Rate Matrix Research Articles

Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching

We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and risky debt under regime switching with a regime switching interest rate. While the value of the firm’s equity is observed continuously, we assume that the total value of the firm is only observed at discrete times, such as the dates of the release of the firm’s annual reports, or quarterly reports. This uncertainty about the true value of the firm results in credit spreads that do not approach zero as the debt approaches maturity, which is a problem with many structural models. The firm’s value is typically decomposed into its equity and debt; however, we consider the asset–to–equity ratio, an accounting ratio used to examine a firm’s financial well-being. The parameters in our model are regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching.

Comparable clinical advantages identification of three formulae on rheumatic disease using a modular-based network proximity approach

Ethnopharmacological relevanceHerbal formulae have been used in China for thousands of years but have unclear clinical positioning and unknown characteristic indications make it difficult to determine their specific application in various diseases, which seriously hamper their clinical value. Identifying the precise clinical positioning and clinical advantages of similar formulae for related diseases is a critical issue. Aim of this studyTo develop a methodology based on modular pharmacology to determine the clinical advantages and precise clinical position of similar formulae. Materials and methodsIn this study, we proposed a modular-based network proximity approach to explore drug repositioning and clinical advantages of three formulae, Shirebi tablets (SRB), Yuxuebi capsules (YXB), and Wangbifukang granules (WBFK), for rheumatic disease. First, we constructed a rheumatology target network, and modules were obtained using the cluster tool molecular complex detection (MCODE). Based on the modular interaction map established by a quantitative approach for inter-module coordination and its transition (IMCC), using a targeting rate (TR) matrix to identify targeted modules of three formulae. Moreover, the network proximity Z-score and Jaccard similarity coefficient were used to identify potential optimal symptomatic indications and related diseases using three formulae. At the same time, the driver genes for SRB and gouty arthritis were identified by flow centrality and shortest distance, and the epresentative driver genes were validated by in vivo experiments. Results32 modules were obtained using the MCODE method. 4, 4, and 14 characteristic targeted modules of SRB, YXB, and WBFK, respectively, were identified using a targeting rate (TR) matrix. Module 2, 16, and 19 were targeted by both SRB and WBFK. The common effects of SRB and WBFK focused on inflammatory response and innate immune response, YXB was found to be involved in the collagen catabolic process, transmembrane receptor protein serine/threonine kinase signaling pathway. Moreover, potential optimal symptomatic indications and representative related diseases were identified for three formulae: SRB was significantly associated with GA (Z = −20.26); YXB was significantly associated with AS (Z = −4.532), MI (Z = −29.11), RhFv (Z = −6.945), OA (Z = −39.97), and GA (Z = −13.03); and WBFK was significantly associated with MI (Z = −205.5), SLE (Z = −37.65), RhFv (Z = −42.45), and GA (Z = −17.24). Finally, 8 driver genes for SRB and gouty arthritis were identified,the representative driver genes TRAF6 and NFE2L1 were validated by in vivo experiments. ConclusionsThe modular-based network proximity approach proposed in this study may provide a new perspective for the precise drug repositioning and clinical advantages of similar formulae in disease treatment.

Efficient occlusion avoidance based on active deep sensing for harvesting robots

With the increasing shortage of agricultural labor, the development of harvesting robots is becoming more and more urgent. Most of them require vision to locate the target, however, occlusion is common in agricultural environment, which restricts the accuracy of visual target recognition, and even leads to failure in serious cases. The active perception method is an effective means, but how to efficiently find the best observation position remains difficult to avoid the waste of time caused by repeated invalid motion. Targeting these problems, an active deep sensing method is proposed for harvesting clustered and single fruits. First, the region of interest of the target is extracted by a segmentation network, and then the occlusion status of it is obtained by image processing methods. Taking the current observation position as the starting point, the camera is moved within a matrix to form confidence and occlusion rate distribution maps. After establishing a series of occlusion rate and confidence matrix datasets, a designed deep network has been trained, which is used to predict the maximum confidence/minimum occlusion rate position after the current occlusion status is estimated. To verify the reliability of the method, laboratory and field experiments were carried out for apples and clustered tomatoes. After 1000 times of verification, results show that the successful pick/recognition rate is increased by 38.7 %, and the average successful recognition time is 5.2 s, which is 63.1 % and 46.4 % faster than that of a fixed movement method and a simple heuristic method.

