Transient behavior of regulated Brownian motion, II: Non-zero initial conditions

1987 ◽  
Vol 19 (3) ◽  
pp. 599-631 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Systems Approach ◽  
Initial Conditions ◽  
Transient Behavior ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Dependent Behavior ◽  
Origin Part ◽  
The Difference ◽  
The Moment

This paper continues an investigation of the time-dependent behavior of regulated or reflecting Brownian motion (RBM). Part I focused on RBM starting at the origin; Part II focuses on RBM starting at a fixed positive state. The first two moments of RBM as functions of time are analyzed by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied in Part I. By appropriate normalization, the two monotone components can be converted into cumulative distribution functions that can be analyzed probabilistically, e.g., their moments can be calculated. Simple approximations are then developed by fitting convenient distributions to these moments. Overall, the analysis yields a better understanding of the way RBM and related stochastic flow systems approach steady state.

Transient behavior of regulated Brownian motion, II: Non-zero initial conditions

1987 ◽  
Vol 19 (03) ◽  
pp. 599-631 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Systems Approach ◽  
Initial Conditions ◽  
Transient Behavior ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Dependent Behavior ◽  
Origin Part ◽  
The Difference ◽  
The Moment

This paper continues an investigation of the time-dependent behavior of regulated or reflecting Brownian motion (RBM). Part I focused on RBM starting at the origin; Part II focuses on RBM starting at a fixed positive state. The first two moments of RBM as functions of time are analyzed by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied in Part I. By appropriate normalization, the two monotone components can be converted into cumulative distribution functions that can be analyzed probabilistically, e.g., their moments can be calculated. Simple approximations are then developed by fitting convenient distributions to these moments. Overall, the analysis yields a better understanding of the way RBM and related stochastic flow systems approach steady state.

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (03) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that thekth-moment c.d.f is thek-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k +1)th moment approaches its steady-state limit more slowly than thekthmoment. It is also easy to derive the asymptotic behavior ast→∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2)c.d.f.&s (mixtures of two exponentials). TheH2approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of theH2approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals The coagulation of smokes and the theory of smoluchowski

1927 ◽  
Vol 116 (775) ◽  
pp. 540-553 ◽  
Keyword(s):  
Experimental Data ◽  
Brownian Motion ◽  
Disperse Systems ◽  
Whole Process ◽  
Rapid Coagulation ◽  
Short Period ◽  
Dispersion Media ◽  
Experimental Errors ◽  
The Difference ◽  
The Moment

In a previous communication it has been shown that smokes are unstable disperse systems which spontaneously coagulate from the moment of formation. In this respect they differ from hydrosols, which are stable normally and coagulate only on the addition of an electrolyte. For these latter systems there is evidence to show that the so-called “rapid coagulation” begins when the complete discharge of the double layer has been effected, and that the particles are brought together mainly by Brownian motion. With smokes, too, chance collision brought about by molecular bombardment is probably the chief factor in aggregation, so that a close analogy between the rapid coagulation of hydrosols and the spontaneous coagulation of smokes would appear to exist. Further, if it is admitted that every chance collision between the particles in the two types of system results in a union, then the equations developed by Smoluchowski for the rate of coagulation of colloids might be expected to be equally valid for smokes after making allowance for the difference in properties of the two dispersion media. Reliable experimental data on sol coagula­tion are not easily obtained, for the process is completed in a relatively short period, but the measurements of Zsigmondy, Westgren, and others show a satisfactory agreement with theory. In a smoke, conditions are more favour­able, for coagulation can be followed at much greater dilutions, so that the whole process is slowed down, and observations may be extended over several hours. On account of the interest presented by the above analogy, and the possibility of testing it quantitatively, we have measured the velocity of coagulation of various smokes by a method which we believe to be free from any serious experimental errors.

Reliable Reservoir Characterization and History Matching Using a Pattern Recognition Based Distance

2016 ◽  
Author(s):  
Jiyoon Lee ◽  
Jonggeun Choe
Keyword(s):  
Pattern Recognition ◽  
History Matching ◽  
Reservoir Characterization ◽  
Correlation Coefficients ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Uncertainty Range ◽  
Box Plots ◽  
Permeability Distribution ◽  
The Difference

A distance is defined as a measure of dissimilarity between two reservoir models. There have been many distances proposed for fast modeling. However, some distances cause distortion or loss in original permeability distribution of models. To avoid such problems, this study proposes a pattern recognition based distance. The distance is defined by the difference of correlation coefficients between ensemble models. From multi-dimensional scaling, initial 400 ensembles are presented on 2D plane using the distance. Then 10 groups are made by K-medoids clustering. After comparing oil production from each centroid and that of the reference field, 100 models are selected around the best centroid. We validate the clustering by comparing the uncertainty range of 100, 50, and 20 ensemble members sampled from the initial 400 models in box plots and cumulative distribution functions. For a history matching and reservoir characterization, ensemble smoother is applied to the 100 models selected. The proposed method takes only 25% time for simulation showing reliable results compared with the initial 400 models.

