A power-law model and other models for long-range dependence

1997 ◽  
Vol 34 (3) ◽  
pp. 657-670 ◽  
Author(s):  
R. J. Martin ◽  
A. M. Walker
Keyword(s):  
Long Range ◽  
Discrete Time ◽  
Power Law ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Power Law Model ◽  
Slow Decay ◽  
Exact Power ◽  
Correlation Structures ◽  
Discrete Time Models

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.

A power-law model and other models for long-range dependence

1997 ◽  
Vol 34 (03) ◽  
pp. 657-670 ◽  
Author(s):  
R. J. Martin ◽  
A. M. Walker
Keyword(s):  
Long Range ◽  
Discrete Time ◽  
Power Law ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Power Law Model ◽  
Slow Decay ◽  
Exact Power ◽  
Correlation Structures ◽  
Discrete Time Models

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.

Power-law correlations and other models with long-range dependence on a lattice

2003 ◽  
Vol 40 (3) ◽  
pp. 690-703 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Long Range ◽  
Power Law ◽  
Correlation Functions ◽  
Short Range ◽  
Long Range Dependence ◽  
Correlation Model ◽  
Double Sequences ◽  
Exact Power ◽  
Stationary Random Field ◽  
Dependent Correlation

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.

Power-law correlations and other models with long-range dependence on a lattice

2003 ◽  
Vol 40 (03) ◽  
pp. 690-703 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Long Range ◽  
Power Law ◽  
Correlation Functions ◽  
Short Range ◽  
Long Range Dependence ◽  
Correlation Model ◽  
Double Sequences ◽  
Exact Power ◽  
Stationary Random Field ◽  
Dependent Correlation

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.

Power-law correlations, related models for long-range dependence and their simulation

2000 ◽  
Vol 37 (04) ◽  
pp. 1104-1109 ◽  
Author(s):  
Tilmann Gneiting
Keyword(s):  
Long Range ◽  
Power Law ◽  
Long Memory ◽  
Stationary Time Series ◽  
Long Range Dependence ◽  
Short Term ◽  
Permissible Range ◽  
Simultaneous Fitting ◽  
Straightforward Proof

Martin and Walker ((1997) J. Appl. Prob. 34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

scholarly journals Variability and Confidence Intervals for the Mean of Climate Data with Short- and Long-Range Dependence

2018 ◽  
Vol 31 (15) ◽  
pp. 6135-6156 ◽  
Author(s):  
Matthew C. Bowers ◽  
Wen-wen Tung
Keyword(s):  
Confidence Intervals ◽  
Long Range ◽  
Long Memory ◽  
Long Range Dependence ◽  
Climate Data ◽  
Temperature State ◽  
Correlation Structures ◽  
The Mean ◽  
Error Bars ◽  
Mean State

This paper presents an adaptive procedure for estimating the variability and determining error bars as confidence intervals for climate mean states by accounting for both short- and long-range dependence. While the prevailing methods for quantifying the variability of climate means account for short-range dependence, they ignore long memory, which is demonstrated to lead to underestimated variability and hence artificially narrow confidence intervals. To capture both short- and long-range correlation structures, climate data are modeled as fractionally integrated autoregressive moving-average processes. The preferred model can be selected adaptively via an information criterion and a diagnostic visualization, and the estimated variability of the climate mean state can be computed directly from the chosen model. The procedure was demonstrated by determining error bars for four 30-yr means of surface temperatures observed at Potsdam, Germany, from 1896 to 2015. These error bars are roughly twice the width as those obtained using prevailing methods, which disregard long memory, leading to a substantive reinterpretation of differences among mean states of this particular dataset. Despite their increased width, the new error bars still suggest that a significant increase occurred in the mean temperature state of Potsdam from the 1896–1925 period to the most recent period, 1986–2015. The new wider error bars, therefore, communicate greater uncertainty in the mean state yet present even stronger evidence of a significant temperature increase. These results corroborate a need for more meticulous consideration of the correlation structures of climate data—especially of their long-memory properties—in assessing the variability and determining confidence intervals for their mean states.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (01) ◽  
pp. 189-220
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (1) ◽  
pp. 189-220 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

scholarly journals Estimation of marginal parameters of SUP-OU processes with long range dependence

2016 ◽  
Vol 4 (2) ◽  
pp. 102
Author(s):  
Emanuele Taufer
Keyword(s):  
Long Range ◽  
Asymptotic Theory ◽  
Marginal Distribution ◽  
Real Data ◽  
Function Technique ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Marginal Distributions ◽  
Non Gaussian ◽  
Basic Features

Superpositions of Ornstein Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non Gaussian marginal distributions. Estimation of the parameters of the marginal distribution is undertaken by means of a characteristic function technique. We provide the relevant asymptotic theory as well as results of simulations and real data applications.

On defining long-range dependence

1997 ◽  
Vol 34 (04) ◽  
pp. 939-944 ◽  
Author(s):  
C. C. Heyde ◽  
Y. Yang
Keyword(s):  
Long Range ◽  
Second Order ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stationary Processes ◽  
Domain Of Applicability

Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.

Sign in / Sign up

Export Citation Format

Share Document