Probability Matrix Research Articles

Computing pth root of a transition matrix with a deep unsupervised learning approach

Transition matrices are a tool used to describe transition probabilities of a Markovian process. From an applied perspective, transition matrices are, for example, used in the fields of credit risk modeling and medical decision analysis. In these domains, it is possible to estimate matrices according to the frequency of data observations, but for modeling purposes, it may be necessary to calculate transition matrices for shorter and more restricted time intervals. From a mathematical perspective, this requirement translates into calculating the pth root of a stochastic matrix through a constrained optimization problem. This article focuses on computing pth root of transition matrices and proposes a novel approach using deep learning-based algorithms, particularly Convolutional Neural Networks. Comparisons with traditional algorithms reveal that, in certain contexts, deep learning models offer computational advantages, especially regarding time efficiency. The applicability of the models is verified through a case study built on Fitch Ratings data.

The Markov chain embedding problem in a one-jump setting

Abstract The embedding problem of Markov chains examines whether a stochastic matrix $\mathbf{P} $ can arise as the transition matrix from time 0 to time 1 of a continuous-time Markov chain. When the chain is homogeneous, it checks if $ \mathbf{P}=\exp{\mathbf{Q}}$ for a rate matrix $ \mathbf{Q}$ with zero row sums and non-negative off-diagonal elements, called a Markov generator. It is known that a Markov generator may not always exist or be unique. This paper addresses finding $ \mathbf{Q}$ , assuming that the process has at most one jump per unit time interval, and focuses on the problem of aligning the conditional one-jump transition matrix from time 0 to time 1 with $ \mathbf{P}$ . We derive a formula for this matrix in terms of $ \mathbf{Q}$ and establish that for any $ \mathbf{P}$ with non-zero diagonal entries, a unique $ \mathbf{Q}$ , called the ${\unicode{x1D7D9}}$ -generator, exists. We compare the ${\unicode{x1D7D9}}$ -generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of $ \mathbf{P}$ in some practical cases.

Time-Lag Transiograms and Their Implications for Landscape Change Characterization

Markov chain transition probability matrices (TPMs) have traditionally been used to characterize land use and land cover (LULC) changes and species succession. However, previous studies relied solely on TPMs or transition area matrices to describe overall class area/proportion changes, overlooking important time correlation features. This study introduces the concept of idealized time-lag transiograms and demonstrates how they can be computed from temporal TPMs, using illustrative examples. The primary objective is to explore the potential value and utility of idealized time-lag transiograms in revealing additional characteristics of landscape change. Specifically, we focus on computing idealized time-lag transiograms with a fixed starting point and highlighting their fundamental features, such as sills, practical correlation ranges, and curve shapes, along with peak positions and peak height ratios of peaked cross-transiograms. These features are identified and discussed in terms of their potential implications for characterizing LULC changes. While idealized time-lag transiograms with a fixed starting point may not precisely predict future LULC changes due to the assumptions of the Markov property and time homogeneity (i.e., stationarity), they provide new insights into future LULC dynamics, revealing aspects that traditional Markov chain analysis has overlooked.

An Earth Mover's Distance-Based Self-Supervised Framework for Cellular Dynamic Grading in Live-Cell Imaging.

Cellular appearance and its dynamics frequently serve as a proxy measurement of live-cell physiological properties. The computational analysis of cell properties is considered to be a significant endeavor in biological and biomedical research. Deep learning has garnered considerable success across various fields. In light of this, various neural networks have been developed to analyze live-cell microscopic videos and capture cellular dynamics with biological significance. Specifically, cellular dynamic grading (CDG) is the task that provides a predefined dynamic grade for a live-cell according to the speed of cellular deformation and intracellular movement. This task involves recording the morphological and cytoplasmic dynamics in live-cell microscopic videos. Similar to other medical image processing tasks, CDG faces challenges in collecting and annotating cellular videos. These deficiencies in medical data limit the performance of deep learning models. In this article, we propose a novel self-supervised framework to overcome these limitations for the CDG task. Our framework relies on the assumption that increasing or decreasing cell dynamic grades is consistent with accelerating or decelerating cell appearance change in videos, respectively. This consistency is subsequently incorporated as a constraint in the loss function for the self-supervised training strategy. Our framework is implemented by formulating a probability transition matrix based on the Earth Mover's Distance and imposing a loss constraint on the elements of this matrix. Experimental results demonstrate that our proposed framework enhances the model's ability to learn spatiotemporal dynamics. Furthermore, our framework outperforms the existing methods on our cell video database.

