Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1

1993 ◽  
Vol 30 (2) ◽  
pp. 315-329 ◽  
Author(s):  
Jean Diebolt ◽  
Dominique Guégan
Keyword(s):  
Markov Chain ◽  
Autoregressive Processes ◽  
Stationary Density ◽  
Tail Behaviour ◽  
Non Linear ◽  
The Common

We examine the main properties of the Markov chain Xt = T(Xt– 1) + σ(Xt– 1)ε t. Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the ε t's.

Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1

1993 ◽  
Vol 30 (02) ◽  
pp. 315-329 ◽  
Author(s):  
Jean Diebolt ◽  
Dominique Guégan
Keyword(s):  
Markov Chain ◽  
Autoregressive Processes ◽  
Stationary Density ◽  
Tail Behaviour ◽  
Non Linear ◽  
The Common

We examine the main properties of the Markov chain Xt = T(Xt – 1) + σ(Xt – 1)ε t . Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt } in terms of the common density of the ε t 's.

Approximations and boundary conditions for continuous-time threshold autoregressive processes

1994 ◽  
Vol 31 (4) ◽  
pp. 1103-1109 ◽  
Author(s):  
Rob J. Hyndman
Keyword(s):  
Time Series ◽  
Boundary Conditions ◽  
Continuous Time ◽  
Linear Time ◽  
Autoregressive Processes ◽  
Threshold Autoregressive ◽  
The Past ◽  
Time Threshold ◽  
Non Linear ◽  
The Way

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

scholarly journals Test sensitivity of mammography and mean sojourn time over 40 years of breast cancer screening in Nijmegen (The Netherlands)

2018 ◽  
Vol 26 (3) ◽  
pp. 147-153 ◽  
Author(s):  
AMWM Aarts ◽  
SW Duffy ◽  
SME Geurts ◽  
DP Vulkan ◽  
JDM Otten ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Cancer Screening ◽  
Linear Regression ◽  
Breast Cancer Screening ◽  
Sojourn Time ◽  
Likelihood Estimation ◽  
Non Linear ◽  
Mean Sojourn Time

Objectives We investigated whether changes in mammographic technique and screening policy have improved mammographic sensitivity, and elongated the mean sojourn time, since the introduction of biennial breast cancer screening in Nijmegen, the Netherlands, in 1975. Methods Maximum likelihood estimation, non-linear regression, and Markov Chain Monte Carlo simulation were used to estimate test sensitivity, mean sojourn time, and underlying breast cancer incidence in four time periods, covering 40 years of breast cancer screening in Nijmegen (1975–2012). Results Maximum likelihood estimation generated an estimated test sensitivity of approximately 90% and a mean sojourn time around three years, while the estimates based on non-linear regression and Markov Chain Monte Carlo simulation were 80% and four years, respectively. All three methods estimated a rise in the underlying breast cancer incidence over time, with approximately one case more per 1000 women per year in the final period compared with the first period. Conclusions The three methods showed a slightly higher mammographic sensitivity and a longer mean sojourn time in the last period, after the introduction of digital mammography. Estimates were more realistic for the more sophisticated methods, non-linear regression and Markov Chain Monte Carlo simulation, while the simple closed form approximation of maximum likelihood estimation led to rather high estimates for sensitivity in the early periods.

The extremal index for a Markov chain

1992 ◽  
Vol 29 (01) ◽  
pp. 37-45 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Extreme Values ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extremal Index ◽  
Scaling Properties ◽  
Continuous State Space ◽  
Continuous State ◽  
Tail Behaviour

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.

Variational Markov chain Monte Carlo for Bayesian smoothing of non-linear diffusions

2011 ◽  
Vol 27 (1) ◽  
pp. 149-176 ◽  
Author(s):  
Yuan Shen ◽  
Dan Cornford ◽  
Manfred Opper ◽  
Cedric Archambeau
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Non Linear ◽  
Linear Diffusions ◽  
Bayesian Smoothing

scholarly journals Rationalitetens indfoldning. Rumopfattelsen i Peter Eisenmans arkitektur

2019 ◽  
Author(s):  
Claus Peder Pedersen
Keyword(s):  
Linear Space ◽  
Gilles Deleuze ◽  
Cultural Space ◽  
Peter Eisenman ◽  
Non Linear ◽  
The Common ◽  
Contemporary Approach ◽  
Pluralistic World

The article considers a contemporary approach to architecture in a pluralistic world. This is done through a discussion of the work of the American architect Peter Eisenman. In his critique of objectivist spatial categories, Eisenman articulates an architecture based on a folded space. Parallels are drawn to the French philosopher Gilles Deleuze and his considerations of Leibniz’s philosophy of folds. The article relates Peter Eisenman’s highly individual architectural method to the common cultural space of architecture. It concludes that Eisenman’s attempt to form non-rationalistic spatial and architectural categories functions only on a metaphorical level. Finally the article describes other approaches to folded and non-linear space, including theories developed by the architect Greg Lynn.

scholarly journals Earthquake sequencing: Chimera states with Kuramoto model dynamics on directed graphs

2015 ◽  
Vol 2 (1) ◽  
pp. 361-398
Author(s):  
K. Vasudevan ◽  
M. Cavers ◽  
A. Ware
Keyword(s):  
Markov Chain ◽  
Stress Field ◽  
Directed Graph ◽  
Directed Graphs ◽  
Kuramoto Model ◽  
Chimera States ◽  
Weakly Coupled ◽  
Non Linear ◽  
Field Fluctuations ◽  
The Kuramoto Model

Abstract. Earthquake sequencing studies allow us to investigate empirical relationships among spatio-temporal parameters describing the complexity of earthquake properties. We have recently studied the relevance of Markov chain models to draw information from global earthquake catalogues. In these studies, we considered directed graphs as graph theoretic representations of the Markov chain model, and analyzed their properties. Here, we look at earthquake sequencing itself as a directed graph. In general, earthquakes are occurrences resulting from significant stress-interactions among faults. As a result, stress-field fluctuations evolve continuously. We propose that they are akin to the dynamics of the collective behaviour of weakly-coupled non-linear oscillators. Since mapping of global stress-field fluctuations in real time at all scales is an impossible task, we consider an earthquake zone as a proxy for a collection of weakly-coupled oscillators, the dynamics of which would be appropriate for the ubiquitous Kuramoto model. In the present work, we apply the Kuramoto model to the non-linear dynamics on a directed graph of a sequence of earthquakes. For directed graphs with certain properties, the Kuramoto model yields synchronization, and inclusion of non-local effects evokes the occurrence of chimera states or the co-existence of synchronous and asynchronous behaviour of oscillators. In this paper, we show how we build the directed graphs derived from global seismicity data. Then, we present conditions under which chimera states could occur and subsequently, point out the role of Kuramoto model in understanding the evolution of synchronous and asynchronous regions.

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