General Probability Distribution Research Articles

Risk Contagion Measurement and Risk Source Tracing between International Commodity Market and Chinese Financial Market

Extreme risk events such as the financial crisis, trade frictions, and public health security have posed huge challenges to the stable development of global financial markets. With the proliferation and spread of the COVID-19 epidemic, international commodity prices fluctuate sharply, and the Chinese financial market is facing great risk shocks. However, previous literatures on risk contagion in financial markets ignore the perspective of extreme risk events. Few scholars measure the risk contagion effect between the international commodity market and the Chinese financial market. At the same time, they cannot trace the source of external risks.This paper uses the DCC-GARCH model and the Granger-Geweke causality test to measure the risk contagion effect between the international commodity market and the Chinese financial market, and traces the external risk sources. It is found that: First, during the period of extreme risk events, the co-movement effect between the international commodity market and the Chinese financial market fluctuates sharply. During the period of the COVID-19 epidemic, the linkage between the international commodity market and the Chinese financial market has increased significantly. Second, under the shock of extreme risk events, the risk contagion effect between the international commodity market and the Chinese financial market has increased. During the period of the COVID-19 epidemic, the interactive shock between the international commodity market and the Chinese financial market is asymmetrical. The impact of Chinese financial market on the international commodity market is very limited. Third, tracing the source of external risks in the Chinese financial market shows that energy, precious metals, and industrial metals have a higher impact on the Chinese financial market than other commodities, and the Chinese stock market and the exchange rate market bear stronger external shocks. This paper provides theoretical support and policy reference for China to prevent international financial risk contagion.The academic value of this paper could be concluded in three aspects: First, during the period of extreme risk events, we measure the general probability distribution of the dynamic condition correlation coefficient between the international commodity market and the Chinese financial market, and identify the general volatility trend of the linkage between the international commodity market and the Chinese financial market. Second, we build the dynamic causal network to measure the general and time-varying characteristics of risk contagion between the international commodity market and the Chinese financial market during the period of extreme risk events. Third, we decompose the external shock of the Chinese financial market, and trace the external risks of the Chinese financial market accurately.

Phase transition in random tensors with multiple independent spikes

Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian p-tensor for p≥3 and signal in the form of a symmetric low-rank random tensor. The latter is defined as a linear combination of k independent symmetric rank-one random tensors, referred to as spikes, with weights referred to as signal-to-noise ratios (SNRs). The entries of the vectors that determine the spikes are i.i.d. sampled from general probability distributions supported on bounded subsets of R. This work focuses on the problem of detecting the presence of these spikes, and establishes the phase transition of this detection problem for any fixed k≥1. In particular, it shows that for a set of relatively low SNRs it is impossible to distinguish between the spiked and nonspiked Gaussian tensors. Furthermore, in the interior of the complement of this set, where at least one of the k SNRs is relatively high, these two tensors are distinguishable by the likelihood ratio test. In addition, when the total number of low-rank components, k, of the p-tensor of size N grows in the order o(N(p−2)/4) as N tends to infinity, the problem exhibits an analogous phase transition. This theory for spike detection is also shown to imply that recovery of the spikes by the minimum mean square error exhibits the same phase transition. The main methods used in this work arise from the study of mean field spin glass models, where the phase transition thresholds are identified as the critical inverse temperatures distinguishing the high and low-temperature regimes of the free energies. In particular, our result formulates the first full characterization of the high temperature regime for vector-valued spin glass models with independent coordinates.

Graph-LSTM with Global Attribute for Scene Graph Generation

Lots of machine learning tasks require dealing with graph data, and among them, scene graph generation is a challenging one that calls for graph neural networks’ potential ability. In this paper, we present a definition of graph neural network (GNN) consists of node, edge and global attribute, as well as their corresponding update and aggregate functions. Based on this, we then propose a realization of GNN model called Graph-LSTM and use it in scene graph generation. The model first extracts the item features in the image as the initial states of the node-LSTM representing subject/object and edge-LSTM representing predicate. Two LSTMs update the states via LSTM’s timestep and aggregate information via message passing. Repeat the update-aggregate until convergence. Meanwhile, the tag feature, i.e., the generated probability distribution of image’s semantic concepts is sent to the LSTM through a semantic compositional network (SCN). The SCN-LSTM is trained in an ensemble style, and hence allows the tag feature to serve as the global attribute providing context information to all individuals. The LSTMs’ final states are input to inference modules and generate the triplet (subject, predicate, object) of the scene graph. Experimental results show that Graph-LSTM outperforms the Message Passing and the attention Graph Covolutional Network methods, proving the effectiveness of the proposed scheme.

