Resulting Probability Distribution Research Articles

Maximum entropy and constraints in composite systems.

The principle of maximum entropy (PME), as expounded by Jaynes, is based on the maximization of the Boltzmann-Gibbs-Shannon (BGS) entropy subject to linear constraints. The resulting probability distributions are of canonical (exponential) form. However, the rationale for linear constraints is nebulous, and probability distributions are not always canonical. Here we show that the correct noncanonical distribution for a system in equilibrium with a finite heat bath is implied by the unconstrained maximization of the total BGS entropy of the system and bath together. This procedure is shown to be equivalent to maximizing the BGS entropy of the system alone subject to a contrived nonlinear constraint which reduces to (a) the usual linear constraint for an infinite heat bath, and (b) a previously enigmatic logarithmic constraint which implies a power-law distribution for a large but finite heat bath. This procedure eliminates the uncertainty as to the proper constraints, and easily generalizes to arbitrary composite systems, for which it provides a simpler alternative to the Jaynes PME.

High frame rate multi-perspective cardiac ultrasound imaging using phased array probes

Ultrasound (US) imaging is used to assess cardiac disease by assessing the geometry and function of the heart utilizing its high spatial and temporal resolution. However, because of physical constraints, drawbacks of US include limited field-of-view, refraction, resolution and contrast anisotropy. These issues cannot be resolved when using a single probe. Here, an interleaved multi-perspective 2-D US imaging system was introduced, aiming at improved imaging of the left ventricle (LV) of the heart by acquiring US data from two separate phased array probes simultaneously at a high frame rate.In an ex-vivo experiment of a beating porcine heart, parasternal long-axis and apical views of the left ventricle were acquired using two phased array probes. Interleaved multi-probe US data were acquired at a frame rate of 170 frames per second (FPS) using diverging wave imaging under 11 angles. Image registration and fusion algorithms were developed to align and fuse the US images from two different probes. First- and second-order speckle statistics were computed to characterize the resulting probability distribution function and point spread function of the multi-probe image data.First-order speckle analysis showed less overlap of the histograms (reduction of 34.4%) and higher contrast-to-noise ratio (CNR, increase of 27.3%) between endocardium and myocardium in the fused images. Autocorrelation results showed an improved and more isotropic resolution for the multi-perspective images (single-perspective: 0.59 mm × 0.21 mm, multi-perspective: 0.35 mm × 0.18 mm). Moreover, mean gradient (MG) (increase of 74.4%) and entropy (increase of 23.1%) results indicated that image details of the myocardial tissue can be better observed after fusion.To conclude, interleaved multi-perspective high frame rate US imaging was developed and demonstrated in an ex-vivo experimental setup, revealing enlarged field-of-view, and improved image contrast and resolution of cardiac images.

Monetary policy in a world of radical uncertainty

Monetary policy in a world of radical uncertainty

Stochastic inversion of magnetotelluric data using deep reinforcement learning

We have adopted a new tool to invert magnetotelluric data for the 1D model based on deep Q-networks (DQN), which works as a stochastic optimization method. By transforming the inversion problem into a Markov decision process, the tool learns by trial and error to find the optimal path for updating the model to fit the observed data. The DQN method converges to the target through different paths (e.g., Bayesian or other stochastic methods) and can partially provide the probability distribution of the inversion results, which can be used for uncertainty estimation. The DQN search space gradually decreases as the learning experience progresses, accelerating the single inversion and approximating the optimal result. To check the effectiveness of the DQN inversion, the five- and eight-layer models were separately designed to test the robustness of the DQN for the initial model and the noise level. A further comparison with Occam’s method and the Bayesian method indicated that our DQN obtained more robust inversion results for data contaminated by different noise levels. The inversion results with the survey data from Zhagaitunuoergong area, Inner Mongolia, China, well reveals the shape of the interface basement.

