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Yu S, Chu R, Wu G, Meng X. A Novel Fractional Brownian Dynamics Method for Simulating the Dynamics of Confined Bottle-Brush Polymers in Viscoelastic Solution. Polymers (Basel) 2024; 16:524. [PMID: 38399901 PMCID: PMC10891538 DOI: 10.3390/polym16040524] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2024] [Revised: 02/11/2024] [Accepted: 02/13/2024] [Indexed: 02/25/2024] Open
Abstract
In crowded fluids, polymer segments can exhibit anomalous subdiffusion due to the viscoelasticity of the surrounding environment. Previous single-particle tracking experiments revealed that such anomalous diffusion in complex fluids (e.g., in bacterial cytoplasm) can be described by fractional Brownian motion (fBm). To investigate how the viscoelastic media affects the diffusive behaviors of polymer segments without resolving single crowders, we developed a novel fractional Brownian dynamics method to simulate the dynamics of polymers under confinement. In this work, instead of using Gaussian random numbers ("white Gaussian noise") to model the Brownian force as in the standard Brownian dynamics simulations, we introduce fractional Gaussian noise (fGn) in our homemade fractional Brownian dynamics simulation code to investigate the anomalous diffusion of polymer segments by using a simple "bottle-brush"-type polymer model. The experimental results of the velocity autocorrelation function and the exponent that characterizes the subdiffusion of the confined polymer segments can be reproduced by this simple polymer model in combination with fractional Gaussian noise (fGn), which mimics the viscoelastic media.
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Affiliation(s)
- Shi Yu
- Department of Chemical Engineering, China University of Mining & Technology, Xuzhou 221116, China; (R.C.); (G.W.); (X.M.)
| | - Ruizhi Chu
- Department of Chemical Engineering, China University of Mining & Technology, Xuzhou 221116, China; (R.C.); (G.W.); (X.M.)
- Key Laboratory of Coal-Based CO2 Capture and Geological Storage, China University of Mining & Technology, Xuzhou 221116, China
| | - Guoguang Wu
- Department of Chemical Engineering, China University of Mining & Technology, Xuzhou 221116, China; (R.C.); (G.W.); (X.M.)
- Key Laboratory of Coal-Based CO2 Capture and Geological Storage, China University of Mining & Technology, Xuzhou 221116, China
| | - Xianliang Meng
- Department of Chemical Engineering, China University of Mining & Technology, Xuzhou 221116, China; (R.C.); (G.W.); (X.M.)
- Key Laboratory of Coal-Based CO2 Capture and Geological Storage, China University of Mining & Technology, Xuzhou 221116, China
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2
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Wunderlich A, Sklar J. Data-driven modeling of noise time series with convolutional generative adversarial networks. Mach Learn Sci Technol 2023; 4:10.1088/2632-2153/acee44. [PMID: 37693073 PMCID: PMC10484071 DOI: 10.1088/2632-2153/acee44] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/12/2023] Open
Abstract
Random noise arising from physical processes is an inherent characteristic of measurements and a limiting factor for most signal processing and data analysis tasks. Given the recent interest in generative adversarial networks (GANs) for data-driven modeling, it is important to determine to what extent GANs can faithfully reproduce noise in target data sets. In this paper, we present an empirical investigation that aims to shed light on this issue for time series. Namely, we assess two general-purpose GANs for time series that are based on the popular deep convolutional GAN architecture, a direct time-series model and an image-based model that uses a short-time Fourier transform data representation. The GAN models are trained and quantitatively evaluated using distributions of simulated noise time series with known ground-truth parameters. Target time series distributions include a broad range of noise types commonly encountered in physical measurements, electronics, and communication systems: band-limited thermal noise, power law noise, shot noise, and impulsive noise. We find that GANs are capable of learning many noise types, although they predictably struggle when the GAN architecture is not well suited to some aspects of the noise, e.g. impulsive time-series with extreme outliers. Our findings provide insights into the capabilities and potential limitations of current approaches to time-series GANs and highlight areas for further research. In addition, our battery of tests provides a useful benchmark to aid the development of deep generative models for time series.