Integral transforms and probability distributions involving a generalized hypergeometric function

Abstract Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

Fuzzy Differential Subordinations Based upon the Mittag-Leffler Type Borel Distribution

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series Bλ,α,β(z) of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function Eα,β(z). Using the above-mentioned operator Bλ,α,β, we also introduce and study a class Mλ,α,βFη of holomorphic and univalent functions in the open unit disk Δ. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function pΨq(z), whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.

Statistical Significance Revisited

Statistical significance measures the reliability of a result obtained from a random experiment. We investigate the number of repetitions needed for a statistical result to have a certain significance. In the first step, we consider binomially distributed variables in the example of medication testing with fixed placebo efficacy, asking how many experiments are needed in order to achieve a significance of 95%. In the next step, we take the probability distribution of the placebo efficacy into account, which to the best of our knowledge has not been done so far. Depending on the specifics, we show that in order to obtain identical significance, it may be necessary to perform twice as many experiments than in a setting where the placebo distribution is neglected. We proceed by considering more general probability distributions and close with comments on some erroneous assumptions on probability distributions which lead, for instance, to a trivial explanation of the fat tail.

Generating probability distributions on intervals and spheres with application to finite element method

This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval [a,b], which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This type of probability has scaling and symmetry feature, and sufficient conditions under which a density function can be generated through discrete polynomial spectrum are given in this work followed by concrete examples. The density function ρ obtained in this way has the advantage of being positive definite. Computer based numerical simulation shows that the theoretically verified criteria for probability distribution are almost optimal with respect to our testing examples. After the establishment of an approximation theorem in L1 space, we propose a probabilistic Galerkin scheme that can be either continuous or discontinuous, which is potentially useful to asymptotically solve some PDEs on the sphere locally and globally.

Spatial prediction of soil calcium carbonate content based on Bayesian maximum entropy using environmental variables

Soil calcium carbonate (CaCO3) content is an important soil property. The prediction of soil CaCO3 content is necessary for the sustainable management of soil fertility. In this work, we attempted to incorporate environmental variables directly and through regression models into the framework of Bayesian maximum entropy (BME) to predict CaCO3 content. Firstly, multiple linear regression (MLR) and geographically weighted regression (GWR) were used to establish a relationship between sampling data and environmental variables, including Digital Elevation Model, pH, temperature, rainfall, and fluvo-aquic soils. Prediction results of MLR and GWR served as soft data and were incorporated into the framework of BME to estimate the CaCO3 content. Secondly, soil samples and environmental variables were combined to generate probability distributions of CaCO3 at unsampled points. These probability distributions were used as soft data for the BME to predict the CaCO3 content. The results showed that the GWR method (r = 0.84, RMSE = 24.0 g kg−1) performed better than the MLR method (r = 0.73, RMSE = 30.1 g kg−1). The BME-GWR method outperformed the BME-EV and BME-MLR methods. The r values of BME-GWR, BME-EV, and BME-MLR methods were 0.87, 0.86, and 0.82, respectively, and the RMSEs of the three methods were 22.2, 23.9, and 25.2 g kg−1, respectively. The spatial distribution of CaCO3 content predicted by the above methods was similar and significantly higher in the southwest than in the northeast.

Fluctuations in irreversible quantum Otto engines

We derive the general probability distribution function of stochastic work for quantum Otto engines in which both the isochoric and driving processes are irreversible due to finite time duration. The time-dependent work fluctuations, average work, and thermodynamic efficiency are explicitly obtained for a complete cycle operating with an analytically solvable two-level system. The effects of the irreversibility originating from finite-time cycle operation on the thermodynamic efficiency, work fluctuations, and relative power fluctuations are discussed.

Expecting the unexpected: Goal recognition for rational and irrational agents

Contemporary cost-based goal-recognition assumes rationality: that observed behaviour is more or less optimal. Probabilistic goal recognition systems, however, explicitly depend on some degree of sub-optimality to generate probability distributions. We show that, even when an observed agent is only slightly irrational (sub-optimal), state-of-the-art systems produce counter-intuitive results (though these may only become noticeable when the agent is highly irrational). We provide a definition of rationality appropriate to situations where the ground truth is unknown, define a rationality measure (RM) that quantifies an agent's expected degree of sub-optimality, and define an innovative self-modulating probability distribution formula for goal recognition. Our formula recognises sub-optimality and adjusts its level of confidence accordingly, thereby handling irrationality—and rationality—in an intuitive, principled manner. Building on that formula, moreover, we strengthen a previously published result, showing that “single-observation” recognition in the path-planning domain achieves identical results to more computationally expensive techniques, where previously we claimed only to achieve equivalent rankings though values differed.

