Probability Matrix Research Articles

Forecasting Using the Markov Chain Transition Matrix for Exchange Rate Fluctuations

The current research is preparing an economic statistical study on exchange rate fluctuations for the period from ( 1996 to mid-2005 ) according to the well-known phrase: (The present is the past and the future is the present ). It is a modest attempt to crystallize this type of price in a scientific-statistic method by adopting the matrix of transitional probabilities in Markov chains, as these chains are among the models that are relied upon in the process of forecasting when the data are in the present time and in three cases: - High - Low - Stability. Between the past, the present and the future, the researcher was able to reach in his research to the analysis by applying the method of the greatest places, and to achieve the main purpose of this research, the researcher dealt with four chapters as follows: The first chapter dealt with the problem of research, its importance, need, purpose and determination of its terms. The second chapter reviewed the previous studies published on the subject, as well as the theoretical aspect, in which the matrix of transitional and stable possibilities, the state of stability and independence of the Markov chains, and the estimation of transitional possibilities in the greatest possible way were presented. As for the third chapter, it included the practical aspect and the field application. The data related to the research procedures were collected from the date of (17/10/2003 to 30/6/2005), in which the Iraqi currency was exchanged from the old to the currency currently in circulation, and in which the situation of the Iraqi market suggested great relative stability at the time. Finally, the fourth chapter of this research included the conclusions reached by the researcher from which it came: The market was characterized by a state of stability after the rise in the exchange rate against the dollar, which confirmed the accuracy of the results of the matrix of transitional possibilities, which was reflected on the local stock exchange, as well as the development of recommendations that can be used by the relevant parties.

Analytic computation for stationary distributions of D-BMAP/D -MSP(a,b)/1 queueing system

ABSTRACT This paper analyses single server correlated batch arrival and correlated batch service queue with innumerable waiting space in discrete-time. The arrival and service processes are governed by discrete batch Markovian arrival process and discrete Markovian service process with general bulk service rule, respectively. Observing two consecutive random epochs, we construct general structure transition probability matrix and it is then reblocked to the desired M/G/1 structure. The UL-type RG-factorization approach combined with spectral expansion method is adopted to unravel the queue length distribution at random epoch. The relationships are undertaken to obtain the queue length distributions at pre-arrival, outside observer's, intermediate and post-departure epochs. Our most challenging effort is to derive the probability mass functions of waiting time and service batch size for arbitrary customer in an arriving batch. We address the warehouse management process for finished products from dealer to retailer as a potential practical use of our batch queue with correlation in both arrival and service, and provide the effect of our queueing behaviour in this application based on our analytical and computational investigation. Based on the implementation of parametrized numeric experiments with numerous categories, we impart our computational results to validate the analytical findings reported in this investigation.

Ergodic theory of diagonal orthogonal covariant quantum channels

We analyse the ergodic properties of quantum channels that are covariant with respect to diagonal orthogonal transformations. We prove that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix. This allows us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels. As an application of our analysis, we study dual unitary brickwork circuits which have recently been proposed as minimal models of quantum chaos in many-body systems. Upon imposing a local diagonal orthogonal invariance symmetry on these circuits, the long-term behaviour of spatio-temporal correlations between local observables in such circuits is completely determined by the ergodic properties of a channel that is covariant under diagonal orthogonal transformations. We utilize this fact to show that such symmetric dual unitary circuits exhibit a rich variety of ergodic behaviours, thus emphasizing their importance.

