Generalized Mittag-Leffler Distributions and Processes for Applications in Astrophysics and Time Series Modeling

Geometric generalized Mittag-Leffler distributions having the Laplace transform $\frac{1}{1+\beta\log(1+t^\alpha)},00$ is introduced and its properties are discussed. Autoregressive processes with Mittag-Leffler and geometric generalized Mittag-Leffler marginal distributions are developed. Haubold and Mathai (2000) derived a closed form representation of the fractional kinetic equation and thermonuclear function in terms of Mittag-Leffler function. Saxena et al (2002, 2004a,b) extended the result and derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions. These results are useful in explaining various fundamental laws of physics. Here we develop first-order autoregressive time series models and the properties are explored. The results have applications in various areas like astrophysics, space sciences, meteorology, financial modeling and reliability modeling.Comment: 12 pages, LaTe

A characterization of random-coefficient AR(1) models

AbstractWe give a characterization of random-coefficient autoregressive processes of order 1, using analytical properties of the transition probabilities. As an example we show that these transition probabilities can be used to find solutions of certain integro-differential equations

Study and simulation of low rate video coding schemes

The semiannual report is included. Topics covered include communication, information science, data compression, remote sensing, color mapped images, robust coding scheme for packet video, recursively indexed differential pulse code modulation, image compression technique for use on token ring networks, and joint source/channel coder design

Synthetic steganography: Methods for generating and detecting covert channels in generated media

Issues of privacy in communication are becoming increasingly important. For many people and businesses, the use of strong cryptographic protocols is sufficient to protect their communications. However, the overt use of strong cryptography may be prohibited or individual entities may be prohibited from communicating directly. In these cases, a secure alternative to the overt use of strong cryptography is required. One promising alternative is to hide the use of cryptography by transforming ciphertext into innocuous-seeming messages to be transmitted in the clear. ^ In this dissertation, we consider the problem of synthetic steganography: generating and detecting covert channels in generated media. We start by demonstrating how to generate synthetic time series data that not only mimic an authentic source of the data, but also hide data at any of several different locations in the reversible generation process. We then design a steganographic context-sensitive tiling system capable of hiding secret data in a variety of procedurally-generated multimedia objects. Next, we show how to securely hide data in the structure of a Huffman tree without affecting the length of the codes. Next, we present a method for hiding data in Sudoku puzzles, both in the solved board and the clue configuration. Finally, we present a general framework for exploiting steganographic capacity in structured interactions like online multiplayer games, network protocols, auctions, and negotiations. Recognizing that structured interactions represent a vast field of novel media for steganography, we also design and implement an open-source extensible software testbed for analyzing steganographic interactions and use it to measure the steganographic capacity of several classic games. ^ We analyze the steganographic capacity and security of each method that we present and show that existing steganalysis techniques cannot accurately detect the usage of the covert channels. We develop targeted steganalysis techniques which improve detection accuracy and then use the insights gained from those methods to improve the security of the steganographic systems. We find that secure synthetic steganography, and accurate steganalysis thereof, depends on having access to an accurate model of the cover media

Modelling Multiple and Structured Oscillatory Phenomena

This work deals with the modelling of multiple and structured oscillatory phenomena. The goal of the thesis is to show how stochastic oscillations can be modelled, and define their elliptical structures as a special class of bivariate time-dependant variation. The central part of the research is the introduction of new multivariate elliptical models and the review of existing definitions. The findings are presented in a table, where the classification is made based on whether the definitions are random or deterministic and whether they are defined in time or frequency domains. The previously introduced ellipse definitions for stochastic processes that have been described in the literature are limited to the frequency domain only. The main contribution of this work is in adding to existing time domain models by defining the description of the autocovariance ellipse and the forecast ellipse. Both of these definitions are non-random. The ellipses are defined from either the autocovariance or the forecast functions of the process as one moves forward in lag-time or forecast-time. In order to illustrate these theoretical concepts and show the usefulness of the new definition we investigate these concepts using a parametric model. Univariate and bivariate, real-valued and complex-valued models are considered, and their representation discussed. The richest model proposed is that of a complex-valued bivariate autoregressive process of order one and this is based on modelling using affine transformation matrices. This model results in a stochastic oscillation and the elliptical definitions proposed are explored in this context. The actual behaviour of the proposed stochastic process is also illustrated on simulated data. Some limitations of this approach are discussed and extensions of this model are presented

A New Laplace Second-Order Autoregressive Time-Series Model-NLAR (2)

The article of record as published may be found at https://doi.org/10.1109/TIT.1985.1057089A time-series model for Laplace (double-exponential) variables having second-order autoregressive structure (NLAR(2)) is presented. The model is Markovian and extends the second-order process in exponential variables, NEAR(2), to the case where the marginal distribution is Laplace. The properties of the Laplace distribution make it useful for modeling in some cases where the normal distribution is not appropriate. The time-series model has four parameters and is easily simulated. The autocorrelation function for the process is derived as well as third-order moments to further explore dependency in the process. The model can exhibit a broad range of positive and negative correlations and is partially time reversible. Joint distributions and the distribution of differences are presented for the first-order case NLAR(1)

A New Second-Order Autoregressive Time-Series Model in Double Exponential (LAPLACE) Variables -NLAR (2)

A time-series model for Laplace (double-exponential) variables having second-order autoregressive structure (NLAR(2)) is presented. The model is Markovian and extends the second-order process in exponential variables, NEAR(2), to the case where the marginal distribution is Laplace. The properties of the Laplace distribution make it useful for modeling in some cases where the normal distribution is not appropriate. The time-series model has four parameters and is easily simulated. The autocorrelation function for the process is derived as well as third-order moments to further explore dependency in the process. The model can exhibit a broad range of positive and negative correlations and is partially time reversible.Office of Naval ResearchNR-42-284Approved for public release; distribution is unlimited