Random Probability Distributions Research Articles

Computational and Mathematical Modeling of Tumor Kinetics and Response to Radiation and Chemotherapy

Computational and Mathematical Modeling of Tumor Kinetics and Response to Radiation and Chemotherapy

Predictability-based extended-range ensemble prediction method and numerical experiments

Ensemble prediction is an effective approach to accounting for uncertainties of initial conditions and model error. By combining the predictability of extended-range, both predictable components and unpredictable random components with different characteristics are treated with different ensemble prediction schemes and strategies. A new predictability-based extended-range ensemble prediction method (PBEP) is proposed. In this method, for predictable component, the uncertainty of model is taken into account through the use of multiple error correction scheme; while the random component probability distribution is obtained from the climate probability distribution of historical data, for the sake of avoiding the influence of model error. Prediction results show that the ensemble prediction method can improve the forecast skill in all regions of the world, and the extents of improvement are different for waves with different spatial scales compared with the operational dynamical extended-range ensemble prediction system of NCC/CMA, exhibiting its potential application perspective to operational extended-range prediction.

Coupling Optional Pólya Trees and the Two Sample Problem

Testing and characterizing the difference between two data samples is of fundamental interest in statistics. Existing methods such as Kolmogorov–Smirnov and Cramer–von Mises tests do not scale well as the dimensionality increases and provide no easy way to characterize the difference should it exist. In this work, we propose a theoretical framework for inference that addresses these challenges in the form of a prior for Bayesian nonparametric analysis. The new prior is constructed based on a random-partition-and-assignment procedure similar to the one that defines the standard optional Pólya tree distribution, but has the ability to generate multiple random distributions jointly. These random probability distributions are allowed to “couple,” that is, to have the same conditional distribution, on subsets of the sample space. We show that this “coupling optional Pólya tree” prior provides a convenient and effective way for both the testing of two sample difference and the learning of the underlying structure of the difference. In addition, we discuss some practical issues in the computational implementation of this prior and provide several numerical examples to demonstrate its work. Supplementary materials for this article are available online.

Circulant type matrices with heavy tailed entries

We study the limiting spectral distribution for a class of circulant type random matrices with heavy tailed input sequence. Unlike the light tailed case where the limit is nonrandom, here the limit is a random probability distribution. We provide an explicit representation of the limit.

Two-Period Production Planning and Inventory Control Model with Demand Forecasts Updating

We develop a stochastic two-period production/inventory planning model, which combines the use of information updating process with the flexibility of different delivery lead-times ordering strategy. Several decision variables are used: two orders are placed at the beginning of the first and second periods respectively and received immediately; another order is placed at the beginning of the first period and received with one period delay. The two different ordering/production modes, with zero and one period delivery lead-time, have different specific costs. The model permits to the retailer to return a certain amount of the available inventory to the supplier at the beginning of each period. Furthermore, a market information permits to update, between successive time periods, the random second period demand probability distribution. Via a dynamic programming approach, we exhibit the structure of the optimal policy, which is partially characterized by threshold levels. Then, via a numerical study, we exhibit the impact of the information quality of the proposed model.

A Markov Process Model of the Number of Years Spent in Major League Baseball

We treat the number of years spent in major league baseball as a random variable and estimate probability distributions for this random variable through the use of recursive formulae. Distributional characteristics, including major league baseball worklife expectancies, are estimated for players by age and current activity status in the major leagues. Data from a recent time period (1977-2007) are used to calculate current characteristics of time spent in major league baseball. However, the contemporaneous nature of our data leads to censoring because many players in our data set had not completed their major league careers by the end of 2007. We deal with censoring through a Markov process model that captures transitions between activity and inactivity in major league baseball.

Structural Reliability Analysis of Composite Advanced Sail for Virginia Class Submarine

A structural reliability model is developed for a large composite submarine sail structure. Random variables include material strength and stiffness properties as well as load. Limit states are formulated in terms of material strength parameters and buckling resistance. A series system reliability model that bases structural performance on first component failure is used. A total of 205 random variables and 117 limit states compose the structural system model. Most component reliability indices ranged from 3 to 7, with overall system failure governed by material in-plane shear with a reliability index of 1.84. A probabilistic sensitivity analysis determined that load and material strength were the most significant random variables, while material stiffness parameters were unimportant. Significant computational effort was saved by reducing the number of random variables to the most influential set. The effects of correlation and random variable probability distribution were explored. It was found that correlation has little effect on results, but probability distribution is significant. Recommendations are made to improve performance.