Cost-effectiveness of iptacopan for paroxysmal nocturnal hemoglobinuria

AbstractIptacopan, a novel oral factor B inhibitor, recently obtained US Food and Drug Administration approval for treating paroxysmal nocturnal hemoglobinuria, a rare blood disorder characterized by persistent complement-mediated hemolytic anemia. The standard-of-care (SOC) has traditionally relied on complement C5 inhibitors eculizumab and ravulizumab, which are limited by persistent anemia from extravascular hemolysis and requirement for intravenous infusion. Recent publication of phase 3 studies in this arena reinforces iptacopan as an effective anticomplement monotherapy compared with SOC. Given ongoing price negotiations and limited literature showing its cost-ineffectiveness in the anti-C5–treated population, we conducted a comprehensive cost-effectiveness analysis of iptacopan monotherapy in anti-C5–treated patients from the societal perspective, as compared with C5 inhibition. The primary outcomes were the incremental net monetary benefit across a lifetime horizon and the cost-effective maximum monthly threshold price of iptacopan monotherapy compared with the SOC. The secondary outcome was time saved for patients and nurses with the use of oral iptacopan therapy. Iptacopan monotherapy and SOC accrued 12.6 and 10.8 quality-adjusted life-years at costs of $9.52 million and $13.5 million, respectively. Iptacopan remained cost saving across extensive sensitivity and all scenario analyses, including alternative parameterization for anemia resolution and aggregated individual-level utilities and transition probability matrix. Across all probabilistic sensitivity analyses, iptacopan therapy was favored over SOC in 100% of 10 000 Monte Carlo iterations. Cost-saving thresholds for iptacopan vs anti-C5 in are ∼1.1, 1.4, and 1.4 in Brazil, Japan, and the United States, respectively. Iptacopan monotherapy can improve quality-adjusted life expectancy for patients while saving health care costs across jurisdictions.

Discrete dynamics in the set of quantum measurements

Abstract A quantum measurement, often referred to as positive operator-valued measurement, is a set of positive operators P j = P j † ⩾ 0 summing to identity, ∑ j P j = 𝟙 . This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe transformations in the set of quantum measurements by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares). Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in the set of quantum measurements.

Forecasting Non-stationary Time Series Using Deep Learning in a Fuzzy Time Series Framework and its Application to Stock Markets

Non-stationary time series prediction is challenging due to its dynamic and complex nature. Fuzzy time series models offer a promising solution for forecasting such data, but a key challenge lies in partitioning the universe of discourse, which significantly impacts forecasting accuracy. Traditional fuzzy time series models often use equal-length interval partitioning, which is more suited for stationary data and limits their adaptability to non-stationary time series. This paper introduces a novel variable-length interval partitioning method designed specifically for non-stationary time series. The developed method combines a Long Short-Term Memory (LSTM) Autoencoder with K-means clustering, enabling dynamic, data-driven partitioning that adapts to the changing characteristics of the data. The LSTM Autoencoder encodes the time series, which is clustered using K-means, and intervals are defined based on cluster centers. Furthermore, the Variable Length Interval Partitioning-based Fuzzy Time Series model (VLIFTS) is developed by incorporating this partitioning method and the concepts of Markov chain and transition probability matrix. In this model, fuzzy sets are viewed as states of a Markov chain, and transition probabilities are used in the forecasting phase. The model is validated on stock market indices Nifty 50, NASDAQ, S&P 500, and Dow Jones. Stationarity and heteroscedasticity are tested using Augmented Dickey-Fuller (ADF) and Levene's tests respectively. Statistical forecast accuracy metrics Root Mean Squared Error (RMSE) and Mean Absolute Percent Error (MAPE) show that VLIFTS significantly improves forecasting accuracy over traditional models. This hybrid approach enhances fuzzy time series modelling and can be applied to various non-stationary time series forecasting problems.

A Distributed Non-Intrusive Load Monitoring Method Using Karhunen–Loeve Feature Extraction and an Improved Deep Dictionary