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Affiliation(s)
- Adam Wunderlich
- Communications Technology Laboratory, National Institute of Standards and Technology, Boulder, CO 80305, United States of America
| | - Jack Sklar
- Communications Technology Laboratory, National Institute of Standards and Technology, Boulder, CO 80305, United States of America
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3
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Janušonis S, Haiman JH, Metzler R, Vojta T. Predicting the distribution of serotonergic axons: a supercomputing simulation of reflected fractional Brownian motion in a 3D-mouse brain model. Front Comput Neurosci 2023; 17:1189853. [PMID: 37265780 PMCID: PMC10231035 DOI: 10.3389/fncom.2023.1189853] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2023] [Accepted: 04/24/2023] [Indexed: 06/03/2023] Open
Abstract
The self-organization of the brain matrix of serotonergic axons (fibers) remains an unsolved problem in neuroscience. The regional densities of this matrix have major implications for neuroplasticity, tissue regeneration, and the understanding of mental disorders, but the trajectories of its fibers are strongly stochastic and require novel conceptual and analytical approaches. In a major extension to our previous studies, we used a supercomputing simulation to model around one thousand serotonergic fibers as paths of superdiffusive fractional Brownian motion (FBM), a continuous-time stochastic process. The fibers produced long walks in a complex, three-dimensional shape based on the mouse brain and reflected at the outer (pial) and inner (ventricular) boundaries. The resultant regional densities were compared to the actual fiber densities in the corresponding neuroanatomically-defined regions. The relative densities showed strong qualitative similarities in the forebrain and midbrain, demonstrating the predictive potential of stochastic modeling in this system. The current simulation does not respect tissue heterogeneities but can be further improved with novel models of multifractional FBM. The study demonstrates that serotonergic fiber densities can be strongly influenced by the geometry of the brain, with implications for brain development, plasticity, and evolution.
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Affiliation(s)
- Skirmantas Janušonis
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Justin H. Haiman
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
- Asia Pacific Center for Theoretical Physics, Pohang, South Korea
| | - Thomas Vojta
- Department of Physics, Missouri University of Science and Technology, Rolla, MO, United States
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4
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Lee C, Zhang Z, Janušonis S. Brain serotonergic fibers suggest anomalous diffusion-based dropout in artificial neural networks. Front Neurosci 2022; 16:949934. [PMID: 36267232 PMCID: PMC9577023 DOI: 10.3389/fnins.2022.949934] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2022] [Accepted: 09/08/2022] [Indexed: 11/13/2022] Open
Abstract
Random dropout has become a standard regularization technique in artificial neural networks (ANNs), but it is currently unknown whether an analogous mechanism exists in biological neural networks (BioNNs). If it does, its structure is likely to be optimized by hundreds of millions of years of evolution, which may suggest novel dropout strategies in large-scale ANNs. We propose that the brain serotonergic fibers (axons) meet some of the expected criteria because of their ubiquitous presence, stochastic structure, and ability to grow throughout the individual's lifespan. Since the trajectories of serotonergic fibers can be modeled as paths of anomalous diffusion processes, in this proof-of-concept study we investigated a dropout algorithm based on the superdiffusive fractional Brownian motion (FBM). The results demonstrate that serotonergic fibers can potentially implement a dropout-like mechanism in brain tissue, supporting neuroplasticity. They also suggest that mathematical theories of the structure and dynamics of serotonergic fibers can contribute to the design of dropout algorithms in ANNs.
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Affiliation(s)
- Christian Lee
- Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Zheng Zhang
- Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Skirmantas Janušonis
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, CA, United States
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Bisewski K, Dȩbicki K, Rolski T. Derivative of the expected supremum of fractional Brownian motion at H = 1. Queueing Syst 2022; 102:53-68. [PMID: 36213862 PMCID: PMC9534823 DOI: 10.1007/s11134-022-09859-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/25/2022] [Revised: 08/08/2022] [Accepted: 08/09/2022] [Indexed: 06/16/2023]
Abstract
The H-derivative of the expected supremum of fractional Brownian motion { B H ( t ) , t ∈ R + } with drift a ∈ R over time interval [0, T] ∂ ∂ H E ( sup t ∈ [ 0 , T ] B H ( t ) - a t ) at H = 1 is found. This formula depends on the quantity I , which has a probabilistic form. The numerical value of I is unknown; however, Monte Carlo experiments suggest I ≈ 0.95 . As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as H ↑ 1 .
