Initial Probability Distribution Research Articles

Quantitative Weakest Hyper Pre: Unifying Correctness and Incorrectness Hyperproperties via Predicate Transformers

We present a novel weakest pre calculus for reasoning about quantitative hyperproperties over nondeterministic and probabilistic programs. Whereas existing calculi allow reasoning about the expected value that a quantity assumes after program termination from a single initial state, we do so for initial sets of states or initial probability distributions. We thus (i) obtain a weakest pre calculus for hyper Hoare logic and (ii) enable reasoning about so-called hyperquantities which include expected values but also quantities (e.g. variance) out of scope of previous work. As a byproduct, we obtain a novel strongest post for weighted programs that extends both existing strongest and strongest liberal post calculi. Our framework reveals novel dualities between forward and backward transformers, correctness and incorrectness, as well as nontermination and unreachability.

Non-Markovian processes on heteroclinic networks.

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values.

Evolution of pore systems in low-maturity oil shales during thermal upgrading—Quantified by dynamic SEM and machine learning

In-situ upgrading by heating is feasible for low-maturity shale oil, where the pore space dynamically evolves. We characterize this response for a heated substrate concurrently imaged by SEM. We systematically follow the evolution of pore quantity, size (length, width and cross-sectional area), orientation, shape (aspect ratio, roundness and solidity) and their anisotropy—interpreted by machine learning. Results indicate that heating generates new pores in both organic matter and inorganic minerals. However, the newly formed pores are smaller than the original pores and thus reduce average lengths and widths of the bedding-parallel pore system. Conversely, the average pore lengths and widths are increased in the bedding-perpendicular direction. Besides, heating increases the cross-sectional area of pores in low-maturity oil shales, where this growth tendency fluctuates at < 300 °C but becomes steady at > 300 °C. In addition, the orientation and shape of the newly-formed heating-induced pores follow the habit of the original pores and follow the initial probability distributions of pore orientation and shape. Herein, limited anisotropy is detected in pore direction and shape, indicating similar modes of evolution both bedding-parallel and bedding-normal. We propose a straightforward but robust model to describe evolution of pore system in low-maturity oil shales during heating.

A flexible framework for coding and predicting acute hypotensive episodes using Markov chains

Over the past decade, the prediction of acute hypotensive episodes (AHEs) using computational models has gained significant attention. AHEs in patients admitted to intensive care units can lead to severe or life-threatening conditions. AHEs are defined as a drop in the mean arterial pressure (MAP) below a threshold during an observation period. However, adaptable AHE modeling approaches are required to account for patient-specific definitions caused by inter-patient variability. This study proposes a flexible three-stage framework for the AHE prediction problem capable of modeling different hypotension states. The initial stage involves encoding the MAP time series using the proposed scheme. A clustering process is then performed to determine the k≥2 hypotension states and transition probabilities between them. The proposed AHE prediction model is built using a cohort of 453 MAP waveforms from MIMIC-II. In the third stage, we use 173 MAP waveforms to validate the proposed prediction method. Given a patient’s MAP time series, we compute the initial probability distribution by considering (a) an encoded observation interval and (b) a similarity measure to determine the closest kth cluster (state) to the encoded interval. Based on the properties of the Markov process, we predict the most probable state for look-ahead times h ranging from 1 to 30 min. The proposed framework offers competitive performance for h=7 with an accuracy of 0.965, which is clinically relevant for medical practitioners. We also provide a comparative analysis to validate the effectiveness of the proposed approach, which exhibits a significant statistical difference (α=0.05) compared with other state-of-the-art methods.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

Geometric conjecture about phase transitions.

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach the steady state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and ε-mixing time of the configuration spaces of hard-disk and hard-sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems and numerical evidence suggests that a discontinuity in the ε-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.

