Random Set Research Articles

Разработка математической модели определения эффективности сельскохозяйственной техники на отдельных уровнях многоуровневой системы оценки качества

The evaluation of a set of quality properties of agricultural machinery requires the application of multi-criteria optimisation models. A possible solution implies trade-offs, because the evaluated index of one of the criteria cannot be varied at the expense of the evaluation indices of other criteria. The mathematical model for determining the effi ciency of objects at individual levels of the multilevel system of agricultural machinery quality assessment implements a universal approach to solving problems with any set of attributes and their varieties. In this case, not only the solutions themselves, but also the algorithms of their search and selection, including procedures for describing con ditions (situations), can be subject to systematisation. The authors propose an analytical defi nition of coeffi cients according to the formula used to study information states based on the principle of decreasing the importance of priorities between the attributes in the matrix. According to the developed mathematical model with a random set of indicators, the authors have calculated relative importance coeffi cients for four investigated objects by four attributes and presented the results of data conversion into relative units and normalisation in the “recogniser”. The calculation results show that the developed mathematical model makes it possible to analytically determine the effi ciency for any number of investigated objects by a suffi ciently large number of attributes and objectively assess their effi ciency for individual “recognisers” in the system of quality assessment of agricultural machinery.

The Risk of Random Sets with Applications to Basket Derivatives

The Risk of Random Sets with Applications to Basket Derivatives

Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

<abstract><p>We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.</p></abstract>

КРИПТОГРАФИЧЕСКОЕ ЗНАЧЕНИЕ МАНУСКРИПТА ВОЙНИЧА

The article analyzes modern approaches to deciphering the text of the Voynich manuscript. The work contains a brief overview in the history of the manuscript discovery and an analysis of the main versions about the authorship and the first owners of the manuscript, the place and time of its creation and its content. The author pays attention to the study of key approaches to the definition of cryptographic methods that could be used by the author or authors of the essay and the analysis of the manuscript text decrypting issues facing modern researchers. The article provides a review of other encrypted manuscripts, the time and place of creation of which allow comparison with the Voynich manuscript according to a number of selected criteria. Particular attention is paid to analyzing the challenges of defining the manuscript language, what involves the choice of one or more natural languages within one or the other language group, an artificial language or “language” consisting of a random set of characters. Difficulties with transliteration of manuscript symbols used to create machine-readable versions of the text are considered. An analysis of the prospects for studying the cryptographic content of the manuscript is carried out. Attention is paid to the content of a number of reports of the international conference on the study of the Voynich manuscript, held by the University of Malta in November–December 2022. The author emphasizes the need for an interdisciplinary approach to the study of the Voynich manuscript, which requires the combined efforts of cryptologists, mathematicians, historians, philologists, linguists to successfully unravel the content of the manuscript Voynich.

Generalization of the Glivenko–Cantelli theorem to finite-dimensional distributions of ergodic homogeneous random fields

Let X(t),t∈Zm×Rk, be an ergodic stationary random processes or an ergodic homogeneous random field (k+m≥1). We prove that the distribution function of each random vector (X(t1),…,X(ts)) can be a.s. arbitrary fine uniformly approximated by the empirical distribution functions, if, e.g., in their construction increasing bounded convex sets Tn with infinitely increasing intrinsic diameters are used. We consider also the case when X is observed on a mixing homogeneous countable random set Sm(ω)⊂Rm (e.g., on a Poisson random set). These results open a way to consistent statistical inference on finite-dimensional distribution functions of ergodic processes and fields.

Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

AbstractWe prove that for any fixed , the probability that a random vertex of a random increasing plane tree is of rank , that is, the probability that a random vertex is at distance from the leaves, converges to a constant as the size of the tree goes to infinity. We prove that , so that the tail of the limiting rank distribution is super‐exponentially narrow. We prove that the latter property holds uniformly for all finite as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution . We compute the exact value of for , demonstrating that the limiting expected fraction of vertices with rank is … We show that with probability the highest rank of a vertex in the tree is sandwiched between and , and that this rank is asymptotic to with probability .