GTRpmix: A Linked General Time-Reversible Model for Profile Mixture Models.

Profile mixture models capture distinct biochemical constraints on the amino acid substitution process at different sites in proteins. These models feature a mixture of time-reversible models with a common matrix of exchangeabilities and distinct sets of equilibrium amino acid frequencies known as profiles. Combining the exchangeability matrix with each profile generates the matrix of instantaneous rates of amino acid exchange for that profile. Currently, empirically estimated exchangeability matrices (e.g. the LG matrix) are widely used for phylogenetic inference under profile mixture models. However, these were estimated using a single profile and are unlikely optimal for profile mixture models. Here, we describe the GTRpmix model that allows maximum likelihood estimation of a common exchangeability matrix under any profile mixture model. We show that exchangeability matrices estimated under profile mixture models differ from the LG matrix, dramatically improving model fit and topological estimation accuracy for empirical test cases. Because the GTRpmix model is computationally expensive, we provide two exchangeability matrices estimated from large concatenated phylogenomic-supermatrices to be used for phylogenetic analyses. One, called Eukaryotic Linked Mixture (ELM), is designed for phylogenetic analysis of proteins encoded by nuclear genomes of eukaryotes, and the other, Eukaryotic and Archaeal Linked mixture (EAL), for reconstructing relationships between eukaryotes and Archaea. These matrices, combined with profile mixture models, fit data better and have improved topology estimation relative to the LG matrix combined with the same mixture models. Starting with version 2.3.1, IQ-TREE2 allows users to estimate linked exchangeabilities (i.e. amino acid exchange rates) under profile mixture models.

Stability and Stabilization of Delayed Fuzzy Semi-Markov Jump Systems with Incomplete Transition Rates and Quadratic Fuzzy Lyapunov Matrix via Quantized Control Design

Stability and Stabilization of Delayed Fuzzy Semi-Markov Jump Systems with Incomplete Transition Rates and Quadratic Fuzzy Lyapunov Matrix via Quantized Control Design

Hybrid algorithms for SAR matrix compression and the impact of post-processing on SAR calculation complexity.

This study proposes faster virtual observation point (VOP) compression as well as post-processing algorithms for specific absorption rate (SAR) matrix compression. Furthermore, it shows the relation between the number of channels and the computational burden for VOP-based SAR calculation. The proposed new algorithms combine the respective benefits of two different criteria for determining upper boundedness of SAR matrices by the VOPs. Comparisons of the old and new algorithms are performed for head coil arrays with various channel counts. The new post-processing algorithm is used to post-process the VOP sets of nine arrays, and the number of VOPs for a fixed median relative overestimation is compared. The new algorithms are faster than the old algorithms by a factor of two to more than 10. The compression efficiency (number of VOPs relative to initial number of SAR matrices) is identical. For a fixed median relative overestimation, the number of VOPs increases logarithmically with the number of RF coil channels when post-processing is applied. The new algorithms are much faster than previous algorithms. Post-processing is very beneficial for online SAR supervision of MRI systems with high channel counts, since for a given number of VOPs the relative SAR overestimation can be lowered.

Noise intensity of a Markov chain.