Online Probabilistic Static Security Assessment for Power Systems Considering High Renewable Penetration

With the rapid development of renewable energy, a large number of renewable energy stations are connected to the power system, which leads to a decrease in the inertia of the power system and an increase in safety risks suffered. Thus, online static security assessment (SSA) is increasingly necessary. However, because of the uncertainty of renewable energy, it is not feasible to check all possible scenarios in online SSA. To reduce the number of calculations and achieve SSA in a short time, a new online SSA method based on scenario clustering for future ultra-short-term security assessment is proposed in this paper. In the offline stage, a key scenario set is constructed by Markov Chain Monte Carlo and K-means with historical data. In the online application, the initial probability distribution of renewable energy outputs is calculated by joint distribution with the output of the previous interval and corrected by weather data. Then load flow calculation with N-1 criteria is executed, and the probability for the safe operation of the system is calculated. The effectiveness of the proposed online SSA scheme has been verified in the IEEE-300 system, where one of the generators is replaced by a renewable energy station.

Heat Flow in a Periodically Forced, Thermostatted Chain II

We derive a macroscopic heat equation for the temperature of a pinned harmonic chain subject to a periodic force at its right side and in contact with a heat bath at its left side. The microscopic dynamics in the bulk is given by the Hamiltonian equation of motion plus a reversal of the velocity of a particle occurring independently for each particle at exponential times, with rate gamma . The latter produces a finite heat conductivity. Starting with an initial probability distribution for a chain of n particles we compute the current and the local temperature given by the expected value of the local energy. Scaling space and time diffusively yields, in the nrightarrow +infty limit, the heat equation for the macroscopic temperature profile T(t, u), t>0, u in [0,1]. It is to be solved for initial conditions T(0, u) and specified T(t,0)=T_-, the temperature of the left heat reservoir and a fixed heat flux J, entering the system at u=1. |J| equals the work done by the periodic force which is computed explicitly for each n.

Quantum Equilibrium in Stochastic de Broglie–Bohm–Bell Quantum Mechanics

This paper investigates dynamical relaxation to quantum equilibrium in the stochastic de Broglie–Bohm–Bell formulation of quantum mechanics. The time-dependent probability distributions are computed as in a Markov process with slowly varying transition matrices. Numerical simulations, supported by exact results for the large-time behavior of sequences of (slowly varying) transition matrices, confirm previous findings that indicate that de Broglie–Bohm–Bell dynamics allows an arbitrary initial probability distribution to relax to quantum equilibrium; i.e., there is no need to make the ad-hoc assumption that the initial distribution of particle locations has to be identical to the initial probability distribution prescribed by the system’s initial wave function. The results presented in this paper moreover suggest that the intrinsically stochastic nature of Bell’s formulation, which is arguable most naturally formulated on an underlying discrete space-time, is sufficient to ensure dynamical relaxation to quantum equilibrium for a large class of quantum systems without the need to introduce coarse-graining or any other modification in the formulation.

Refined probability distribution module for fine-grained visual categorization

Fine-grained visual categorization is an important task in computer vision. Prior works on fine-grained visual categorization have paid much attention to addressing intra-class variation and inter-class similarity. However, they rarely study that task from the perspective of probability distribution. In this paper, we propose a novel refined probability distribution module based on deep convolutional neural network. Our module computes the probability of an image by fully utilizing the similarity information between images. Firstly, we use deep neural networks to obtain the initial probability distribution and extract features. Then, we build a network whose inputs are features for calculating image-to-image similarity scores. Finally, our module refines the initial probability distribution based on an effective batch random walk operation with similarity scores. Our module can be plugged into many deep convolutional neural networks. Experimental results show that our approach outperforms state-of-the-art methods on the CUB-200–2011, FGVC-Aircraft and Stanford Cars datasets respectively.

Convex Approach to Covariance Control with Application to Stochastic Low-Thrust Trajectory Optimization

This paper outlines a novel approach to the design of optimal space trajectories under significant uncertainty. Finite-horizon covariance control, i.e., the steering of a system from an initial probability distribution to a desired one at a prescribed time, is employed to plan an optimal nominal path along with a robust feedback controller that compensates for exogenous in-flight disturbances. A mindful convexification strategy is devised to recast the nonlinear covariance control problem as a deterministic convex optimization problem. The convexification is based on a convenient change of variables that allows for relaxing the covariance matrix discrete-time propagation into a set of semidefinite cone constraints. While featuring a larger feasible space, the relaxed problem shares the same optimal solution as the original one. The lossless property of the relaxation is analytically proven by leveraging optimal control theory and discussed on the basis of numerical experiments. Monte Carlo campaigns are carried out to validate the in-flight performance of the attained control policies.