Genomic prediction of tocochromanols in exotic-derived maize.

Tocochromanols (vitamin E) are an essential part of the human diet. Plant products, including maize (Zea mays L.) grain, are the major dietary source of tocochromanols; therefore, breeding maize with higher vitamin content (biofortification) could improve human nutrition. Incorporating exotic germplasm in maize breeding for trait improvement including biofortification is a promising approach and an important research topic. However, information about genomic prediction of exotic-derived lines using available training data from adapted germplasm is limited. In this study, genomic prediction was systematically investigated for nine tocochromanol traits within both an adapted (Ames Diversity Panel [AP]) and an exotic-derived (Backcrossed Germplasm Enhancement of Maize [BGEM]) maize population. Although prediction accuracies up to 0.79 were achieved using genomic best linear unbiased prediction (gBLUP) when predicting within each population, genomic prediction of BGEM based on an AP training set resulted in low prediction accuracies. Optimal training population (OTP) design methods fast and unique representative subset selection (FURS), maximization of connectedness and diversity (MaxCD), and partitioning around medoids (PAM) were adapted for inbreds and, along with the methods mean coefficient of determination (CDmean) and mean prediction error variance (PEVmean), often improved prediction accuracies compared with random training sets of the same size. When applied to the combined population, OTP designs enabled successful prediction of the rest of the exotic-derived population. Our findings highlight the importance of leveraging genotype data in training set design to efficiently incorporate new exotic germplasm into a plant breeding program.

Multiple extended target tracking based on gamma box particle and labeled random finite sets

A gamma box particle (GBP) implementation of the extended target generalized labeled multi-Bernoulli (ET-GLMB) filter is proposed in this paper, named the ET-GBP-GLMB filter. The filter can simultaneously estimate the unknown number, measurement rate states, kinematics states, extension states and tracks of extended targets in the presence of clutter, missed detection and data association uncertainty. In order to obtain the GLMB recursion of the ET-GBP-GLMB filter, an appropriate measurement likelihood function that comprehensively considers the measurement rate state, kinematics state and extension state of each extended target is introduced for a given measurement partition cell, and the specific implementation steps of the filter are derived with necessary interval operations and approximations. Further, an extended target GBP labeled multi-Bernoulli (ET-GBP-LMB) filter is proposed as an effective approximation of the ET-GBP-GLMB filter. The advantages and disadvantages of the two proposed filters are presented in simulation examples. The simulation results show that, compared with the gamma sequential Monte Carlo (GSMC) implementation of the ET-GLMB filter, the GBP implementation of the ET-GLMB filter can effectively reduce the number of particles and the amount of computation. Compared with the ET-GBP-GLMB filter, the ET-GBP-LMB filter can further reduce the computational burden. Compared with the extended target box particle probability hypothesis density (ET-BP-PHD) filter, the two proposed filters can obtain better estimation results.

Asymptotics of a time bounded cylinder model

AbstractOne way to model telecommunication networks are static Boolean models. However, dynamics such as node mobility have a significant impact on the performance evaluation of such networks. Consider a Boolean model in $\mathbb {R}^d$ and a random direction movement scheme. Given a fixed time horizon $T>0$, we model these movements via cylinders in $\mathbb {R}^d \times [0,T]$. In this work, we derive central limit theorems for functionals of the union of these cylinders. The volume and the number of isolated cylinders and the Euler characteristic of the random set are considered and give an answer to the achievable throughput, the availability of nodes, and the topological structure of the network.

Iterated tour partitioning for Euclidean capacitated vehicle routing

AbstractWe give a probabilistic analysis of the unit‐demand Euclidean capacitated vehicle routing problem in the random setting. The objective is to visit all customers using a set of routes of minimum total length, such that each route visits at most customers. The best known polynomial‐time approximation is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the solution obtained by the ITP algorithm is arbitrarily close to the optimum when is either or , and they asked whether the ITP algorithm was “also effective in the intermediate range”. In this work, we show that the ITP algorithm is at best a ‐approximation, for some positive constant , and is at worst a 1.915‐approximation.

