Probabilities Of Deviations Research Articles

Deviation probabilities for arithmetic progressions and other regular discrete structures

AbstractLet the random variable count the number of edges of a hypergraph induced by a random m element subset B of its vertex set. Focussing on the case that satisfies some regularity condition we prove bounds on the probability that X is far from its mean. It is possible to apply these results to discrete structures such as the set of k‐term arithmetic progressions in the cyclic group . Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case is generated by including each vertex independently with probability p.

Large deviations of branching process in a random environment

Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Y n+1=A n Y n + B n , where A 1, A 2, … are independent identically distributed random variables and B n may depend on { ( A k , B k ) , 0 ⩽ k < n } $ \{(A_k,B_k),0\leqslant k \lt n\} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.

Probabilistic Mesoscale Analysis of Concrete Beams Subjected to Flexure

In this paper, the probabilistic behavior of plain concrete beams subjected to flexure is studied using a continuous mesoscale model. The model is two-dimensional where aggregate and mortar are treated as separate constituents having their own characteristic properties. The aggregate is represented as ellipses and generated under prescribed grading curves. Ellipses are randomly placed so it requires probabilistic analysis for model using the Monte Carlo simulation with 20 realizations to represent geometry uncertainty. The nonlinear behavior is simulated with an isotropic damage model for the mortar, while the aggregate is assumed to be elastic. The isotropic damage model softening behavior is defined in terms of fracture mechanics parameters. This damage model is compared with the fixed crack model in macroscale study before using it in the mesoscale model. Then, it is used in the mesoscale model to simulate flexure test and compared to experimental data and shows a good agreement. The probabilistic behavior of the model response is presented through the standard deviation, moment parameters and cumulative probability density functions in different loading stages. It shows variation of the probabilistic characteristics between pre-peak and post-peak behaviour of load-CMOD curves.

Development of a DSO support tool for congestion forecast

This paper presents a novel DSO support tool with visualisation capability for forecasting network congestion in distribution systems with a high level of renewables. To incorporate the uncertainties in the distribution systems, the probabilistic power flow framework has been utilised. An advanced photovoltaic production forecast based on sky images and a load forecast using an artificial neural network is used as the input to the tool. In addition, advanced load models and operating modes of photovoltaic inverters have been incorporated into the tool. The tool has been applied in case studies to perform congestion forecasts for two real distribution systems to validate its usability and scalability. The results from case studies demonstrated that the tool performs satisfactorily for both small and large networks and is able to visualise the cumulative probabilities of nodes voltage deviation and network components (branches and transformers) congestion for a variety of forecast horizons as desired by the DSO. The results have also shown that explicit inclusion of load-voltage dependency models would improve the accuracy of the congestion forecast. For demonstrating the applicability of the tool, it has been integrated into an existing distribution management system via the IoT platform of a DMS vendor, Atos Worldgrid.

Multimodal Machine Learning Using Visual Fields and Peripapillary Circular OCT Scans in Detection of Glaucomatous Optic Neuropathy

