scholarly journals Rarefied particle motions on hillslopes – Part 3: Entropy

2021 ◽  
Vol 9 (3) ◽  
pp. 615-628
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Individual Particle ◽  
Energetic Cost ◽  
Isothermal Conditions ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
Travel Distances

Abstract. Theoretical and experimental work (Furbish et al., 2021a, b) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating, due to conversion of potential to kinetic energy, and frictional cooling, due to particle–surface collisions; it varies from a bounded form associated with rapid thermal collapse to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates – the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost – the generalized Pareto distribution represents the most probable arrangement. Because this idea applies equally to the accessible microstates associated with net cooling, isothermal conditions and net heating, the fixed energetic cost provides a unifying interpretation for these distinctive behaviors, including the abrupt transition in the form of the generalized Pareto distribution in crossing isothermal conditions. The analysis therefore represents a novel generalization of an energy-based constraint in using the maximum entropy method to infer non-exponential distributions of particle motions. Moreover, the energetic costs of individual particle motions follow an extreme-value distribution that is heavy-tailed for net cooling and light-tailed for net heating. The relative contribution of different travel distances to the total energetic cost is reflected by the product of the travel distance distribution and the cost of individual particle motions – effectively a frequency–magnitude product.

scholarly journals Rarefied particle motions on hillslopes: 3. Entropy

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Individual Particle ◽  
Energetic Cost ◽  
Isothermal Conditions ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
Travel Distances

Abstract. Theoretical and experimental work (Furbish et al., 2020a, 2020b) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating, due to conversion of potential to kinetic energy, and frictional cooling, due to particle-surface collisions; it varies from a bounded form associated with rapid thermal collapse to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic cost – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates – the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost – the generalized Pareto distribution represents the most probable arrangement. Because this idea applies equally to the accessible microstates associated with net cooling, isothermal conditions and net heating, the fixed energetic cost provides a unifying interpretation for these distinctive behaviors, including the abrupt transition in the form of the generalized Pareto distribution in crossing isothermal conditions. The analysis therefore represents a novel generalization of an energy-based constraint in using the maximum entropy method to infer non-exponential distributions of particle motions. Moreover, the energetic costs of individual particle motions follow an extreme-value distribution that is heavy-tailed for net cooling and light-tailed for net heating. The relative contribution of different travel distances to the total energetic cost is reflected by the product of the travel distance distribution and the cost of individual particle motions – effectively a frequency-magnitude product.

scholarly journals Rarefied particle motions on hillslopes: 4. Philosophy

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Particle Energy ◽  
Controlling Factors ◽  
Rate Of Change ◽  
Energetic Cost ◽  
Generalized Pareto

Abstract. Theoretical and experimental work (Furbish et al., 2020a, 2020b, 2020c) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle-surface collisions. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic cost – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. The analyses leading to these results provide an ideal case study for highlighting three key elements of a statistical mechanics framework for describing sediment particle motions and transport: the merits of probabilistic versus deterministic descriptions of sediment motions; the implications of rarefied versus continuum transport conditions; and the consequences of increasing uncertainty in descriptions of sediment motions and transport that accompany increasing length and time scales. We use the analyses of particle energy extraction, the spatial evolution of particle energy states, and the maximum entropy method applied to the generalized Pareto distribution as examples to illustrate the mechanistic yet probabilistic nature of the approach. These examples highlight the idea that the endeavor is not simply about adopting theory or methods of statistical mechanics off the shelf, but rather involves appealing to the style of thinking of statistical mechanics while tailoring the analysis to the process and scale of interest. Under rarefied conditions, descriptions of the particle flux and its divergence pertain to ensemble conditions involving a distribution of possible outcomes, each realization being compatible with the controlling factors. When these factors change over time, individual outcomes reflect a legacy of earlier conditions that depends on the rate of change in the controlling factors relative to the intermittency of particle motions. The implication is that landform configurations and associated particle fluxes reflect an inherent variability (weather) that is just as important as the expected (climate) conditions in characterizing system behavior.

