Initial Probability Distribution Research Articles

A linear iterative unfolding method

A frequently faced task in experimental physics is to measure the probability distribution of some quantity. Often this quantity to be measured is smeared by a non-ideal detector response or by some physical process. The procedure of removing this smearing effect from the measured distribution is called unfolding, and is a delicate problem in signal processing, due to the well-known numerical ill behavior of this task. Various methods were invented which, given some assumptions on the initial probability distribution, try to regularize the unfolding problem. Most of these methods definitely introduce bias into the estimate of the initial probability distribution. We propose a linear iterative method (motivated by the Neumann series / Landweber iteration known in functional analysis), which has the advantage that no assumptions on the initial probability distribution is needed, and the only regularization parameter is the stopping order of the iteration, which can be used to choose the best compromise between the introduced bias and the propagated statistical and systematic errors. The method is consistent: "binwise" convergence to the initial probability distribution is proved in absence of measurement errors under a quite general condition on the response function. This condition holds for practical applications such as convolutions, calorimeter response functions, momentum reconstruction response functions based on tracking in magnetic field etc. In presence of measurement errors, explicit formulae for the propagation of the three important error terms is provided: bias error (distance from the unknown to-be-reconstructed initial distribution at a finite iteration order), statistical error, and systematic error. A trade-off between these three error terms can be used to define an optimal iteration stopping criterion, and the errors can be estimated there. We provide a numerical C library for the implementation of the method, which incorporates automatic statistical error propagation as well. The proposed method is also discussed in the context of other known approaches.

Global sensitivity analysis for stochastic ground motion modeling in seismic-risk assessment

Seismic risk assessment requires adoption of appropriate models for the earthquake hazard, the structural system and for its performance, and quantification of the uncertainties involved in these models through appropriate probability distributions. Characterization of the seismic hazard comprises undoubtedly the most critical component of this process, the one associated with the largest amount of uncertainty. For applications involving dynamic analysis this hazard is frequently characterized through stochastic ground motion models. This paper discusses a novel, global sensitivity analysis for the seismic risk with emphasis on such a stochastic ground motion modeling. This analysis aims to identify the overall (i.e. global) importance of each of the uncertain model parameters, or of groups of them, towards the total risk. The methodology is based on definition of an auxiliary density (distribution) function, proportional to the integrand of the integral quantifying seismic risk, and on comparison of this density to the initial probability distribution for the model parameters of interest. Uncertainty in the rest of the model parameters is explicitly addressed through integration of their joint auxiliary distribution to calculate the corresponding marginal distributions. The relative information entropy is used to quantify the difference between the compared density functions and an efficient approach based on stochastic sampling is introduced for estimating this entropy for all quantities of interest. The framework is illustrated in an example that adopts a source-based stochastic ground motion model, and valuable insight is provided for its implementation within structural engineering applications.

LANGEVIN MOLECULAR DYNAMICS DERIVED FROM EHRENFEST DYNAMICS

Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a Kac–Zwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small [Formula: see text] friction and o(M-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuation–dissipation relation.

MATHEMATICAL EXAMPLES OF SPACE-TIME PHASES

The space-time phases of a complex dynamic system are the probability distributions for state as a function of space and time which arise by evolving initial probability distributions from the distant past. Toom proved important results about space-time phases for a class of majority voter probabilistic cellular automata (PCA). Here, variants of the majority voter PCA are presented which are proved to exhibit a variety of types of space-time phase. These examples are expected to serve as useful steps on the way to a general theory of space-time phases.

On the statistical estimation of the initial probability distribution based on the observations of dynamics at the end of an interval

We consider the problem of estimation of density of a random variable playing the role of initial value for a certain dynamics. The dynamics is defined by a differential equation whose solution is observable at the end of an interval. This problem is called the problem of estimation according to indirect observations. We propose a procedure for the estimation of density based on the method of transformation of measure along the integral curve in combination with kernel estimates.