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Affiliation(s)
- Krzysztof Bisewski
- Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
| | - Krzysztof Dȩbicki
- Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
| | - Tomasz Rolski
- Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
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6
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Michel E, Néel MC, Capowiez Y, Sammartino S, Lafolie F, Renault P, Pelosi C. Making Waves: Modeling bioturbation in soils - are we burrowing in the right direction? Water Res 2022; 216:118342. [PMID: 35349922 DOI: 10.1016/j.watres.2022.118342] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2021] [Revised: 03/15/2022] [Accepted: 03/19/2022] [Indexed: 06/14/2023]
Abstract
The burrowing, feeding and foraging activities of terrestrial and benthic organisms induce displacements of soil and sediment materials, leading to a profound mixing of these media. Such particle movements, called "sediment reworking" in aquatic environments and "bioturbation" in soils, have been thoroughly studied and modeled in sediments, where they affect organic matter mineralization and contaminant fluxes. In comparison, studies characterizing the translocation, by soil burrowers, of mineral particles, organic matter and adsorbed contaminants are paradoxically fewer. Nevertheless, models borrowed from aquatic ecology are used to predict the impact of bioturbation on organic matter turnover and contaminant transport in the soil. However, these models are based on hypotheses that have not been tested with adequate observations in soils, and may not necessarily reflect the actual impact of soil burrowers on particle translocation. This paper aims to (i) highlight the possible shortcomings linked to the current use of sediment reworking models for soils, (ii) identify how recent progresses in aquatic ecology could help to circumvent these limitations, and (iii) propose key steps to ensure that soil bioturbation models are built on solid foundations: more accurate models of organic matter turnover, soil evolution and contaminant transport in the soil are at stake.
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Affiliation(s)
- Eric Michel
- EMMAH, INRAE, Avignon Université, 84000 Avignon, France.
| | | | - Yvan Capowiez
- EMMAH, INRAE, Avignon Université, 84000 Avignon, France
| | | | | | | | - Céline Pelosi
- EMMAH, INRAE, Avignon Université, 84000 Avignon, France.
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7
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Szarek D, Sikora G, Balcerek M, Jabłoński I, Wyłomańska A. Fractional Dynamics Identification via Intelligent Unpacking of the Sample Autocovariance Function by Neural Networks. Entropy (Basel) 2020; 22:e22111322. [PMID: 33287087 PMCID: PMC7712253 DOI: 10.3390/e22111322] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/31/2020] [Accepted: 11/18/2020] [Indexed: 12/20/2022]
Abstract
Many single-particle tracking data related to the motion in crowded environments exhibit anomalous diffusion behavior. This phenomenon can be described by different theoretical models. In this paper, fractional Brownian motion (FBM) was examined as the exemplary Gaussian process with fractional dynamics. The autocovariance function (ACVF) is a function that determines completely the Gaussian process. In the case of experimental data with anomalous dynamics, the main problem is first to recognize the type of anomaly and then to reconstruct properly the physical rules governing such a phenomenon. The challenge is to identify the process from short trajectory inputs. Various approaches to address this problem can be found in the literature, e.g., theoretical properties of the sample ACVF for a given process. This method is effective; however, it does not utilize all of the information contained in the sample ACVF for a given trajectory, i.e., only values of statistics for selected lags are used for identification. An evolution of this approach is proposed in this paper, where the process is determined based on the knowledge extracted from the ACVF. The designed method is intuitive and it uses information directly available in a new fashion. Moreover, the knowledge retrieval from the sample ACVF vector is enhanced with a learning-based scheme operating on the most informative subset of available lags, which is proven to be an effective encoder of the properties inherited in complex data. Finally, the robustness of the proposed algorithm for FBM is demonstrated with the use of Monte Carlo simulations.
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Affiliation(s)
- Dawid Szarek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland; (D.S.); (G.S.); (M.B.)
| | - Grzegorz Sikora
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland; (D.S.); (G.S.); (M.B.)
| | - Michał Balcerek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland; (D.S.); (G.S.); (M.B.)
| | - Ireneusz Jabłoński
- Department of Electronics, Wroclaw University of Science and Technology, B. Prusa 53/55, 50-317 Wroclaw, Poland;
| | - Agnieszka Wyłomańska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland; (D.S.); (G.S.); (M.B.)
