Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

1992 ◽  
Vol 29 (4) ◽  
pp. 896-903 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Time Series ◽  
Random Number ◽  
Autoregressive Process ◽  
Noise Component ◽  
Random Number Generators ◽  
Order Linear ◽  
First Order ◽  
Uniformly Distributed Sequence ◽  
Non Linear ◽  
Scatter Plots

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk-valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators

1992 ◽  
Vol 29 (04) ◽  
pp. 896-903 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Time Series ◽  
Random Number ◽  
Autoregressive Process ◽  
Noise Component ◽  
Random Number Generators ◽  
Order Linear ◽  
First Order ◽  
Uniformly Distributed Sequence ◽  
Non Linear ◽  
Scatter Plots

The work is concerned with the first-order linear autoregressive process which has a rectangular stationary marginal distribution. A derivation is given of the result that the time-reversed version is deterministic, with a first-order recursion function of the type used in multiplicative congruential random number generators, scaled to the unit interval. The uniformly distributed sequence generated is chaotic, giving an instance of a chaotic process which when reversed has a linear causal and non-chaotic structure. An mk -valued discrete process is then introduced which resembles a first-order linear autoregressive model and uses k-adic arithmetic. It is a particular form of moving-average process, and when reversed approximates in m a non-linear discrete-valued process which has the congruential generator function as its deterministic part, plus a discrete-valued noise component. The process is illustrated by scatter plots of adjacent values, time series plots and directed scatter plots (phase diagrams). The behaviour very much depends on the adic number, with k = 2 being very distinctly non-linear and k = 10 being virtually indistinguishable from independence.

Linear Stochastic Difference Equations

2013 ◽  
Author(s):  
Lars Peter Hansen ◽  
Thomas J. Sargent
Keyword(s):  
Time Series ◽  
Difference Equation ◽  
Difference Equations ◽  
Martingale Difference ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
Order Linear ◽  
Economic Agents ◽  
First Order ◽  
Competitive Equilibria

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive equilibria. The vector first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric application. Ease of analysis has prompted the adoption of economic specifications that cause competitive equilibria to have representations as vector first-order linear stochastic difference equations. Because it expresses next period's vector of state variables as a linear function of this period's state vector and a vector of random disturbances, a vector first-order vector stochastic difference equation is recursive. Disturbances that form a “martingale difference sequence” are basic building blocks used to construct time series. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions of time series.

scholarly journals Quantifiers for randomness of chaotic pseudo-random number generators

2009 ◽  
Vol 367 (1901) ◽  
pp. 3281-3296 ◽  
Author(s):  
L. De Micco ◽  
H. A. Larrondo ◽  
A. Plastino ◽  
O. A. Rosso
Keyword(s):  
Time Series ◽  
Comparative Analysis ◽  
Invariant Measure ◽  
Random Number ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
General Purpose ◽  
Statistical Characteristics ◽  
Random Number Generators ◽  
Pseudo Random Number

We deal with randomness quantifiers and concentrate on their ability to discern the hallmark of chaos in time series used in connection with pseudo-random number generators (PRNGs). Workers in the field are motivated to use chaotic maps for generating PRNGs because of the simplicity of their implementation. Although there exist very efficient general-purpose benchmarks for testing PRNGs, we feel that the analysis provided here sheds additional didactic light on the importance of the main statistical characteristics of a chaotic map, namely (i) its invariant measure and (ii) the mixing constant. This is of help in answering two questions that arise in applications: (i) which is the best PRNG among the available ones? and (ii) if a given PRNG turns out not to be good enough and a randomization procedure must still be applied to it, which is the best applicable randomization procedure? Our answer provides a comparative analysis of several quantifiers advanced in the extant literature.

