Stilian Stoev

On the wavelet spectrum diagnostic for Hurst parameter estimation in the analysis of Internet traffic

The fluctuations of Internet traffic possess an intricate structure which cannot be simply explained by long-range dependence and self-similarity. In this work, we explore the use of the wavelet spectrum, whose slope is commonly used to estimate the Hurst parameter of long-range dependence. We show that much more than simple slope estimates are needed for detecting important traffic features. In particular, the multi-scale nature of the traffic does not admit simple description of the type attempted by the Hurst parameter. By using simulated examples, we demonstrate the causes of a number of interesting effects in the wavelet spectrum of the data. This analysis leads us to a better understanding of several challenging phenomena observed in real network traffic. Although the wavelet analysis is robust to many smooth trends, high-frequency oscillations and non-stationarities such as abrupt changes in the mean have an important effect. In particular, the breaks and level-shifts in the local mean of the traffic rate can lead one to overestimate the Hurst parameter of the time series. Novel statistical techniques are required to address such issues in practice.

Simulation methods for linear fractional stable motion and farima using the fast fourier transform

We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks. We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error. We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustr...

LASS: a tool for the local analysis of self-similarity

The Hurst parameter H characterizes the degree of long-range dependence (and asymptotic self-similarity) in stationary time series. Many methods have been developed for the estimation of H from data. In practice, however, the classical estimation techniques can be severely affected by non-stationary artifacts in the time series. In fact, the assumption that the data can be modeled by a stationary process with a single Hurst exponent H may be unrealistic. This work focuses on practical issues associated with the detection of long-range dependence in Internet traffic data and proposes two tools that can be used to address some of these issues. The first is an animation tool which is used to visualize the local dependence structure. The second is a statistical tool for the local analysis of self-similarity (LASS). The LASS tool is designed to handle time series that have long-range dependence and are long enough that some parts are essentially stationary, while others exhibit non-stationarity, which is either deterministic or stochastic in nature. The tool exploits wavelets to analyze the local dependence structure in the data over a set of windows. It can be used to visualize local deviations from self-similar, long-range dependence scaling and to provide reliable local estimates of the Hurst exponents. The tool, which is illustrated by using a trace of Internet traffic measurements, can also be applied to economic time series. In addition, a median-based wavelet spectrum is introduced. It yields robust local or global estimates of the Hurst parameter that are less susceptible to local non-stationarity. The software tools are freely available and their use is described in an appendix.

On the structure and representations of max-stable processes

We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles. We propose a general classification strategy for measurable max-stable processes based on the notion of co-spectral functions. In particular, we discuss the spectrally continuous-discrete, the conservative-dissipative, and the positive-null decompositions. For stationary max-stable processes, the latter two decompositions arise from connections to nonsingular flows and are closely related to the classification of stationary sum-stable processes. The interplay between the introduced decompositions of max-stable processes is further explored. As an example, the Brown-Resnick stationary processes, driven by fractional Brownian motions, are shown to be dissipative.

Conditional sampling for spectrally discrete max-stable random fields

Max-stable random fields play a central role in modeling extreme value phenomena. We obtain an explicit formula for the conditional probability in general max-linear models, which include a large class of max-stable random fields. As a consequence, we develop an algorithm for efficient and exact sampling from the conditional distributions. Our method provides a computational solution to the prediction problem for spectrally discrete max-stable random fields. This work offers new tools and a new perspective to many statistical inference problems for spatial extremes, arising, for example, in meteorology, geology, and environmental applications.

Stochastic properties of the linear multifractional stable motion

We study a family of locally self-similar stochastic processes Y = {Y(t)} t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

Estimating Heavy-Tail Exponents Through Max Self–Similarity

In this paper, a novel approach to the problem of estimating the heavy-tail exponent α >; 0 of a distribution is proposed. It is based on the fact that block-maxima of size m scale at a rate m1/α for independent, as well as for a number of dependent data. This scaling rate can be captured well by the max-spectrum plot of the data that leads to regression based estimators for α. Consistency and asymptotic normality of these estimators is established for independent data under mild conditions on the behavior of the tail of the distribution. The proposed estimators have an important computational advantage over existing methods; namely, they can be calculated and updated sequentially in an on-line fashion without having to store the entire data set. Practical issues on the automatic selection of tuning parameters for the estimators and corresponding confidence intervals are also addressed. Extensive numerical simulations show that the proposed method is competitive for both small and large sample sizes, robust to contaminants and continues to work under the presence of substantial amount of dependence. The proposed estimators are used to illustrate the close connection between long-range dependence and heavy tails over an Internet traffic trace.

Visualization and inference based on wavelet coefficients, SiZer and SiNos

SiZer (SIgnificant ZERo crossing of the derivatives) and SiNos (SIgnificant NOn-Stationarities) are scale-space based visualization tools for statistical inference. They are used to discover meaningful structure in data through exploratory analysis involving statistical smoothing techniques. Wavelet methods have been successfully used to analyze various types of time series. In this paper, we propose a new time series analysis approach, which combines the wavelet analysis with the visualization tools SiZer and SiNos. We use certain functions of wavelet coefficients at different scales as inputs, and then apply SiZer or SiNos to highlight potential non-stationarities. We show that this new methodology can reveal hidden local non-stationary behavior of time series, that are otherwise difficult to detect.

Path Properties of the Linear Multifractional Stable Motion

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t). We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Holder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Holder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Holder exponents. The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Holder regularity of their paths.

Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions

We establish characterization results for the ergodicity of stationary symmetric