Level-crossing Probabilities Research Articles

APPROXIMATE LEVEL-CROSSING PROBABILITIES FOR INTERACTIVE VISUALIZATION OF UNCERTAIN ISOCONTOURS

A major method for quantitative visualization of a scalar field is depiction of its isocontours. If the scalar field is afflicted with uncertainties, uncertain counterparts to isocontours have to be extracted and depicted. We consider the case where the input data is modeled as a discretized Gaussian field with spatial correlations. For this situation we want to compute level-crossing probabilities that are associated to grid cells. To avoid the high computational cost of Monte Carlo integrations and direction dependencies of raycasting methods, we formulate two approximations for these probabilities that can be utilized during rendering by looking up precomputed univariate and bivariate distribution functions. The first method, called maximum edge crossing probability, considers only pairwise correlations at a time. The second method, called linked-pairs method, considers joint and conditional probabilities between vertices along paths of a spanning tree over the n vertices of the grid cell; with each possible tree an n-dimensional approximate distribution is associated; the choice of the distribution is guided by minimizing its Bhattacharyya distance to the original distribution. We perform a quantitative and qualitative evaluation of the approximation errors on synthetic data and show the utility of both approximations on the example of climate simulation data.

Analysis of porosity distribution of large-scale porous media and their reconstruction by Langevin equation

Several methods have been developed in the past for analyzing the porosity and other types of well logs for large-scale porous media, such as oil reservoirs, as well as their permeability distributions. We developed a method for analyzing the porosity logs ϕ(h) (where h is the depth) and similar data that are often nonstationary stochastic series. In this method one first generates a new stationary series based on the original data, and then analyzes the resulting series. It is shown that the series based on the successive increments of the log y(h)=ϕ(h+δh)-ϕ(h) is a stationary and Markov process, characterized by a Markov length scale h(M). The coefficients of the Kramers-Moyal expansion for the conditional probability density function (PDF) P(y,h|y(0),h(0)) are then computed. The resulting PDFs satisfy a Fokker-Planck (FP) equation, which is equivalent to a Langevin equation for y(h) that provides probabilistic predictions for the porosity logs. We also show that the Hurst exponent H of the self-affine distributions, which have been used in the past to describe the porosity logs, is directly linked to the drift and diffusion coefficients that we compute for the FP equation. Also computed are the level-crossing probabilities that provide insight into identifying the high or low values of the porosity beyond the depth interval in which the data have been measured.

Landau-Zener problem in a three-level neutrino system with nonlinear time dependence

We consider the level-crossing problem in a three-level system with nonlinearly time-varying Hamiltonian (time-dependence ${t}^{\ensuremath{-}3}$). We study the validity of the so-called independent crossing approximation in the Landau-Zener model by making comparison with results obtained numerically in the density matrix approach. We also demonstrate the failure of the so-called ``nearest zero'' approximation of the Landau-Zener level-crossing probability integral.

Level-crossing probabilities and first-passage times for linear processes

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

Level-crossing probabilities and first-passage times for linear processes

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

SN 1987A and the status of oscillation solutions to the solar neutrino problem

We study neutrino oscillations and the level-crossing probability PLZ in power-law potential profiles A(r)\propto r^n. We give local and global adiabaticity conditions valid for all mixing angles theta and discuss different representations for PLZ. For the 1/r^3 profile typical of supernova envelopes we compare our analytical to numerical results and to earlier approximations used in the literature. We then perform a combined likelihood analysis of the observed SN1987A neutrino signal and of the latest solar neutrino data, including the recent SNO CC measurement. We find that, unless all relevant supernova parameters (released binding energy, \bar\nu_e and \bar\nu_{\mu,\tau} temperatures) are near their lowest values found in simulations, the status of large mixing type solutions deteriorates considerably compared to fits using only solar data. This is sufficient to rule out the vacuum-type solutions for most reasonable choices of astrophysics parameters. The LOW solution may still be acceptable, but becomes worse than the SMA-MSW solution which may, in some cases, be the best combined solution. On the other hand the LMA-MSW solution can easily survive as the best overall solution, although its size is generally reduced when compared to fits to the solar data only.

Nonadiabatic level crossing in resonant and nonresonant neutrino oscillations

We study neutrino oscillations and the level-crossing probability ${P}_{\mathrm{LSZ}}=\mathrm{exp}(\ensuremath{-}{\ensuremath{\gamma}}_{n}{\mathcal{F}}_{n}\ensuremath{\pi}/2)$ (LSZ stands for Landau-St\"uckelberg-Zener) in power-law-like potential profiles $A(r)\ensuremath{\propto}{r}^{n}.$ After showing that the resonance point coincides only for a linear profile with the point of maximal violation of adiabaticity, we point out that the ``adiabaticity'' parameter ${\ensuremath{\gamma}}_{n}$ can be calculated at an arbitrary point if the correction function ${\mathcal{F}}_{n}$ is rescaled appropriately. We present a new representation for the level-crossing probability, ${P}_{\mathrm{LSZ}}=\mathrm{exp}(\ensuremath{-}{\ensuremath{\kappa}}_{n}{\mathcal{G}}_{n}),$ which allows a simple numerical evaluation of ${P}_{\mathrm{LSZ}}$ in both the resonant and nonresonant cases, and where ${\mathcal{G}}_{n}$ contains the full dependence of ${P}_{\mathrm{LSZ}}$ on the mixing angle $\ensuremath{\vartheta}.$ As an application we consider the case $n=\ensuremath{-}3$ important for oscillations of supernova neutrinos.