In recent years, the non-invasive load monitoring (NILM) method based on sparse coding has shown promising research prospects. This type of method learns a sparse dictionary for each monitoring target device, and it expresses load decomposition as a problem of signal reconstruction using dictionaries and sparse vectors. The existing NILM methods based on sparse coding have problems such as inability to be applied to multi-state and time-varying devices, single-load characteristics, and poor recognition ability for similar devices in distributed manners. Using the analysis above, this paper focuses on devices with similar features in households and proposes a distributed non-invasive load monitoring method using Karhunen–Loeve (KL) feature extraction and an improved deep dictionary. Firstly, Karhunen–Loeve expansion (KLE) is used to perform subspace expansion on the power waveform of the target device, and a new load feature is extracted by combining singular value decomposition (SVD) dimensionality reduction. Afterwards, the states of all the target devices are modeled as super states, and an improved deep dictionary based on the distance separability measure function (DSM-DDL) is learned for each super state. Among them, the state transition probability matrix and observation probability matrix in the hidden Markov model (HMM) are introduced as the basis for selecting the dictionary order during load decomposition. The KL feature matrix of power observation values and improved depth dictionary are used to discriminate the current super state based on the minimum reconstruction error criterion. The test results based on the UK-DALE dataset show that the KL feature matrix can effectively reduce the load similarity of devices. Combined with DSM-DDL, KL has a certain information acquisition ability and acceptable computational complexity, which can effectively improve the load decomposition accuracy of similar devices, quickly and accurately estimating the working status and power demand of household appliances.

A Stochastic Dynamic Operator framework that improves the precision of analysis and prediction relative to the classical spike-triggered average method, extending the toolkit.

Here we test the Stochastic Dynamic Operator (SDO) as a new framework for describing physiological signal dynamics relative to spiking or stimulus events. The SDO is a natural extension of existing spike-triggered average (STA) or stimulus-triggered average techniques currently used in neural analysis. It extends the classic STA to cover state-dependent and probabilistic responses where STA may fail. In simulated data, SDO methods were more sensitive and specific than the STA for identifying state-dependent relationships. We have tested SDO analysis for interactions between electrophysiological recordings of spinal interneurons, single motor units, and aggregate muscle electromyograms (EMG) of major muscles in the spinal frog hindlimb. When predicting target signal behavior relative to spiking events, the SDO framework outperformed or matched classical spike-triggered averaging methods. SDO analysis permits more complicated spike-signal relationships to be captured, analyzed, and interpreted visually and intuitively. SDO methods can be applied at different scales of interest where spike-triggered averaging methods are currently employed, and beyond, from single neurons to gross motor behaviors. SDOs may be readily generated and analyzed using the provided SDO Analysis Toolkit We anticipate this method will be broadly useful for describing dynamical signal behavior and uncovering state-dependent relationships of stochastic signals relative to discrete event times.Significance Statement Here the authors introduce new tools and demonstrate data analysis using a new probabilistic and state-dependent technique, which is an expansion and extension of the classical spike-triggered average, the Stochastic Dynamic Operator. Stochastic Dynamic Operator methods extend application into domains where classical spike-triggered averages fail, capture more information on spike correlations, and match or outperform the spike-triggered average when generating predictions of signal amplitude based on spiking events. A data and code package toolkit for utilizing and interpreting Stochastic Dynamic Operator methods is provided together with example simulated and physiological data analyses. Both the method and the associated toolkit are expected to be broadly useful in research domains where the spike triggered average is currently used for analysis, and beyond.

Количественная оценка степени сформированности навыков командной деятельности

<p>The article presents a method for quantifying teamwork skills formation using a triangular scheme and mathematical models based on transition probability matrices. The study focuses on analyzing individual and jointly coordinated activities of participants under experimental conditions, involving working on a simulator for jointly controlling an aircraft. Markov chains are used to analyze the results, allowing for detailed analysis of task completion dynamics and identification of key aspects of participant interaction. The findings confirm the effectiveness of the proposed approach to analyzing and improving team coordination.</p>

Powers of Karpelevič arcs and their sparsest realising matrices

The region in the complex plane containing the eigenvalues of all n×n stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order n to be a power of another stochastic matrix.

Developing high-precision battery electric forklift driving cycle with variable cargo weight

Driving cycles are essential for assessing vehicle energy demand, estimating driving range, and evaluating environmental impacts. Numerous driving cycles have been developed for passenger cars and buses. However, tailored driving cycles for logistics vehicles, especially forklifts, remains limited. Therefore, we introduce high-precision driving cycles for battery electric forklifts, which include profiles of velocity and cargo mass. The construction of driving cycles involves route selection, data acquisition, micro-trip segmentation, characteristic parameters selection, driving pattern categorization, transition probability matrix development, and driving cycle construction and evaluation. The methods proposed for constructing driving cycles are based on Markov Chain, Micro-trips combinations, and genetic algorithms. The constructed driving cycles are evaluated using relative error analysis and a simulation model. The results confirm that these cycles accurately reflect actual forklift operations and can be utilized to estimate their energy consumption.