Strategic Interactions and Uncertainty in Decisions to Curb Greenhouse Gas Emissions

This paper examines the strategic interactions of two large regions making choices about greenhouse gas emissions in the face of rising global temperatures. Three central features are highlighted: uncertainty, the incentive for free riding, and asymmetric characteristics of decision makers. Optimal decisions are modelled in a fully dynamic, feedback Stackelberg pollution game. Global average temperature is modelled as a mean reverting stochastic process. A numerical solution of a coupled system of Hamilton-Jacobi-Bellman equations is implemented and the probability distribution of outcomes is illustrated with Monte Carlo simulation. When players are identical, the outcome of the game is much worse than the social planner’s outcome. An increase in temperature volatility reduces player utility, making cooperative action through a social planner more urgent. Asymmetric damages or asymmetric preferences for emissions reductions are shown to have important effffects on the strategic interactions of players.

Assessing the sequestration time scales of some ocean-based carbon dioxide reduction strategies

Ocean-based carbon dioxide (CO2) removal (CDR) strategies are an important part of the portfolio of approaches needed to achieve negative greenhouse gas emissions. Many ocean-based CDR strategies rely on injecting CO2 or organic carbon (that will eventually become CO2) into the ocean interior, or enhancing the ocean’s biological pump. These approaches will not result in permanent sequestration, because ocean currents will eventually return the injected CO2 back to the surface, where it will be brought into equilibrium with the atmosphere. Here, a model of steady state global ocean circulation and mixing is used to assess the time scales over which CO2 injected in the ocean interior remains sequestered from the atmosphere. There will be a distribution of sequestration times for any single discharge location due to the infinite number of pathways connecting a location at depth with the sea surface. The resulting probability distribution is highly skewed with a long tail of very long transit times, making mean sequestration times much longer than typical time scales. Deeper discharge locations will sequester purposefully injected CO2 much longer than shallower ones and median sequestration times are typically decades to centuries, and approach 1000 years in the deep North Pacific. Large differences in sequestration times occur both within and between the major ocean basins, with the Pacific and Indian basins generally having longer sequestration times than the Atlantic and Southern Oceans. Assessments made over a 50 year time horizon illustrates that most of the injected carbon will be retained for injection depths greater than 1000 m, with several geographic exceptions such as the Western North Atlantic. Ocean CDR strategies that increase upper ocean ecosystem productivity with the goal of exporting more carbon to depth will have mainly a short-term influence on atmospheric CO2 levels because ∼70% will be transported back to the surface ocean within 50 years. The results presented here will help plan appropriate ocean CDR strategies that can help limit climate damage caused by fossil fuel CO2 emissions.

Fast Algorithms for General Spin Systems on Bipartite Expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander Δ-regular graphs, including the canonical class of bipartite random Δ-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Roughly, this guarantees that the spin system is in the so-called low-temperature regime. Our approach generalises the techniques of Jenssen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to “bicliques” of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in Õ( n 2 ) time, where n is the size of the graph.

Probability distribution for heat exchange in plastic deformation.

Fluctuation theorems allow one to obtain equilibrium information from nonequilibrium experiments. The probability distribution function of the relevant magnitude measured along the irreversible nonequilibrium trajectories is an essential ingredient of fluctuation theorems. In small systems, where fluctuations can be larger than average values, probability distribution functions often deviate from being Gaussian, showing long tails, mostly exponential, and usually strongly asymmetric. Recently, the probability distribution function of the van Hove correlation function of the relevant magnitude was calculated, instead of that of the magnitude itself. The resulting probability distribution function is highly symmetric, obscuring the application of fluctuation theorems. Here, the discussion is illustrated with the help of results for the heat exchanged during plastic deformation of aluminum nanowires, obtained from molecular dynamics calculations. We find that the probability distribution function for the heat exchanged is centrally Gaussian, with asymmetric exponential tails further out. By calculating the symmetry function we show that this distribution is consistent with fluctuation theorems relating the differences between two equilibrium states to an infinite number of nonequilibrium paths connecting those two states.