Study of Variability of the Crack Growth Rate by Applying a Crack Growth Model to Sequences of Extrema Modeled by Markov Matrices

Study of Variability of the Crack Growth Rate by Applying a Crack Growth Model to Sequences of Extrema Modeled by Markov Matrices

Robust Adaptive Transition Probability Matrix in Interacting Multiple Model With Polynomial Functions and Feedback Structure

Robust Adaptive Transition Probability Matrix in Interacting Multiple Model With Polynomial Functions and Feedback Structure

CC4S: Encouraging Certainty and Consistency in Scribble-Supervised Semantic Segmentation.

Deep learning-based solutions have achieved impressive performance in semantic segmentation but often require large amounts of training data with fine-grained annotations. To alleviate such requisition, a variety of weakly supervised annotation strategies have been proposed, among which scribble supervision is emerging as a popular one due to its user-friendly annotation way. However, the sparsity and diversity of scribble annotations make it nontrivial to train a network to produce deterministic and consistent predictions directly. To address these issues, in this paper we propose holistic solutions involving the design of network structure, loss and training procedure, named CC4S to improve Certainty and Consistency for Scribble-Supervised Semantic Segmentation. Specifically, to reduce uncertainty, CC4S embeds a random walk module into the network structure to make neural representations uniformly distributed within similar semantic regions, which works together with a soft entropy loss function to force the network to produce deterministic predictions. To encourage consistency, CC4S adopts self-supervision training and imposes the consistency loss on the eigenspace of the probability transition matrix in the random walk module (we named neural eigenspace). Such self-supervision inherits the category-level discriminability from the neural eigenspace and meanwhile helps the network focus on producing consistent predictions for the salient parts and neglect semantically heterogeneous backgrounds. Finally, to further improve the performance, CC4S uses the network predictions as pseudo-labels and retrains the network with an extra color constraint regularizer. From comprehensive experiments, CC4S achieves comparable performance to those from fully supervised methods and shows promising robustness under extreme supervision cases.

Associative algebras generated by regular Volterra operators

In this work, we consider genetic algebras, generated by Volterra quadratic stochastic operators, and establish relation between regularity of Volterra operators and associativity of corresponding genetic algebra.

Multisource sensors navigation methodology applied to ADAS testing platform vehicle

This study addresses sensor signal loss in Advanced Driver Assistance Systems (ADAS) testing when switching between open and tunnel test sites. Current methods require pausing and switching sensor devices, disrupting the testing process. This study proposes a multisource sensor integrated navigation algorithm based on the Interacting Multiple Model (IMM) framework, using GNSS/INS error state Kalman filter (ESKF) positioning in open areas and UWB/INS ESKF positioning in GNSS failure scenarios, like tunnels. An Interacting Multiple Model based on Adaptive Transition Probability Matrix (ATPM-IMM) algorithm is introduced, enabling adaptive Markov jump probability for target tracking. The method ensures positioning accuracy using low-cost UWB and IMU and achieves seamless sensor source switching. Compared with traditional GNSS/INS and UWB/INS methods, the proposed approach improves accuracy by 31.17% and 41.23%, respectively, providing a reference for future multisource sensor fusion positioning research.

Completion problems and sparsity for Kemeny’s constant

Abstract For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny’s constant. We prove that for any partially specified stochastic matrix for which the problem is well defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny’s constant in two special cases: when the diagonal has been specified and when all specified entries lie in a common row.

Joint Classification of Hyperspectral and LiDAR Data via Multiprobability Decision Fusion Method

With the development of sensor technology, the sources of remotely sensed image data for the same region are becoming increasingly diverse. Unlike single-source remote sensing image data, multisource remote sensing image data can provide complementary information for the same feature, promoting its recognition. The effective utilization of remote sensing image data from various sources can enhance the extraction of image features and improve the accuracy of feature recognition. Hyperspectral remote sensing (HSI) data and light detection and ranging (LiDAR) data can provide complementary information from different perspectives and are frequently combined in feature identification tasks. However, the process of joint use suffers from data redundancy, low classification accuracy and high time complexity. To address the aforementioned issues and improve feature recognition in classification tasks, this paper introduces a multiprobability decision fusion (PRDRMF) method for the combined classification of HSI and LiDAR data. First, the original HSI data and LiDAR data are downscaled via the principal component–relative total variation (PRTV) method to remove redundant information. In the multifeature extraction module, the local texture features and spatial features of the image are extracted to consider the local texture and spatial structure of the image data. This is achieved by utilizing the local binary pattern (LBP) and extended multiattribute profile (EMAP) for the two types of data after dimensionality reduction. The four extracted features are subsequently input into the corresponding kernel–extreme learning machine (KELM), which has a simple structure and good classification performance, to obtain four classification probability matrices (CPMs). Finally, the four CPMs are fused via a multiprobability decision fusion method to obtain the optimal classification results. Comparison experiments on four classical HSI and LiDAR datasets demonstrate that the method proposed in this paper achieves high classification performance while reducing the overall time complexity of the method.