Event-Triggered Reduced-Order Filtering for Continuous Semi-Markov Jump Systems With Imperfect Measurements.

This article conducts the issue of event-triggered reduced-order filtering for continuous-time semi-Markov jump systems with imperfect measurements as well as randomly occurring uncertainties (ROUs). Specifically, the sojourn-time-dependent transition probability matrix (TPM) is presumed to be polytopic and a quantizer is introduced to quantize output signals aiming to reflect the reality. Both ROUs and sensor failures are generated by individual random variables belonging to be mutually independent Bernoulli-distributed white sequences. First, sufficient conditions for the existence of the event-triggered reduced-order filter are obtained by utilizing the dissipativity-based technique to ensure the asymptotical stability with a strictly dissipative performance of the filtering error system. The time-varying TPM is then fractionalized, which enhances the results as stated. Furthermore, the required reduced-order filter parameters are obtained by introducing slack symmetric matrix as well as cone complementarity linearization algorithm. The effectiveness of the suggested event-triggered reduced-order filter design method is shown through simulation results.

On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for k∈{2,3,4,5}. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.

The usage of stochastic matrices while learning the topic “Eigenvalues and eigenvectors of a matrix” in the course of Higher Mathematics

Abstract The article looks into teaching students mathematical modeling during the mastering of the topic “Eigenvalues and eigenvectors of a matrix” in the Linear Algebra. The modeling is based on the usage of matrices built based on Markov chains. The analysis of the researchers’ papers shows that the usage of matrices of this type can be implemented while teaching students of different university specialties. The analysis of the prerequisites for the involvement of stochastic matrices in the consideration of students has helped to determine additional concepts and theorems that should be considered while learning “Eigenvalues and eigenvectors of a matrix”. The theorem on the presence of stochastic vectors in a stochastic matrix can serve as a formula for systematizing problems that can present applications of eigenvalues and eigenvectors of the matrix in real-life situations. The results of the experiment conducted using the developed system of problems for students of different University specialties show the improvement of skills provided by mastering the topic “Eigenvalues and eigenvectors of a matrix”.

Generalized matrix functions and some refinement inequalities

In this short paper, we provide some refinement inequalities on generalized matrix functions. In particular, permanent inequalities concerning doubly stochastic positive semidefinite matrices are also included.

Event-Based Asynchronous H∞ Control for Nonhomogeneous Markov Jump Systems With Imperfect Transition Probabilities.

The event-based H∞ control problem is investigated for a class of nonhomogeneous Markov jump systems (MJSs) with partially unknown transition probabilities (TPs). The MJS is characterized by a piecewise nonhomogeneous Markovian chain, where the switching of the system TP matrix is governed by a higher-level chain. A hidden Markov model (HMM) is employed to observe the system mode, which cannot always be correctly detected in practice. Under this framework, the partially unknown TPs existing in both higher-level TPs (HTPs) and conditional TPs (CTPs) are taken into account for practical consideration. Additionally, an observed-mode-dependent event-triggered mechanism (ETM) is employed to design an asynchronous controller, which is expected to alleviate the burden of the communication network. Evidently, the considered scenario is fairly general and covers some special cases. With the above consideration, sufficient conditions are established to guarantee stochastic stability of the resulting closed-loop system with a prescribed H∞ performance. Finally, two examples are presented to demonstrate the effectiveness and applicability of the proposed method.