Stochastic transitions between discrete microscopic states play an important role in many physical and biological systems. Often these transitions lead to fluctuations on a macroscopic scale. A classic example from neuroscience is the stochastic opening and closing of ion channels and the resulting fluctuations in membrane current. When the microscopic transitions are fast, the macroscopic fluctuations are nearly uncorrelated and can be fully characterized by their mean and noise intensity. We show how, for an arbitrary Markov chain, the noise intensity can be determined from an algebraic equation, based on the transition rate matrix; these results are in agreement with earlier results from the theory of zero-frequency noise in quantum mechanical and classical systems. We demonstrate the validity of the theory using an analytically tractable two-state Markovian dichotomous noise, an eight-state model for a calcium channel subunit (De Young-Keizer model), and Markov models of the voltage-gated sodium and potassium channels as they appear in a stochastic version of the Hodgkin-Huxley model.

Steady-state probabilities for Markov jump processes in terms of powers of the transition rate matrix.

Several types of dynamics at stationarity can be described in terms of a Markov jump process among a finite number N of representative sites. Before dealing with the dynamical aspects, one basic problem consists in expressing the a priori steady-state occupation probabilities of the sites. In particular, one wishes to go beyond the mere black-box computational tools and find expressions in which the jump rate constants appear explicitly, therefore allowing for a potential design/control of the network. For strongly connected networks admitting a unique stationary state with all sites populated, here we express the occupation probabilities in terms of a formula that involves powers of the transition rate matrix up to order N - 1. We also provide an expression of the derivatives with respect to the jump rate constants, possibly useful in sensitivity analysis frameworks. Although we refer to dynamics in (bio)chemical networks at thermal equilibrium or under nonequilibrium steady-state conditions, the results are valid for any Markov jump process under the same assumptions.

A Study of Replenishment And Pricing Strategies For Vegetable Commodities Based On Multiple Linear Regression

Vegetables, which can provide the human body with rich nutrients such as vitamins, minerals and dietary fiber, are indispensable necessities in people's daily life. Therefore, on the one hand, in order to ensure the quality of vegetables to provide fresh vegetables, superstores need to replenish goods in a timely manner; on the other hand, in order to improve the revenue, superstores need to reasonable pricing. This paper takes 251 kinds of vegetables sold in fresh food superstores as the research object, firstly, using Pearson's correlation coefficient to explore the correlation between the sales volume of vegetable single product and category, and obtains that the strongest correlation in vegetable category is cauliflower and edible mushrooms, with a correlation coefficient of -0.9989, and that the strongest correlation in vegetable single product is Hericium erinaceus and fresh zongzi leaves, with a correlation coefficient of 0.9973. Then, this paper set up a multivariate linear regression model to solve the problem for each vegetable. Linear regression model to solve the relationship between total sales volume and cost-plus pricing for each category. Finally, for the total daily replenishment, through the total daily replenishment = total predicted sales / (1 - wastage rate) processing to get the total daily replenishment of vegetables for 7 days, to obtain the cost matrix and markup rate matrix, and then get the pricing of vegetables for 7 days. For example, the replenishment of cauliflower on Monday should be 268.7238 kg, and the pricing is $2.5179/kg.

A study on the influence of closure model and numerical technology on the stability of two-fluid model

The influence of closure model and numerical technology on two-fluid model’s stability is studied. A stability analysis method for two-fluid model with interfacial drag term is proposed. The stability depends on the eigenvalue spectral radius of a perturbation growth rate matrix. By analyzing eigenvalue spectral radius, the effect of interfacial drag, cell size, time step and time-smoothing is studied. The main findings are: Besides interfacial drag coefficient’s magnitude, the mathematic form of drag coefficient model also has large effect. Increase of cell size, decrease of time step and time-smoothing do not necessarily lead to stability enhancement. The effectiveness of stability enhancement technology depends on relationship between spectral radius and cell size, time step and time-smoothing coefficient.

Avoiding matrix exponentials for large transition rate matrices.

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge-Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N-reasonable conditions for rate matrices often exponentiated-computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N.