Prediction of Attack Frequency in Migraine Treatment

Usually limited information about the frequency of migraine episodes is derived from acute migraine trials. However, the design of some studies is such that they also provide relevant information about the attack frequency without the bias associated with patient expectations of treatment effect between attacks during prophylaxis trials. Using clinical data from repeated migraine attacks treated with placebo, naratriptan 2.5 mg or sumatriptan 100 mg, we show that attack and interictal periods can be described by a random probability distribution. Based on a gamma distribution, the mean interval between attacks was estimated to be 24 (17-34) days for placebo, 23 (18-29) days for naratriptan 2.5 mg and 22 (17-28) for sumatriptan 100 mg. These findings suggest that the interictal interval is not affected by abortive treatment with triptans. Interpretation of these results may be limited by the study type, yet the method represents a new tool for the evaluation of disease dynamics and treatment effect in the prophylaxis of migraine.

The Convergence-Guaranteed Random Walk and Its Applications in Peer-to-Peer Networks

Network structure construction and global state maintenance are expensive in large-scale dynamic peer-to-peer (P2P) networks. With inherent topology independence and low state maintenance overhead, random walk is an excellent tool in such network environments. However, the current uses are limited to unguided or heuristic random walks with no guarantee on their converged node visitation probability distribution. Such a convergence guarantee is essential for strong analytical properties and high performance of many P2P applications. In this paper, we investigate an approach for random walks to converge to application-desired node visitation probability distributions while only requiring information about the direct neighbors of each peer. Our approach is guided by the Metropolis-Hastings algorithm for Monte Carlo Markov Chain sampling. Our contributions are threefold. First, we analyze the convergence time of the random walk node visitation probability distribution on common P2P network topologies. Second, we analyze the fault tolerance of our random walks in dynamic networks with potential walker losses. Third, we present the effectiveness of random walks in assisting three realistic network applications: random membership subset management, search, and load balancing. Both search and load balancing desire random walks with biased node visitation distributions to achieve application-specific goals. Our analysis, simulations, and Internet experiment demonstrate the advantage of our random walks compared with alternative topology-independent index-free approaches.

국내 단지도로에서 도로교통소음에 대한 통계학적 특성연구 -도로교통 소음원이 수음점에 미치는 수평.수직 음원 분포특성 중심으로-

This paper represents the statistical characteristics of sources and receptors for road traffic noise in apartment complex. Also we suggested that the site-specific characteristics of vertical and horizontal distributions in the complex apartment have been derived using a different analysis for evaluating levels of Leq1h by the apartment floor at a given distance from a road in terms of the flow rate, the mean speed of the traffic, and the percentage of the type vehicles in the day and night periods. As a result, the contribution orders of traffic quantity by the type of vehicles showed as followed: light vehicle>medium vehicle>heavy vehicle>motorcycle. Especially, the mixing ratio of entering the road on the heavy vehicle in the daytime was two times higher than that of in the night. The speed in all types of vehicle is in the range of 41 and 81 km/hr and noise level was not significantly different in day and night-time. The sources of road traffic noise had different variations and uncertainties using a random variable and probability distribution. The sound distribution to receptors by the apartment floor showed S curve between 1st floor and floor With the normality test, the normal distributions using Anderson-Darling Test followed and , floor in the daytime and all floors except floor in the night (p>0.05). And also, the value of the pearson correlation coefficient (r) obtained in this study have significantly different at the range of floor. In conclusion, the results suggested that the distribution characteristics of levels of on the sources and receptors of road traffic noise in apartment complex must be improved and developed on the guideline of regulation of road traffic noise.

Investigation of reliability method formulations in DAKOTA/UQ

Reliability methods are probabilistic algorithms for quantifying the effect of simulation input uncertainties on response metrics of interest. In particular, they compute approximate response function distribution statistics (probability, reliability and response levels) based on specified input random variable probability distributions. In this paper, a number of algorithmic variations are explored for both the forward reliability analysis of computing probabilities for specified response levels (the reliability index approach (RIA)) and the inverse reliability analysis of computing response levels for specified probabilities (the performance measure approach (PMA)). These variations include limit state linearizations, probability integrations, warm starting and optimization algorithm selections. The resulting RIA/PMA reliability algorithms for uncertainty quantification are then employed within bi-level and sequential reliability-based design optimization approaches. Relative performance of these uncertainty quantification and reliability-based design optimization algorithms are presented for a number of computational experiments performed using the DAKOTA/UQ software.

Bayesian Density Regression

SummaryThe paper considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a non-parametric mixture of regression models, with the mixture distribution changing with predictors. A class of weighted mixture of Dirichlet process priors is proposed for the uncountable collection of mixture distributions. It is shown that this specification results in a generalized Pólya urn scheme, which incorporates weights that are dependent on the distance between subjects’ predictor values. To allow local dependence in the mixture distributions, we propose a kernel-based weighting scheme. A Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated by using simulated data examples and an epidemiologic application.