Estimating the state of the COVID-19 epidemic in France using a model with memory.

In this paper, we use a deterministic epidemic model with memory to estimate the state of the COVID-19 epidemic in France, from early March until mid-December 2020. Our model is in the SEIR class, which means that when a susceptible individual (S) becomes infected, he/she is first exposed (E), i.e. not yet contagious. Then he/she becomes infectious (I) for a certain length of time, during which he/she may infect susceptible individuals around him/her, and finally becomes removed (R), that is, either immune or dead. The specificity of our model is that it assumes a very general probability distribution for the pair of exposed and infectious periods. The law of large numbers limit of such a model is a model with memory (the future evolution of the model depends not only upon its present state, but also upon its past). We present theoretical results linking the (unobserved) parameters of the model to various quantities which are more easily measured during the early stages of an epidemic. We then apply these results to estimate the state of the COVID-19 epidemic in France, using available information on the infection fatality ratio and on the distribution of the exposed and infectious periods. Using the hospital data published daily by Santé Publique France, we gather some information on the delay between infection and hospital admission, intensive care unit (ICU) admission and hospital deaths, and on the proportion of people who have been infected up to the end of 2020.

Least singular value and condition number of a square random matrix with i.i.d. rows

Introducing a new method for studying general probability distributions on Rn, we generalize some results about the least singular value and the condition number of random matrices with i.i.d. Gaussian entries to the whole class of random matrices with i.i.d. rows.

Average search time bounds in cue-based searches.

In this work we consider search problems that evaluate the probability distribution of finding the source at each step in the search. We start with a sample strategy where the movement at each time step is in the immediate neighborhood. The jump probability is taken to be proportional to the normalized difference between the probability of finding the source at the jump location with the probability of finding the source at the present location. We evaluate a lower bound on the average search time for a searcher using this strategy. We next consider the problem of evaluating the lower bound on the search time for a generic strategy which would utilize the source probability distribution to figure out the position of the source. We derive an expression for the lower bound on the search time. We present an analytic expression for this lower bound in a case in which the particles emitted by the source diffuse in a homogeneous manner. For a general probability distribution with entropy E, we find that the lower bound goes as e^{E/2}.

FlexSketch: Estimation of Probability Density for Stationary and Non-Stationary Data Streams.

Efficient and accurate estimation of the probability distribution of a data stream is an important problem in many sensor systems. It is especially challenging when the data stream is non-stationary, i.e., its probability distribution changes over time. Statistical models for non-stationary data streams demand agile adaptation for concept drift while tolerating temporal fluctuations. To this end, a statistical model needs to forget old data samples and to detect concept drift swiftly. In this paper, we propose FlexSketch, an online probability density estimation algorithm for data streams. Our algorithm uses an ensemble of histograms, each of which represents a different length of data history. FlexSketch updates each histogram for a new data sample and generates probability distribution by combining the ensemble of histograms while monitoring discrepancy between recent data and existing models periodically. When it detects concept drift, a new histogram is added to the ensemble and the oldest histogram is removed. This allows us to estimate the probability density function with high update speed and high accuracy using only limited memory. Experimental results demonstrate that our algorithm shows improved speed and accuracy compared to existing methods for both stationary and non-stationary data streams.

Joint reserve stock and just-in-time inventory under regular preventive maintenance and random disruptions

ABSTRACT We study a repairable production-inventory system with a constant demand rate subject to random disruptions, which supplies an assembly line according to a just-in-time configuration. Preventive maintenance (PM) actions are regularly performed on the production unit at fixed time instants. We assume that the time between breakdowns is constant and the PM and corrective actions have random durations that follow a general probability distribution. We propose the following policy to cope with production interruptions: A buffer stock is built up at the maximum production rate at the beginning of each PM cycle. At the same time, a reserve stock is maintained to hedge against random disruptions and shortages during planned maintenance. The joint convexity of the cost function is established and optimality conditions for the optimal policy parameters are derived. We perform numerical and sensitivity analyses under the exponential and Asymmetric Laplace distributions. The results suggest that: (i) The proposed policy performs better than traditional models that consider these two strategies separately; (ii) As the shortage cost increases, it is optimal to increase the reserve level; (iii) Higher reserve stock levels allow to hedge against the increased duration and rate of preventive maintenance and random disruptions.