scholarly journals Rarefied particle motions on hillslopes – Part 4: Philosophy

2021 ◽  
Vol 9 (3) ◽  
pp. 629-664
Author(s):  
David Jon Furbish ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Particle Energy ◽  
Controlling Factors ◽  
Rate Of Change ◽  
Energetic Cost ◽  
Generalized Pareto

Abstract. Theoretical and experimental work (Furbish et al., 2021a, b, c) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle–surface collisions. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. The analyses leading to these results provide an ideal case study for highlighting three key elements of a statistical mechanics framework for describing sediment particle motions and transport: the merits of probabilistic versus deterministic descriptions of sediment motions, the implications of rarefied versus continuum transport conditions, and the consequences of increasing uncertainty in descriptions of sediment motions and transport that accompany increasing length scales and timescales. We use the analyses of particle energy extraction, the spatial evolution of particle energy states, and the maximum entropy method applied to the generalized Pareto distribution as examples to illustrate the mechanistic yet probabilistic nature of the approach. These examples highlight the idea that the endeavor is not simply about adopting theory or methods of statistical mechanics “off the shelf” but rather involves appealing to the style of thinking of statistical mechanics while tailoring the analysis to the process and scale of interest. Under rarefied conditions, descriptions of the particle flux and its divergence pertain to ensemble conditions involving a distribution of possible outcomes, each realization being compatible with the controlling factors. When these factors change over time, individual outcomes reflect a legacy of earlier conditions that depends on the rate of change in the controlling factors relative to the intermittency of particle motions. The implication is that landform configurations and associated particle fluxes reflect an inherent variability (“weather”) that is just as important as the expected (“climate”) conditions in characterizing system behavior.

scholarly journals Rarefied particle motions on hillslopes – Part 1: Theory

2021 ◽  
Vol 9 (3) ◽  
pp. 539-576
Author(s):  
David Jon Furbish ◽  
Joshua J. Roering ◽  
Tyler H. Doane ◽  
Danica L. Roth ◽  
Sarah G. W. Williams ◽  
...  
Keyword(s):  
Kinetic Energy ◽  
Particle Surface ◽  
Generalized Pareto Distribution ◽  
Planck Equation ◽  
Particle Energy ◽  
Finite Variance ◽  
Isothermal Conditions ◽  
Scale Parameters ◽  
Size Sorting ◽  
Heavy Tailed

Abstract. We describe the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle–surface collisions, and an apparent heating associated with preferential deposition of low-energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states whose spatial evolution is described by a Fokker–Planck equation. The Kirkby number Ki – defined as the ratio of gravitational heating to frictional cooling – sets the basic deposition behavior and the form of the probability distribution fr(r) of particle travel distances r, a generalized Pareto distribution. The shape and scale parameters of the distribution are well-defined mechanically. For isothermal conditions where frictional cooling matches gravitational heating plus the apparent heating due to deposition, the distribution fr(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution fr(r) becomes heavy-tailed and represents net particle heating. It may possess a finite mean and finite variance, or the mean and variance may be undefined with sufficiently large Ki. The formulation provides key elements of the entrainment forms of the particle flux and the Exner equation, and it clarifies the mechanisms of particle-size sorting on large talus and scree slopes. Namely, with conversion of translational to rotational kinetic energy, large spinning particles are less likely to be stopped by collisional friction than are small or angular particles for the same surface roughness.

scholarly journals The Degree Distribution of Human Brain Functional Connectivity is Generalized Pareto: A Multi-Scale Analysis

2019 ◽  
Author(s):  
Riccardo Zucca ◽  
Xerxes D. Arsiwalla ◽  
Hoang Le ◽  
Mikail Rubinov ◽  
Antoni Gurguí ◽  
...  
Keyword(s):  
Functional Connectivity ◽  
Human Brain ◽  
Power Law ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Cross Model ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
And Function ◽  
Brain Functional Connectivity

