Marina Knight

Adaptive lifting for nonparametric regression

Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2J 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.

Implications of changing the minimal survival benefit in liver transplantation

The limited availability of livers donated by deceased donors for transplantation means that not everyone who might benefit from the procedure can receive a graft, so any selection and allocation system must have clearly defined goals. The United Kingdom, in common with many other countries, has adopted a minimum benefit criterion of a greater than 50% probability of survival 5 years after transplantation. We investigated the impact of changing this minimum benefit criterion on a case mix of listed patients. The analysis was based on 5330 adult elective patients who underwent transplantation with livers from donation after brain death donors between January 1994 and December 2007. We examined the impact of balancing the number of registrations on the list with the number of available donor livers while allowing a 10% mortality rate and found that this would require a survival threshold of at least 74% at 5 years. According to historical data, the application of this more stringent criterion would significantly reduce the eligibility of older and nonwhite patients and patients with hepatocellular carcinoma or hepatitis C virus infections. Thus, if such undesirable restrictions on access to liver transplantation are to be avoided, we must consider alternative strategies such as the acceptance of higher transplant list mortality. Liver Transpl, 2012.

A `nondecimated' lifting transform

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.

Complex-Valued Wavelet Lifting and Applications

ABSTRACT Signals with irregular sampling structures arise naturally in many fields. In applications such as spectral decomposition and nonparametric regression, classical methods often assume a regular sampling pattern, thus cannot be applied without prior data processing. This work proposes new complex-valued analysis techniques based on the wavelet lifting scheme that removes “one coefficient at a time.” Our proposed lifting transform can be applied directly to irregularly sampled data and is able to adapt to the signal(s)’ characteristics. As our new lifting scheme produces complex-valued wavelet coefficients, it provides an alternative to the Fourier transform for irregular designs, allowing phase or directional information to be represented. We discuss applications in bivariate time series analysis, where the complex-valued lifting construction allows for coherence and phase quantification. We also demonstrate the potential of this flexible methodology over real-valued analysis in the nonparametric regression context. Supplementary materials for this article are available online.

A wavelet lifting approach to long-memory estimation

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 even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.

Modelling and Prediction of Time Series Arising on a Graph

Time series that arise on a graph or network arises in many scientific fields. In this paper we discuss a method for modelling and prediction of such time series with potentially complex characteristics. The method is based on the lifting scheme first proposed by Sweldens, a multiscale transformation suitable for irregular data with desirable properties. By repeated application of this algorithm we can transform the original network time series data into a simpler, lower dimensional time series object which is easier to forecast. The technique is illustrated with a data set arising from an energy time series application.

Long memory estimation for complex-valued time series

Long memory has been observed for time series across a multitude of fields, and the accurate estimation of such dependence, for example via the Hurst exponent, is crucial for the modelling and prediction of many dynamic systems of interest. Many physical processes (such as wind data) are more naturally expressed as a complex-valued time series to represent magnitude and phase information (wind speed and direction). With data collection ubiquitously unreliable, irregular sampling or missingness is also commonplace and can cause bias in a range of analysis tasks, including Hurst estimation. This article proposes a new Hurst exponent estimation technique for complex-valued persistent data sampled with potential irregularity. Our approach is justified through establishing attractive theoretical properties of a new complex-valued wavelet lifting transform, also introduced in this paper. We demonstrate the accuracy of the proposed estimation method through simulations across a range of sampling scenarios and complex- and real-valued persistent processes. For wind data, our method highlights that inclusion of the intrinsic correlations between the real and imaginary data, inherent in our complex-valued approach, can produce different persistence estimates than when using real-valued analysis. Such analysis could then support alternative modelling or policy decisions compared with conclusions based on real-valued estimation.