Non-homogeneous Markov Chain Model Research Articles

Non-homogeneous continuous time Markov chain model for information dissemination on Indonesian Twitter users

Nonhomogeneous Continuous-Time Markov Chain (NH-CTMC) is a stochastic process that can be used to model problems where the future state depends only on the current state and is independent of the past. The transition intensity in NH-CTMC is not constant but is a function of time. In this paper, NH-CTMC is employed to model information dissemination on Twitter, where transitions occur only from followee groups to follower groups. Information is considered spread on Twitter when followers retweet posts from their followees. The tweet-retweet process on Twitter satisfies the Markov property, as a retweet from a follower depends only on the tweet posted just before by the corresponding followee. The probability of a tweet spreading is determined by the transition intensity, assumed to be a Sigmoid function whose parameters are estimated using Maximum Likelihood Estimation (MLE). This method is applied to Twitter data from Indonesia related to discussions on Covid-19 vaccination. The results indicate that information about Covid-19 vaccination on Twitter spreads rapidly from followees to followers in the first 20 hours, and then slows down after 40 hours. The NH-CTMC model outperforms the Homogeneous Continuous-Time Markov Chain (H-CTMC) approach, where the transition intensity (tweet spreading intensity) is assumed to be constant.

An exponentially weighted moving average control chart based on signed ranks for finite horizon processes

Distribution-free control charts are an efficient quality monitoring tool to inspect lots of parts manufactured within a finite production horizon. In this work, the performance of the Exponentially Weighted Moving Average chart based on the Wilcoxon signed rank statistic is investigated for on-line monitoring of finite production runs. The chart’s on-target performance is evaluated through a specific non-homogeneous Markov chain model under different process scenarios. A numerical analysis is conducted for determining its optimal design and a performance comparison with other available schemes is presented for different symmetric distributions of observations. Finally, an illustrative example is presented to show a practical implementation of the investigated chart.

Calibration of Transition Intensities for a Multistate Model: Application to Long-Term Care

We consider a non-homogeneous continuous time Markov chain model for Long-Term Care with five states: the autonomous state, three dependent states of light, moderate and severe dependence levels and the death state. For a general approach, we allow for non null intensities for all the returns from higher dependence levels to all lesser dependencies in the multi-state model. Using data from the 2015 Portuguese National Network of Continuous Care database, as the main research contribution of this paper, we propose a method to calibrate transition intensities with the one step transition probabilities estimated from data. This allows us to use non-homogeneous continuous time Markov chains for modeling Long-Term Care. We solve numerically the Kolmogorov forward differential equations in order to obtain continuous time transition probabilities. We assess the quality of the calibration using the Portuguese life expectancies. Based on reasonable monthly costs for each dependence state we compute, by Monte Carlo simulation, trajectories of the Markov chain process and derive relevant information for model validation and premium calculation.

Detection of Factors Affecting State Transition Based on Non-Homogeneous Markov Chain Model

The dependent and independent variables in traditional linear regression models are continuous numerical variables. When the dependent variable or independent variable is a discrete variable, the traditional linear regression model can no longer be used to analyze. To solve this problem, this article introduces the non-homogeneous Markov chain model. It introduces the mathematical definition of the non-homogeneous Markov chain model. And then this article uses Bayesian estimation method to derive posterior distribution of model parameters. Through the MCMC algorithm, we simulate an experiment, posterior means value of the parameters is estimated, and the estimation effect is found to be better. Finally, we analyze the impact of learning state transition about college students on the non-homogeneous Markov chain model. Influencing factors include whether to receive a scholarship and whether to serve as a class leader. In this paper, non-homogeneous Markov chain model is used to analyze and detect the impact of discrete variables on dependent variables. This is the major innovation in this article.

