Probabilistic Insight Research Articles

Rankability and linear ordering problem: Probabilistic insight and algorithms

The linear ordering problem (LOP), which consists in ordering M objects from their pairwise comparisons, is commonly applied in many areas of research. While efforts have been made to devise efficient LOP algorithms, verification of whether the data is rankable, that is, if the LOP solutions have a meaningful interpretation, received much less attention. To address this problem, we adopt a probabilistic perspective where the results of pairwise comparisons are modeled as Bernoulli variables with a common parameter, and we estimate the latter from the observed data. The enumeration over the space of permutations, required to solve the problem, has a prohibitive complexity of O(M!) if implemented explicitly. We thus reformulate the problem and introduce a concept of the Slater spectrum that generalizes the Slater index, and then devise an algorithm to find the spectrum with complexity O(M32M) that is manageable for moderate values of M. Furthermore, with a minor modification of the algorithm, we are able to find all solutions of the LOP with the complexity O(M2M). Numerical examples are shown on synthetic and real-world data, and the algorithms are publicly available.

A convolution formula for the local time of an Itô diffusion reflecting at 0 and a generalized Stroock–Williams equation

A new probabilistic insight into the structure of local time is presented. A convolution formula for the local time at 0 of Itô diffusions reflecting at 0 is obtained. A simple integro-differential equation for the cumulative distribution function of the local time is given. Finally, a probabilistic representation of a generalized Stroock–Williams equation is presented.

Metal‐poor stars observed with the automated planet finder telescope. III. CEMP‐no stars are the descendant of population III stars

AbstractThis study reports a probabilistic insight into the stellar‐mass and supernovae (SNe) explosion energy of five CEMP‐no stars' possible progenitors. As such, a direct comparison between the abundance ratios [X/Fe] of the light‐elements and the predicted nucleosynthetic yields of SN of high‐mass metal‐free stars has been performed. This comparison suggests that the possible progenitors have a stellar‐mass range of 11 − 22 M⊙ and explosion energies of 0.3 − 1.8 × 1051 erg. In addition, we investigate the kinematic signatures, derived from Gaia DR2, to conclude that these five CEMP‐no stars have halo‐like kinematic and do not enter the outer‐halo region. In addition, we link the abundance patterns with kinematic signatures to investigate the Gaia‐Sausage and Gaia‐Sequoia memberships. This chemo‐dynamical analysis suggests that these CEMP‐no stars are neither Gaia‐Sausage nor Gaia‐Sequoia remnant stars, but another accretion event might be responsible for the contribution of these stars to the Galactic Halo.

Day-Ahead Energy and Reserve Dispatch Problem under Non-Probabilistic Uncertainty

The current energy transition and the underlying growth in variable and uncertain renewable-based energy generation challenge the proper operation of power systems. Classical probabilistic uncertainty models, e.g., stochastic programming or robust optimisation, have been used widely to solve problems such as the day-ahead energy and reserve dispatch problem to enhance the day-ahead decisions with a probabilistic insight of renewable energy generation in real-time. By doing so, the scheduling of the power system becomes, production and consumption of electric power, more reliable (i.e., more robust because of potential deviations) while minimising the social costs given potential balancing actions. Nevertheless, these classical models are not valid when the uncertainty is imprecise, meaning that the system operator may not rely on a unique distribution function to describe the uncertainty. Given the Distributionally Robust Optimisation method, our approach can be implemented for any non-probabilistic, e.g., interval models rather than only sets of distribution functions (ambiguity set of probability distributions). In this paper, the aim is to apply two advanced non-probabilistic uncertainty models: Interval and ϵ-contamination, where the imprecision and in-determinism in the uncertainty (uncertain parameters) are considered. We propose two kinds of theoretical solutions under two decision criteria—Maximinity and Maximality. For an illustration of our solutions, we apply our proposed approach to a case study inspired by the 24-node IEEE reliability test system.

Mehler’s Formula, Branching Process, and Compositional Kernels of Deep Neural Networks

–We use a connection between compositional kernels and branching processes via Mehler’s formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. We study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases. We investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample size, dimensionality, and nonlinearity of the activation. Explicit formulas on the eigenvalues of the compositional kernel are provided, which quantify the complexity of the corresponding reproducing kernel Hilbert space. On the methodological front, we propose a new random features algorithm, which compresses the compositional layers by devising a new activation function. Supplementary materials for this article are available online.