ABSTRACTAre degree distributions of human brain functional connectivity networks heavy-tailed? Initial claims based on least-square fitting suggested that brain functional connectivity networks obey power law scaling in their degree distributions. This interpretation has been challenged on methodological grounds. Subsequently, estimators based on maximum-likelihood and non-parametric tests involving surrogate data have been proposed. No clear consensus has emerged as results especially depended on data resolution. To identify the underlying topological distribution of brain functional connectivity calls for a closer examination of the relationship between resolution and statistics of model fitting. In this study, we analyze high-resolution functional magnetic resonance imaging (fMRI) data from the Human Connectome Project to assess its degree distribution across resolutions. We consider resolutions from one thousand to eighty thousand regions of interest (ROIs) and test whether they follow a heavy or short-tailed distribution. We analyze power law, exponential, truncated power law, log-normal, Weibull and generalized Pareto probability distributions. Notably, the Generalized Pareto distribution is of particular interest since it interpolates between heavy-tailed and short-tailed distributions, and it provides a handle on estimating the tail’s heaviness or shortness directly from the data. Our results show that the statistics support the short-tailed limit of the generalized Pareto distribution, rather than a power law or any other heavy-tailed distribution. Working across resolutions of the data and performing cross-model comparisons, we further establish the overall robustness of the generalized Pareto model in explaining the data. Moreover, we account for earlier ambiguities by showing that down-sampling the data systematically affects statistical results. At lower resolutions models cannot easily be differentiated on statistical grounds while their plausibility consistently increases up to an upper bound. Indeed, more power law distributions are reported at low resolutions (5K) than at higher ones (50K or 80K). However, we show that these positive identifications at low resolutions fail cross-model comparisons and that down-sampling data introduces the risk of detecting spurious heavy-tailed distributions. This dependence of the statistics of degree distributions on sampling resolution has broader implications for neuroinformatic methodology, especially, when several analyses rely on down-sampled data, for instance, due to a choice of anatomical parcellations or measurement technique. Our findings that node degrees of human brain functional networks follow a short-tailed distribution have important implications for claims of brain organization and function. Our findings do not support common simplistic representations of the brain as a generic complex system with optimally efficient architecture and function, modeled with simple growth mechanisms. Instead these findings reflect a more nuanced picture of a biological system that has been shaped by longstanding and pervasive developmental and architectural constraints, including wiring-cost constraints on the centrality architecture of individual nodes.

scholarly journals EXPECTED SHORTFALL DENGAN PENDEKATAN GLOSTEN-JAGANNATHAN-RUNKLE GARCH DAN GENERALIZED PARETO DISTRIBUTION

2020 ◽  
Vol 9 (4) ◽  
pp. 505-514
Author(s):  
Lina Tanasya ◽  
Di Asih I Maruddani ◽  
Tarno Tarno
Keyword(s):  
Time Series ◽  
Stock Return ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Garch Model ◽  
Value Theory ◽  
Expected Shortfall ◽  
Series Data ◽  
Generalized Pareto ◽  
Heavy Tailed

Stock is a type of investment in financial assets that are many interested by investors. When investing, investors must calculate the expected return on stocks and notice risks that will occur. There are several methods can be used to measure the level of risk one of which is Value at Risk (VaR), but these method often doesn’t fulfill coherence as a risk measure because it doesn’t fulfill the nature of subadditivity. Therefore, the Expected Shortfall (ES) method is used to accommodate these weakness. Stock return data is time series data which has heteroscedasticity and heavy tailed, so time series models used to overcome the problem of heteroscedasticity is GARCH, while the theory for analyzing heavy tailed is Extreme Value Theory (EVT). In this study, there is also a leverage effect so used the asymmetric GARCH model with Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model and the EVT theory with Generalized Pareto Distribution (GPD) to calculate ES of the stock return from PT. Bank Central Asia Tbk for the period May 1, 2012-January 31, 2020. The best model chosen was ARIMA(1,0,1) GJR-GARCH(1,2). At the 95% confidence level, the risk obtained by investors using a combination of GJR-GARCH and GPD calculations for the next day is 0.7147% exceeding the VaR value of 0.6925%. 

scholarly journals GENERALIZED PARETO DISTRIBUTION UNTUK PENGUKURAN VALUE AT RISK PADA PORTOFOLIO SAHAM SYARIAH DAN APLIKASINYA MENGGUNAKAN GUI MATLAB

2018 ◽  
Vol 7 (3) ◽  
pp. 224-235
Author(s):  
Desi Nur Rahma ◽  
Di Asih I Maruddani ◽  
Tarno Tarno
Keyword(s):  
At Risk ◽  
Capital Market ◽  
Value At Risk ◽  
Pareto Distribution ◽  
Extreme Values ◽  
Generalized Pareto Distribution ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
Is Value

