Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2 J for some J . These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data.A key lifting component is the "predict" step where a prediction of a data point is made. The residual from the prediction is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing the kind of prediction according to a criterion. In this way the smoothness of the underlying 'wavelet' can be adapted to the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods.We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting on the real inductance plethysmography and motorcycle data.

Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors

ArticleAbstract. Rhythmic processes are found at all biological and ecological scales, and are fun-5 damental to the efficient functioning of living systems in changing environments. The biochemical 6 mechanisms underpinning these rhythms are therefore of importance, especially in the context of 7 anthropogenic challenges such as pollution or changes in climate and land use. Here we develop and 8 test a new method for clustering rhythmic biological data with a focus on circadian oscillations. The 9 method combines locally stationary wavelet time series modelling with functional principal compo-10 nents analysis and thus extracts the time-scale patterns arising in a range of rhythmic data. We 11 demonstrate the advantages of our methodology over alternative approaches, by means of a simula-12 tion study and real data applications, using both a published circadian dataset and a newly generated 13 one. The new dataset records plant response to various levels of stress induced by a soil pollutant, a 14 biological system where existing methods which assume stationarity are shown to be inappropriate. 15Our method successfully clusters the circadian data in an interesting way, thereby facilitating wider 16 ranging analyses of the response of biological rhythms to environmental changes. 17Key words. evolutionary wavelet spectrum, nondecimated wavelet transform, nonstationary 18 processes, unsupervised learning, plant circadian clock 19

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 behaviour. 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.

This article introduces the GNAR package, which fits, predicts, and simulates from a powerful new class of generalized network autoregressive processes. Such processes consist of a multivariate time series along with a real, or inferred, network that provides information about inter-variable relationships. The GNAR model relates values of a time series for a given variable and time to earlier values of the same variable and of neighboring variables, with inclusion controlled by the network structure. The GNAR package is designed to fit this new model, while working with standard 'ts' objects and the igraph package for ease of use.