Risk-factor model for postpartum hemorrhage after cesarean delivery: a retrospective study based on 3498 patients

This study aimed to investigate the risk factors of patients with postpartum hemorrhage (PPH) after cesarean delivery (CD) and to develop a risk-factor model for PPH after CD. Patients were selected from seven affiliated medical institutions of Chongqing Medical University from January 1st, 2015, to January 1st, 2020. Continuous and categorical variables were obtained from the hospital’s electronic medical record systems. Independent risk factors were identified by univariate analysis, least absolute shrinkage and selection operator and logistic regression. Furthermore, logistic, extreme gradient boosting, random forest, classification and regression trees, as well as an artificial neural network, were used to build the risk-factor model. A total of 701 PPH cases after CD and 2797 cases of CD without PPH met the inclusion criteria. Univariate analysis screened 28 differential indices. Multi-variable analysis screened 10 risk factors, including placenta previa, gestational age, prothrombin time, thrombin time, fibrinogen, anemia before delivery, placenta accreta, uterine atony, placental abruption and pregnancy with uterine fibroids. Areas under the curve by random forest for the training and test sets were 0.957 and 0.893, respectively. The F1 scores in the random forest training and test sets were 0.708. In conclusion, the risk factors for PPH after CD were identified, and a relatively stable risk-factor model was built.

Use of Neural Networks and Computer Vision for Spill and Waste Detection in Port Waters: An Application in the Port of Palma (MaJorca, Spain)

Water quality and pollution is the main environmental concern for ports and adjacent coastal waters. Therefore, the development of Port Environmental Management systems often relies on water pollution monitoring. Computer vision is a powerful and versatile tool for an exhaustive and systematic monitoring task. An investigation has been conducted at the Port of Palma de Mallorca (Spain) to assess the feasibility and evaluate the main opportunities and difficulties of the implementation of water pollution monitoring based on computer vision. Experiments on surface slicks and marine litter identification based on random image sets have been conducted. The reliability and development requirements of the method have been evaluated, concluding that computer vision is suitable for these monitoring tasks. Several computer vision techniques based on convolutional neural networks were assessed, finding that Image Classification is the most adequate for marine pollution monitoring tasks due to its high accuracy rates and low training requirements. Image set size for initial training and the possibility to improve accuracy through retraining with increased image sets were considered due to the difficulty in obtaining port spill images. Thus, we have found that progressive implementation can not only offer functional monitoring systems in a shorter time frame but also reduce the total development cost for a system with the same accuracy level.

Permutation Jensen–Shannon divergence for Random Permutation Set

Random Permutation Set (RPS) considers the permutation of elements for a certain set, which is an efficient tool for dealing with uncertainty with ordered information. An important feature of RPS theory is that in the fusion rule of RPS sources, the fusion order has a great impact on fusion results. However, how to determine the fusion order has not yet been discussed. To address this problem, this paper first proposes Permutation Jensen–Shannon (PJS) divergence for measuring the distance between two RPSs. Based on PJS divergence, a new Reliability Assessment algorithm, named RAPJS, is then presented for determining the fusion order of RPSs. The proposed PJS divergence satisfies the properties of non–degeneracy, boundary, and symmetry, and has desirable compatibility with Belief Jensen–Shannon divergence and Jensen–Shannon divergence under certain conditions. The presented RAPJS makes use of the divergence information to calculate the reliability degree of RPS sources, the RPS with a higher reliability is fused first. Experiment results in threat assessment reveal that the presented RAPJS algorithm can determine the fusion order reasonably and effectively. The assessment results using the proposed RAPJS algorithm has the highest target recognition rate compared to other results under different fusion orders.