Reliability and Performance Evaluation of Safety-Critical Instrumentation and Control Systems of Nuclear Power Plant

Instrumentation and control systems are nervous systems of nuclear power plant (NPP). These systems interact with several safety-critical components of the NPP, such as actuators, transformers, control valves, sensors, circuit breakers, signal processing units, controllers, and heat exchangers. Therefore, the failure of these systems could result in significant financial loss, harm to human resources, or environmental damage. As a result, these systems need to be highly reliable and accurate. In this article, we suggest a framework, based on the batch deterministic and stochastic Petri nets (BDSPNs) to measure the reliability and performance of safety-critical system. The framework consists of three phases. In the first phase, the system is modeled using the BDSPN to derive the transition rate among the system states. In phase 2, the transition rate matrix is utilized to compute the steady-state probability values, which help to evaluate the response time of the system using Little's law. The third phase uses a transition probability matrix to assess the reliability of the system. The technique is illustrated on shutdown system of NPP and is validated on the operational profile data. The obtained accuracy of 99.9905% in measurement of reliability validates the approach.

A novel bearing current signal diagnosis method combining variational modal decomposition and improved random forests.

A new fault diagnosis approach based on bearing current signals is proposed in this paper. First, in view of strong background noise of the current signal, the variational modal decomposition method is applied to decompose the bearing current signal to obtain multiple intrinsic mode functions, and then the intrinsic mode functions are constructed as the input feature vector according to the kurtosis. Second, to avoid the influence of random forest parameters on the random forest classifier, a random forest faulty bearing diagnostic model optimized by the whale algorithm is established. Finally, the accuracy rate and confusion matrix are adopted to evaluate the prediction effects of both established and traditional models. The classification accuracy of the real damaged bearing fault type can reach 95.11%. The fault diagnosis accuracy of manually damaged bearings can reach 93.83%. The results show that the method proposed in this paper has high accuracy and good generalization ability for bearing fault diagnosis.

Thermodynamic bound on spectral perturbations, with applications to oscillations and relaxation dynamics

In discrete-state Markovian systems, many important properties of correlation functions and relaxation dynamics depend on the spectrum of the rate matrix. Here we demonstrate the existence of a universal trade-off between thermodynamic and spectral properties. We show that the entropy production rate, the fundamental thermodynamic cost of a nonequilibrium steady state, bounds the difference between the eigenvalues of a nonequilibrium rate matrix and a reference equilibrium rate matrix. Using this result, we derive thermodynamic bounds on the spectral gap, which governs autocorrelation times and the speed of relaxation to a steady state. We also derive the thermodynamic bounds on the imaginary eigenvalues, which govern the speed of oscillations. We illustrate our approach using a simple model of biomolecular sensing. Published by the American Physical Society 2024

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the derivative's extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a "blessing of dimensionality" result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology.

Performance approximations of Markov Modulated Poisson Process arrival queue with Markovian Service using matrix geometric approach

Queue analysis of correlated work of the arrival and the service process are not available much in literature. This research paper applies matrix geometric approach to study the MMPP/MSP/1 queuing model under the quasi-birth death(QBD) process. Customers admitted in the system are having two different intensity of arrival followed by Markov Modulated Poisson Process(MMPP). MMPP is a versatile class of MAP and it is necessary to adapt the first characterization of the queuing model. In service mechanism Markov Service Process(MSP) is adopted. As like MMPP, MSP is also a versatile and can capture the correlation of the modulated arrivals. The methodology presented in this research work calculates the metric performances not depending on the series of infinite state of the system. The assumed model is formulated as QBD process to obtain the steady state probability of the system. The complexity model formulated involves structures matrices with appropriate dimension. Neut?s pioneered the matrix geometric approach to such complex models which involves the rate computation of rate matrix R. The rate matrix R is approximated to derive the performance metrics of the system and then compute the cost for the underlying Markov chain. To attain this approximation the sensitivity analysis is done with numerical results.

Study on the response of ecological sensitivity to land use and land cover changes in Jinzhai, China

Changes in land use types are often influenced by human activities, natural factors, and other factors. To explore the changes in land use types, in this paper, we focused on Jinzhai County in Anhui Province. Landsat series data from 2000 to 2020 were utilized as the primary remote sensing data source and combined with land use investigations, and the land use types were classified to better analyze their changes in Jinzhai County from 2000 to 2020. The main calculation indicators, such as the land use change rate and land use transfer matrix, were used to study the proportions of the different land use types during the past two decades, the comprehensive land use dynamics, and the directions of change of the different land use types.

