Probabilistic Sense Research Articles

Assessment of the decisions made by the marine vessel navigator

The paper deals with interdisciplinary scientific research that lies in cognitive psychology, information technology and navigation, and is devoted to the relevant topic of navigation - vessel navigator’s decision-making. It reflects the results of theoretical research on the formalization of certain aspects of human mental activity in relation to solving navigation problems. The specific nature of this paper lies in navigator’s inclusion into the ship management which leads to the need in simultaneous consideration of psychological characteristics and ship manageability characteristics. Thus, navigator’s decisions are subjective and consequently difficult to formalize. The purpose of the research is to formalize, i.e. to mathematically describe the quantitative assessment of navigator’s decision-making. Author’s method of probabilistic sense evaluation is used in this research. The semantic space structure, which channels navigator’s decision-making, is represented as the model of a semantic prism. The results are adapted to the concepts of navigation and may be of interest to specialists in ship traffic control automation, artificial intelligence, creation of intelligent control systems and navigation safety.

A Witness Function Based Construction of Discriminative Models Using Hermite Polynomials

In machine learning, we are given a dataset of the form $\{(x_j,y_j)\}_{j=1}^M$, drawn as i.i.d. samples from an unknown probability distribution $\mu$; the marginal distribution for the $x_j$'s being $\mu^*$, and the marginals of the $k^{th}$ class $\mu_k^*(x)$ possibly overlapping. We address the problem of detecting, with a high degree of certainty, for which $x$ we have $\mu_k^*(x)>\mu_i^*(x)$ for all $i\neq k$. We propose that rather than using a positive kernel such as the Gaussian for estimation of these measures, using a non-positive kernel that preserves a large number of moments of these measures yields an optimal approximation. We use multi-variate Hermite polynomials for this purpose, and prove optimal and local approximation results in a supremum norm in a probabilistic sense. Together with a permutation test developed with the same kernel, we prove that the kernel estimator serves as a ``witness function'' in classification problems. Thus, if the value of this estimator at a point $x$ exceeds a certain threshold, then the point is reliably in a certain class. This approach can be used to modify pretrained algorithms, such as neural networks or nonlinear dimension reduction techniques, to identify in-class vs out-of-class regions for the purposes of generative models, classification uncertainty, or finding robust centroids. This fact is demonstrated in a number of real world data sets including MNIST, CIFAR10, Science News documents, and LaLonde data sets.

Probabilistic seismic analysis of single-layer reticulated shell structures controlled by viscoelastic dampers with an effective placement

In this paper, an effective placement of viscoelastic dampers on single-layer reticulated shell structures is proposed to mitigate seismic response. For comparison, an existing placement of viscoelastic dampers is also investigated. The effectiveness of the proposed placement of viscoelastic dampers is investigated from both deterministic and probabilistic perspectives. The single-layer reticulated shell structure is driven by the stochastic seismic ground motions, which are modeled via the random function based spectral representation method. The probability density evolution method is then implemented for probabilistic response and reliability analyses of the shell structure. A series of deterministic numerical simulations are first implemented under different earthquake excitations, including both actual and artificial seismic ground motions, to compare the control effects in deterministic sense. Then, the control effects of the proposed placement and the existing placement are compared in probabilistic sense, where the probabilistic responses, equivalent extreme value distributions and failure probabilities are extracted. Numerical results demonstrate that the seismic performance of the shell structure can be greatly improved by arranging the viscoelastic dampers with the proposed placement.

Linearized Wave Turbulence Convergence Results for Three-Wave Systems

We consider stochastic and deterministic three-wave semi-linear systems with bounded and almost continuous set of frequencies. Such systems can be obtained by considering nonlinear lattice dynamics or truncated partial differential equations on large periodic domains. We assume that the nonlinearity is small and that the noise is small or void and acting only in the angles of the Fourier modes (random phase forcing). We consider random initial data and assume that these systems possess natural invariant distributions corresponding to some Rayleigh-Jeans stationary solutions of the wave kinetic equation appearing in wave turbulence theory. We consider random initial modes drawn with probability laws that are perturbations of theses invariant distributions. In the stochastic case, we prove that in the asymptotic limit (small nonlinearity, continuous set of frequency and small noise), the renormalized fluctuations of the amplitudes of the Fourier modes converge in a weak sense towards the solution of the linearized wave kinetic equation around these Rayleigh-Jeans spectra. Moreover, we show that in absence of noise, the deterministic equation with the same random initial condition satisfies a generic Birkhoff reduction in a probabilistic sense, without kinetic description at least in some regime of parameters.