On Extended Class of Totally Ordered Interval-Valued Convex Stochastic Processes and Applications

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping γ, and center-radius ordering relations to establish a class of extended cr-interval-valued convex stochastic processes. This class yields a combination of innovative convex and non-convex stochastic processes. We characterize our class by illustrating its relationships with other classes as well as certain key attributes and sufficient conditions for this class of processes. Additionally, leveraging Riemann–Liouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensen’s inequality, Hermite–Hadamard’s inequality, Fejer’s inequality, and product Hermite–Hadamard’s like inequality. We establish an interesting relation between means by means of Hermite–Hadamard’s inequality. We utilize the numerical and graphical approaches to showcase the significance and effectiveness of primary findings. Also, the proposed results are powerful tools to evaluate the bounds for stochastic Riemann–Liouville fractional operators in different scenarios for a larger space of processes.

Seismic vulnerability analysis of hospitals and school buildings considering the Gaussian regression algorithm

This study combines the Gaussian regression algorithm to propose a nonlinear regression model that can be used to evaluate the seismic vulnerability of regional hospitals and school buildings. The failure features and disaster mechanisms of hospital and school buildings affected by the Jishishan (JSS) earthquake on December 18, 2023, and the Wenchuan (WC) earthquake on May 12, 2008, were reported. The samples of 252 damaged buildings (37 hospitals and 215 schools) in Dujiangyan city affected by the Wenchuan earthquake for which the author participated in the survey were collected and counted. Hospitals and school buildings were classified according to the types of reinforced concrete (RC) and masonry structures, and earthquake vulnerability probability matrices for hospital and school building clusters were established considering the actual failure probabilities. A total of 2103,213 acceleration records (from the Wenchuan earthquake) monitored by 12 real stations were selected. Using dynamic time history and response spectrum analysis methods, time history and spectrum curves were generated, considering the influence of different seismic directions. Using the developed Gaussian regression model, vulnerability comparison curves and parameter matrices considering multiple regression algorithms were generated. This paper considers the effects of the construction year and number of floors on the seismic vulnerability of hospital and school buildings. Seismic vulnerability curves and failure probability matrices were generated and established on the basis of actual structural failure probabilities. According to regression and vulnerability surface analysis, the evaluation results of the proposed model are highly consistent with actual field observations.

The Geometry of Dynamic Time-Dependent Best–Worst Choice Pairs

There has been increasing interest in best–worst discrete choice experiments (BWDCEs) in health economics, transportation research, and other fields over the last few years. BWDCEs have distinct advantages compared to other measurement approaches in discrete choice experiments (DCEs). A systematic study of best–worst (BW) choice pairs can be traced back to the 1990s. Recently, new ideas have been introduced to the subject. Calculating utility helps measure the attractiveness of BW choices. The goal of this paper is twofold. First, we extend the idea of the BW choice pair to include dynamic, time-dependent transition probability and capture utility at each time and for each choice pair. Second, we used the geometry of BW choice pairs to capture the correlations among them and to characterize and clarify the BW choice pairs in the network, where properties can be derived within each class. This paper discusses BWDCEs, the probability transition matrix of choices over time, and the utility function. The proposed network classification for BW choice pairs is laid out. A detailed simulated example is presented, and the results are compared with the classical K-means classification.

Decomposition and graphical correspondence analysis of checkerboard copulas

Abstract We analyze optimal low-rank approximations and correspondence analysis of the dependence structure given by arbitrary bivariate checkerboard copulas. Methodologically, we make use of the truncation of singular value decompositions of doubly stochastic matrices representing the copulas. The resulting (truncated) representations of the dependence structures are sparse, in particular, compared to the number of squares on the checkerboard. The additive structure of the decomposition carries through to statistical functionals of the copula, such as Kendall’s τ \tau or Spearman’s ρ \rho , and also motivates similarity measures for checkerboard copulas. We link our analysis to continuous decompositions of copula densities and copula-generating algorithms and discuss further general properties of the decomposition and its truncation. For example, truncated series might lack nonnegativity, and approximation errors increase for monotonicity-like copulas. We provide algorithms and extensions that account for and counteract these properties. The low-rank representation is illustrated for various copula examples, and some analytical results are derived. The resulting correspondence analysis profile plots are analyzed, providing graphical insights into the dependence structure implied by the copula. An illustration is provided with an empirical data set on fuel injector spray characteristics in jet engines.