Detailed-level modelling of influence spreading on complex networks

The progress in high-performance computing makes it increasingly possible to build detailed models to investigate spreading processes on complex networks. However, current studies have been lacking detailed computational methods to describe spreading processes in large complex networks. To fill this gap we present a new modelling approach for analysing influence spreading via individual nodes and links on various network structures. The proposed influence-spreading model uses a probability matrix to capture the spreading probability from one node to another in the network. This approach enables analysing network characteristics in a number of applications and spreading processes using metrics that are consistent with the quantities used to model the network structures. In addition, this study combines sub-models and offers a comprehensive look at different applications and metrics previously discussed in cases of social networks, community detection, and epidemic spreading. Here, we also note that the centrality measures based on the probability matrix are used to identify the most significant nodes in the network. Furthermore, the model can be expanded to include additional properties, such as introducing individual breakthrough probabilities for the nodes and specific temporal distributions for the links.

Mapping the Trade Direction of Indian Rice with Markov Chain Analysis

Agriculture is a cornerstone of India’s economy, with rice being a critical crop that ranks second globally, feeding over half of the world’s population and serving as a staple in Southeast Asia. This study explores the direction of India’s rice trade using Markov Chain Analysis, following an examination of the compound annual growth rate (CAGR) in rice exports to various global regions. Results reveal a 2.26 per cent CAGR in rice production, while export values surged by 16.8 per cent annually. Specifically, the CAGR for rice exports to Iraq and Iran reached 58.6 per cent and 46.47 per cent, respectively. In quantity, India’s rice exports grew by 10.82 per cent, with Iraq and Iran at 44 per cent and 34 per cent. The Transitional Probability Matrix for 2001-2022 indicates strong regional retention, with Africa maintaining 78 per cent of trade value and Asia 93.7 per cent, and similar trends in quantity, where Asia retained 92.6 per cent and Africa 89.1 per cent. These findings highlight India’s potential to diversify rice exports and strengthen its global competitive edge, urging a focus on enhancing processing, transportation, and quality standards for wider market appeal.

A novel operational reliability and state prediction method based on degradation hidden Markov model with random threshold

AbstractBearings are widely used as a common mechanical component and significantly impacts the reliability and safety of various engineering equipment. In the field of large‐scale equipment such as wind turbines, statistical data for reliability and state prediction of bearings is usually limited, leading to the inapplicability of traditional statistical methods. Besides, individual differences commonly exist among bearings, and the appropriate failure threshold for a specific bearing is difficult to be determined due to the individual uncertainty, which may induce prediction errors. To address these issues, a novel operational reliability and state prediction method based on random threshold degradation hidden Markov model was proposed in the study. The traditional hidden Markov model was improved by considering the effect of performance degradation on state transition probability matrix. Moreover, random failure thresholds of performance degradation were employed to describe individual differences between bearings. The operational reliability and state of bearings was obtained by comprehensive analysis of prediction results under different failure thresholds. Verification of the proposed operational reliability and state prediction method were conducted using PHM2012 bearing operation data. The results underscored the effectiveness of the proposed method in achieving a comparatively high accuracy in operational reliability prediction without the prerequisite of an exact failure threshold, when contrasted with several established reliability and life prediction techniques.

Convergence of the trajectories of a non-Volterra quadratic stochastic operator

In the present paper we consider non-Volterra quadratic stochastic operators defined on the two-dimensional simplex depending on a parameter \alpha . We show that such an operator has a unique fixed point and all the trajectories converge to this unique fixed point.