Risk assessment for financial accounting: modeling probability of default

Purpose This paper aims to introduce and tests models for point-in-time probability of default (PD) term structures as required by international accounting standards. Corresponding accounting standards prescribe that expected credit losses (ECLs) be recognized for the impairment of financial instruments, for which the probability of default strongly embodies the included default risk. This paper fills the research gap resulting from a lack of models that expand upon existing risk management techniques, link PD term structures of different risk classes and are compliant with accounting standards, e.g. offering the flexibility for business cycle-related variations. Design/methodology/approach The author modifies the non-homogeneous continuous-time Markov chain model (NHCTMCM) by Bluhm and Overbeck (2007a, 2007b) and introduces the generalized through-the-cycle model (GTTCM), which generalizes the homogeneous Markov chain approach to a point-in-time model. As part of the overall ECL estimation, an empirical study using Standard and Poor’s (S&P) transition data compares the performance of these models using the mean squared error. Findings The models can reflect observed PD term structures associated with different time periods. The modified NHCTMCM performs best at the expense of higher complexity and only its cumulative PD term structures can be transferred to valid ECL-relevant unconditional PD term structures. For direct calibration to these unconditional PD term structures, the GTTCM is only slightly worse. Moreover, it requires only half of the number of parameters that its competitor does. Both models are useful additions to the implementation of accounting regulations. Research limitations/implications The tests are only carried out for 15-year samples within a 35-year span of available S&P transition data. Furthermore, a point-in-time forecast of the PD term structure requires a link to the business cycle, which seems difficult to find, but is in principle necessary corresponding to the accounting requirements. Practical implications Research findings are useful for practitioners, who apply and develop the ECL models of financial accounting. Originality/value The innovative models expand upon the existing methodologies for assessing financial risks, motivated by the practical requirements of new financial accounting standards.

An adaptive Markov Chain approach for probabilistic forecasting of categorical events

Abstract Categorical probabilistic prediction is widely used for terrestrial and space weather forecasting as well as for other environmental forecasts. One example is a warning system for geomagnetic disturbances caused by space weather, which are often classified on a 10-level scale. The simplest approach assumes that the transition probabilities are stationary in time – the Homogeneous Markov Chain (HMC). We extend this approach by developing a flexible Non-Homogeneous Markov Chain (NHMC) model using Bayesian non-parametric estimation to describe the time-varying transition probabilities. The transition probabilities are updated using a modified Bayes’ rule that gradually forgets transitions in the distant past, with a tunable memory parameter. The approaches were tested by making daily geomagnetic state forecasts at lead times of 1-4 days and verified over the period 2000-2019 using the Rank Probability Score (RPS). Both HMC and NHMC models were found to be skilful at all lead times when compared with climatological forecasts. The NHMC forecasts with an optimal memory parameter of ~100 days were found to be substantially more skilful than the HMC forecasts, with an RPS skill for the NHMC of 10.5% and 5.6% for lead times of 1 and 4 days ahead, respectively. The NHMC is thus a viable alternative approach for forecasting geomagnetic disturbances, and could provide a new benchmark for producing operational forecasts. The approach is generic and applicable to other forecasts including discrete weather regimes or hydrological conditions, e.g. wet and dry days.

Degradation Dynamics Cognition and Prediction of Li-Ion Battery: An Integrated Methodology for Alleviating Range Anxiety

We report an integrated methodology (FMC-XGBoost) that mainly consists of a five-state nonhomogeneous Markov chain model (FMC) and XGBoost model. Unlike those existing methods in which capacity fading processes are assumed to be irreversible, the proposed integrated methodology can combine user-specific driving patterns (UDP) and capacity recovery effects (CRE) to predict battery fading dynamics even with partially available data for an individual battery. The parameters of the constructed FMC model are linked to the known physicochemical and material properties of Li-ion battery fading dynamics, which aims to cognize and predict the primary fading dynamics, and the proposed XGBoost model is to cognizes and predicts the fluctuation dynamics regarding UDP & CRE. To comprehensively verify the capabilities of the proposed integrated methodology, a series of cases and comparisons are conducted and analysed based on partial available fading data by selecting batteries to simulate situations of individual differences and different UDPs & CREs. The averages of MAE, MRE and RMSE are approximately 0.0128, 0.9251%, and 0.0153 respectively even when only 60% of the data are available. All verifications and comparison analyses reveal that the proposed integrated methodology provides an accurate, robust, stable, and general way to cognize and predict battery fading dynamics during usage, and subsequently to alleviate range anxiety for batteries in real applications.

Monthly precipitation modeling using Bayesian Non-homogeneous Hidden Markov Chain

Abstract Monthly precipitation modeling is important in various applications, e.g. streamflow forecasts and water resources management. This paper develops an operational precipitation forecasting scheme, using Bayesian Non-homogeneous Hidden Markov Chain (NHMM) model and teleconnection index. Although the Hidden Markov Chain model has been investigated before in similar studies, the NHMM algorithm employed in this study allows modeling both non-stationary transition probabilities and emission matrix. Climatic teleconnection that affect precipitation is used to drive changes in transition probabilities of different states in the Markov model. The proposed framework is illustrated for multiple-station precipitation analysis in NingXiang County, a southern inland area in China with a high population density. A simulation model is constructed to examine the model's capacity in capturing variabilities and temporal-spatial characteristics exhibiting in monthly precipitation data during 1961–2013. Results indicate that the proposed NHMM model captures the precipitation characteristics at different stations well. Spearman correlation between conditional mean of simulated ensembles and observed data is 0.87–0.9, with few variations at distinct stations. The proposed framework has general applications and can be applied to simulate and generate stochastic monthly precipitation. Further application of the method is also described in the paper.

