Ten lectures on wavelets
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Adaptive lifting for nonparametric regression
Statistics and Computing
Real-parameter genetic algorithms for finding multiple optimal solutions in multi-modal optimization
GECCO'03 Proceedings of the 2003 international conference on Genetic and evolutionary computation: PartI
Adaptive lifting schemes with perfect reconstruction
IEEE Transactions on Signal Processing
Time-invariant orthonormal wavelet representations
IEEE Transactions on Signal Processing
Nonlinear wavelet transforms for image coding via lifting
IEEE Transactions on Image Processing
Spectral estimation for locally stationary time series with missing observations
Statistics and Computing
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Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new `nondecimated' lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a `best' representation, which we shall explore.