A mathematical analysis of EDAs with distance-based exponential models

Estimation of Distribution Algorithms have been successfully used to solve permutation-based Combinatorial Optimization Problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models are distance-based exponential models, and one example of this class is the Mallows model. In spite of its practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based Combinatorial Optimization Problems has not been developed as extensively as it has been for binary problems. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on Mallows models. The model removes the randomness of the algorithm in order to associate a dynamical system to it. Several scenarios of increasing complexity with different fitness functions and initial probability distributions are analyzed. The obtained results show: a) the strong dependence of the final results on the initial population, and b) the possibility to converge to non-degenerate distributions even in very simple scenarios, which has not been reported before in the literature.

Sensitivity analysis and uncertainty propagation for SMA-TB potentials

Uncertainty propagation and sensitivity analysis have been applied to SMA-TB interatomic potentials that are often used in atomistic simulations to study metals and alloys. Usually, the parameters of potentials are estimated by a minimization method to best fit a set of experimental or ab initio computational data, such as bulk properties (e.g., cohesive energy and elastic constants) without considering the uncertainty in these data. Our goal is to perform uncertainty propagation from the bulk properties to extract uncertainties and correlations of SMA-TB ξ,A,q,p potential parameters to generate valid parameter sets. Using a statistical framework, we estimate initial probability distributions that allow us to determine the uncertainties for each potential parameter of the SMA-TB. We show that many sets of potential parameters lead to bulk properties included in the initial probability distributions. Local and global Sobol’ sensitivity analysis methods, which entails no additional computational cost, show how physical properties can be adjusted using interatomic potentials.

Obtaining the long-term behavior of master equations with finite state space from the structure of the associated state transition network

The master equation describes the time evolution of the probabilities of a system with a discrete state space. This time evolution approaches for long times a stationary state that will in general depend on the initial probability distribution. Conditions under which the stationary state is unique are usually given as remarks appended to more comprehensive theories in the mathematical literature. We provide a direct and complete derivation of a necessary and sufficient criterion for when this steady state is unique. We translate this problem into the language of graph theory and show that there is a one-to-one correspondence between minimal absorbing sets within the state-transition network and linearly independent stationary states of the master equation.

Predication and Photon Statistics of a Three-Level System in the Photon Added Negative Binomial Distribution

Statistical and artificial neural network models are applied to forecast the quantum scheme of a three-level atomic system (3LAS) and field, initially following a photon added negative binomial distribution (PANBD). The Mandel parameter is used to detect the photon statistics of a radiation field. Explicit forms of the PANBD are given. The prediction of the Mandel parameter, atomic probability of the 3LAS in the upper state, and von Neumann entropy are obtained using time series and artificial neural network methods. The influence of probability success photons and the number of added photons to the NBD are examined. The total density matrix is used to compute and analyze the time evolution of the initial photonic negative binomial probability distribution that governs the 3LAS–field photon entanglement behavior. It is shown that the statistical quantities are strongly affected by probability success photons and the number of added photons to the NBD. Also, the prediction of quantum entropy is achieved by the time series and neural network.

DPNPED: Dynamic Perception Network for Polysemous Event Trigger Detection

Event detection is the process of analyzing event streams to detect the occurrences of events and categorize them. General methods for solving this problem are to identify and classify event triggers. Most previous works focused on improving the recognition and classification networks which neglected the representation of polysemous event triggers. Polysemy is habitually somewhat confusing in semantic understanding and hard to detect. To improve polysemous trigger detection, this paper proposes a novel framework called DPNPED, which dynamically adjusts the network depth between polysemous and common words. Firstly, to measure the polysemy, the difficulty factor is devised based on the frequency of a word as an event trigger. Secondly, the DPNPED utilizes a confidence measure to automatically adjust the network depth by comparing the predicted and initial probability distribution. Finally, our model applies focal loss to dynamically integrate the difficulty factor and confidence measure to enhance the learning of polysemous triggers. The experimental results show that our method achieves a noticeable improvement in polysemous event trigger detection.

