1 University of Cambridge, Addenbrookes Hospital, Cambridge CB2 2QQ, UK

1
Wavelet generalised least squares (WLS) a new
BLU estimator for regression models with long
memory errors
(1) University of Cambridge, Addenbrooke?s
Hospital, Cambridge CB2 2QQ, UK (2) GREYC-ISMRA,
Caen, France
E-mail etb23_at_cam.ac.uk
A brief summary of WLS from start to finish?.
Figure 1 Functional MRI data acquired at 1.5T
and 3.0T under the null hypothesis (no
experimental effect) show long-memory
autocorrelational structure
Box 1 Classical BLU estimation of parameters of
GLM, Y aX e, assumes that errors are
naturally (or after pre-whitening) uncorrelated,
i.e., ei N (0,Is), where I is identity matrix
and s is error variance. BLU estimation by WLS in
the wavelet domain assumes Y aX e, ei N
(0,S(H)), where S is the error covariance matrix
and H is the Hurst exponent or long-memory
parameter of the noise.
Figure 4 Parameter maps estimated by WLS for a
single slice of fMRI data acquired during
periodic visual stimulation. Above - maps of
error variance and Hurst exponent. Note that many
voxels show large H (long memory) especially in
cortex. Below - activation maps generated by
probabilistic thresholding of OLS and WLS
estimates of a vector
Figure 2 (above) Unbiased and efficient WLS
estimation of signal and noise parameters in
simulated fMRI signals solid line indicates
perfectly unbiased estimation, dashed line
indicates Cramer-Rao lower bounds on parameter
variance. Figure 3 (below) Good type 1 error
control by WLS, relative to OLS and ARLS,
calibrated by analysis of several ?null? 3.0T
fMRI datasets acquired at rest. Each blue line
indicates the type 1 error curve for a single
subject. Valid tests will generate curves lying
at or below the line of identity
Some useful general references on
wavelets? Wornell GW (1992) Wavelet-based
representations for the 1/f family of fractal
processes. Proc IEEE 81, 1428-1450 Mallat SG
(1998) A Wavelet Tour of Signal Processing,
Academic Press ?and on model estimation for
fMRI Bullmore ET et al (2001) Colored noise and
computational inference in neurophysiological
(fMRI) time series analysis Resampling methods
in time and wavelet domains. Human Brain Mapping
12, 61-78