Ecological risk assessment to substantiate the efficiency of the economy and the organization of construction

The purpose of this article is to assess risk in order to substantiate the economic and organizational efficiency of housing and industrial construction. The authors study methods related to quantifying environmental risk and making decisions under conditions of uncertainty. An algorithm for making sound decisions is proposed as a result of a quantitative analysis of the identified risks. A quantitative risk assessment of pessimistic and optimistic options for preventing environmental damage was made using the Monte Carlo method. The results of simulation modeling clearly illustrate the choice of the general principle of assessment and adoption of the optimal decision. The results of a sensitivity analysis are also presented to prove the hypothesis. This risk analysis technique has a computer implementation. The result of Monte Carlo modeling is a graphical representation of the log-normal probability distribution of possible outcomes of the preset emergency scenarios in the form of a curve on a scalable field of the estimated consequences. The model has reproduced the distribution derived from the evidence-based data selection. The Monte Carlo method is one of the scientifically based methods for analyzing the characteristics of a random process and making optimal decisions in order to reduce risk and minimize environmental damage. It could become the desired basis for interaction between the State Environmental Expert Review (SEER) and environmental impact assessment (EIA) in order to make sound decisions and actions in conditions of uncertainty and choosing environmentally friendly options for economic projects.

Flexural Capacity Prediction Model For Steel Fibre-Reinforced Concrete Beams

Steel fibre (SF) reinforcement has been shown to improve the ductility of high strength concrete (HSC), which is known to be brittle. Research conducted to date on steel fibre reinforced concrete and its effects have emphasised post-failure performance and cracking mechanism. The difficulty in predicting the behaviour of fibres is due to the randomly distributed nature of the material within the matrix leading to a probability distribution of results. Published literature has shown a benefit of adding steel fibres in terms of the ductility performance of structures. Clearly, there is a potential for such material as replacement of conventional steel reinforcement. This study proposes a theoretical model of evaluating the potential of using steel fibres as a replacement material to conventional steel reinforcement bars based on the case study, laboratory and theoretical methodologies. The compressive strength of the concrete at key dates, the effective fibre cross-sectional were measured, and a prediction model was created based on the measurement parameters. The use of four-point flexural testing, standard compressive testing and software image modelling provided the study with relevant data used to analyse and compare to the prediction. Greater ductility performance and toughness were observed with increased fibre volumes, confirming proposed predictions and conclusion drawn from published literature. No consistent or conclusive correlations between fibre volumes and the compressive strength of concrete were found. A relationship between fibre volumes and predicted moment capacities of steel fibre reinforced concrete beams was found based on the proposed theoretical flexural analysis method.

Remaining useful life prediction of lithium-ion batteries based on Monte Carlo Dropout and gated recurrent unit

Lithium-ion batteries have been widely used for energy storage systems and vehicle industries, highly accurate remaining useful life (RUL) prediction of lithium-ion batteries is one of the key technologies on prognostics and health management. However, the uncertainty quantification and reliability of RUL prediction is ignored. To describe the uncertainty of the RUL prediction and avoid the over-fitting phenomenon, a model combining Monte Carlo Dropout (MC_dropout) and gated recurrent unit (GRU) is proposed. Firstly, the indirect health indicator is extracted and gray relation analysis (GRA) is used to analyze the relation with capacity Then, dropout method is combined with GRU model to avoid the phenomenon of gradient disappearance and over-fitting. The probability distribution of the prediction results and 95% confidence interval is obtained through the MC_dropout method, which established the uncertainty of the prediction model. Finally, propagation neural network based on particle swarm optimization (BPNN-PSO), the least-squares support vector machine (LS-SVM) is compared with the proposed method. The results show that the proposed model not only has superiority point prediction of RUL compared with other methods, but also can describe the uncertainty of RUL prediction.

Bound Entanglement from Randomized Measurements.

If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present systematic methods to analyze the different forms of entanglement with these moments in an optimized manner. First, we find the optimal criteria for different forms of multiparticle entanglement in three-qubit systems using the second moments of randomized measurements. Second, we present the optimal inequalities if entanglement in a bipartition of a multiqubit system shall be analyzed in terms of these moments. Finally, for higher-dimensional two-particle systems and higher moments, we provide criteria that are able to characterize various examples of bound entangled states, showing that detection of such states is possible in this framework.