- Correspondence:
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8
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Janušonis S, Detering N, Metzler R, Vojta T. Serotonergic Axons as Fractional Brownian Motion Paths: Insights Into the Self-Organization of Regional Densities. Front Comput Neurosci 2020; 14:56. [PMID: 32670042 PMCID: PMC7328445 DOI: 10.3389/fncom.2020.00056] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2019] [Accepted: 05/19/2020] [Indexed: 01/03/2023] Open
Abstract
All vertebrate brains contain a dense matrix of thin fibers that release serotonin (5-hydroxytryptamine), a neurotransmitter that modulates a wide range of neural, glial, and vascular processes. Perturbations in the density of this matrix have been associated with a number of mental disorders, including autism and depression, but its self-organization and plasticity remain poorly understood. We introduce a model based on reflected Fractional Brownian Motion (FBM), a rigorously defined stochastic process, and show that it recapitulates some key features of regional serotonergic fiber densities. Specifically, we use supercomputing simulations to model fibers as FBM-paths in two-dimensional brain-like domains and demonstrate that the resultant steady state distributions approximate the fiber distributions in physical brain sections immunostained for the serotonin transporter (a marker for serotonergic axons in the adult brain). We suggest that this framework can support predictive descriptions and manipulations of the serotonergic matrix and that it can be further extended to incorporate the detailed physical properties of the fibers and their environment.
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Affiliation(s)
- Skirmantas Janušonis
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Nils Detering
- Department of Statistics and Applied Probability, University of California, Santa Barbara, Santa Barbara, CA, United States
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany
| | - Thomas Vojta
- Department of Physics, Missouri University of Science and Technology, Rolla, MO, United States
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9
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Wu L. A Note on Wavelet-Based Estimator of the Hurst Parameter. Entropy (Basel) 2020; 22:E349. [PMID: 33286123 DOI: 10.3390/e22030349] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/07/2020] [Revised: 03/15/2020] [Accepted: 03/15/2020] [Indexed: 01/28/2023]
Abstract
The signals in numerous fields usually have scaling behaviors (long-range dependence and self-similarity) which is characterized by the Hurst parameter H. Fractal Brownian motion (FBM) plays an important role in modeling signals with self-similarity and long-range dependence. Wavelet analysis is a common method for signal processing, and has been used for estimation of Hurst parameter. This paper conducts a detailed numerical simulation study in the case of FBM on the selection of parameters and the empirical bias in the wavelet-based estimator which have not been studied comprehensively in previous studies, especially for the empirical bias. The results show that the empirical bias is due to the initialization errors caused by discrete sampling, and is not related to simulation methods. When choosing an appropriate orthogonal compact supported wavelet, the empirical bias is almost not related to the inaccurate bias correction caused by correlations of wavelet coefficients. The latter two causes are studied via comparison of estimators and comparison of simulation methods. These results could be a reference for future studies and applications in the scaling behavior of signals. Some preliminary results of this study have provided a reference for my previous studies.
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Nicolis O, Mateu J, Contreras-Reyes JE. Wavelet-Based Entropy Measures to Characterize Two-Dimensional Fractional Brownian Fields. Entropy (Basel) 2020; 22:e22020196. [PMID: 33285971 PMCID: PMC7516627 DOI: 10.3390/e22020196] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/25/2019] [Revised: 01/23/2020] [Accepted: 01/27/2020] [Indexed: 11/29/2022]
Abstract
The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282–288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is estimated by the regression of the logarithm of the square coefficients over the levels of resolutions. Using the same methodology. we also defined two other entropies in 2D: Tsallis and the Rényi entropies. A simulation study was performed for showing the ability of the method to characterize 2D (in this case, α=2) self-similar processes.
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Affiliation(s)
- Orietta Nicolis
- Facultad de Ingenieria, Universidad Andres Bello, Viña del Mar 2520000, Chile
- Correspondence:
| | - Jorge Mateu
- Department of Mathematics, Universitat Jaume I, E-12071 Castellon, Spain
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11
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Abstract
Experimental and theoretical research suggests that serotonergic axons (fibers) can be modeled as random walks or stochastic processes. This rigorous approach can support descriptive methods and dynamic control of the ascending reticular activating system, at the level of individual fiber trajectories.
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Affiliation(s)
- Skirmantas Janušonis
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, California 93106-9660, United States
| | - Kasie C. Mays
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, California 93106-9660, United States
| | - Melissa T. Hingorani
- Department of Psychological and Brain Sciences, University of California, Santa Barbara, California 93106-9660, United States
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12
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Abstract
Time-of-Use (TOU) pricing is an important strategy for electricity providers to manage supply and hence making the grid more efficient and for consumers to manage their costs. In this paper, we discuss a general stochastic modeling framework for consumer's power demand based on which the TOU contract characteristics can be selected, so as to minimize the mean electricity price paid by the customer. We exploit the characteristics of power demand observed in real grids to propose to model it during homogeneous peak periods as a constant level with fluctuations described by a scaled fractional Brownian motion. We analyze the exceedance process over pre-specified thresholds and use this information for formulating an optimization problem to determine the key features of the TOU contract. Due to the analytical intractability of certain expressions with the exception of short-range dependence fluctuations, the solution of the posited optimization problem requires using techniques such as Monte Carlo simulation and numerical search. The methodology for two pricing schemes is illustrated using real data.