Determination of the initial density in nonlocal diffusion from final time measurements

&lt;p style='text-indent:20px;'&gt;This paper is concerned with an inverse problem related to a fractional parabolic equation. We aim to reconstruct an unknown initial condition from noise measurement of the final time solution. It is a typical nonlinear and ill-posed inverse problem related to a nonlocal operator. The considered problem is motivated by a probabilistic framework when the initial condition represents the initial probability distribution of the position of a particle. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. In this work, we have discussed some theoretical and practical issues related to the considered problem. The existence, uniqueness, and stability of the optimization problem solution have been proved. The conjugate gradient method combined with Morozov's discrepancy principle are exploited for building an iterative reconstruction process. Some numerical examples are carried out showing the accuracy and efficiency of the proposed method.&lt;/p&gt;

Improved Fusion of Spatial Information into Hyperspectral Classification through the Aggregation of Constrained Segment Trees: Segment Forest

Compared with traditional optical and multispectral remote sensing images, hyperspectral images have hundreds of bands that can provide the possibility of fine classification of the earth’s surface. At the same time, a hyperspectral image is an image that coexists with the spatial and spectral. It has become a hot research topic to combine the spatial spectrum information of the image to classify hyperspectral features. Based on the idea of spatial–spectral classification, this paper proposes a novel hyperspectral image classification method based on a segment forest (SF). Firstly, the first principal component of the image was extracted by the process of principal component analysis (PCA) data dimension reduction, and the data constructed the segment forest after dimension reduction to extract the non-local prior spatial information of the image. Secondly, the images’ initial classification results and probability distribution were obtained using support vector machine (SVM), and the spectral information of the images was extracted. Finally, the segment forest constructed above is used to optimize the initial classification results and obtain the final classification results. In this paper, three domestic and foreign public data sets were selected to verify the segment forest classification. SF effectively improved the classification accuracy of SVM, and the overall accuracy of Salinas was enhanced by 11.16%, WHU-Hi-HongHu by 15.89%, and XiongAn by 19.56%. Then, it was compared with six decision-level improved space spectrum classification methods, including guided filtering (GF), Markov random field (MRF), random walk (RW), minimum spanning tree (MST), MST+, and segment tree (ST). The results show that the segment forest-based hyperspectral image classification improves accuracy and efficiency compared with other algorithms, proving the algorithm’s effectiveness.

Reliability modelling for linear and circular [formula omitted]-out-of-[formula omitted]: F systems with shared components

Reliability analysis for k-out-of-n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k-out-of-n: F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the nith step transition rate matrix is calculated by using the Markov chain technique, for i=1,2,…,m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method.

Improving stochastic local search for uniform k‐SAT by generating appropriate initial assignment

AbstractStochastic local search (SLS) algorithms are well known for their ability to efficiently find models of random instances of the SAT problem, especially for uniform random k‐SAT instances. Two processes affect most SLS solvers—the initial assignment of the variables and the heuristics that select which variable to flip. In the last few years, the work on generating the appropriate initial assignment has not been paid much attention or seen much progress, while most SLS solvers focused on the heuristic algorithm. The present work aims to improve SLS algorithms on uniform random k‐SAT instances by developing effective methods for generating the initial assignment of variables in a controlled way. First, the allocation strategy introduced recently for 3‐SAT instances is extended to initialize the initial assignment on random k‐SAT instances. Then a concept of an initial probability distribution of the clause‐to‐variable ratio of the instance is introduced to determine the parameters of the allocation strategy. This combined method is added to the beginning of six state‐of‐the‐art SLS algorithms in order to generate initial assignments of variables in a controlled way instead of generating them randomly, resulting in six extended SLS algorithms named WalkSATlm_E, DCCASat_E, Score2SAT_E, CSCCSat_E, Probsat_E, and Sparrow_E, respectively. They are then evaluated in terms of their capabilities and efficiency on uniform random k‐SAT instance from the random track of SAT Competitions in 2016, 2017, and 2018. Experimental results show that these improved SLS solvers outperform their original performance, especially WalkSAT_E, Score2SAT_E, and CSCCSat_E outperform the winner of the random track of SAT competition in 2017. In addition, based on the initial probability distribution method, the present work proposes a parameter tuning and analysis of random 3‐SAT instances and provides an additional comparative analysis with the state‐of‐the‐art random SLS solvers based on large‐scale experiments.