Probability of heavy traffic period in third generation CDMA mobile communication

The paper considers a problem of deriving the multidimensional distribution of a segment of a long-range dependent traffic in the third generation mobile communication network. An exact expression for the probability is found when a self-similar process from [8] models the traffic. the probability of heavy-traffic period, the outage probability, and the level-crossing probability are found. It is shown that the level crossing probability depends on the average call length only. Further, this probability for traffic with dependent samples is lower than for traffic with independent samples. Also, it is shown that there is a linear dependence between the average heavy traffic interval and the average call length.

SIMULATING LEVEL CROSSING PROBABILITIES BY IMPORTANCE SAMPLING FOR NON-DECREASING COMPOUND POISSON PROCESSES WITH BOUNDED JUMPS AND A NEGATIVE DRIFT

Article SIMULATING LEVEL CROSSING PROBABILITIES BY IMPORTANCE SAMPLING FOR NON-DECREASING COMPOUND POISSON PROCESSES WITH BOUNDED JUMPS AND A NEGATIVE DRIFT was published on February 1, 2001 in the journal Statistics & Risk Modeling (volume 19, issue 2).

Rough limit results for level-crossing probabilities

Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

Rough limit results for level-crossing probabilities

Let Y 1, Y 2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn ).

Nonadiabatic neutrino oscillations reexamined.

Employing the Feynman procedure of ordered exponential operators and the stationary phase method to evaluate the multiple integrals involved, we calculate the level-crossing probability and analyze the role of a resonance in the evolution of a two-level neutrino system. We compare this procedure with more conventional ones, such as Landau's method and the ansatz of Kuo and Pantaleone and Petcov. We verify that our results reproduce the correct extreme nonadiabatic limit and give the standard solutions in the adiabatic regime for any arbitrary matter density distributions. We discuss in particular the case of solar neutrino propagation using the standard solar model predictions for the matter distribution in the Sun.

Simulating level-crossing probabilities by importance sampling

Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.

Simulating level-crossing probabilities by importance sampling

Let X 1, X 2, · ·· be independent and identically distributed random variables such that ΕΧ 1 < 0 and P (X 1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X 1 + X 2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P (T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P (T <∞) and . The paper ends with an example.

Nonadiabatic neutrino oscillations in matter.

We examine the classical survival probability for neutrino propagation through matter of arbitrary density distributions. It is shown that the level-crossing probability can be calculated approximately for any density distribution by Landau's method. Exact solutions are discussed and new results are presented for a 1/r density distribution. From an examination of the exact solutions, we present an ansatz for generalizing Landau's method to a correct treatment of the extreme nonadiabatic limit. Our two-flavor results easily generalize to three flavors.

Aperture-averaged level-crossing probability in an atmospheric communication link

The intensity statistics of light pulses in an atmospheric communication link are studied here for the case of a finite (nonpoint) detector aperture. The calculation of the statistics of the pulse intensities averaged over the aperture is performed by using a discrete aperture model and applying the multidimensional ognormal distribution. The results of the numerical calculations demonstrate the dependence of the level-crossing probability on aperture radius, number of pulses, and the length of a pulse sequence. The level-crossing probabilities were measured experimentally and found to be in close agreement with the theoretical values.

Poisson models and mean-squared error for correlator estimator of time delay

A method for modeling large errors in correlation-based time-delay estimation is developed in terms of level-crossing probabilities. The level-crossing interpretation for peak ambiguity leads directly to an exact expression for the probability of large error involving the hazard function associated with the level-crossing process. Two models for the distribution of the error over the level-crossing time yield approximations to the mean-square error (MSE) that involve the low-order ( >

False alarm and correct detection probabilities over a time interval for restricted classes of failure detection algorithms

The statistical analysis of failure detection decisions in terms of the instantaneous probabilities of false alarm and correct detection for a specified failure magnitude at each check-time have previously been performed for several different failure detection techniques that utilize a Kalman filter. By performing a discrete-time specialization of a result of Gallager and Helstrom on a tightened upper bound for continuous-time level-crossing probabilities, upper bounds on the probabilities of false alarm and correct detection over a time interval have been obtained for the specific technique of CR2 tailnre detection (to allow an accounting for the effect of time correlations of the filter estimates). When these upper bounds are optimized to be as tight as possible to the desired probabilities, the resulting optimization problem for discrete-time is a collection of quadratic programming (QP) problems, which may easily be solved exactly without recourse to approximate solutions as were resorted to in the continuous-time formulation. This technique for evaluating tightened upper bounds on the false alarm and correct detection probabilities may be of general interest, since it can be applied to any failure detection technique or signal detection technique that can relate an exceeding of the deterministic decision threshold by the test statistic directly to a deterministic level being exceeded by a scalar Gaussian random process.