Multi-sourcing under supply uncertainty and Buyer's risk aversion

• Supplier selection and ordering decision under yield uncertainty is discussed. • We model it as a stochastic program with recourse. • We develop a scalable solution technique. • We also incorporate buyer’s risk aversion to shortage in procured quantity. We address the combined problem of supplier (or vendor) selection and ordering decision when a buyer can choose to procure from multiple suppliers whose yields are uncertain and potentially correlated. We model this problem as a stochastic program with recourse in which the buyer purchases from the suppliers in the first period and, if needed, chooses to purchase from the spot market or from the suppliers with excess supply, whichever is beneficial, in the second period in order to meet the target procurement quantity. We solve the above problem using sample average approximation (SAA) technique that enables us to solve the problem easily in practice. We compare the performance of our solution with the certainty equivalent problem, which is practiced widely and which we use as the benchmark, to evaluate the efficacy of our approach. Next, we extend our model to incorporate buyer’s risk aversion with respect to the quantity procured. We reformulate the multi-sourcing problem as a mixed integer linear program (MILP) and adopt a statistical approach to account for buyer’s risk aversion. Thus, we design a simple computational technique that provides an optimal sourcing policy from a set of suppliers when each supplier’s yield is uncertain with a generic probability distribution.

Closing Gaps of a Quantum Advantage with Short-Time Hamiltonian Dynamics.

Demonstrating a quantum computational speed-up is a crucial milestone for near-term quantum technology. Recently, sampling protocols for quantum simulators have been proposed that have the potential to show such a quantum advantage, based on commonly made assumptions. The key challenge in the theoretical analysis of this scheme-as of other comparable schemes such as boson sampling-is to lessen the assumptions and close the theoretical loopholes, replacing them by rigorous arguments. In this work, we prove two open conjectures for a simple sampling protocol that is based on the continuous time evolution of a translation-invariant Ising Hamiltonian: anticoncentration of the generated probability distributions and average-case hardness of exactly evaluating those probabilities. The latter is proven building upon recently developed techniques for random circuit sampling. For the former, we exploit the insight that approximate 2-designs for the unitary group admit anticoncentration. We then develop new techniques to prove that the 2D time evolution of the protocol gives rise to approximate 2-designs. Our work provides the strongest theoretical evidence to date that Hamiltonian quantum simulators are classically intractable.

Intraspecific variation and energy channel coupling within a Chilean kelp forest.

The widespread importance of variable types of primary production, or energy channels, to consumer communities has become increasingly apparent. However, the mechanisms underlying this "multichannel" feeding remain poorly understood, especially for aquatic ecosystems that pose unique logistical constraints given the diversity of potential energy channels. Here, we use bulk tissue isotopic analysis along with carbon isotope (δ13 C) analysis of individual amino acids to characterize the relative contribution of pelagic and benthic energy sources to a kelp forest consumer community in northern Chile. We measured bulk tissue δ13 C and δ15 N for >120 samples; of these we analyzed δ13 C values of six essential amino acids (EAA) from nine primary producer groups (n=41) and 11 representative nearshore consumer taxa (n=56). Using EAA δ13 C data, we employed linear discriminant analysis (LDA) to assess how distinct EAA δ13 C values were between local pelagic (phytoplankton/particulate organic matter), and benthic (kelps, red algae, and green algae) endmembers. With this model, we were able to correctly classify nearly 90% of producer samples to their original groupings, a significant improvement on traditional bulk isotopic analysis. With this EAA isotopic library, we then generated probability distributions for the most important sources of production for each individual consumer and species using a bootstrap-resampling LDA approach. We found evidence for multichannel feeding within the community at the species level. Invertebrates tended to focus on either pelagic or benthic energy, deriving 13-67% of their EAA from pelagic sources. In contrast, mobile (fish) taxa at higher trophic levels used more equal proportions of each channel, ranging from 19% to 47% pelagically derived energy. Within a taxon, multichannel feeding was a result of specialization among individuals in energy channel usage, with 37 of 56 individual consumers estimated to derive >80% of their EAA from a single channel. Our study reveals how a cutting-edge isotopic technique can characterize the dynamics of energy flow in coastal food webs, a topic that has historically been difficult to address. More broadly, our work provides a mechanism as to how multichannel feeding may occur in nearshore communities, and we suggest this pattern be investigated in additional ecosystems.

Universal properties of a run-and-tumble particle in arbitrary dimension.

We consider an active run-and-tumble particle (RTP) in d dimensions, starting from the origin and evolving over a time interval [0,t]. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are noninstantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the x component does not change sign up to time t, showing that it does not depend on d. As a consequence of this result, we compute exactly other x-component properties, namely, the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e., they do not depend on d. Moreover, we show that these universal results hold also if the speed v of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)10.1103/PhysRevLett.124.090603].

Sampling the feasible sets of SDPs and volume approximation

We present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to dimension 200. We illustrate its efficiency on various data sets.