Probabilistic loss assessment of a seismic retrofit technique for medium- and high-voltage transformer bushing systems in high seismicity regions

Transformer bushing systems have exhibited vulnerability during past earthquakes. Based on past qualification tests, it was found that the seismic performance of transformer bushing systems is improved when the bushings are mounted on a rigid base, as opposed to when mounted on more flexible cover plates of transformers. The addition of flexural stiffeners on the transformer cover plates has been proven to stiffen the base of the bushings and mitigate their seismic vulnerability. In this study, the Performance-Based Earthquake Engineering framework, introduced by the Pacific Earthquake Engineering Research (PEER) center, is used to evaluate in a probabilistic sense the earthquake-induced economic losses of transformer bushing systems both for as-installed and retrofitted mounting conditions. The results are presented in the form of expected losses conditioned on seismic intensity as well as expected annual losses for locations in Southern and Northern California. The results demonstrate the ability of the retrofit approach of adding flexural stiffeners on the transformer cover plates to reduce earthquake-induced economic losses compared to the as-installed conditions/design and further enhance community recovery in the occurrence of a strong seismic event.

Predictability of the early winter Arctic oscillation from autumn Eurasian snowcover in subseasonal forecast models

The response of the early winter Northern Hemisphere stratospheric polar vortex and tropospheric Arctic Oscillation to anomalous autumn snow cover in Eurasia is evaluated in four operational subseasonal forecasting models. Of these four models, the two with finer stratospheric resolution simulate a weakened vortex for hindcasts initialized with more extensive snow as compared to those with less extensive snow, consistent with the observed effect, though the modeled effect is significantly weaker than that observed. The other two models fail to capture the local Western Eurasian ridge in response to enhanced snow, and hence their failure to simulate a stratospheric response may be due to biases in representing surface–atmosphere coupling rather than their coarser stratospheric resolution per se. There is no evidence of a tropospheric Arctic Oscillation response in early winter in any of these models, which may be related to the weakness of the stratospheric response or (in one model) to too-weak coupling from the stratosphere down to the surface. Overall, the possible contribution of autumn snowcover over Eurasia to improved predictability of the winter Arctic Oscillation in subseasonal forecast models has not yet been realized even in a probabilistic sense.

The real tau‐conjecture is true on average

Koiran's real τ‐conjecture claims that the number of real zeros of a structured polynomial given as a sum of m products of k real sparse polynomials, each with at most t monomials, is bounded by a polynomial in mkt. This conjecture has a major consequence in complexity theory since it would lead to superpolynomial lower bounds for the arithmetic circuit size of the permanent. We confirm the conjecture in a probabilistic sense by proving that if the coefficients involved in the description of f are independent standard Gaussian random variables, then the expected number of real zeros of f is .

Storyline description of Southern Hemisphere midlatitude circulation and precipitation response to greenhouse gas forcing

As evidence of climate change strengthens, knowledge of its regional implications becomes an urgent need for decision making. Current understanding of regional precipitation changes is substantially limited by our understanding of the atmospheric circulation response to climate change, which to a high degree remains uncertain. This uncertainty is reflected in the wide spread in atmospheric circulation changes projected in multimodel ensembles, which cannot be directly interpreted in a probabilistic sense. The uncertainty can instead be represented by studying a discrete set of physically plausible storylines of atmospheric circulation changes. By mining CMIP5 model output, here we take this broader perspective and develop storylines for Southern Hemisphere (SH) midlatitude circulation changes, conditioned on the degree of global-mean warming, based on the climate responses of two remote drivers: the enhanced warming of the tropical upper troposphere and the strengthening of the stratospheric polar vortex. For the three continental domains in the SH, we analyse the precipitation changes under each storyline. To allow comparison with previous studies, we also link both circulation and precipitation changes with those of the Southern Annular Mode. Our results show that the response to tropical warming leads to a strengthening of the midlatitude westerly winds, whilst the response to a delayed breakdown (for DJF) or strengthening (for JJA) of the stratospheric vortex leads to a poleward shift of the westerly winds and the storm tracks. However, the circulation response is not zonally symmetric and the regional precipitation storylines for South America, South Africa, South of Australia and New Zealand exhibit quite specific dependencies on the two remote drivers, which are not well represented by changes in the Southern Annular Mode.

On the Success of Seriously Mishandling Euclid’s Lemma

In the November 2016 issue of the Monthly, A. W. Dudek showed that the implication over positive integers is almost always false in a certain probabilistic sense. By considering two variations on this idea, we show that this fact continues to hold when ab is replaced by a product with a fixed number of factors, but a nontrivial frequency develops when r is assumed to have a fixed number of prime divisors, including multiplicity.