An Efficient and Numerically Stable Method for Computing Bounds for the Interval Availability Distribution

This paper is concerned with the computation of the interval availability (proportion of time in a time interval in which the system is up) distribution of a fault-tolerant system modeled by a finite (homogeneous) continuous-time Markov chain (CTMC). General-purpose methods for performing that computation tend to be very expensive when the CTMC and the time interval are large. Based on a previously available method (regenerative transformation) for computing the interval availability complementary distribution, we develop a method called bounding regenerative transformation for the computation of bounds for that measure. Similar to regenerative transformation, bounding regenerative transformation requires the selection of a regenerative state. The method is targeted at a certain class of models, including both exact and bounding failure/repair models of fault-tolerant systems with increasing structure function, with exponential failure and repair time distributions and repair in every state with failed components having failure rates much smaller than repair rates (F/R models), with a “natural” selection for the regenerative state. The method is numerically stable and computes the bounds with well-controlled error. For models in the targeted class and the natural selection for the regenerative state, computational cost should be traded off with bounds tightness through a control parameter. For large models in the class, the version of the method that should have the smallest computational cost should have small computational cost relative to the model size if the value above which the interval availability has to be guaranteed to be is close to 1. In addition, under additional conditions satisfied by F/R models, the bounds obtained with the natural selection for the regenerative state by the version that should have the smallest computational cost seem to be tight for all time intervals or not small time intervals, depending on whether the initial probability distribution of the CTMC is concentrated in the regenerative state or not. CORRECTED VERSION OF RECORD, SEE LAST PAGE OF ARTICLE

Low-Rank Unscented Kalman Filter for Freeway Traffic Estimation Problems

Data assimilation techniques are a process of using measurements combined with model predictions to (optimally) estimate the value of a certain set of variables. These techniques belong to a group of model-based estimation approaches. In traffic science, the Kalman filter and its family, namely, the extended Kalman filter (EKF), have been widely applied to problems of traffic state estimation. The EKF is successful for computation and provides some reasonably accurate results. However, traffic systems are generally so nonlinear that the linearization used in the EKF is not always straightforward. To avoid such linearization, another family of Kalman filter, the unscented Kalman filter (UKF), has been designed to account for the nonlinear system in which a set of deterministic sample points is chosen to capture the initial probability distribution. These sample points are then propagated through the nonlinear system, and the probability density function of the actual state is approximated by the ensemble of the estimates. In the UKF, the number of sample points is determined by the dimension of the states to be estimated so that it becomes computationally expensive when the size of traffic networks is increased. Thus an effort was made to approximate a UKF in which the required sample points were reduced. Numerical experiments were carried out to assess the relevance of the proposed approach with real traffic data, and comparisons with the UKF for the level of accuracy and computational time were made.

A Novel Embedded Statistical Scheduler for Realising High System-QoS

A Novel Embedded Statistical Scheduler for Realising High System-QoS

Imprecise Markov chains with absorption

We consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. We prove that under conditioning on non-absorption the imprecise conditional probabilities converge independently of the initial imprecise probability distribution if some regularity conditions are assumed. This is a generalisation of a known result from the classical theory of Markov chains by Darroch and Seneta [6].

Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition.

Probabilistic Study of the Speed of Approach to Equilibrium for an Inelastic Kac Model

This paper deals with a one-dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter p>0. In particular, the paper provides bounds for certain distances—such as specific weighted χ-distances and the Kolmogorov distance—between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent α=2/(1+p). With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, close to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function.

Developing an Event Tree for probabilistic hazard and risk assessment at Vesuvius

Probabilistic characterizations of possible future eruptive scenarios at Vesuvius volcano are elaborated and organized within a risk-based framework. In the EXPLORIS project, a wide variety of topics relating to this basic problem have been pursued: updates of historical data, reinterpretation of previous geological field data and the collection of new fieldwork results, the development of novel numerical modelling codes and of risk assessment techniques have all been completed. To achieve coherence, many diverse strands of evidence had to be unified within a formalised structure, and linked together by expert knowledge. For this purpose, a Vesuvius ‘Event Tree’ (ET) was created to summarise in a numerical-graphical form, at different levels of detail, all the relative likelihoods relating to the genesis and style of eruption, development and nature of volcanic hazards, and the probabilities of occurrence of different volcanic risks in the next eruption crisis. The Event Tree formulation provides a logical pathway connecting generic probabilistic hazard assessment to quantitative risk evaluation. In order to achieve a complete parameterization for this all-inclusive approach, exhaustive hazard and risk models were needed, quantified with comprehensive uncertainty distributions for all factors involved, rather than simple ‘best-estimate’ or nominal values. Thus, a structured expert elicitation procedure was implemented to complement more traditional data analysis and interpretative approaches. The structure of the Vesuvius Event Tree is presented, and some of the data analysis findings and elicitation outcomes that have provided initial indicative probability distributions to be associated with each of its branches are summarized. The Event Tree extends from initiating volcanic eruption events and hazards right through to human impact and infrastructure consequences, with the complete tree and its parameterisation forming a quantitative synoptic framework for comprehensive hazard evaluation and mapping of risk impacts. The organization of the Event Tree allows easy updating, as and when new information becomes available.