The capital market is one of long-term investment alternative. One of the traded products is stock, including sharia stock. The risk measurement is an important thing for investor in other that can decrease investment loss. One of the popular methods now is Value at Risk (VaR). There are many financial data that have heavy tailed, because of extreme values, so Value at Risk Generalized Pareto Distribution is used for this case. This research also result a Matlab GUI programming application that can help users to measure the VaR. The purpose of this research is to analyze VaR with GPD approach with GUI Matlab for helping the computation in sharia stock. The data that is used in this case are PT XL Axiata Tbk, PT Waskita Karya (Persero) Tbk, dan PT Charoen Pokphand Indonesia Tbk on January, 2nd 2017 until May, 31st 2017. The results of VaRGPD are: EXCL single stock VaR 8,76% of investment, WSKT single stock VaR 4% of investment, CPIN single stock VaR 5,86% of investment, 2 assets portfolio (EXCL and WSKT) 4,09% of investment, 2 assets portfolio (EXCL and CPIN) 5,28% of investment, 2 assets portfolio (WSKT and CPIN) 3,68% of investment, and 3 assets portfolio (EXCL, WSKT, and CPIN) 3,75% of investment. It can be concluded that the portfolios more and more, the risk is smaller. It is because the possibility of all stocks of the company dropped together is small. Keywords: Generalized Pareto Distribution, Value at Risk, Graphical User Interface, sharia stock

Heavy-tailed features and dependence in limit order book volume profiles in futures markets

2015 ◽  
Vol 02 (03) ◽  
pp. 1550033 ◽  
Author(s):  
Kylie-Anne Richards ◽  
Gareth W. Peters ◽  
William Dunsmuir
Keyword(s):  
Heavy Tails ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Limit Order Book ◽  
Order Book ◽  
Limit Order ◽  
Generalized Pareto ◽  
Futures Exchanges ◽  
Asset Class ◽  
Heavy Tailed

This paper investigates fundamental stochastic attributes of the random structures of the volume profiles of the limit order book. We find statistical evidence that heavy-tailed sub-exponential volume profiles occur on the limit order book and these features are best captured via the generalized Pareto distribution MLE method. In futures exchanges, the heavy tail features are not asset class dependent and occur on ultra or mid-range high frequency. Volume forecasting models should account for heavy tails, time varying parameters and long memory. In application, utilizing the generalized Pareto distribution to model volume profiles allows one to avoid over-estimating the round trip cost of trading.

scholarly journals Rarefied particle motions on hillslopes: 1. Theory

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Joshua J. Roering ◽  
Tyler H. Doane ◽  
Danica L. Roth ◽  
Sarah G. W. Williams ◽  
...  
Keyword(s):  
Kinetic Energy ◽  
Particle Surface ◽  
Generalized Pareto Distribution ◽  
Planck Equation ◽  
Particle Energy ◽  
Finite Variance ◽  
Isothermal Conditions ◽  
Scale Parameters ◽  
Size Sorting ◽  
Heavy Tailed

Abstract. We describe the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle-surface collisions, and an apparent heating associated with preferential deposition of low energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states whose spatial evolution is described by a Fokker-Planck equation. The Kirkby number Ki – defined as the ratio of gravitational heating to frictional cooling – sets the basic deposition behavior and the form of the probability distribution fr(r) of particle travel distances r, a generalized Pareto distribution. The shape and scale parameters of the distribution are well-defined mechanically. For isothermal conditions where frictional cooling matches gravitational heating plus the apparent heating due to deposition, the distribution fr(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution fr(r) becomes heavy-tailed and represents net particle heating. It may possess a finite mean and finite variance, or the mean and variance may be undefined with sufficiently large Ki. The formulation provides key elements of the entrainment forms of the particle flux and the Exner equation, and it clarifies the mechanisms of particle-size sorting on large talus and scree slopes. Namely, with conversion of translational to rotational kinetic energy, large spinning particles are less likely to be stopped by collisional friction than are small or angular particles for the same surface roughness.

Sign in / Sign up

Export Citation Format

Share Document