Unit Root Tests with Wavelets

This paper develops a wavelet (spectral) approach to test the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations

Unit Root Tests with Wavelets

This paper develops a wavelet (spectral) approach to test the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations

Unit Root Tests with Wavelets

This paper develops a wavelet (spectral) approach to test the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations

Multiresolution methods for materials modeling via coarse-graining

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2005.Includes bibliographical references (p. 209-222).(cont.) time, while obtaining useful information about the thermodynamic behavior of the system. We show how statistical mechanics can be formulated using the wavelet transform as a coarse-graining technique. For small systems in which exact enumerations of all states is possible, we illustrate how the method recovers reasonably good estimates for physical properties (errors no more than 10%) with several orders of magnitude fewer operations than are required for an exact enumeration. In addition, we illustrate that errors introduced by the wavelet transform vanish in the neighborhood of fixed points of systems as determined by RG theory. Using scaling results from simulations at different length scales, we estimate the thermodynamic behavior of the original system without performing simulations on the full original system. In addition, we make the method adaptive by using fluctuation properties of the system to set criteria under which further coarse graining or refinement of the system is required. We demonstrate our method for the Ising universality class of problems. We also examine the applicability of the WAMC framework to polymer chains. Polymers are quintessential examples of the need for simulations at multiple scales: at one end, we can study short chains using quantum chemistry methods; yet polymers can have relaxation times on the order of seconds or longer, and molecular weights of 10⁶ or more. Even with modern computational resources, simulating behavior at long times or for long chains is still prohibitively expensive ...Multiscale modeling of physical systems often requires the use of multiple types of simulations to bridge the various length scales that. need to be considered: for example, a density-functional theory at the electronic scale will be combined with a molecular-dynamics simulation at the atomistic level, and with a finite-element method at the macroscopic level. An improvement to this scheme would be a method which is capable of consistently simulating a system at multiple levels of resolution without passing from one simulation type to another, so that different simulations can be studied at a common length scale by appropriate coarse-graining or refinement of a given model. We introduce the wavelet transform as the basis for a new coarse-graining framework. A family of orthonormal basis, the wavelet transform separates data sets, such as spatial coordinates or signal strengths, into subsets representing local averages and local differences. The wavelet transform has several desirable properties for coarse-graining: it is hierarchical, compact, and has natural applications to approximating physical data sets. As a hierarchical method, it can be used to rescale a Hamiltonian to a desired length scale, and at the same time also rescales the particles of the system by creating "blocked" particles in the spirit of renor-malization group (RG) calculations. The wavelet-accelerated Monte Carlo (WAMC) framework performs a Monte Carlo simulations on a small system which will be transformed into a block particle to obtain the probability distribution of the blocked particle; a Monte Carlo simulation is then performed on the resulting system of blocked particles. This method, which can be repeated as needed, can achieve significant speed-ups in computationalby Ahmed E. Ismail.Ph.D

Unit Root Tests with Wavelets

This paper develops a wavelet (spectral) approach to test the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

A wavelet approach to modelling the evolutionary dynamics across ordered replicate time series

Experimental time series data collected across a sequence of ordered replicates often crop up in many fields, from neuroscience to circadian biology. In practice, it is natural to observe variability across time in the dynamics of the underlying process within a single replicate and wavelets are essential in analysing nonstationary behaviour. Additionally, signals generated within an experiment may also exhibit evolution across replicates even for identical stimuli. We propose the Replicate-Evolving Locally Stationary Wavelet process (REv-LSW) which gives a stochastic wavelet representation of the replicate time series. REv-LSW yields a natural desired time- and replicate-localisation of the process dynamics, capturing nonstationary behaviour both within and across replicates, while accounting for between-replicate correlation. Firstly, we rigorously develop the associated wavelet spectral estimation framework along with its asymptotic properties for the particular case that replicates are uncorrelated. Next, we crucially develop the framework to allow for dependence between replicates. By means of thorough simulation studies, we demonstrate the theoretical estimator properties hold in practice. Finally, it is unreasonable to make the typical assumption that all replicates stem from the same process if a replicate spectral evolution exists. Thus, we propose two novel tests that assess whether a significant replicate-effect is manifest across the replicate time series. Our modelling framework uses wavelet multiscale constructions that mitigate against the potential nonstationarities, across both times and replicates. Thorough simulation studies prove both tests to be flexible tools and allow the analyst to accordingly tune their subsequent analysis. Throughout this thesis, our work is motivated by an investigation into the evolutionary dynamics of brain processes during an associative learning experiment. The neuroscience data analysis illustrates the utility of our proposed methodologies and demonstrates the wider experimental data analysis achievable that is also of benefit to other experimental fields, e.g. circadian biology, and not just the neurosciences