Moving object detection and trajectory prediction based on image processing

For moving object detection and trajectory prediction in video images, it is necessary to perform image processing, feature extraction, and localization of the object. Therefore, this paper designs an optimized Kalman-Elman (KE) algorithm for trajectory prediction. In order to remove the noise points on the measured values in the Kalman filter algorithm and to solve the problem of the random setting of the initial weights and thresholds of the Elman neural network, we encode the above parameters and improve the two algorithms by using Particle Swarm Optimization (PSO). Quantitative values of the object feature extraction are used as input parameters of the Elman neural network. After a large amount of training, we obtain the predicted position of the moving object finally. The experimental results show that the prediction error of this method is significantly smaller when it is compared with previous methods.

The Stochasticity Parameter of Quadratic Residues

Abstract Following V. I. Arnold, we define the stochasticity parameter $S(U)$ of a subset $U$ of ${\mathbb {Z}}/M{\mathbb {Z}}$ to be the sum of squares of the consecutive distances between elements of $U$. In this paper, we study the stochasticity parameter of the set $R_{M}$ of quadratic residues modulo $M$. We present a method that allows to find the asymptotics of $S(R_{M})$ for a set of $M$ of positive density. In particular, we obtain the following two corollaries. Denote by $s(k)=s(k,{\mathbb {Z}}/M{\mathbb {Z}})$ the average value of $S(U)$ over all subsets $U\subseteq {\mathbb {Z}}/M{\mathbb {Z}}$ of size $k$, which can be thought of as the stochasticity parameter of a random set of size $k$. Let ${\mathfrak {S}}(R_{M})=S(R_{M})/s(|R_{M}|)$. We show that (a) $\varliminf _{M\to \infty }{\mathfrak {S}}(R_{M})<1<\varlimsup _{M\to \infty }{\mathfrak {S}}(R_{M})$; (b) the set $\{ M\in {\mathbb {N}}: {\mathfrak {S}}(R_{M})<1 \}$ has positive lower density.

Joint DOA tracking and spectral signature estimation of wideband sources using a multi-spectral TBD-CPHD filter

In this paper, a novel method for direction of arrival (DOA) tracking of multiple unknown time-varying number of wideband emitters is presented using a random finite set (RFS) framework. This method combines multiple narrowband covariance-based measurements to form a wideband cardinalized probability hypothesis density (CPHD) filter using the parallel combination (PC) Lemma generally used in multisensor fusion. By doing this, each narrowband frequency bin acts as a separate virtual spectral sensor. The covariance-based measurements, represented by complex Wishart random matrices, also allow the method to have a track-before-detect (TBD) property, which has shown to be advantageous in very low signal-to-noise ratios (SNR). Furthermore, the proposed method can simultaneously monitor the spectrum of the environment and extract the spectral characteristics of the active sources, i.e. can work as some sort of cognitive radio (CR). The suggested algorithm is implemented using an auxiliary particle filter. The simulation results and the comparison with the known state-of-the-art approaches confirm the superiority of the proposed method in negative SNR scenarios and low number of snapshots.

Posterior Cruciate Ligament and Patellar Tendon Can Predict Anterior Cruciate Ligament Size for Planning During ACL Reconstruction in Pediatric Patients

To establish correlations between the anterior cruciate ligament (ACL), posterior cruciate ligament (PCL), and patellar tendon in normal pediatric knees to inform surgical planning for ACL reconstruction graft size. Magnetic resonance imaging scans of patients ages 8 to 18 years were assessed. Measurements included ACL and PCL length, thickness, and width, and ACL footprint thickness and width at the tibial insertion. Interrater reliability was assessed with a random set of 25 patients. Pearson correlation coefficients were used to assess the correlation between ACL, PCL, and patellar tendon measurements. Linear regression models were used to test whether the relationships differed by sex or age. Magnetic resonance imaging scans of 540 patients were assessed. Interrater reliability was high for all measurements except PCL thickness at midsubstance. Sample equations for estimating ACL size are as follows: ACL length= 22.61+ 1.55∗PCL origin width (R2= 0.46; 8- to 11-year-old male patients), ACL length= 12.37+ 0.58∗PCL length+ 2.29∗PCL origin thickness - 0.90∗PCL insertion width (R2= 0.68; 8- to 11-year-old female patients), ACL midsubstance thickness= 4.95+ 0.25∗PCL midsubstance thickness+ 0.04∗PCL insertion thickness - 0.08∗PCL insertion width (R2= 0.12; 12- to 18-year-old male patients), and ACL midsubstance width= 0.57+ 0.23∗PCL midsubstance thickness+ 0.07∗PCL midsubstance width+ 0.16∗PCL insertion width (R2= 0.24; 12- to 18-year-old female patients). We found correlations between ACL, PCL, and patellar tendon measurements that can be used to create equations that predict ACL size in various dimensions based on PCL and patellar tendon measurements. There is a lack of consensus on the ideal ACL graft diameter for pediatric ACL reconstruction. The findings from this study can assist orthopaedic surgeons to individualize ACL graft size for specific patients.

