Wavelet theory constitutes one of the most significant mathematical advances for signal processing, and thus presents a great interest in point process analysis. For instance, numerous approaches have been explored to estimate
the first-order intensity of a point process with wavelets. In this thesis, we intend to consolidate existing results in wavelet-based linear estimation and investigate further applications of wavelets on point processes in the context
of multiresolution analysis.
We initially take this wavelet-based approach to estimate the first and higher-order intensities of a point process in any finite dimension and under a continuous spatial setting. We perform a statistical study of wavelet linear estimators when the observed events are located in a hyperrectangle of R^d. It is notably shown that the linear estimator of the complete k-th order intensity is the product of k linear estimators of the first-order intensity.
Such wavelet modelling also motivates the construction of a first-order multiresolution analysis, through the definition of properties at different scales, termed J-th level homogeneity and L-th level innovation. Likelihood ratio
tests for these properties are provided and studied under Poisson processes and Haar wavelets. A key result is that the applicability of these tests is linked to the product of the number of realizations and the expectation measure of the process. This means that one can use the asymptotic distributions of the test statistics from a single point pattern if the expectation measure is itself sufficiently high.
The hypothesis test for L-th level innovation is then used to design new data-driven thresholding strategies, each based on a different grouping of wavelet coefficients. Our thresholding methods are studied through extensive simulations and applied to NetFlow data to exhibit the differences between human and automated behaviour.
Since wavelet estimation of Cox processes has received very little treatment to this day, we provide new developments in this topic essentially through the wavelet-based estimation of the pointwise probability density or mass function for the intensity field. The behaviour of this estimator is studied with example Cox process models for different wavelets and resolutions, followed by an application to firing patterns from virtual reality military training.
We eventually extend the idea of J-th level homogeneity to Cox processes by re-defining it through the mean intensity field, which we then test with a method based on Hotelling’s t-squared statistic.