ARRIVAL PATTERNS AND TRAFFIC FLOW CHARACTERISTICS AT SIGNALIZED INTERSECTIONS

Traffic arrivals at signal intersection approaches is inherently stochastic. This variability is typically reflected by I-ratio and there is a general consensus that the presence or absence of nearby upstream signal affects Variance to Mean Ratio (I-ratio). However, the effect of time resolution on arrival variability and the interaction effect between upstream signal and time resolution is yet to be examined in detail. This can lead to model misspecification and invariably, erroneous outcomes. This work examines the effect of time resolution and intersection type and their interaction on I-ratio and the resultant probability distributions. Traffic arrivals were measured at high time resolution- 10 seconds interval and then aggregated to lower time resolutions (30-150 seconds) at six intersections. Spectral density analysis showed statistically significant periodicity, specifically at 30 seconds interval with p-values < 0.0001 at all connected intersections while observations at isolated intersections lacked periodicity. Two-way ANOVA using I-ratio as the dependent variable and intersection type and time-resolution as the independent variables was performed. Statistically significant effect with F-value 8.606 at p-value < 0.0001 and R2 value 0.32 were observed. Intersection type, time resolution and the interaction between them were statistically significant, with p-values 0.002, < 0.0001 and 0.000 respectively. The combined effect of these factors led to a wide I-ratio range of 0.37-9.2. Negative Binomial, Poisson, and Binomial distributions represented 76.4, 20.4 and 4.2% of all I-ratios observed. Therefore, in contrast to literature which recommends Poisson, Negative Binomial may be a better suited probability distribution for traffic arrivals.

Using expert knowledge to support Endangered Species Act decision-making for data-deficient species.

Many questions relevant to conservation decision‐making are characterized by extreme uncertainty due to lack of empirical data and complexity of the underlying ecologic processes, leading to a rapid increase in the use of structured protocols to elicit expert knowledge. Published ecologic applications often employ a modified Delphi method, where experts provide judgments anonymously and mathematical aggregation techniques are used to combine judgments. The Sheffield elicitation framework (SHELF) differs in its behavioral approach to synthesizing individual judgments into a fully specified probability distribution for an unknown quantity. We used the SHELF protocol remotely to assess extinction risk of three subterranean aquatic species that are being considered for listing under the U.S. Endangered Species Act. We provided experts an empirical threat assessment for each known locality over a video conference and recorded judgments on the probability of population persistence over four generations with online submission forms and R‐shiny apps available through the SHELF package. Despite large uncertainty for all populations, there were key differences between species’ risk of extirpation based on spatial variation in dominant threats, local land use and management practices, and species’ microhabitat. The resulting probability distributions provided decision makers with a full picture of uncertainty that was consistent with the probabilistic nature of risk assessments. Discussion among experts during SHELF's behavioral aggregation stage clearly documented dominant threats (e.g., development, timber harvest, animal agriculture, and cave visitation) and their interactions with local cave geology and species’ habitat. Our virtual implementation of the SHELF protocol demonstrated the flexibility of the approach for conservation applications operating on budgets and time lines that can limit in‐person meetings of geographically dispersed experts.

Time evolution with symmetric stochastic action

Quantum dynamical time-evolution of bosonic fields is shown to be equivalent to a stochastic trajectory in space-time, corresponding to samples of a statistical mechanical steady-state in a higher dimensional quasi-time. This is proved using the Q-function of quantum theory with time-symmetric diffusion, that is equivalent to a forward-backward stochastic process in both the directions of time. The resulting probability distribution has a positive, time-symmetric action principle and path integral, whose solution corresponds to a classical field equilibrating in an additional dimension. Comparisons are made to stochastic quantization and other higher dimensional physics proposals. Five-dimensional space-time was originally introduced by Kaluza and Klein, and is now widely proposed in cosmology and particle physics. Time-symmetric action principles for quantum fields are also related to electrodynamical absorber theory, which is known to be capable of violating a Bell inequality. We give numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time-dimension, giving agreement with exact solutions for soluble boson field quantum dynamics. This approach may lead to useful computational techniques for quantum field theory, as the action principle is real, as well as to ontological models of physical reality.

Generalized stochastic microdosimetric model: The main formulation.

The present work introduces a rigorous stochastic model, called the generalized stochastic microdosimetric model (GSM^{2}), to describe biological damage induced by ionizing radiation. Starting from the microdosimetric spectra of energy deposition in tissue, we derive a master equation describing the time evolution of the probability density function of lethal and potentially lethal DNA damage induced by a given radiation to a cell nucleus. The resulting probability distribution is not required to satisfy any a priori conditions. After the initial assumption of instantaneous irradiation, we generalized the master equation to consider damage induced by a continuous dose delivery. In addition, spatial features and damage movement inside the nucleus have been taken into account. In doing so, we provide a general mathematical setting to fully describe the spatiotemporal damage formation and evolution in a cell nucleus. Finally, we provide numerical solutions of the master equation exploiting Monte Carlo simulations to validate the accuracy of GSM^{2}. Development of GSM^{2} can lead to improved modeling of radiation damage to both tumor and normal tissues, and thereby impact treatment regimens for better tumor control and reduced normal tissue toxicities.