Uncertainty Quantification in Planetary Thermal History Models: Implications for Hypotheses Discrimination and Habitability Modeling

Abstract Multiple hypotheses/models have been put forward regarding Earth’s cooling history. Searching for life beyond Earth has brought these models into a new light as they connect to an energy source that life can tap. Discriminating between different cooling models and adapting them to aid in the assessment of planetary habitability has been hampered by a lack of uncertainty quantification. Here, we provide an uncertainty quantification that accounts for a range of interconnected model uncertainties. This involved calculating over a million individual model evolutions to determine uncertainty metrics. Accounting for uncertainties means that model results must be evaluated in a probabilistic sense, even though the underlying models are deterministic. The uncertainty analysis was used to quantify the degree to which different models can satisfy observational constraints on the Earth’s cooling. For the Earth’s cooling history, uncertainty leads to ambiguity—multiple models, based on different hypotheses, can match observations. This has implications for using such models to forecast conditions for exoplanets that share Earth characteristics but are older than the Earth, i.e., ambiguity has implications for modeling the long-term life potential of terrestrial planets. Even for the most earthlike planet we know of, the Earth itself, model uncertainty and ambiguity leads to large forecast spreads. Given that Earth has the best data constraints, we should expect larger spreads for models of terrestrial planets, in general. The uncertainty analysis provided here can be expanded by coupling planetary cooling models to climate models and propagating uncertainty between them to assess habitability from a probabilistic view.

On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of weak (distributional) solutions. Specifically, we find a sequence $\tau_M \to \infty$ of positive stopping times for which the Euler system admits infinitely many solutions originating from the same initial data. The solutions are weak in the PDE sense but strong in the probabilistic sense, meaning, they are defined on an {\it a priori} given stochastic basis and adapted to the driving stochastic process.

Probabalistic modelling of the results of economic activity as a function of random values

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors.Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

Chance-constrained Model Predictive Control for Near Rectilinear Halo Orbit spacecraft rendezvous

This work presents a robust Model Predictive Controller (MPC) to solve the problem of spacecraft rendezvous in the context of the restricted three-body problem (R3BP) as will be required to dock with space stations in cislunar space. The employed methodology is both valid for chemical and electric thrusters. By exploiting the state transition matrix and using a chance-constrained approach, the robust MPC assures constraints satisfaction under the presence of disturbances in a probabilistic sense. The perturbations parameters are computed on-line using a disturbance estimator. The robust controller is tested for a rendezvous scenario with a target placed in an Earth-Moon L2 Near-Rectilinear Halo Orbit. Numerical results are shown and discussed.

Newton’s method for nonlinear stochastic wave equations

Abstract We consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Computing the allowable uncertainty of sparse control configurations

Given a generic Control Configuration Selection (CCS) protocol and a plant uncertainty description isomorph to a unit ball in a finite-dimensional Lp space, we search for the largest perturbation radius for which the nominal configuration remains the preferred one. To this aim, we develop a randomized search algorithm based on sampling the uncertain plants and characterize its statistical performance. By adopting an intuitive accuracy measure that relates to the volume of points in which the preferred configuration differs from the nominal one, we devise a generally applicable strategy that allows for arbitrarily accurate estimates, in a specific probabilistic sense, depending on the number of uncertain plants that are sampled. We benchmark the proposed algorithm using examples from the literature and in a data center flow provisioning problem. In the latter setting, we identify the uncertainty description with the space of controls and sample the “uncertain” plants from an underlying nonlinear model.

Effect of Noise and Nonlinearities on Thermoacoustics of Can-Annular Combustors

Abstract Can-annular combustors consist of N distinct cans setup symmetrically around the axis of the gas turbine. Each can is connected to the turbine inlet by means of a transition duct. At the turbine inlet, a small gap between the neighboring transition ducts allows acoustic communication between the cans. Thermoacoustic pulsations in the cans are driven by the respective flames, but also the communication between neighboring cans through the gap plays a significant role. In this study, we focus on the effect of the background noise intensity and of the nonlinear flame saturation. We predict how usually clusters of thermoacoustic modes are unstable in the linear regime and compete with each other in the nonlinear regime, each cluster consisting of axial, azimuthal and push-pull modes. Since linear theory cannot predict the nonlinear solution, stochastic simulations are run to study the nonlinear solution in a probabilistic sense. One outcome of these simulations is the various pulsation patterns, which are in principle different from one can to the next. We recover how not only a stronger flame response in one can gives rise to the phenomenon of mode localization, but also how the nonlinearity of the flame saturation and the competition between modes have an effect on the nonlinear mode shape. We finally predict the coherence and phase between cans on the linearized system subject to noise, and compare the predictions with engine measurements, in terms of spectra of amplitude in each can and coherence and phase, observing a good match.