Mehler’s Formula, Branching Process, and Compositional Kernels of Deep Neural Networks

We utilize a connection between compositional kernels and branching processes via Mehler’s formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. We study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases. We investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample size, dimensionality, and non-linearity of the activation. Explicit formulas on the eigenvalues of the compositional kernel are provided, which quantify the complexity of the corresponding reproducing kernel Hilbert space. On the methodological front, we propose a new random features algorithm, which compresses the compositional layers by devising a new activation function.

The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra

In Ekhad and Zeilberger (Personal J Shalosh B Ekhad and Doron Zeilberger, 2020) an exciting case study has been initiated in which experimental mathematics and symbolic computation are utilized to discover new properties concerning the so-called Absent-Minded Passengers Problem. Based on these results, Doron Zeilberger raised some challenging tasks to gain further probabilistic insight. In this note we report on this enterprise. In particular, we demonstrate how the computer algebra packages of RISC can be used to carry out the underlying heavy calculations.

3D Channel Tracking for UAV-Satellite Communications in Space-Air-Ground Integrated Networks

The space-air-ground integrated network (SAGIN) aims to provide seamless wide-area connections, high throughput and strong resilience for 5G and beyond communications. Acting as a crucial link segment of the SAGIN, unmanned aerial vehicle (UAV)-satellite communication has drawn much attention. However, it is a key challenge to track dynamic channel information due to the low earth orbit (LEO) satellite orbiting and three-dimensional (3D) UAV trajectory. In this paper, we explore the 3D channel tracking for a Ka-band UAV-satellite communication system. We firstly propose a statistical dynamic channel model called 3D two-dimensional Markov model (3D-2D-MM) for the UAV-satellite communication system by exploiting the probabilistic insight relationship of both hidden value vector and joint hidden support vector. Specifically, for the joint hidden support vector, we consider a more realistic 3D support vector in both azimuth and elevation direction. Moreover, the spatial sparsity structure and the time-varying probabilistic relationship between degree patterns named the spatial and temporal correlation, respectively, are studied for each direction. Furthermore, we derive a novel 3D dynamic turbo approximate message passing (3D-DTAMP) algorithm to recursively track the dynamic channel with the 3D-2D-MM priors. Numerical results show that our proposed algorithm achieves superior channel tracking performance to the state-of-the-art algorithms with lower pilot overhead and comparable complexity.

A flexible approach to defining weather patterns and their application in weather forecasting over Europe

ABSTRACTA method is presented for deriving weather patterns objectively over an area of interest, in this case the UK and surrounding European area. A set of 30 and eight patterns are derived through k‐means clustering of daily mean sea level pressure (MSLP) data (1850–2003). These patterns have been designed for the purpose of post‐processing forecast output from ensemble prediction systems and understanding how forecast models perform under different circulation types. The 30 weather patterns are designed for use in the medium‐range and the eight weather patterns are designed for use in the monthly and seasonal timescales, or when there is low forecast confidence in the medium‐range. Weather patterns are numbered according to their annual historic occurrences, with lower numbered patterns occurring most often. Lower numbered patterns occur more in summer (with weak MSLP anomalies) and higher numbered patterns occur more in winter (with strong MSLP anomalies). Weather patterns have been applied in a weather forecasting context, whereby ensemble members are assigned to the closest matching pattern definition. This provides a probabilistic insight into which patterns are most likely within the forecast range and summarises key aspects from the large volumes of data which ensembles provide. Verification of European Centre for Medium‐Range Weather Forecasts medium‐range ensemble forecasts for the set of eight weather patterns shows small forecast biases annually with some large variations seasonally. The most prominent seasonal variation shows the westerly (NAO+) pattern to over‐forecast in summer and under‐forecast in winter. Forecast skill was found to be better in winter than summer for most patterns.

Probabilistic insight to possibility of new metal filament nucleation during repeated cycling of conducting bridge memory

The question of whether resistance switching in conducting bridge random access memory (CBRAM) devices occurs through a single filament which repeatedly undergoes nucleation and rupture or through different filaments for different cycles is difficult to ascertain using conventional electrical tests on metal–insulator–metal (MIM) capacitor structures. In order to profile the spatial location of the conductive filament during multiple switching cycles, we make use of the Ni–HfO2–Si based metal–insulator–semiconductor (MIS) stack with a transistor test structure so that the lateral location of the filament along the source to drain can be probed electrically by considering the weighted ratio of source and drain currents measured. Our analysis reveals that filaments can evolve in spatially uncorrelated locations and switching is not always caused by the same filament over and over again. A simple statistical model is also provided to justify the inferences of the electrical study. The probability of a new filament nucleating elsewhere in the dielectric is a strong function of the oxide barrier thickness as well as the curvature radii of the previously ruptured metal filament edges.