Exploiting Hierarchical Features for Crop Yield Prediction Based on 3-D Convolutional Neural Networks and Multikernel Gaussian Process

Accurate and timely prediction of crop yield based on remote sensing data is important for food security. However, crop growth is a complex process, which makes it quite difficult to achieve better performance. To address this problem, a novel 3-D convolutional neural multikernel network is proposed to capture hierarchical features for predicting crop yield. First, a full 3-D convolutional neural network is constructed to maximally explore deep spatial–spectral features from multispectral images. Then, a multikernel learning (MKL) approach is proposed for fusion of intraimage deep spatial–spectral features and intersample spatial consistency features. Specifically, we assign a group of nonlinear kernels for each feature in the MKL framework, which provides a robust way to fit features extracted from different domains. Finally, the probability distribution of prediction results is obtained by a kernel-based method. We evaluate the performance of the proposed method on China wheat yield prediction and offer detailed and systematic analyses of the performance of the proposed method. In addition, our method is compared with several competing methods. Experimental results demonstrate that the proposed method has certain advantages and can provide better prediction performance than the competitive methods.

Quantum dynamics on a lossy non-Hermitian lattice* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11404199 and 11674201), Natural Science Foundation of Shanxi Province, China (Grant No. 1331KSC), Natural Science Foundation for Youths of Shanxi Province, China (Grant No. 2015021012), and Research Initiation Funds from SXU (Grant No. 216533801001).

We investigate quantum dynamics of a quantum walker on a finite bipartite non-Hermitian lattice, in which the particle can leak out with certain rate whenever it visits one of the two sublattices. Quantum walker initially located on one of the non-leaky sites will finally totally disappear after a length of evolution time and the distribution of decay probability on each unit cell is obtained. In one regime, the resultant distribution shows an expected decreasing behavior as the distance from the initial site increases. However, in the other regime, we find that the resultant distribution of local decay probability is very counterintuitive, in which a relatively high population of decay probability appears on the edge unit cell which is the farthest from the starting point of the quantum walker. We then analyze the energy spectrum of the non-Hermitian lattice with pure loss, and find that the intriguing behavior of the resultant decay probability distribution is intimately related to the existence and specific property of the edge states, which are topologically protected and can be well predicted by the non-Bloch winding number. The exotic dynamics may be observed experimentally with arrays of coupled resonator optical waveguides.

Data-driven inference, reconstruction, and observational completeness of quantum devices

The range of a quantum measurement is the set of outcome probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of outcome distributions, infers the quantum measurement which is, i) consistent with the data, in the sense that its range contains all the distributions observed, and, ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of outcome distributions. We show that data-driven inference is able to return a unique measurement for any data set if and only if the inference adopts a (hyper)-spherical state space (for example, the classical or the quantum bit). In analogy to informational completeness for quantum tomography, we define observational completeness as the property of any set of states that, when fed into any given measurement, produces a set of outcome distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)-spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.

Using a stochastic economic scenario generator to analyse uncertain superannuation and retirement outcomes

AbstractThe retirement systems in many developed countries have been increasingly moving from defined benefit towards defined contribution system. In defined contribution systems, financial and longevity risks are shifted from pension providers to retirees. In this paper, we use a probabilistic approach to analyse the uncertainty associated with superannuation accumulation and decumulation. We apply an economic scenario generator called the Simulation of Uncertainty for Pension Analysis (SUPA) model to project uncertain future financial and economic variables. This multi-factor stochastic investment model, based on the Monte Carlo method, allows us to obtain the probability distribution of possible outcomes regarding the superannuation accumulation and decumulation phases, such as relevant percentiles. We present two examples to demonstrate the implementation of the SUPA model for the uncertainties during both phases under the current superannuation and Age Pension policy, and test two superannuation policy reforms suggested by the Grattan Institute.