An Efficient Labeled/Unlabeled Random Finite Set Algorithm for Multiobject Tracking

In this article, we propose an efficient random finite set (RFS)-based algorithm for multiobject tracking, in which the object states are modeled by a combination of a labeled multi-Bernoulli (LMB) RFS and a Poisson RFS. The less computationally demanding Poisson part of the algorithm is used to track potential objects whose existence is unlikely. Only if a quantity characterizing the plausibility of object existence is above a threshold, a new labeled Bernoulli component is created, and the object is tracked by the more accurate but more computationally demanding LMB part of the algorithm. Conversely, a labeled Bernoulli component is transferred back to the Poisson RFS if the corresponding existence probability falls below another threshold. Contrary to existing hybrid algorithms based on multi-Bernoulli and Poisson RFSs, the proposed method facilitates track continuity and implements complexity-reducing features. Simulation results demonstrate a large complexity reduction relative to other RFS-based algorithms with comparable performance.

Fusion of Labeled RFS Densities With Different Fields of View

In this article, the interest is on the fusion of labeled random finite set (LRFS) densities computed by sensors having different fields of view (FoVs). In order to deal with different FoVs of the local densities, the global label space is divided into disjoint subspaces, which represent the exclusive FoVs and the common FoV of the agents, and a method based on the minimization of the Kullback–Leilber divergence from the resulting factorized density to the original one is proposed to decompose each local density into the subdensities defined in the corresponding subspaces. Then, fusion is performed according to a constrained minimum information loss criterion: the local subdensities are fused, subspace by subspace, into global ones, and the global density is obtained by multiplying the global subdensities. The application of the proposed approach to two important types of LRFS density, i.e., marginalized <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -generalized labeled multi-Bernoulli and labeled multi-Bernoulli densities, is developed in this article. Furthermore, in order to tackle the label mismatching issue arising in practical applications, an assignment optimization of a suitably defined cost is carried out so as to match labels from different agents. Finally, the performance of the proposed fusion approach is assessed via simulation experiments.

Optimization hierarchy for fair statistical decision problems

Data-driven decision making has drawn scrutiny from policy makers due to fears of potential discrimination, and a growing literature has begun to develop fair statistical techniques. However, these techniques are often specialized to one model context and based on ad hoc arguments, which makes it difficult to perform theoretical analysis. This paper develops an optimization hierarchy, which is a sequence of optimization problems with an increasing number of constraints, for fair statistical decision problems. Because our hierarchy is based on the framework of statistical decision problems, this means it provides a systematic approach for developing and studying fair versions of hypothesis testing, decision making, estimation, regression, and classification. We use the insight that qualitative definitions of fairness are equivalent to statistical independence between the output of a statistical technique and a random variable that measures attributes for which fairness is desired. We use this insight to construct an optimization hierarchy that lends itself to numerical computation, and we use tools from variational analysis and random set theory to prove that higher levels of this hierarchy lead to consistency in the sense that it asymptotically imposes this independence as a constraint in corresponding statistical decision problems. We demonstrate numerical effectiveness of our hierarchy using several data sets, and we use our hierarchy to fairly perform automated dosing of morphine.