Product autoregression: a time-series characterization of the gamma distribution

1982 ◽  
Vol 19 (2) ◽  
pp. 463-468 ◽  
Author(s):  
Ed Mckenzie
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Gamma Distribution ◽  
Stationary Stochastic Process ◽  
Correlation Structure ◽  
First Order ◽  
Non Linear ◽  
Autoregressive Correlation

A non-linear stationary stochastic process {Xt} is derived and shown to have the property that both the processes {Xt} and {log Xt} have the same correlation structure, viz. the Markov or first-order autoregressive correlation structure. The generation of such processes is discussed briefly and a characterization of the gamma distribution is obtained.

scholarly journals A new approach to model the counts of earthquakes: INARPQX(1) process

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Emrah Altun ◽  
Deepesh Bhati ◽  
Naushad Mamode Khan
Keyword(s):  
Time Series ◽  
Estimation Procedure ◽  
Autoregressive Process ◽  
Estimation Methods ◽  
Important Process ◽  
New Approach ◽  
Time Series Of Counts ◽  
First Order ◽  
Conditional Maximum ◽  
New Distribution

AbstractThis paper introduces a first-order integer-valued autoregressive process with a new innovation distribution, shortly INARPQX(1) process. A new innovation distribution is obtained by mixing Poisson distribution with quasi-xgamma distribution. The statistical properties and estimation procedure of a new distribution are studied in detail. The parameter estimation of INARPQX(1) process is discussed with two estimation methods: conditional maximum likelihood and Yule-Walker. The proposed INARPQX(1) process is applied to time series of the monthly counts of earthquakes. The empirical results show that INARPQX(1) process is an important process to model over-dispersed time series of counts and can be used to predict the number of earthquakes with a magnitude greater than four.

scholarly journals Time Series Analysis: A New Methodology for Comparing the Temporal Variability of Air Temperature

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Piia Post ◽  
Olavi Kärner
Keyword(s):  
Time Series ◽  
Random Walk ◽  
Temporal Variability ◽  
Temperature Time Series ◽  
Noise Component ◽  
First Order ◽  
Global Era ◽  
Temperature Time ◽  
White Noises ◽  
Different Levels

Temporal variability of three different temperature time series was compared by the use of statistical modeling of time series. The three temperature time series represent the same physical process, but are at different levels of spatial averaging: temperatures from point measurements, from regional Baltan65+, and from global ERA-40 reanalyses. The first order integrated average model IMA(0, 1, 1) is used to compare the temporal variability of the time series. The applied IMA(0, 1, 1) model is divisible into a sum of random walk and white noise component, where the variances for both white noises (one of them serving as a generator of the random walk) are computable from the parameters of the fitted model. This approach enables us to compare the models fitted independently to the original and restored series using two new parameters. This operation adds a certain new method to the analysis of nonstationary series.

scholarly journals Random Number Generators Based on EEG Non-linear and Chaotic Characteristics

2017 ◽  
Author(s):  
Dang Nguyen ◽  
Dat Tran ◽  
Wanli Ma ◽  
Dharmendra Sharma
Keyword(s):  
Random Number ◽  
Brain Activity ◽  
Eeg Signal ◽  
Success Rates ◽  
Random Number Generators ◽  
Non Linear ◽  
Number Sequences ◽  
Level Of Significance ◽  
High Level

Current electroencephalogram (EEG)-based methods in security have been mainly used for person authentication and identification purposes only. The non-linear and chaotic characteristics of EEG signal have not been taken into account. In this paper, we propose a new method that explores the use of these EEG characteristics in generating random numbers. EEG signal and its wavebands are transformed into bit sequences that are used as random number sequences or as seeds for pseudo-random number generators. EEG signal has the following advantages: 1) it is noisy, complex, chaotic and non-linear in nature, 2) it is very difficult to mimic because similar mental tasks are person dependent, and 3) it is almost impossible to steal because the brain activity is sensitive to the stress and the mood of the person and an aggressor cannot force the person to reproduce his/her mental pass-phrase. Our experiments were conducted on the four EEG datasets: AEEG, Alcoholism, DEAP and GrazA 2008. The randomness of the generated bit sequences was tested at a high level of significance by comprehensive battery of tests recommended by the National Institute of Standard and Technology (NIST) to verify the quality of random number generators, especially in cryptography application. Our experimental results showed high average success rates for all wavebands and the highest rate is 99.17% for the gamma band.  

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