In this paper, we study the complexity of a well-studied class of graphs, exchangeable graphs, and quantify a graph’s complexity via the graphon entropy. Instead of selecting a particular graph descriptor that is only appropriate for one graph-generating mechanism, we estimate the entropy of the generating mechanism of a given graph, assumed to be exchangeable. We naturally take into account a graph’s key graph-theoretic and topological properties. We develop estimators of graphon entropy as measures of complexity for real-world graphs under an increasingly complex range of generating processes. We provide the convergence rate of the entropy estimator for any exchangeable graph and conclude by discussing how to use them to describe evolving real-world graphs.

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 letter, 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.

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.

We propose a novel algorithm for testing the hypothesis of nonstationarity in complex-valued signals. The implementation uses both the bootstrap and the Fast Fourier Transform such that the algorithm can be efficiently implemented in O(NlogN) time, where N is the length of the observed signal. The test procedure examines the second-order structure and contrasts the observed power variance -- i.e. the variability of the instantaneous variance over time -- with the expected characteristics of stationary signals generated via the bootstrap method. Our algorithmic procedure is capable of learning different types of nonstationarity, such as jumps or strong sinusoidal components. We illustrate the utility of our test and algorithm through application to turbulent flow data from fluid dynamics.

Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or nondirectional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2-D signal into local amplitude, local orientation, and local phase. We investigate the second-order statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images and determine a threshold to distinguish between unidirectional and nonunidirectional random fields, which allows the automatic classification of images.

The monogenic signal allows us to decompose a two-dimensional real signal into a local amplitude, a local orientation, and a local phase. In this paper, we introduce the random monogenic signal and study its second-order statistical properties. The monogenic signal may be represented as a quaternion-valued signal. We show that for homogeneous random fields, we need exactly two quaternion-valued covariance functions for a complete second-order description. We also introduce a stochastic model for unidirectional signals and a measure of unidirectionality.

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of nonnegligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.

In this paper we propose a new method for estimating the ambiguity function (AF) of a random process with limited spreading support. The observed process is modelled as the aggregation of a non-stationary signal of interest and noise. As the AF has limited spreading, thresholding is a suitable estimation procedure. Some key stochastic properties of the empirical ambiguity function are derived to obtain a suitable threshold. Based on a median absolute deviation estimator for the variance, we derive a suitable threshold, which forms the basis for our proposed estimator. The estimator is tested on both artificial and real signals, and our results demonstrate a remarkably high resolution and reduced variance.

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.

A new thresholding criterion is introduced, and combined with a local function decomposition, to form an estimate of a deterministic image immersed in white noise. The purpose of the new threshold is to construct image estimates exhibiting continuity jointly across spatial location and spatial frequency. The new threshold criterion is based on the magnitudes of the local decomposition coefficients of the image and phase-shifted versions of the image. Two possible sets of phase-shifted versions are used, and suitable universal thresholds noted. The additional computational cost of the proposed procedure is minor, and thus it makes an attractive improvement to existing thresholding methodology. Examples are given where the reconstructed image, using the new threshold, has improved visual characteristics, as well as a reduced mean square error.

Complex random variables arise naturally in many settings and their properties are of general interest. Past work on complex variables has mainly focused on their second-order structure, as well as that of their conjugates, whereas the main purpose of this correspondence is to clarify the concept of a density function for a complex random variable, and to discuss its properties. Two different functions play the role that the density of a real univariate random variable holds. Only one of these two functions can be correctly interpreted as a density, but both functions clarify the nature of a complex variable. The role played by the complex conjugate of the variable in this formulation is clarified, and the complex scalar nature of Z is discussed. As the properties of complex random variables are most naturally specified in terms of the complex quantities directly, and given in terms of the distribution of the complex variables rather than formulated in terms of the real and imaginary parts, ensuring that an interpretable complex formulation exists is important