An EWMA sign control chart with varying control limits for finite horizon processes

AbstractThe sign SN distribution‐free control charts are efficient and simple online tools to perform quality monitoring when the normality of observations cannot be assumed a priori. The possibility of designing an SN control chart whichever is the distribution of observations allows it to be implemented with success in processes with finite production horizon (FPH). Today, FPH processes are widespread in industry. Recently, Shewhart SN control charts have been investigated in literature for this kind of quality control problem. In this paper, we investigate the implementation of an exponentially weighted moving average (EWMA) SN control chart for FPH processes. The property to have varying control limits at each inspection allows the EWMA SN control chart to be very sensitive to shifts in the location. A nonhomogeneous Markov chain model is used to evaluate the statistical properties of the EWMA SN control chart with varying control limits. An illustrative example shows the implementation of the EWMA SN control chart with observations collected from a quality monitoring procedure in a company bottling soft drinks.

Non-Homogeneous Markov Wind Speed Time Series Model Considering Daily and Seasonal Variation Characteristics

Wind speed model lays the foundation of wind power simulation and is crucial to the analysis of wind power integrated into power systems. This paper proposes a non-homogeneous Markov chain (NHMC) wind speed model that takes the daily and seasonal characteristics of wind speed variation into account. An optimal partition method is adopted to divide the wind speed time series into several segments affected by seasonal changes. A seasonal index is introduced before modeling to reduce the impact of seasonal variation. A time-related variable is also introduced to describe the daily periodic variation of wind speed. Evaluation on the probability distribution, autocorrelation function, and power spectral density of NHMC model and commonly used wind speed models is conducted. Moreover, the number of NHMC states on model performance is investigated. Simulation results demonstrate that the proposed approach offers excellent fitness on overall statistical properties of the wind speed time series.

Li-ion battery capacity cycling fading dynamics cognition: A stochastic approach

Li-ion batteries have been commercially used for many years in small portable devices and electrical vehicles. Here, a five-state nonhomogeneous Markov chain model is introduced, which can assist to saving time and costs incurred by large amounts of cycling life tests with various cell formulations in battery design stage. The model is designed to have five states that belong to three phases: storage phase, active phase, and absorbing phase. The storage phase has one storage state that could be transformed into the third active state; the active phase is comprised of a stable state, an inherent unstable state, and a state that is transformed from the storage phase; the absorbing phase, which is converted from the active phase. The verification results suggest that the proposed model provides an accurate and effective way to cognizing the capacity cycling fading dynamics of various Li-ion batteries with different anode materials even under different working conditions (The cognition accuracy of R-Square can reach 0.999). Furthermore, this method would be a promising way to evaluate the features and performance of Li-ion batteries made of different formulations in the design stage, which could provide valuable information for battery manufacturers to accelerate battery design process.

Lithium-ion battery capacity fading dynamics modelling for formulation optimization: A stochastic approach to accelerate the design process

A five-state nonhomogeneous Markov chain model, which is an effective and promising way to accelerate the Li-ion battery design process by investigating the capacity fading dynamics of different formulations during the battery design phase, is reported. The parameters of this model are linked to known physicochemical degradation dynamics and material properties. Herein, the states and behaviors of the active materials in Li-ion batteries are modelled. To verify the efficiency of the proposed model, a dataset from approximately 3years of cycling capacity fading experiments of various formulations using several different materials provided by Contemporary Amperex Technology Limited (CATL), as well as a NASA dataset, are employed. The capabilities of the proposed model for different amounts (50%, 70%, and 90%) of available experimental capacity data are tested and analyzed to assist with the final design determination for manufacturers. The average relative errors of life cycling prediction acquired from these tests are less than 2.4%, 0.8%, and 0.3%, even when only 50%, 70%, and 90% of the data, respectively, is available for different anode materials, electrolyte materials, and individual batteries. Furthermore, the variance is 0.518% when only 50% of the data are available; i.e., one can save at least 50% of the total experimental time and cost with an accuracy greater than 97% in the design phase, which demonstrates an effective and promising way to accelerate the Li-ion battery design process. The qualitative and quantitative analyses conducted in this study suggest that the proposed model provides an accurate, robust, and simple way to accelerate the Li-ion battery design process for battery manufacturers, thereby enabling rapid market capture.

