The Debiased Spatial Whittle likelihood.

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well-known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalize the approach to flexibly allow for significant volumes of missing data including those with lower-dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non-Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O ( n log n ) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle-based methods for large data sets. We validate our procedure over a range of simulated and realworld settings, and compare with state-of-the-art alternatives, demonstrating the enduring practical appeal of Fourier-based methods, provided they are corrected by the procedures developed in this paper.

Network histograms and universality of blockmodel approximation.

In this paper we introduce the network histogram, a statistical summary of network interactions to be used as a tool for exploratory data analysis. A network histogram is obtained by fitting a stochastic blockmodel to a single observation of a network dataset. Blocks of edges play the role of histogram bins and community sizes that of histogram bandwidths or bin sizes. Just as standard histograms allow for varying bandwidths, different blockmodel estimates can all be considered valid representations of an underlying probability model, subject to bandwidth constraints. Here we provide methods for automatic bandwidth selection, by which the network histogram approximates the generating mechanism that gives rise to exchangeable random graphs. This makes the blockmodel a universal network representation for unlabeled graphs. With this insight, we discuss the interpretation of network communities in light of the fact that many different community assignments can all give an equally valid representation of such a network. To demonstrate the fidelity-versus-interpretability tradeoff inherent in considering different numbers and sizes of communities, we analyze two publicly available networks--political weblogs and student friendships--and discuss how to interpret the network histogram when additional information related to node and edge labeling is present.

A DNA methylation network interaction measure, and detection of network oncomarkers.

Epigenetic processes--including DNA methylation--are increasingly seen as having a fundamental role in chronic diseases like cancer. DNA methylation patterns offer a route to develop prognostic measures based directly on DNA measurements, rather than less-stable RNA measurements. A novel DNA methylation-based measure of the co-ordinated interactive behaviour of genes is developed, in a network context. It is shown that this measure reflects well the co-regulatory behaviour linked to gene expression (at the mRNA level) over the same network interactions. This measure, defined for pairs of genes in a single patient/sample, associates with overall survival outcome independent of known prognostic clinical features, in several independent data sets relating to different cancer types. In total, more than half a billion CpGs in over 1600 samples, taken from nine different cancer entities, are analysed. It is found that groups of gene-pair interactions which associate significantly with survival identify statistically significant subnetwork modules. Many of these subnetwork modules are shown to be biologically relevant by strong correlation with pre-defined gene sets, such as immune function, wound healing, mitochondrial function and MAP-kinase signalling. In particular, the wound healing module corresponds to an increase in co-ordinated interactive behaviour between genes for worse prognosis, and the immune module corresponds to a decrease in co-ordinated interactive behaviour between genes for worse prognosis. This measure has great potential for defining DNA-based cancer biomarkers. Such biomarkers could naturally be developed further, by drawing on the rapidly expanding knowledge base of network science.

Corruption of the intra-gene DNA methylation architecture is a hallmark of cancer.

Epigenetic processes--including DNA methylation--are increasingly seen as having a fundamental role in chronic diseases like cancer. It is well known that methylation levels at particular genes or loci differ between normal and diseased tissue. Here we investigate whether the intra-gene methylation architecture is corrupted in cancer and whether the variability of levels of methylation of individual CpGs within a defined gene is able to discriminate cancerous from normal tissue, and is associated with heterogeneous tumour phenotype, as defined by gene expression. We analysed 270985 CpGs annotated to 18272 genes, in 3284 cancerous and 681 normal samples, corresponding to 14 different cancer types. In doing so, we found novel differences in intra-gene methylation pattern across phenotypes, particularly in those genes which are crucial for stem cell biology; our measures of intra-gene methylation architecture are a better determinant of phenotype than measures based on mean methylation level alone (K-S test [Formula: see text] in all 14 diseases tested). These per-gene methylation measures also represent a considerable reduction in complexity, compared to conventional per-CpG beta-values. Our findings strongly support the view that intra-gene methylation architecture has great clinical potential for the development of DNA-based cancer biomarkers.

The memory of spatial patterns: changes in local abundance and aggregation in a tropical forest.

The current spatial pattern of a population is the result of previous individual birth, death, and dispersal events. We present a simple model followed by a comparative analysis for a species-rich plant community to show how the current spatial aggregation of a population may hold information about recent population dynamics. Previous research has shown how locally restricted seed dispersal often leads to stronger aggregation in less abundant populations than it does in more abundant populations. In contrast, little is known about how changes in the local abundance of a species may affect the spatial distribution of individuals. If the level of aggregation within a species depends to some extent on the abundance of the species, then changes in abundance should lead to subsequent changes in aggregation. However, an overall change of spatial pattern relies on many individual birth and death events, and a surplus of deaths or births may have short-term effects on aggregation that are opposite to the long-term change predicted by the change in abundance. The change in aggregation may therefore lag behind the change in abundance, and consequently, the current aggregation may hold information about recent population dynamics. Using an individual-based simulation model with local dispersal and density-dependent competition, we show that, on average, recently growing populations should be more aggregated than shrinking populations of the same current local abundance. We tested this hypothesis using spatial data on individuals from a long-term tropical rain forest plot, and find support for this relationship in canopy trees, but not in understory and shrub species. On this basis we argue that current spatial aggregation is an important characteristic that contains information on recent changes in local abundance, and may be applied to taxonomic groups where dispersal is limited and within-species aggregation is observed.

Hyperanalytic denoising.

A new threshold rule for the estimation of a deterministic image immersed in noise is proposed. The full estimation procedure is based on a separable wavelet decomposition of the observed image, and the estimation is improved by introducing the new threshold to estimate the decomposition coefficients. The observed wavelet coefficients are thresholded, using the magnitudes of wavelet transforms of a small number of "replicates" of the image. The "replicates" are calculated by extending the image into a vector-valued hyperanalytic signal. More than one hyperanalytic signal may be chosen, and either the hypercomplex or Riesz transforms are used, to calculate this object. The deterministic and stochastic properties of the observed wavelet coefficients of the hyperanalytic signal, at a fixed scale and position index, are determined. A "universal" threshold is calculated for the proposed procedure. An expression for the risk of an individual coefficient is derived. The risk is calculated explicitly when the "universal" threshold is used and is shown to be less than the risk of "universal" hard thresholding, under certain conditions. The proposed method is implemented and the derived theoretical risk reductions substantiated.

Noise reduction in directional signals using multiple morse wavelets illustrated on quadrature Doppler ultrasound.

The use of multiple complex-valued Morse wavelets for the scalogram study of signals which are unidirectional at any time, but are bidirectional overall is considered. These wavelets are well-suited to identifying the forward and reverse components. Scalogram averaging which is possible due to the multiplicity of the complex-valued wavelets leads to a scalogram with reduced noise. Information from positive and negative scales can then be used to estimate a final "cleaned" scalogram. Quadrature Doppler ultrasound blood flow in the femoral artery is taken as an example to clearly illustrate the noise reduction.