Discrete-Time continuous-dilation construction of linear scale-invariant systems and multi-dimensional self-similar signals

This dissertation presents novel models for purely discrete-time self-similar processes and scale- invariant systems. The results developed are based on the definition of a discrete-time scaling (dilation) operation through a mapping between discrete and continuous frequencies. It is shown that it is possible to have continuous scaling factors through this operation even though the signal itself is discrete-time. Both deterministic and stochastic discrete-time self-similar signals are studied. Conditions of existence for self-similar signals are provided. Construction of discrete-time linear scale-invariant (LSI) systems and white noise driven models of self-similar stochastic processes are discussed. It is shown that unlike continuous-time self-similar signals, a wide class of non-trivial discrete-time self-similar signals can be constructed through these models. The results obtained in the one-dimensional case are extended to multi-dimensional case. Constructions of discrete-space self-similar ran dom fields are shown to be potentially useful for the generation, modeling and analysis of multi-dimensional self-similar signals such as textures. Constructions of discrete-time and discrete-space self-similar signals presented in the dissertation provide potential tools for applications such as image segmentation and classification, pattern recognition, image compression, digital halftoning, computer vision, and computer graphics. The other aspect of the dissertation deals with the construction of discrete-time continuous-dilation wavelet transform and its existence condition, based on the defined discrete-time continuous-dilation scaling operator

High frequency surface estimation

Mountain generation can be considered a part of the general theory of surface estimation; In this thesis, two methods have been presented to generate fractals--fast Fourier method and a new generalized stochastic subdivision method. Also, a new surface estimation method has been introduced that deals with points of unequal powers. The uniqueness of this method is the usage of splines to calculate the arc lengths between the points, as opposed to Euclidean distances used in Kriging. The fast Fourier technique has been used to generate mountains in particular; also, some extensions have been suggested, whereby different sets of mountains can be obtained by modifying some parameters. This method is global and has the advantages of simplicity and efficiency; it also provides exact spectral control. The search for a more localized method resulted in the new generalized stochastic subdivision technique. The choice of an autocorrelation function is pivotal here. The only significant differences between the fractal subdivision method and this new technique are the increased neighborhood size, boundary conditions and the need to solve a system of equations for each subdivision level; The source code for these techniques was implemented on SGI machines, using C with GL as a graphics standard

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Level crossings and other level functionals of stationary Gaussian processes

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general $m$-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.Comment: Published at http://dx.doi.org/10.1214/154957806000000087 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local estimation of the Hurst index of multifractional Brownian motion by Increment Ratio Statistic method

We investigate here the Central Limit Theorem of the Increment Ratio Statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer-Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator

Modeling operating system crash behavior through multifractal analysis, long range dependence and mining of memory usage patterns

Software Aging is a phenomenon where the state of the operating systems degrades over a period of time due to transient errors. These transient errors can result in resource exhaustion and operating system hangups or crashes.;Three different techniques from fractal geometry are studied using the same datasets for operating system crash modeling and prediction. Holder Exponent is an indicator of how chaotic a signal is. M5 Prime is a nominal classification algorithm that allows prediction of a numerical quantity such as time to crash based on current and previous data. Hurst exponent measures the self similarity and long range dependence or memory of a process or data set and has been used to predict river flows and network usage.;For each of these techniques, a thorough investigation was conducted using crash, hangup and nominal operating system monitoring data. All three approaches demonstrated a promising ability to identify software aging and predict upcoming operating system crashes. This thesis describes the experiments, reports the best candidate techniques and identifies the topics for further investigation

Some Advances in Nonlinear Speech Modeling Using Modulations, Fractals, and Chaos

In this paper we briefly summarize our on-going work on modeling nonlinear structures in speech signals, caused by modulation and turbulence phenomena, using the theories of modulation, fractals, and chaos as well as suitable nonlinear signal analysis methods. Further, we focus on two advances: i) AM-FM modeling of fricative sounds with random modulation signals of the 1/f-noise type and ii) improved methods for speech analysis and prediction on reconstructed multidimensional attractors. 1

Use of wavelet-packet transforms to develop an engineering model for multifractal characterization of mutation dynamics in pathological and nonpathological gene sequences

This study uses dynamical analysis to examine in a quantitative fashion the information coding mechanism in DNA sequences. This exceeds the simple dichotomy of either modeling the mechanism by comparing DNA sequence walks as Fractal Brownian Motion (fbm) processes. The 2-D mappings of the DNA sequences for this research are from Iterated Function System (IFS) (Also known as the Chaos Game Representation (CGR)) mappings of the DNA sequences. This technique converts a 1-D sequence into a 2-D representation that preserves subsequence structure and provides a visual representation. The second step of this analysis involves the application of Wavelet Packet Transforms, a recently developed technique from the field of signal processing. A multi-fractal model is built by using wavelet transforms to estimate the Hurst exponent, H. The Hurst exponent is a non-parametric measurement of the dynamism of a system. This procedure is used to evaluate gene-coding events in the DNA sequence of cystic fibrosis mutations. The H exponent is calculated for various mutation sites in this gene. The results of this study indicate the presence of anti-persistent, random walks and persistent sub-periods in the sequence. This indicates the hypothesis of a multi-fractal model of DNA information encoding warrants further consideration.;This work examines the model\u27s behavior in both pathological (mutations) and non-pathological (healthy) base pair sequences of the cystic fibrosis gene. These mutations both natural and synthetic were introduced by computer manipulation of the original base pair text files. The results show that disease severity and system information dynamics correlate. These results have implications for genetic engineering as well as in mathematical biology. They suggest that there is scope for more multi-fractal models to be developed