Wavelet-Generalized Least Squares: A New BLU Estimator of Linear Regression Models with 1/f Errors

Abstract

Long-memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors. Classical autoregressive moving average (ARMA) methods can adequately address the problem of linear time invariant, short-memory errors but may be inefficient and/or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function. Here we introduce a novel method, called wavelet-generalized least squares (WLS), which is (to a good approximation) the best linear unbiased (BLU) estimator of regression model parameters in the context of long-memory errors. The method also provides maximum likelihood (ML) estimates of the Hurst exponent (which can be readily translated to the fractal dimension or spectral exponent) characterizing the correlational structure of the errors, and the error variance. The algorithm exploits the whitening or Karhunen-Loé ve-type property of the discrete wavelet transform to diagonalize the covariance matrix of the errors generated by an iterative fitting procedure after both data and design matrix have been transformed to the wavelet domain. Properties of this estimator, including its Cramè r-Rao bounds, are derived theoretically and compared to its empirical performance on a range of simulated data. Compared to ordinary least squares and ARMA-based estimators, WLS is shown to be more efficient and to give excellent type 1 error control. The method is also applied to some real (neurophysiological) data acquired by functional magnetic resonance imaging (fMRI) of the human brain. We conclude that wavelet-generalized least squares may be a generally useful estimator of regression models in data complicated by long-memory or fractal noise. © 2002 Elsevier Science

WSPM: Wavelet-based statistical parametric mapping

“…One shortcoming of this model is that it may produce overestimates when there is serial correlation in the data that has not been dealt with correctly. One possibility could be to deploy more advanced methods to deal with colored noise; e.g., Bullmore et al (2001) and Fadili and Bullmore (2002).…”

Spatiotemporal wavelet analysis for functional MRI

“…Detail on the relationship between these models is given by the Wold Decomposition Theorem (Wold, 1954). Finally, we note that more complex long-term 1/f noise processes have been suggested for fMRI noise, for example, Fadili and Bullmore (2002), that may yield equally compact and perhaps more powerful representations of brain noise. Indeed, these methods have been shown in some cases to better constrain Type-I errors in resting data compared to AR( p) models.…”

“…A temporal wavelet denoising was subsequently performed on the spatial presmoothed data that included prior knowledge about the frequency structure of the experimental response. Temporal applications of wavelets in fMRI have included Bullmore et al (2002), Fadili and Bullmore (2002), and Long et al (2001). In Bullmore et al (2002), temporal wavelets were used to serially decorrelate the data, enabling the use of resampling schemes to accurately ascertain null distributions of activation statistics, and better control Type-I error.…”

“…In Bullmore et al (2002), temporal wavelets were used to serially decorrelate the data, enabling the use of resampling schemes to accurately ascertain null distributions of activation statistics, and better control Type-I error. In Fadili and Bullmore (2002), a least-squares likelihood function was optimized in the wavelet domain under 1/f assumptions about the noise. In earlier work, we (Long et al, 2001) developed wavelet packet time-series analysis in a correlated noise setting that constructed basis expansions using time -frequency properties of the experimental stimulus.…”

“…Temporal applications of wavelets in fMRI have included Bullmore et al (2002) who utilized the decorrelating properties of wavelets to control Type-1 error. Also, Fadili and Bullmore (2002) constructed a temporal maximum-likelihood framework for estimating signal in the presence of 1/f noise.…”