regression involving functional data with (typically) nonparametrically represented coefficients Types of Functional regression:
– MALDI-MS: protein peaks; fMRI: brain regions; 2dGE: spots
2. Identify which are related to factors of interest (FDR)

Alternative: Functional Data Analysis
Robust Regression for Functional and Image Data using Functional Mixed Models
Jeffrey S. Morris Department of Biostatistics
The University of Texas MD Anderson Cancer Center
Gaussian Functional Mixed Model
N T N m U l ~ MVN (0, Q) Y B Z U X Ei ~ MVN (0, S ) E p T mT N T N p
Random effect and residual error functions assumed to be mean-zero Gaussians Q and S are covariance matrices (T T) describing curve-to-curve deviations
– Too high dimension to leave fully unstructured
– Irregular functional data: Q and S will contain many nonstationarities (heteroscedasticity, varying smoothness), so important to allow flexible representations; i.e. not assume something naï ve and simple. – Morris and Carroll (2006) model in wavelet space; make parsimonious assumptions in there that accommodates nonstationary covariance structures for Q,S
N T
ห้องสมุดไป่ตู้
N m Y B Z U X E p T mT
N p
N T
Columns index grid points on function t
Image FMM can also be written in this form
– Vectorize each T1xT2 image: Yi=vec{Yi} – Stack into a single response matrix Y (T=T1×T2 cols), with each row a vectorized image
Gaussian Functional Mixed Models
– Functional Mixed Models – Gaussian Wavelet-Based Functional Mixed Models
Robust Functional Mixed Models
– Robust Functional Mixed Models – Robustness and Adaptive Properties – Simulation study results
• Ba(t) summarizes partial effect of Xa on Yi(t) • Ul(t) are mean-zero random effect functions • Ei(t) are mean-zero residual error functions
• Key to flexibility: forms of Ba(t), Ul(t) and Ei(t) left unspecified • Assumptions needed on distributions and covariance structure of U,E
Gaussian Wavelet-Based Functional Mixed Models (G-WFMM)
1. Project observed functions Y into wavelet space; Apply DWT to each observed function (D=YW’)
Wavelets
Wavelet representation: Yi(t)=Sdi jkyjk (t) Benefits of Using Wavelet Bases
1. Compact support allows efficient representations of local features and discontinuities 2. Sparsity property useful for compression and denoising 3. Whitening property allows parsimonious yet flexible representations of Q and S

Quantitative Image Analysis:
– – – –
Statistical Modeling of Functions/Images
Standard Approach: Feature Extraction
1. Detect and quantify information for each feature
Discrete Version of FMM
Functional Data on Fine Grid: Suppose each observed curve is sampled on a common equally-spaced grid of length T, with T very large (hundreds to millions)
– Function as response: Yi(t) ~ Xi – Function as predictor: Yi ~ Xi(t) – Functional response and predictor: Yi(t) ~ Xi(s)
Our focus here: function as response
Functional Mixed Model (FMM)
Idea: Relate functional response to scalar predictors through functional coefficients, while adjusting for possible design-induced correlation between functions Suppose we observe sample of N functions, Yi(t), i=1, …,N, on a closed interval T, along with p predictors
– Requires very flexible modeling techniques to capture complex local features in data. – Methods must be computationally efficient enough to handle extremely high dimensions of these data – Should borrow strength spatially (i.e. smooth, preferably adaptively) in performing inference – Must find way to adjust for multiple testing (FDR) – Not used for 2dGE or MALDI-MS; approach used in fMRI, but few methods integrate smoothing, inference, and FDR – Functional Regression commonly used to relate functions to other outcomes/groups