Cross Sections for Charge Exchange and Ionization of High-Energy Ions in Noble Gases

Interactions of charge exchange and ionization of fast, low-charged heavy ions are very important in heavy ion beam inertial confinement fusion. These effects are crucial indetermination of the final focusing in the chamber. However, corresponding cross section data is very limited and/or not accurate over the entire range of energies and ions of interest. This paper reports on our recent studies of cross sections for interactions of heavy ions with noble gases. Since a quantum mechanical treatment encounters a complex many-body problem, a classical trajectory Monte Carlo method is employed. The distribution of inner electrons is estimated by a modified Hartree-Fock model for the purpose of decreasing the number of electron orbits calculated, a micro-canonical ensemble for the initial electron probability distribution is introduced to describe quantum mechanical uncertainty. Cross sections are evaluated over a limited energy range; then scaling laws are developed to reflect the change probability for the beam charge state over a larger energy range.

Niche Construction and Cognitive Evolution

Despite the fact that animal behavior involves a particularly powerful form of niche construction, few researchers have considered how the environmental impact of behavior may feed back to influence the evolution of the cognitive underpinnings of behavior. I explore a model that explicitly incorporates niche construction while tracking cognitive evolution. Agents and their stimuli are modeled as coevolving populations. The agents are born with “weights” attached to behaviors in a repertoire. Further, these agents are able to change these weights based on previous success and an inherited learning parameter. Both the agent and the stimulus receive payoffs through a behavioral interaction (where the payoff structure is influenced by the “genotype” of the stimulus). The behaving agent exhibits niche construction through its effects on stimuli (the “environment”), which can feed back to influence the value of different cognitive strategies. Here I focus on two forms of niche construction: (1) the stimulus and responding agent have common interests (positive niche construction) and (2) the stimulus and agent have dissimilar interests (negative niche construction). The form of niche construction qualitatively affects cognitive evolution (i.e., the initial behavioral probability distribution and the value of the learning parameter). Given a mutualism between the stimulus and responding agent, rapid learning and “fixed” behavioral distributions (i.e., most of the weight on a single behavior) evolve. Given an antagonism between the stimulus and agent, slower learning and “flexible” behavioral distributions (i.e., equal weight on different behaviors) evolve. I discuss these results in light of findings from the fields of ethology, psychology, and evolutionary ecology.

Coherence and Intermittency of Electron Density in Small‐Scale Interstellar Turbulence

Spatial intermittency in decaying kinetic Alfven wave turbulence is investigated to determine if it produces non Gaussian density fluctuations in the interstellar medium. Non Gaussian density fluctuations have been inferred from pulsar scintillation scaling. Kinetic Alfven wave turbulence characterizes density evolution in magnetic turbulence at scales near the ion gyroradius. It is shown that intense localized current filaments in the tail of an initial Gaussian probability distribution function possess a sheared magnetic field that strongly refracts the random kinetic Alfven waves responsible for turbulent decorrelation. The refraction localizes turbulence to the filament periphery, hence it avoids mixing by the turbulence. As the turbulence decays these long-lived filaments create a non Gaussian tail. A condition related to the shear of the filament field determines which fluctuations become coherent and which decay as random fluctuations. The refraction also creates coherent structures in electron density. These structures are not localized. Their spatial envelope maps into a probability distribution that decays as density to the power -3. The spatial envelope of density yields a Levy distribution in the density gradient.