Predicting the Downward and Surface Influence of the February 2018 and January 2019 Sudden Stratospheric Warming Events in Subseasonal to Seasonal (S2S) Models.

Using the real‐time predictions from 11 models, this study analyzes the prediction of the downward propagation and surface impact of the 2018 and 2019 sudden stratospheric warmings (SSWs). These two SSWs differed both in their morphology types (2018: split; 2019: displacement followed by split) and magnitudes (the former being stronger). With a large sample size (>2,200) of multimodel ensemble forecasts, it is revealed that the strength of the SSW is more important than the vortex morphology in determining the magnitude of its downward impact, with strong SSWs more likely to propagate downward than weak SSWs. Therefore, based on the probabilistic forecasts, the observed strong SSW in February 2018 was more likely to have a downward and surface impact than the January 2019 SSW. The relationship between the 10‐hPa dominant wave number and the 100‐hPa polar cap height (or the Northern Annular Mode) is weak, implying that the dominant wave number might not be the primary factor determining the downward propagation of SSWs in the two analyzed cases. Hence, the high polar cap height (or negative Northern Annular Mode) response in the lower stratosphere and troposphere following the February 2018 SSW is mainly attributed to its strong intensity rather than the split morphology. Further, the 2‐m temperature anomaly pattern following the January 2019 SSW is not forecasted due to its weak downward propagation, whereas the 2‐m temperature in North Eurasia, Middle East, south China, and eastern United States could be forecasted for the downward propagating February 2018 SSW. However, regional rainfall anomalies are poorly forecasted (both in a deterministic and probabilistic sense) for both SSWs.

A stochastic compartmental model to simulate the Covid-19 epidemic spread on a simple network.

The recent Covid-19 epidemic has pointed out the inadequacy of the plans applied by industrial countries to limit the epidemic spread and frailty of the global economy to cope with a pandemic. Many countries were forced to a global lockdown with a great socio-economic impact. In Italy, one of the problems was the complex mobility network structure of the Northern regions that made ineffective the attempts to isolate the initial hotspots. In the paper we study a simple model that simulates the epidemic spread on a community network that may exchange population according to a daily mobility rate. In each community the epidemic evolution is provided by a stochastic compartmental model whose parameters are tuned to reproduce the Covid-19 evolution observed in Italy before the global lockdown policies. We initially study the delay in the epidemic spread due to the finite local mobility by proposing a power law relation for the increasing of the infection peak time in each node and the network distance from the initial node where the epidemic starts. We consider two scenarios to study the effectiveness of local lockdown policies: the presence of two clusters weakly connected by the mobility or a homogeneous chain of communities that exchange the population at a fixed rate. In both cases we show the existence of a threshold effect, in a probabilistic sense, for the effectiveness of lockdown policies as a function of the delay time at which such policies are applied, or of the network distance from the outbreak node.

Robust Hierarchical-Optimization RLS Against Sparse Outliers

This letter fortifies the recently introduced hierarchical-optimization recursive least squares (HO-RLS) against outliers which contaminate infrequently linear-regression models. Outliers are modeled as nuisance variables and are estimated together with the linear filter/system variables via a sparsity-inducing (non-)convexly regularized least-squares task. The proposed outlier-robust HO-RLS builds on steepest-descent directions with a constant step size (learning rate), needs no matrix inversion (lemma), accommodates colored nominal noise of known correlation matrix, exhibits small computational footprint, and offers theoretical guarantees, in a probabilistic sense, for the convergence of the system estimates to the solutions of a hierarchical-optimization problem: Minimize a convex loss, which models a-priori knowledge about the unknown system, over the minimizers of the classical ensemble LS loss. Extensive numerical tests on synthetically generated data in both stationary and non-stationary scenarios showcase notable improvements of the proposed scheme over state-of-the-art techniques.

A Novel Model of Mimic Defense Based on Minimal L-Order Error Probability

Mimic defense is an active defense theory, which aims to fundamentally change the “easy to attack and difficult to defend” situation of network security. In this paper, we propose an evaluation method based on the probability of being attack successfully, and improve the evaluation scheme of historical confidence. We combine the two evaluation schemes with the TOPSIS (technique for order performance by similarity to ideal solution) algorithm, and finally form a complete heterogeneous variant dynamic scheduling model. Different from traditional multi-mode voting algorithms, the effect of the heterogeneous degree in voting is considered, and we use Bayesian estimation to obtain the optimal result in the probabilistic sense. Finally, simulation results show that the proposed algorithm can effectively enhance the dynamic and security of the mimic defense model, and give full play to the characteristics of mimic defense.