scholarly journals Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 196 ◽  
Author(s):  
Jocelyn Sabatier ◽  
Christophe Farges ◽  
Vincent Tartaglione
Keyword(s):  
Power Law ◽  
Long Memory ◽  
Granular Matter ◽  
Dynamical Behaviour ◽  
Sand Pile ◽  
Alternative Models ◽  
Fractional Models ◽  
Alternative Solutions

The paper first describes a process that exhibits a power law-type long memory behaviour: the dynamical behaviour of the heap top of falling granular matter such as sand. Fractional modelling is proposed for this process, and some drawbacks and difficulties associated to fractional models are reviewed and illustrated with the sand pile process. Alternative models that solve the drawbacks and difficulties mentioned while producing power law-type long memory behaviours are presented.

scholarly journals Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems

2019 ◽  
Vol 4 (1) ◽  
pp. 1 ◽  
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Time Delay ◽  
Power Law ◽  
Long Memory ◽  
Distributed Delay ◽  
Lithium Ion ◽  
Power Laws ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Fractional Models ◽  
Distributed Time Delay

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell.

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.

Single step estimation of ARMA roots for non-fundamental nonstationary fractional models

2022 ◽  
Author(s):  
Ignacio N Lobato ◽  
Carlos Velasco
Keyword(s):  
Stock Market ◽  
Strong Evidence ◽  
Long Memory ◽  
Trading Volume ◽  
Moving Average ◽  
Single Step ◽  
Arma Process ◽  
Dow Jones Industrial Average ◽  
Fractional Models ◽  
Stock Market Trading

Abstract We propose a single step estimator for the autoregressive and moving-average roots (without imposing causality or invertibility restrictions) of a nonstationary Fractional ARMA process. These estimators employ an efficient tapering procedure, which allows for a long memory component in the process, but avoid estimating the nonstationarity component, which can be stochastic and/or deterministic. After selecting automatically the order of the model, we robustly estimate the AR and MA roots for trading volume for the thirty stocks in the Dow Jones Industrial Average Index in the last decade. Two empirical results are found. First, there is strong evidence that stock market trading volume exhibits non-fundamentalness. Second, non-causality is more common than non-invertibility.

Power-law shot noise and its relationship to long-memory α-stable processes

2000 ◽  
Vol 48 (7) ◽  
pp. 1883-1892 ◽  
Author(s):  
A.P. Petropulu ◽  
J.-C. Pesquet ◽  
Xueshi Yang
Keyword(s):  
Power Law ◽  
Shot Noise ◽  
Long Memory ◽  
Stable Processes

Random coefficient autoregression, regime switching and long memory

2003 ◽  
Vol 35 (03) ◽  
pp. 737-754 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Renewal Process ◽  
Long Memory ◽  
Regime Switching ◽  
Covariance Function ◽  
Random Coefficient ◽  
Levy Process ◽  
Partial Sums ◽  
Stable Lévy Process

We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Lévy process.

Where grains and fluids meet: the complex physics of wet granular matter

2017 ◽  
Author(s):  
S. Herminghaus
Keyword(s):  
Mechanical Properties ◽  
Energy Consumption ◽  
Raw Materials ◽  
Granular Matter ◽  
Deep Understanding ◽  
Dynamical Behaviour ◽  
Practical Significance ◽  
Granular Systems ◽  
Snow Avalanches ◽  
Granular Form

In this chapter, the physics of wet granular matter is discussed. The practical significance of wet granular matter goes of course well beyond the construction of sand sculptures. Most industrial raw materials are solids and come in granular form, and the processes into which they feed involve their being mixed with liquids and agglomerated, conveyed, kneaded, or cast in moulds. For appropriately engineering these processes, including the minimization of energy consumption, a deep understanding of the mechanical properties of this class of materials is indispensable. Furthermore, if we want to mitigate, or even reliably predict, such devastating events as land slides or mud flows, we need to study the dynamical behaviour of wet granular matter in detail. This applies as well to other, similar systems of relevance, such as ice and snow avalanches, which can be modelled as wet granular systems as well.

A Day at the Beach: Human Agents Self-Organizing on the Sand Pile

1999 ◽  
Vol 02 (01) ◽  
pp. 37-63 ◽  
Author(s):  
Hiroshi Ishii ◽  
Scott E. Page ◽  
Niniane Wang
Keyword(s):  
Power Law ◽  
Critical State ◽  
Sand Pile ◽  
Self Organized ◽  
Ability To Act ◽  
Sand Land ◽  
Social Scientific ◽  
Pile Model ◽  
Power Law Distributions ◽  
Self Organizing

In this paper, we analyze the sand pile model of self-organized criticallity from a social scientific perspective. In the sand pile model, particles of sand land at random locations on a square table and self-organize into a critical state: a conical pile. Thereafter, the size of avalanches satisfies a power law. This empirical fact has led some to claim that self-organizing criticality explains power law distributions that occur in human systems. However, unlike grains of sand, people possess both preferences and the ability to act purposefully given those preferences. We find that by including purposive agents and allowing heterogeneity of purposes, the sand pile need not become critical. We also show that if we allow institutions to moderate actions that we can create any distribution of avalanches.

Power-Law and Long-Memory Characteristics of the Atmospheric General Circulation

2009 ◽  
Vol 22 (11) ◽  
pp. 2890-2904 ◽  
Author(s):  
Dmitry I. Vyushin ◽  
Paul J. Kushner
Keyword(s):  
Climate Variability ◽  
Power Law ◽  
Long Memory ◽  
General Circulation ◽  
Fluctuation Analysis ◽  
Temperature Record ◽  
Power Law Model ◽  
Large Power ◽  
Internal Climate Variability ◽  
Power Law Function

Abstract The question of which statistical model best describes internal climate variability on interannual and longer time scales is essential to the ability to predict such variables and detect periodicities and trends in them. For over 30 yr the dominant model for background climate variability has been the autoregressive model of the first order (AR1). However, recent research has shown that some aspects of climate variability are best described by a “long memory” or “power-law” model. Such a model fits a temporal spectrum to a single power-law function, which thereby accumulates more power at lower frequencies than an AR1 fit. In this study, several power-law model estimators are applied to global temperature data from reanalysis products. The methods employed (the detrended fluctuation analysis, Geweke–Porter-Hudak estimator, Gaussian semiparametric estimator, and multitapered versions of the last two) agree well for pure power-law stochastic processes. However, for the observed temperature record, the power-law fits are sensitive to the choice of frequency range and the intrinsic filtering properties of the methods. The observational results converge once frequency ranges are made consistent and the lowest frequencies are included, and once several climate signals have been filtered. Two robust results emerge from the analysis: first, that the tropical circulation features relatively large power-law exponents that connect to the zonal-mean extratropical circulation; and second, that the subtropical lower stratosphere exhibits power-law behavior that is volcanically forced.

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

2000 ◽  
Vol 37 (4) ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document