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...genvector determination (O(n3)), loess is quadratic in storage and some of the above algorithms rely on variants of MCMC, which do not scale well to large problems. (b) our methods are rotationally invariant. Some of the above methods are not. (c) our methods are easily extendable to smoother ‘predict’ and ‘update’ steps (see later for an explanation of these). For methods such as triograms extensions to smoother basis functions are not trivial, see Hansen et al. (1998). Moreover, our methods can even be further developed to adapt to local smoothness conditions by use of adaptive lifting, see Nunes et al. (2006) for this in 1D. (d) on a range of real and simulated examples reported in Section 7, our methods work well. The examples include both discontinuous and smooth functions. It is reassuring that a method developed to allow for possible discontinuities also work well in the smoother case. The main disadvantage is that, apart from analogies with regular wavelets, there is currently no substantial body of theory behind our methods. We discuss the reasons for this in section 8, but some theoretical remarks are addressed in section 5. 4 Stefan Wager 200 220 240 260 280 300 320 340 360 X (km) 20 40 60...

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... 325 378 and compare it with the ratios from 10,000 random clustering solutions with the same size allocation as the current one. From this we obtain an empirical significance probability .0001 by calculating the proportion of clustering solutions with a higher ratio than the current one. Since parts of the data are irregularly spaced due to missing epochs, we interpolate the missing values by applying the local linear smoothing to Y ∗i (t) in Step E4. Also, since the numbers of epochs are not the power of two, we use the adaptive lifting scheme in nonparametric wavelet regression proposed by Nunes et al. (2006). Hart et al. (2007) find that 56 out of the 378 stars have significant trends in the times between maximum brightness, controlling the pFDR level at 0.05 for multiplicity adjustment. For a fair comparison, however, we apply the FDR method at the level of 0.05 to their p-values and find 53 significant trends. In our analysis, on the other hand, we find that 24 stars have significant trends. Table 6 summarizes the difference between our results and those found by Hart et al. (2007). Figure 5 shows some curves for which Hart et al. (2007) and the proposed method do not agree. Figures 5(a) and (b...

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...g an empirical Bayesian procedure, implemented in EBayesThresh (Johnstone and Silverman, 2004a,b, 2005), with posterior median threshold, adapted to suit the characteristics of the lifting algorithm (=-=Nunes et al., 2006-=-). Following the shrinkage procedure, each algorithm is inverted, and the averaged estimator is computed. 4.2.2 Assessing performance For assessing the performance of our algorithm, we repeat our deno...

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...elty: the scheme serves as a general framework for the design of new types of wavelet transforms. First, the filter operations in a lifting scheme can easily be made nonlinear, and even data adaptive =-=[3, 4, 17, 18, 16, 14, 12]-=-. The most well known example of a nonlinear lifting scheme is the wavelet transforms that maps integers onto integers [2]. Second, the data that are analysed need not be sampled on equidistant interv...