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (3) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that the kth-moment c.d.f is the k-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k + 1)th moment approaches its steady-state limit more slowly than the kth moment. It is also easy to derive the asymptotic behavior as t →∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2) c.d.f.&s (mixtures of two exponentials). The H2 approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of the H2 approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

scholarly journals Non-uniform Estimates in the Approximation by the Irwin Law

2016 ◽  
Vol 55 (1) ◽  
pp. 112-118
Author(s):  
Kazimieras Padvelskis ◽  
Ruslan Prigodin
Keyword(s):  
Distribution Function ◽  
Remainder Term ◽  
Cumulative Distribution Function ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Type Condition ◽  
Uniform Estimates ◽  
Cumulative Distribution Functions ◽  
Cumulant Method ◽  
The Difference

We consider an approximation of a cumulative distribution function F(x) by the cumulative distributionfunction G(x) of the Irwin law. In this case, a function F(x) can be cumulative distribution functions of sums (products) ofindependent (dependent) random variables. Remainder term of the approximation is estimated by the cumulant method.The cumulant method is used by introducing special cumulants, satisfying the V. Statulevičius type condition. The mainresult is a nonuniform bound for the difference |F(x)-G(x)| in terms of special cumulants of the symmetric cumulativedistribution function F(x).

scholarly journals A Note on Confidence Interval for the Power of the One Sample Test

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
A. Wong
Keyword(s):  
Confidence Interval ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Sample Mean ◽  
Unknown Variance ◽  
Power Of The Test ◽  
Sample Test ◽  
The Difference ◽  
The One

In introductory statistics texts, the power of the test of a one-sample mean when the variance is known is widely discussed. However, when the variance is unknown, the power of the Student's -test is seldom mentioned. In this note, a general methodology for obtaining inference concerning a scalar parameter of interest of any exponential family model is proposed. The method is then applied to the one-sample mean problem with unknown variance to obtain a 100% confidence interval for the power of the Student's -test that detects the difference . The calculations require only the density and the cumulative distribution functions of the standard normal distribution. In addition, the methodology presented can also be applied to determine the required sample size when the effect size and the power of a size test of mean are given.

scholarly journals Impact of Soil Moisture Assimilation on Land Surface Model Spinup and Coupled Land–Atmosphere Prediction

2016 ◽  
Vol 17 (2) ◽  
pp. 517-540 ◽  
Author(s):  
Joseph A. Santanello ◽  
Sujay V. Kumar ◽  
Christa D. Peters-Lidard ◽  
Patricia M. Lawston
Keyword(s):  
Soil Moisture ◽  
Land Surface ◽  
Initial Conditions ◽  
Weather Prediction ◽  
Wrf Model ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Satellite Monitoring ◽  
Surface Model ◽  
The Impact

Abstract Advances in satellite monitoring of the terrestrial water cycle have led to a concerted effort to assimilate soil moisture observations from various platforms into offline land surface models (LSMs). One principal but still open question is that of the ability of land data assimilation (LDA) to improve LSM initial conditions for coupled short-term weather prediction. In this study, the impact of assimilating Advanced Microwave Scanning Radiometer for EOS (AMSR-E) soil moisture retrievals on coupled WRF Model forecasts is examined during the summers of dry (2006) and wet (2007) surface conditions in the southern Great Plains. LDA is carried out using NASA’s Land Information System (LIS) and the Noah LSM through an ensemble Kalman filter (EnKF) approach. The impacts of LDA on the 1) soil moisture and soil temperature initial conditions for WRF, 2) land–atmosphere coupling characteristics, and 3) ambient weather of the coupled LIS–WRF simulations are then assessed. Results show that impacts of soil moisture LDA during the spinup can significantly modify LSM states and fluxes, depending on regime and season. Results also indicate that the use of seasonal cumulative distribution functions (CDFs) is more advantageous compared to the traditional annual CDF bias correction strategies. LDA performs consistently regardless of atmospheric forcing applied, with greater improvements seen when using coarser, global forcing products. Downstream impacts on coupled simulations vary according to the strength of the LDA impact at the initialization, where significant modifications to the soil moisture flux–PBL–ambient weather process chain are observed. Overall, this study demonstrates potential for future, higher-resolution soil moisture assimilation applications in weather and climate research.

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