Fuzzy-Random FOSM and its Application in Low Cycle Fatigue Life Reliability Analysis of an Aeronautical Engine Turbine Disk

An approximate analytical method is presented to analyze reliability for the structure with fuzzy-random uncertainty in basic variables. On the basis of equivalent transformation from fuzzy possibility distribution (FPD) to random probability distribution (RPD), this contribution expands first order and second moment method (FOSM) for random reliability to that for fuzzy-random reliability. The expanded FOSM is illustrated by the way of the fuzzy-random low fatigue life reliability analysis about an aeronautical engine disk affected by fuzzy-random uncertainty. Comparison between the FOSM and the numerical simulation for the fuzzy-random reliability demonstrates the precision of the presented computational model.

Infinitely Divisible Random Probability Distributions with an Application to a Random Motion in a Random Environment

The infinite divisibility of probability distributions on the space $P (R )$ of probability distributions on $R$ is defined and related fundamental results such as the Levy-Khintchin formula, representation of Ito type of infinitely divisible RPD, stable RPD and Levy processes on $P (R )$ are obtained. As an application we investigate limiting behaviors of a simple model of a particle motion in a random environment

The hyper-Dirichlet process and its discrete approximations: The butterfly model

The aim of this paper is the study of some random probability distributions, called hyper-Dirichlet processes. In the simplest situation considered in the paper these distributions charge the product of three sample spaces, with the property that the first and the last component are independent conditional to the middle one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper-Markov combination, introduced in [A.P. Dawid, S.L. Lauritzen, Hyper-Markov laws in the statistical analysis of decomposable graphical models, Ann. Statist. 21 (3) (1993) 1272–1317], of these two Dirichlet processes. The processes constructed in this way in fact are in fact generalizations of the hyper-Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the hyper-Dirichlet process of the sequence of hyper-Dirichlet laws associated to finer and finer “discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction.

Normalized random measures driven by increasing additive processes

This paper introduces and studies a new class of nonparametric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increasing additive processes. In particular, we present results for the distribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the well-known and widely used mixture of a Dirichlet process.

An ANOVA Model for Dependent Random Measures

We consider dependent nonparametric models for related random probability distributions. For example, the random distributions might be indexed by a categorical covariate indicating the treatment levels in a clinical trial and might represent random effects distributions under the respective treatment combinations. We propose a model that describes dependence across random distributions in an analysis of variance (ANOVA)-type fashion. We define a probability model in such a way that marginally each random measure follows a Dirichlet process (DP) and use the dependent Dirichlet process to define the desired dependence across the related random measures. The resulting probability model can alternatively be described as a mixture of ANOVA models with a DP prior on the unknown mixing measure. The main features of the proposed approach are ease of interpretation and computational simplicity. Because the model follows the standard ANOVAstructure, interpretation and inference parallels conventions for ANOVA models. This includes the notion of main effects, interactions, contrasts, and the like. Of course, the analogies are limited to structure and interpretation. The actual objects of the inference are random distributions instead of the unknown normal means in standard ANOVA models. Besides interpretation and model structure, another important feature of the proposed approach is ease of posterior simulation. Because the model can be rewritten as a DP mixture of ANOVAmodels, it inherits all computational advantages of standard DP mixture models. This includes availability of efficient Gibbs sampling schemes for posterior simulation and ease of implementation of even high-dimensional applications. Complexity of implementing posterior simulation is—at least conceptually—dimension independent.

Weak Convergence of a Dirichlet-Multinomial Process

Abstract We present a random probability distribution which approximates, in the sense of weak convergence, the Dirichlet process and supports a Bayesian resampling plan called a proper Bayesian bootstrap.

A unified approach to estimation and prediction under simple random sampling

We consider a probability model where the design-based approach to inference under simple random sampling of a finite population encompasses a simple random permutation super-population model. The model consists of an expanded set of random variables following a random permutation probability distribution that keeps track of both the units’ labels and positions in the permutation. In particular, since we keep track of the labels, the model allows us to attack the problem of estimation of a unit's parameter. While some linear combinations of the expanded set of random variables correspond to linear combinations of the unit parameters, other linear combinations correspond to random variables known as random effects. Using a prediction technique similar to that employed under the model-based approach, we develop optimum estimators of the linear combinations of the unit parameters and optimum predictors of the random effects. The unbiased minimum variance estimator of the population mean is the sample mean and of a unit parameter is the Horvitz–Thompson estimator if the unit is included in the sample, and zero otherwise. The predictor of the random variable at a given position in the permutation is the realized unit's parameter for positions in the sample, and the sample mean for other positions. For other linear functions, unique minimum variance unbiased estimators may not exist.

Moments of the Mean of Dubins–Freedman Random Probability Distributions

Explicit formulas are given to recursively generate the moments of the mean M for Dubins–Freedman random distribution functions with arbitrary base measure μ. Using a standard inversion formula for moments of a distribution on the unit interval, the distribution of M is approximated for several natural choices of μ. The support of the mean is also considered. It is shown that the support of M is connected whenever μ is concentrated on the vertical bisector of the unit square S, but may have arbitrarily many gaps otherwise.