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Regularized nonstationary regression (Fomel, 2009) is based on the
following simple model. Let represent the data as a
function of data coordinates , and
,
, represent a collection of basis functions. The goal
of stationary regression is to estimate coefficients ,
such that the prediction error
|
(1) |
is minimized in the least-squares sense. In the case of regularized
nonstationary regression (RNR), the coefficients become
variable,
|
(2) |
The problem in this case is underdetermined but can be constrained by regularization (Engl et al., 1996). I use
shaping
regularization (Fomel, 2007) to implement an explicit control on the resolution and variability of regression coefficients.
Shaping regularization applied to RNR
amounts to linear inversion,
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(3) |
where is a vector composed of
,
the elements of vector are
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(4) |
the elements of matrix are
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(5) |
is a scaling coefficient, and represents a
shaping (typically smoothing) operator. When inversion in
equation 3 is implemented by an iterative method, such
as conjugate gradients, strong smoothing makes
close to identity and easier (taking less iterations) to
invert, whereas weaker smoothing slows down the inversion but allows for
more details in the solution. This intuitively logical behavior
distinguishes shaping regularization from alternative methods (Fomel, 2009).
Regularized nonstationary autoregression (RNAR) corresponds to the
case of basis functions being causal translations of the input data
itself. In 1D, with , this condition implies
.
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| Seismic data decomposition into spectral components using regularized nonstationary autoregression | |
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Next: Autoregressive spectral analysis
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Previous: Introduction
2013-10-09