Applied statistics in ecology: common pitfalls and simple solutions

The most common statistical pitfalls in ecological research are those associated with data exploration, the logic of sampling and design, and the interpretation of statistical results. Although one can find published errors in calculations, the majority of statistical pitfalls result from incorrect logic or interpretation despite correct numerical calculations. There are often simple solutions to avoiding these problems that require only patience, clarity of thinking, probabilistic insight, and a reduced reliance on out‐of‐the‐box approaches. Some of these trouble spots are inherent to all statistical analyses and others are particularly insidious in ecology where true controls or replication are challenging, small sample sizes are common, and correctly linking mathematical constructs and ecological ideas is essential. Here we summarize the most common statistical pitfalls observed over nearly a century of combined consulting and research experience in ecological statistics. We provide short, simple solutions.

On the probabilistic approach to heterogeneous structure interactions in agent-based computational models

The paper suggests a unified stochastic method in order to simulate multi-agent interactions in heterogeneous complex systems. This approach (which appears phenomenological if it is not agent-based) can describe precisely the average outcomes of multiple agent-based computations (where there is a lack of probabilistic insight). The recent discussions about the gap between agent-based computational models (ABM) and stochastic analytical models have stimulated the present research (Feng et al., 2012 [1] and Kononovicius and Gontis, 2013 [2]). In general, agent-based modeling is currently a technique widely used to simulate complex systems in computer science and social sciences. The present developments in the physics of complex systems, in particular the structural relaxation of supercooled liquids and glasses, are also discussed by using such probabilistic approach. It is possible to depict the impact of interface between the nucleus considered as a cluster of a certain number of molecules and the liquid phase for the enhancement of the overall nucleation process. In this study, a useful aggregation procedure for representing the three-dimensional distribution of states and a general equation that describes clustering process among interacting agents in heterogeneous populations are proposed.

A simple probabilistic approach to evaluate radioiodine behavior at severe accidents: Application to PHEBUS test FPT1

A simple probabilistic approach is proposed to quantify the relative importance of selected processes and mechanisms which govern the behavior of radioiodine under conditions pertaining to a postulated severe accident. In principle, these are sensitivity studies of a dedicated, small event tree which describes the iodine behavior at an accident. The emphasis is put on the processes taking place in containment at a late phase of the accident. Here, volatile species of iodine (volatile iodine: I 2 and various organic iodides) could form from the species which were transported from RCS to containment originally as non-volatile. The relative importance of the processes leading to volatile iodine formation in containment is, for a long time, a subject of some dispute among the international research communities. The proposed probabilistic insight could be one of the many ways to address these complicated questions. In this work, the probabilistic approach was tested against the results of the well-documented PHEBUS FPT1 experiment. The small containment iodine event tree was processed by the EVNTRE code with a direct call from within the tree to the detailed iodine chemistry program calculating the speciation of iodine in containment. The French program IODE, as a part of the ASTEC suite of codes, has been used for this purpose. The results of this sensitivity study of PHEBUS FPT1 are promising as for the possibility to use such (or similar) probabilistic method for thorough analyses of uncertainties linked with the iodine behavior in containment at an accident. Similar approach could be used even for the iodine behavior modeling in whole-plant PRA analyses.

A Note on Insensitivity in Stochastic Networks

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

A Note on Insensitivity in Stochastic Networks

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

On the asymptotic relation between equilibrium density and exit measure in the exit problem

We consider the diffusion dx(t) = b{x(t))dt-\-y/sa(x(t))dw in a domain D which is contained in the domain of attraction of an asymptotically stable critical point of x = b(x). Using a formula of Hasminskii for the equilibrium measure of x(-) we show that the asymptotic behaviors of the exit measure P[x?{i:D)edy] and the equilibrium density pE(y) are connected. The formula connecting the two is essentially the same as one derived by Matkowsky and Schuss using formal methods. The treatment here provides probabilistic insight into the Matkowsky-Schuss formula.