DUALITY: DETECTABILITY VERSUS STABILIZABILITY IN THE STOCHASTIC CONTEXT

The aim of this paper is to extend to the stochastic framework the well known duality relation between the detectability property and the stabilizability property of a finite dimensional, linear time invariant, deterministic system. We consider continuous time linear stochastic systems having the state space representation described by a system of Itˆo differential equations with periodic coefficients possibly subject to some stochastic perturbations modeled by a standard homogeneous Markov process with a finite number of states. It will be seen that in the case when the given system is affected by jump Markov perturba tions a state space representation of the dual triple may be rigorously defined if and only if the transition probability matrix of the Markov process is double stochastic (in the sense that the sums of all elements from its each row and each column are equal 1).

ADP-Based Decentralized Load Frequency Control Schemes to Multiarea Asynchronous Markov Jumping Power Systems With Experience Replay.

This article investigates the problem of robust decentralized load frequency control (LFC) in multiarea interconnection power systems, in the presence of the external disturbances and stochastic abrupt variations, such as component failures and different load demands. To capture the different operating conditions of the load in the multiarea power systems, a Markov superposition technique is skillfully employed to model the system's component matrices. To solve the Nash equilibrium solution of the zero-sum differential game problem, an improved online adaptive dynamic programming (ADP) algorithm based on the experience replay technique (ERT) is developed, which addresses the nonlinear coupling difficulties encountered in solving the game algebraic Riccati equations (Game AREs). The proposed algorithm weakens the condition that traditional policy iteration online solutions of zero-sum games require an initial stabilizing control policy pair, and considers the fact that only the transition probability matrix is known, while the prior knowledge of the dynamics is completely unknown. The algorithm is used to obtain both a minimum LFC and maximum disturbance policy for a given disturbance rejection level. Additionally, a safer LFC controller of the power systems with the Markov jumping parameters is designed. The stability and convergence of the proposed algorithm are demonstrated. Finally, the effectiveness and good performance of the proposed design method are validated using a two-area four-mode simulation example.

Attribute enhanced random walk for community detection in attributed networks

Traditional community detection methods often rely solely on network topology, potentially overlooking the influence of attributes. However, community detection in attributed networks presents a unique challenge due to the complex interplay between network topology and node attributes. In this paper, we present AERW, a novel approach that integrates both network topology and node attributes for effective community detection. AERW leverages the principle of homophily by aggregating neighbor attributes within triangles to capture higher-order structural information. Then constructing a bipartite graph to effectively integrate topological and attribute information, ensuring that both aspects are equally represented in the community detection process. Next AERW utilizes hierarchical clustering applied to the probability transition matrix derived from the random walk, enabling the identification of community structures without needing prior knowledge of the number of communities. Extensive experimentation across synthetic and real-world networks underscores AERW’s efficacy, establishing it as a superior approach in community detection within attributed networks.

Floating wave energy farms: How energy calculations shape economic feasibility?

The aim of this study is to use two different methodologies for calculating the energy produced by a total wave energy farm in the Cantabrian Sea (North of Spain). First method studies the energy produced by the farm taking into account the wave energy converter power matrix and the probability matrix of sea conditions for a particular location. On the other hand, second method is a more simplified methodology in which the wave energy converter power matrix is not taken into account, but it considers the density of the water, gravity, period and height of the wave, as well as the percentage of efficiency, among others. Finally, the energy generated by the farm and the economic parameters that allow calculating its economic feasibility are studied, such as Levelized Cost of Energy, Net Present Value and Internal Rate of Return. Results indicate the importance of using a particular method for calculating the energy produced by a wave energy farm in terms of the variation of its economic feasibility. In this sense, it is shown that despite the fact that the first method is more detailed, the variations between both methods are very small, therefore the simplest model can be used in the future studies, reducing the calculation process in the future.

Structured multi-view k-means clustering

K-means is a very efficient clustering method and many multi-view k-means clustering methods have been proposed for multi-view clustering during the past decade. However, since k-means have trouble uncovering clusters of varying sizes and densities, these methods suffer from the same performance issues as k-means. Improving the clustering performance of multi-view k-means has become a challenging problem. In this paper, we propose a new multi-view k-means clustering method that is able to uncover clusters in arbitrary sizes and densities. The new method simultaneously performs three tasks, i.e., sparse connection probability matrices learning, prototypes aligning, and cluster structure learning. We evaluate the proposed new method by 5 benchmark datasets and compare it with 11 multi-view clustering methods. The experimental results on both synthetic and real-world experiments show the superiority of our proposed method.