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Affiliation(s)
- Ying-Chao Hung
- Department of Statistics, National Chengchi University, Taipei, 11605 Taiwan
| | - George Michailidis
- Informatics Institute and Departmenst of Statistics and Computer Science, University of Florida, Gainesville, FL 32611, USA
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13
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Vitali S, Sposini V, Sliusarenko O, Paradisi P, Castellani G, Pagnini G. Langevin equation in complex media and anomalous diffusion. J R Soc Interface 2018; 15:20180282. [PMID: 30158182 PMCID: PMC6127165 DOI: 10.1098/rsif.2018.0282] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2018] [Accepted: 07/30/2018] [Indexed: 11/30/2022] Open
Abstract
The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
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Affiliation(s)
- Silvia Vitali
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Vittoria Sposini
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Oleksii Sliusarenko
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Paolo Paradisi
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- ISTI-CNR, Institute of Information Science and Technologies 'A. Faedo' (Consiglio Nazionale delle Ricerche), Via Moruzzi 1, 56124 Pisa, Italy
| | - Gastone Castellani
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Gianni Pagnini
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- Ikerbasque-Basque Foundation for Science, Calle de María Díaz de Haro 3, 48013 Bilbao, Basque Country, Spain
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14
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Ding XL, Nieto JJ. Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications. Entropy (Basel) 2018; 20:E63. [PMID: 33265151 DOI: 10.3390/e20010063] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Revised: 01/03/2018] [Accepted: 01/11/2018] [Indexed: 11/17/2022]
Abstract
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.
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15
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Lai D, Zhang Q, Yamal JM, Einhorn PT, Davis BR. Conditional moving linear regression: modeling the recruitment process for ALLHAT. COMMUN STAT-THEOR M 2017; 46:8943-8951. [PMID: 30906106 PMCID: PMC6430572 DOI: 10.1080/03610926.2016.1197251] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2014] [Accepted: 05/31/2016] [Indexed: 10/21/2022]
Abstract
Effective recruitment is a prerequisite for successful execution of a clinical trial. ALLHAT, a large hypertension treatment trial (N = 42, 418), provided an opportunity to evaluate adaptive modeling of recruitment processes using conditional moving linear regression. Our statistical modeling of recruitment, comparing Brownian and fractional Brownian motion, indicates that fractional Brownian motion combined with moving linear regression is better than classic Brownian motion in terms of higher conditional probability of achieving a global recruitment goal in four week ahead projections. Further research is needed to evaluate how recruitment modeling can assist clinical trialists in planning and executing clinical trials. Clinical Trial Registration: www.clinicaltrials.gov NCT00000542.
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Affiliation(s)
- Dejian Lai
- Coordinating Center for Clinical Trials, Biostatistics Division, The University of Texas School of Public Health, Houston, TX, United States
| | - Qiang Zhang
- NRG Oncology Statistics and Data Management Center, Philadelphia, PA, United States
| | - Jose-Miguel Yamal
- Coordinating Center for Clinical Trials, Biostatistics Division, The University of Texas School of Public Health, Houston, TX, United States
| | - Paula T. Einhorn
- Division of Cardiovascular Sciences, National Heart, Lung, and Blood Institute, Bethesda, MD, United States
| | - Barry R. Davis
- Coordinating Center for Clinical Trials, Biostatistics Division, The University of Texas School of Public Health, Houston, TX, United States
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Abstract
This paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process, when evaluated at a "sensitive time point", is associated with a scalar response. The proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years. The trajectories are assumed to have fractal (self-similar) properties in common with a fractional Brownian motion with an unknown Hurst exponent. Bootstrap confidence intervals based on the least-squares estimator of the sensitive time point are developed. Misspecification of the point impact model by a functional linear model is also investigated. Non-Gaussian limit distributions and rates of convergence determined by the Hurst exponent play an important role.
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Affiliation(s)
- Ian W McKeague
- Department of Biostatistics, Columbia University, 722 West 168th Street, 6th Floor, New York, NY 10032,
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