To develop and validate a multimodal artificial intelligence algorithm, FusionNet, using the pattern deviation probability plots from visual field (VF) reports and circular peripapillary OCT scans to detect glaucomatous optic neuropathy (GON). Cross-sectional study. Two thousand four hundred sixty-three pairs of VF and OCT images from 1083 patients. FusionNet based on bimodal input of VF and OCT paired data was developed to detect GON. Visual field data were collected using the Humphrey Field Analyzer (HFA). OCT images were collected from 3 types of devices (DRI-OCT, Cirrus OCT, and Spectralis). Two thousand four hundred sixty-three pairs of VF and OCT images were divided into 4 datasets: 1567 for training (HFA and DRI-OCT), 441 for primary validation (HFA and DRI-OCT), 255 for the internal test (HFA and Cirrus OCT), and 200 for the external test set (HFA and Spectralis). GON was defined as retinal nerve fiber layer thinning with corresponding VF defects. Diagnostic performance of FusionNet compared with that of VFNet (with VF data as input) and OCTNet (with OCT data as input). FusionNet achieved an area under the receiver operating characteristic curve (AUC) of 0.950 (0.931-0.968) and outperformed VFNet (AUC, 0.868 [95% confidence interval (CI), 0.834-0.902]), OCTNet (AUC, 0.809 [95% CI, 0.768-0.850]), and 2 glaucoma specialists (glaucoma specialist 1: AUC, 0.882 [95% CI, 0.847-0.917]; glaucoma specialist 2: AUC, 0.883 [95% CI, 0.849-0.918]) in the primary validation set. In the internal and external test sets, the performances of FusionNet were also superior to VFNet and OCTNet (FusionNet vs VFNet vs OCTNet: internal test set 0.917 vs 0.854 vs 0.811; external test set 0.873 vs 0.772 vs 0.785). No significant difference was found between the 2 glaucoma specialists and FusionNet in the internal and external test sets, except for glaucoma specialist 2 (AUC, 0.858 [95% CI, 0.805-0.912]) in the internal test set. FusionNet, developed using paired VF and OCT data, demonstrated superior performance to both VFNet and OCTNet in detecting GON, suggesting that multimodal machine learning models are valuable in detecting GON.

False Positive Responses in Standard Automated Perimetry

To analyze the relationship between rates of false positive (FP) responses and standard automated perimetry results. Prospective multicenter cross-sectional study. One hundred twenty-six patients with manifest or suspect glaucoma were tested with Swedish Interactive Thresholding Algorithm (SITA) Standard, SITA Fast, and SITA Faster at each of 2 visits. We calculated intervisit differences in mean deviation (MD), visual field index (VFI), and number of statistically significant test points as a function of FP rates and also as a function of general height (GH). Increasing FP values were associated with higher MD values for all 3 algorithms, but the effects were small, 0.3 dB to 0.6 dB, for an increase of 10 percentage points of FP rate, and for VFI even smaller (0.6%-1.4%). Only small parts of intervisit differences were explained by FP (r2 values 0.00-0.11). The effects of FP were larger in severe glaucoma, with MD increases of 1.1 dB to 2.0 dB per 10 percentage points of FP, and r2values ranging from 0.04 to 0.33. The numbers of significantly depressed total deviation points were affected only slightly, and pattern deviation probability maps were generally unaffected. GH was much more strongly related to perimetric outcomes than FP. Across 3 different standard automated perimetry thresholding algorithms, FP rates showed only weak associations with visual field test results,except in severe glaucoma. Current recommendations regarding acceptable FP ranges may require revision. GH or other analyses may be better suited than FP rates for identifying unreliable results in patients who frequently press the response button without having perceived stimuli.

On large deviations for combinatorial sums

We investigate an asymptotic behaviour of probabilities of large deviations for normalized combinatorial sums. We find a zone in which these probabilities are equivalent to the tail of the standard normal law. Our conditions are similar to the classical Bernstein conditions. The range of the zone of the normal convergence can be of a power order.

On the Probabilities of Large Deviations of Chi-square and Log-likelihood Ratio Statistics

On the Probabilities of Large Deviations of Chi-square and Log-likelihood Ratio Statistics

Visual Field Endpoints Based on Subgroups of Points May Be Useful in Glaucoma Clinical Trials: A Study With the Humphrey Field Analyzer and Compass Perimeter.