Hyperspectral Band Selection From Statistical Wavelet Models

High spectral resolution brings hyperspectral images with large amounts of information, which makes these images more useful in many applications than images obtained from traditional multispectral scanners with low spectral resolution. However, the high data dimensionality of hyperspectral images increases the burden on data computation, storage, and transmission; fortunately, the high redundancy in the spectral domain allows for significant dimensionality reduction. Band selection provides a simple dimensionality reduction scheme by discarding bands that are highly redundant, thereby preserving the structure of the data set. This paper proposes a new criterion for pointwise-ranking-based band selection that uses a nonhomogeneous hidden Markov chain (NHMC) model for redundant wavelet coefficients of each hyperspectral signature. The model provides a binary multiscale label that encodes semantic features that are useful to discriminate spectral types. A band ranking score considers the average correlation among the average NHMC labels for each band. We also test richer discrete-valued label vectors that provide a more finely grained quantization of spectral fluctuations. In addition, since band selection methods based on band ranking often ignore correlations in selected bands, we study the effect of redundancy elimination, applied on the selected features, on the performance of an example classification problem. Our experimental results also include an optional redundancy elimination step and test their effect on classification performance that is based on the selected bands. The experimental results also include a comparison with several relevant supervised band selection techniques.

Semisupervised Endmember Identification in Nonlinear Spectral Mixtures via Semantic Representation

This paper proposes a new hyperspectral unmixing method for nonlinearly mixed hyperspectral data using a semantic representation in a semi-supervised fashion, assuming the availability of a spectral reference library. Existing semi-supervised unmixing algorithms select members from an endmember library that are present at each of the pixels; most such methods assume a linear mixing model. However, those methods will fail in the presence of nonlinear mixing among the observed spectra. To address this issue, we develop an endmember selection method using a recently proposed semantic spectral representation obtained via non-homogeneous hidden Markov chain (NHMC) model for a wavelet transform of the spectra. The semantic representation can encode spectrally discriminative features for any observed spectrum and, therefore, our proposed method can perform endmember selection without any assumption on the mixing model. Experimental results show that in the presence of sufficiently nonlinear mixing our proposed method outperforms dictionary-based sparse unmixing approaches based on linear models.

Performance evaluation of corrosion-affected reinforced concrete bridge girders using Markov chains with fuzzy states

A methodology for performance evaluation of reinforced concrete bridge girders in corrosive environments is proposed. The methodology uses the concept of performability and considers both serviceability- and ultimate-limit states. The serviceability limit states are defined based on the degree of cracking (characterized by crack width) in the girder due to chloride induced corrosion of reinforcement, and the ultimate limit states are defined based on the flexural load carrying capacity of the girder (characterized in terms of rating factor using the load and resistance factor rating method). The condition of the bridge girder is specified by the assignment of a condition state from a set of predefined condition states. Generally, the classification of condition states is linguistic, while the condition states are considered to be mutually exclusive and collectively exhaustive. In the present study, the condition states of the bridge girder are also represented by fuzzy sets to consider the ambiguities arising due to the linguistic classification of condition states. A non-homogeneous Markov chain (MC) model is used for modeling the condition state evolution of the bridge girder with time. The usefulness of the proposed methodology is demonstrated through a case study of a severely distressed beam of the Rocky Point Viaduct. The results obtained using the proposed approach are compared with those obtained using conventional MC model. It is noted that the use of MC with fuzzy states leads to conservative decision making for the problem considered in the case study.

Discrete-Time Markov Chain Model for Transport of Mixed-Size Sediment Particles under Unsteady Flow Conditions

AbstractA nonhomogeneous discrete-time three-state Markov chain model is developed in this study to quantify the bedload and suspended load discharge under unsteady flow for mixed size sediment particles. When flow is subject to sudden changes, the particle holding time, defined as the amount of time for a sediment particle staying on the bed or in the moving state, needs to be carefully evaluated. The time step used in this study for single-step motion in the discrete-time Markov chain is represented by a characteristic timescale for particle motion. The transition probabilities are functions of flow conditions and particle properties. Specifically, the likelihood of particle movement between the bedload layer and the bed surface is evaluated by the entrainment probability. Exchange of sediment particles between the bedload layer and suspended load layer is quantified by the suspension probability. A nonhomogeneous Markov chain ensures the transition probabilities are time dependent as they are a functio...