Quantum probabilities as degrees of belief

I outline an argument for a subjective Bayesian interpretation of quantum probabilities as degrees of belief distributed subject to consistency constraints on a quantum rather than a classical event space. I show that the projection postulate of quantum mechanics can be understood as a noncommutative generalization of the classical Bayesian rule for updating an initial probability distribution on new information, and I contrast the Bayesian interpretation of quantum probabilities sketched here with an alternative approach defended by Chris Fuchs.

Enabling access-privacy for random walk based data analysis applications

Random walk graph and Markov chain based models are used heavily in many data and system analysis domains, including web, bioinformatics, and queuing. These models enable the description and analysis of various behaviors of stochastic systems. If the system being modeled has certain properties, such as if it is irreducible and aperiodic, close form formulations corresponding to its stationary behavior can be used to analyze its behavior. However, if the system does not have these properties or if the user is not interested in the stationary behavior, then an iterative approach needs to be used to determine potential outcomes based on the initial probability distribution inputs to the model. In this paper, we focus on access-privacy enabled outsourced Markov chain based data analysis applications, where a non-trusted service provider takes (hidden) user queries that are described in terms of initial state distributions, and evaluates them iteratively in an oblivious manner. We show that this iterative process can leak information regarding the possible values of the hidden input if the server has a priori knowledge about the underlying Markovian process. Hence as opposed to simple obfuscation mechanisms, we develop an algorithm based on methodical addition of extra states, which guarantees unbounded feasible regions for the inputs, thus preventing a malicious host from having an informed guess regarding the inputs. In particular, we show that if the underlying transition matrix is diagonalizable then we can compute the exact number of states needed for access-privacy, while if the matrix is non-diagonalizable, then only a lower-bound can be computed.

Entropy conserving probability transforms and the entailment principle

Our main result here is the development of a general procedure for transforming some initial probability distribution into a new probability distribution in a way that the resulting distribution has entropy at least as great as the original distribution.A significant aspect of our approach is that it makes use of Zadeh's entailment principle which is itself a general procedure for going from an initial possibility distribution to a new possibility distribution so that the resulting possibility has an uncertainty at least as great as the original.

Decentralized State Initialization with Delay Compensation for Multi-modal Sensor Networks

Decentralized processing algorithms are attractive alternatives to centralized algorithms for target tracking applications in smart sensor networks since they provide the ability to scale, reduce vulnerability, reduce communication, and share processing responsibilities among individual nodes. Sharing the processing responsibilities allows parallel processing of raw data at the individual nodes. However, this introduces other difficulties in multi-modal smart sensor networks, such as non-observability of the targets' states at any individual node and various delays such as varying processing delays, communication delays and signal propagation delays for the different modalities. In this paper, we provide a novel algorithm to determine the initial probability distribution of multiple targets' states in a decentralized manner. The targets' state vectors consist of the targets' positions and velocities on the 2D plane. Our approach can determine the state vector distribution even if the individual sensors alone are not capable of observing it. Our approach can also compensate for varying delays among the assorted modalities. The resulting distribution can be used to initialize various tracking algorithms. Our approach is based on Monte Carlo methods, where the state distributions are represented by a weighted set of discrete state realizations. A robust weighting strategy is formulated to account for missed detections, clutter and estimation delays. To demonstrate the effectiveness of the algorithm, we simulate a network with direction-of-arrival nodes and range-Doppler nodes.

2305 維持規格の現場適用の高能率化に関する研究(OS5-2 設計と最適化II)

Codes for nuclear power generation facilities (rules on fitness for service for nuclear power plants) has been developed by the JSME committee in 2004. However, JSME does not endorse any facility in accordance with this code. Users must understand interpret of this code and accept responsibility for using the code. The calculation program for the crack propagation of fatigue and stress corrosion crack (SCC) has been developed in accordance with the operating procedure of the code. The calculation time for out put the life of fatigue (or SCC) is a few minutes. The initial crack depth probability distribution and the probability of detection for crack in pipes have been investigated for support of management of plant pipe life. The developed support program can estimate the reasonable replace time of pipe by using of Probabilistic fracture mechanics (PFM).