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

Multiple multidimensional morse wavelets

This paper defines a set of operators that localize a radial image in space and radial frequency simultaneously. The eigenfunctions of the operator are determined and a nonseparable orthogonal set of radial wavelet functions are found. The eigenfunctions are optimally concentrated over a given region of radial space and scale space, defined via a triplet of parameters. Analytic forms for the energy concentration of the functions over the region are given. The radial function localization operator can be generalised to an operator localizing any L-2(R-2) function. It is demonstrated that the latter operator, given an appropriate choice of localization region, approximately has the same radial eigenfunctions as the radial operator. Based on a given radial wavelet function a quaternionic wavelet is defined that can extract the local orientation of discontinuous signals as well as amplitude, orientation and phase structure of locally oscillatory signals. The full set of quaternionic wavelet functions are component by component orthogonal; their statistical properties are tractable, and forms for the variability of the estimators of the local phase and orientation are given, as well as the local energy of the image. By averaging estimators across wavelets, a substantial reduction in the variance is achieved

Generalized morse wavelets

Published versio

Local directional denoising

Published versio

A method to detect sub-communities from multivariate spatial associations

1.Species are seldom distributed randomly across a community, but instead show spatial structure that is determined by environmental gradients and/or biotic interactions. Analysis of the spatial co-associations of species may therefore reveal information on the processes that helped to shape those patterns. 2.We propose a multivariate approach that uses the spatial co-associations between all pairs of species to find sub-communities of species whose distribution in the study area are positively correlated. Our method, which begins with the patterns of individuals, is particularly well-suited for communities with large numbers of species, and gives rare species an equal weight. We propose a method to quantify a maximum number of sub-communities that are significantly more correlated than expected under a null model of independence. 3.Using data on the distribution of tree and shrub species from a 50 ha forest plot on Barro Colorado Island (BCI), Panama, we show that our method can be used to construct biologically meaningful sub-communities that are linked to the spatial structure of the plant community. As an example, we construct spatial maps from the sub-communities that closely follow habitats based on environmental gradients (such as slope) as well as different biotic conditions (such as canopy gaps). 4.We discuss extensions and adaptations to our method that might be appropriate for other types of spatially referenced data and for other ecological communities. We make suggestions for other ways to interpret the sub-communities using phylogenetic relationships, biological traits, and environmental variables as covariates, and note that sub-communities that are hard to interpret may suggest groups of species and/or regions of the landscape that warrant further attention

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

Visualizing the Wavenumber Content of a Point Pattern

Spatial point patterns are a commonly recorded form of data in ecology, medicine, astronomy, criminology, epidemiology and many other application fields. One way to understand their second order dependence structure is via their spectral density function. However, unlike time series analysis, for point patterns such approaches are currently underutilized. In part, this is because the interpretation of the spectral representation of the underlying point processes is challenging. In this paper, we demonstrate how to band-pass filter point patterns, thus enabling us to explore the spectral representation of point patterns in space by isolating the signal corresponding to certain sets of wavenumbers

The growing ubiquity of algorithms in society: implications, impacts and innovations

The growing ubiquity of algorithms in society raises a number of fundamental questions concerning governance of data, transparency of algorithms, legal and ethical frameworks for automated algorithmic decision-making and the societal impacts of algorithmic automation itself. This article, an introduction to the discussion meeting issue of the same title, gives an overview of current challenges and opportunities in these areas, through which accelerated technological progress leads to rapid and often unforeseen practical consequences. These consequences—ranging from the potential benefits to human health to unexpected impacts on civil society—are summarized here, and discussed in depth by other contributors to the discussion meeting issue. This article is part of a discussion meeting issue ‘The growing ubiquity of algorithms in society: implications, impacts and innovations'

The future of statistics and data science

The ubiquity of sensing devices, the low cost of data storage, and the commoditization of computing have together led to a big data revolution. We discuss the implication of this revolution for statistics, focusing on how our discipline can best contribute to the emerging field of data science

What is the Fourier Transform of a Spatial Point Process?

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fourth order moments of the Fourier transform using Campbell’s theorem. Third we determine how to implement tapering, an important component for spectral analysis of other stochastic processes. Fourth we answer the question of how to produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process