ABSTRACT
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.
ABSTRACT
In this article we propose a novel framework for the modelling of non-stationary multivariate lattice processes. Our approach extends the locally stationary wavelet paradigm into the multivariate two-dimensional setting. As such the framework we develop permits the estimation of a spatially localised spectrum within a channel of interest and, more importantly, a localised cross-covariance which describes the localised coherence between channels. Associated estimation theory is also established which demonstrates that this multivariate spatial framework is properly defined and has suitable convergence properties. We also demonstrate how this model-based approach can be successfully used to classify a range of colour textures provided by an industrial collaborator, yielding superior results when compared against current state-of-the-art statistical image processing methods.
ABSTRACT
How best to summarize large and complex datasets is a problem that arises in many areas of science. We approach it from the point of view of seeking data summaries that minimize the average squared error of the posterior distribution for a parameter of interest under approximate Bayesian computation (ABC). In ABC, simulation under the model replaces computation of the likelihood, which is convenient for many complex models. Simulated and observed datasets are usually compared using summary statistics, typically in practice chosen on the basis of the investigator's intuition and established practice in the field. We propose two algorithms for automated choice of efficient data summaries. Firstly, we motivate minimisation of the estimated entropy of the posterior approximation as a heuristic for the selection of summary statistics. Secondly, we propose a two-stage procedure: the minimum-entropy algorithm is used to identify simulated datasets close to that observed, and these are each successively regarded as observed datasets for which the mean root integrated squared error of the ABC posterior approximation is minimized over sets of summary statistics. In a simulation study, we both singly and jointly inferred the scaled mutation and recombination parameters from a population sample of DNA sequences. The computationally-fast minimum entropy algorithm showed a modest improvement over existing methods while our two-stage procedure showed substantial and highly-significant further improvement for both univariate and bivariate inferences. We found that the optimal set of summary statistics was highly dataset specific, suggesting that more generally there may be no globally-optimal choice, which argues for a new selection for each dataset even if the model and target of inference are unchanged.