Visual field (VF) endpoints based on average deviation of specific subsets of points rather than all points may offer a more homogeneous data set without necessarily worsening test-retest variability and so may be useful in clinical trials. The purpose of this study was to characterize the outcome measures encompassing particular subsets of VF points and compare them as obtained with Humphrey [Humphrey visual field analyser (HVF)] and Compass perimeters. Thirty patients with imaging-based glaucomatous neuropathy performed a pair of 24-2 tests with each of 2 perimeters. Nonweighted mean deviation (MD) was calculated for the whole field and separate vertical hemifields, and again after censoring of points with low sensitivity (MDc) and subsequently including only "abnormal" points with a total deviation probability of <5% (MDc5%) or <2% (MDc2%). Test-retest variability was assessed using Bland-Altman 95% limits of agreement (95%LoA). For the whole field, using HVF, MD was -7.5±6.9 dB, MDc -3.6±2.8 dB, MDc5% -6.4±1.7 dB, and MDc2% -7.3±1.5 dB. With Compass the MD was -7.5±6.6, MDc -2.9±1.7 dB, MDc5% -6.3±1.5, and MDC2% -7.9±1.6. The respective 95%LoA were 5.5, 5.3, 4.6, and 5.6 with HVF, and 4.8, 3.7, 7.1, and 7.1 with Compass. The respective number of eligible points were 52, 42±12, 20±11, and 15±9 with HVF, and 52, 41.2±12.6, 10±7, and 7±5 with Compass. With both machines, SD and 95%LoA increased in hemifields compared with the total field, but this increase was mitigated after censoring. Restricting analysis to particular subsets of points of interest in the VF after censoring points with low sensitivity, as compared with using the familiar total field MD, can provide outcome measures with a broader range of MD, a markedly reduced SD and therefore more homogeneous data set, without necessarily worsening test-retest variability.

Lower deviations for supercritical branching processes with immigration

for a supercritical branching processes with immigration {Zn}, it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of ℙ(Zn = kn) with kn = o(mn) as n → ∞. We give asymptotic behavior of such lower deviation probabilities in both Schroder and Bottcher cases, unifying and extending the previous results for Galton-Watson processes in literature.

Lower deviations in β-ensembles and law of iterated logarithm in last passage percolation

For last passage percolation (LPP) on ℤ2 with exponential passage times, let Tn denote the passage time from (1, 1) to (n,n). We investigate the law of iterated logarithm of the sequence {Tn}n≥1; we show that $$\lim \,{\inf _{n \to \infty }}{{{T_n} - 4n} \over {{n^{1/3}}{{\left( {\log \,\log \,n} \right)}^{1/3}}}}$$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight shift in perspective from point-to-point passage times to considering point-to-line passage times instead, and exploiting the correspondence of the latter to the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient, which is of independent interest, is a new lower bound of lower tail deviation probability of the largest eigenvalue of β-Laguerre ensembles, which extends the results proved in the context of the β-Hermite ensembles by Ledoux and Rider (2010).

Avoiding mask-related artefacts in visual field tests during the COVID-19 pandemic

AimsTo assess visual field (VF) pseudoprogression related to face mask use.MethodsWe reviewed a total of 307 VFs performed with a face mask (FPP2/KN95 or surgical masks) and compared them with...

Correction The probabilities of large deviations for a certain class of statistics associated with multinomial distributions

Correction The probabilities of large deviations for a certain class of statistics associated with multinomial distributions

Numerical Simulation of a Nonlinear Parabolic Equation for Analyzing The Perceived Loudness Statistics of Sonic Boom Wave after Propagation Through Atmospheric Turbulent Layer

Soniс boom propagation in the atmospheric turbulent layer, represented by a model of homogeneous isotropic turbulence, is investigated numerically in two-dimensional geometry using a KZK-type nonlinear parabolic equation for an inhomogeneous moving medium with relaxation. The mean value, standard deviation, and cumulative probabilities of the sonic boom amplitude, shock front steepness, and the Perceived Loudness Mark VII metric of subjective loudness of an impulse noise are analyzed at different propagation distances traveled by the sonic boom wave in the turbulent layer and for different initial amplitudes of the wave.

Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories

Extensive time-series encoding the position of particles such as viruses, vesicles, or individual proteins are routinely garnered in single-particle tracking experiments or supercomputing studies. They contain vital clues on how viruses spread or drugs may be delivered in biological cells. Similar time-series are being recorded of stock values in financial markets and of climate data. Such time-series are most typically evaluated in terms of time-averaged mean-squared displacements (TAMSDs), which remain random variables for finite measurement times. Their statistical properties are different for different physical stochastic processes, thus allowing us to extract valuable information on the stochastic process itself. To exploit the full potential of the statistical information encoded in measured time-series we here propose an easy-to-implement and computationally inexpensive new methodology, based on deviations of the TAMSD from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion (BM) to check the applicability of this approach to simulated and real data sets. By comparing the probability of deviations for different data sets, we demonstrate how the theoretical bound for BM reveals additional information about observed stochastic processes. We apply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracer beads measured in mucin hydrogels, and of geographic surface temperature anomalies. Our analysis shows how the large-deviation properties can be efficiently used as a simple yet effective routine test to reject the BM hypothesis and unveil relevant information on statistical properties such as ergodicity breaking and short-time correlations.