Enhancement of electron transport mobility in GaAs/InGaAs asymmetrically doped narrow quantum well pHEMT structure

We study the effect of asymmetric doping concentrations on the electron mobility μ in GaAs/InGaAs-based single quantum well (SQW) as well as double quantum well (DQW) pseudomorphic high electron mobility transistor (pHEMT) structures. Unequal doping in the substrate and surface barriers (n d1 and n d2 ) causes asymmetric distributions of subband wave functions, ψ 0 and ψ 1, which influence the subband scattering rate matrix elements (SSRME), thereby affecting the subband mobility μ n . For narrow well widths (w w ), in SQW structures, mostly a single subband is occupied. We show that an increase in n d2 , keeping n d1 fixed, enhances μ nonlinearly. The interface roughness (ir-) scattering mostly dominates μ in thin wells (w w < 70 Å), while generally, μ is determined by ionized impurity (ii-) scattering and to some extent by alloy disorder (ad-) scattering. The influence of ir-scattering enhances, while ad-scattering diminishes, by reducing n d2 . For DQW, a double subband is occupied. In a symmetric DQW structure at resonance, n d1 = n d2 , ψ 0 and ψ 1 equally extend into both the wells. For a minor variation, say n d1 > n d2 , ψ 0 mostly lies in one well while ψ 1 is in the other well. In the case of n d1 < n d2 , the distribution reverts. The substantial changes in ψ 0 and ψ 1 influence the intra- and inter-SSRME differently through intersubband effects, leading to nonlinear μ n as a function of n d2 . Taking n d1 + n d2 = 3 × 1018 cm−3, we show that for w w1 = w w2 = 80 Å, a shallow dip in μ occurs at n d1 = n d2 = 1.5 × 1018 cm−3. Whereas, for w w1 = 60 Å and w w2 = 100 Å, the dip in μ occurs near the corresponding resonance, n d1 = 2.3 × 1018 cm−3 and n d2 = 0.7 × 1018 cm−3. Our results of nonlinear μ can be utilized for performance analysis of pHEMT.

Exact confidence intervals for population growth rate, longevity and generation time

By quantifying key life history parameters in populations, such as growth rate, longevity, and generation time, researchers and administrators can obtain valuable insights into its dynamics. Although point estimates of demographic parameters have been available since the inception of demography as a scientific discipline, the construction of confidence intervals has typically relied on approximations through series expansions or computationally intensive techniques. This study introduces the first mathematical expression for calculating confidence intervals for the aforementioned life history traits when individuals are unidentifiable and data are presented as a life table. The key finding is the accurate estimation of the confidence interval for r, the instantaneous growth rate, which is tested using Monte Carlo simulations with four arbitrary discrete distributions. In comparison to the bootstrap method, the proposed interval construction method proves more efficient, particularly for experiments with a total offspring size below 400. We discuss handling cases where data are organized in extended life tables or as a matrix of vital rates. We have developed and provided accompanying code to facilitate these computations.

Monte Carlo Calculation Method for Reactor Period Utilizing the Differential Operator Sampling Technique

The inverse reactor period α is a fundamental mode eigenvalue of the α-mode nonlinear Boltzmann eigenvalue equation that considers delayed neutron contributions. Thus far, several Monte Carlo methods, including the α-k, weight balancing, and transition rate matrix methods, have been developed to calculate α. This study presents a new Monte Carlo method for predicting α by using the derivatives of the k-eigenvalue with respect to α. Formulae are derived to calculate the first and second derivatives using the differential operator sampling method. The key feature of the new proposed method is its ability to estimate the uncertainty of the predicted α by considering the uncertainty of the k-eigenvalue and its derivatives with respect to α.