Deviations for random sums indexed by the generations of a branching process

Applying the results about harmonic moments of classical Galton-Watson process, we obtain the deviations for random sums indexed by the generations of a branching process. Our results show that the decay rates of large deviations and moderate deviations depend heavily on the degree of the heavy tail and the asymptotic distributions depend heavily on the normalizing constants. If the underlying Galton-Watson process belongs to the Schr?der case, both large deviation and moderate deviation probabilities show three decay rates, where the critical case depends heavily on the Schr?der index. Else if the Galton-Watson process belongs to the B?ttcher case, there are only two decay rate for both large deviation and moderate deviation probabilities. Simulations are also given to illustrate our results.

Secretary problem with vanishing objects

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

Analytical Framework for Space Debris Collision Avoidance Maneuver Design

An analytical formulation for collision avoidance maneuvers involving a spacecraft and a space debris is presented, including solutions for the maximum deviation and minimum collision probability cases. Gauss’s planetary equations and relative motion equations are used to map maneuvers at a given time to displacements at the predicted close approach. The model is then extended to map changes in state between two times, allowing one to propagate covariance matrices. The analytical formulation reduces the optimization problem to an eigenproblem, for both maximum deviation and minimum collision probability. Two maximum deviation cases, total deviation and impact parameter, are compared for a large set of spacecraft–debris conjunction geometries derived from European Space Agency’s Meteoroid and Space Debris Terrestrial Environment Reference (MASTER-2009) model. Moreover, the maximum impact parameter and minimum collision probability maneuvers are compared assuming covariances known at the maneuver time, to evaluate the net effect of lead time in collision probability. In all cases, solutions are analyzed in the b-plane to leverage its natural separation of phasing and geometry change effects. Both uncertainties and maximum deviation grow along the time axis for long lead times, limiting the reduction in collision probability.

СТАТИСТИЧНИЙ АНАЛІЗ РЕЗУЛЬТАТІВ ГЕОДЕЗИЧНИХ ЗЙОМОК КОЛІЙ МОСТОВИХ КРАНІВ У ВИРОБНИЧИХ БУДІВЛЯХ

Abstract. The studies were carried out in order to develop effective methodology for the statistical analysis of the results of geodetic surveys of the rails of overhead cranes. The findings will provide generalization of the measurement results, demonstration of their changes over time, as well as comparison with the permissible values of deviations from the design position. The analysis includes the results of geodetic surveys of the position of the rails of overhead cranes in machine-building shops during four years. The position of the heads of the crane rails was determined using a level, a transit theodolite, and a laser sight. In accordance with the requirements of regulatory documents for the operation of overhead cranes, three parameters of the position of the crane rail heads were controlled: the height difference on adjacent pillars of the same row; height difference on pillars of different rows in one transverse frame; deviation of the distance between the rails from the design value. The controlled parameters are random variables that should be analyzed by statistical methods. Deviations of the distance between the axes of the crane rails from the design value are presented in the form of a random variable with the normal distribution law. The differences in the elevation marks of the heads of the crane rails, which are monitored without taking into account the sign, are described by the distribution law of the modulus of a random variable. As a generalized characteristic of the state of crane rails, it is proposed to use the probabilities of horizontal and vertical deviations of the crane rails beyond maximum permissible values established by the norms. Based on the results of the research, conclusions were drawn about the need for systematic control of the position of overhead crane rails by geodetic methods, the applicability of the above distributions for the probabilistic description of deviations of crane rails from the design position, as well as the feasibility of using the probability of deviations of the crane rails geometry beyond the permissible values established by the norms.

Stability of functionals of perturbed Markov chains under the condition of uniform minorization

Abstract In this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, L 2 {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.