Short-term droughts forecast using Markov chain model in Victoria, Australia

A comprehensive risk management strategy for dealing with drought should include both short-term and long-term planning. The objective of this paper is to present an early warning method to forecast drought using the Standardised Precipitation Index (SPI) and a non-homogeneous Markov chain model. A model such as this is useful for short-term planning. The developed method has been used to forecast droughts at a number of meteorological monitoring stations that have been regionalised into six (6) homogenous clusters with similar drought characteristics based on SPI. The non-homogeneous Markov chain model was used to estimate drought probabilities and drought predictions up to 3 months ahead. The drought severity classes defined using the SPI were computed at a 12-month time scale. The drought probabilities and the predictions were computed for six clusters that depict similar drought characteristics in Victoria, Australia. Overall, the drought severity class predicted was quite similar for all the clusters, with the non-drought class probabilities ranging from 49 to 57 %. For all clusters, the near normal class had a probability of occurrence varying from 27 to 38 %. For the more moderate and severe classes, the probabilities ranged from 2 to 13 % and 3 to 1 %, respectively. The developed model predicted drought situations 1 month ahead reasonably well. However, 2 and 3 months ahead predictions should be used with caution until the models are developed further.

Paleo-reconstructed net basin supply scenarios and their effect on lake levels in the upper great lakes

Paleo-reconstructed hydrologic records offer the potential to evaluate water resources system performance under conditions that may be more extreme than seen in the historical record. This study uses a stochastic simulation framework consisting of a non-homogeneous Markov chain model (NHMM) to simulate the climate state using Palmer Drought Severity Index (PDSI)-reconstructed data, and K-nearest neighbor (K-NN) to resample observational net basin supply magnitudes for the Great Lakes of North America. The method was applied to generate 500 plausible simulations, each with 100 years of monthly net basin supply for the Upper Great Lakes, to place the observed data into a longer temporal context. The range of net basin supply sequences represents what may have occurred in the past 1,000 years and which can occur in future. The approach was used in evaluation of operational plans for regulation of Lake Superior outflows with implications for lake levels of Superior, Michigan, Huron and Erie, and their interconnecting rivers. The simulations generally preserved the statistics of the observed record while providing new variability statistics. The framework produced a variety of high and low net basin supply sequences that provide a broader estimate of the likelihood of extreme lake levels and their persistence than with the historical record. The method does not rely on parametrically generated net basin supply values unlike parametric stochastic simulation techniques, yet still generates new variability through the incorporation of the paleo-record. The process described here generated new scenarios that are plausible based on the paleo and historic record. The evaluation of Upper Great Lakes regulation plans, subject to these scenarios, was used to evaluate robustness of the regulation plans. While the uncertain future climate cannot be predicted, one can evaluate system performance on a wide range of plausible climate scenarios.

Markov chain applications in modelling facility condition deterioration

Condition states of civil infrastructure such as pavements and bridges are usually indexed on discrete scales. Historical condition data is modelled using Markov chain to estimate transition probabilities from one condition state to another, the rate of change and the time spent in any given state. The usefulness of such models is a function of the completeness of the available records and underlying assumptions of homogeneity. However, complete sets of condition data are not always easily available. In addition, the transition probabilities between states are assumed to be homogeneous, even though they tend not to be. Therefore, the objective of this study is two-fold: First, to maximise the usage of limited available data in estimating transition probabilities between condition states; and second to assess the sensitivity of model predictions to variations in transition probabilities between condition states. The paper presents a novel method to estimate transition probabilities based on the simulation of long term behaviour of a Markov chain model. Next, building on existing research, a Monte Carlo simulation of a non-homogeneous Markov chain model is used to explicitly consider heterogeneity in transition probabilities.

Identification of a Multistate Continuous-Time Nonhomogeneous Markov Chain Model for Patients with Decreased Renal Function

Markov chain models are frequently used to study the clinical course of chronic diseases. The aim of this article is to adopt statistical methods to describe the time dynamics of chronically ill patients when 2 kinds of data sets--fully and partially observable data are available. We propose a 6-state continuous-time Markov chain model for the progression of chronic kidney disease (CKD), where little is known about the transitions between the disease stages. States 1 to 3 of the model correspond to stages III to V of chronic kidney disease in the Kidney Disease Outcomes Quality Initiative (KDOQI) CKD classification. States 4 and 5 relate to dialysis and transplantation (renal replacement therapy), respectively. Death is the (absorbing) state 6. The model can be investigated and identified using Kolmogorov's forward equations and the methods of survival analysis. Age dependency, covariates in the form of the Cox regression, and unobservable risks of transition (frailties) can be included in the model. We applied our model to a data set consisting of all 2097 patients from all renal centers in a region in North Rhine-Westphalia (Germany) in 2005-2010. We compared transitions and relative risks to the few data published and found them to be reasonable. For example, patients with diabetes had a significantly higher risk for disease progression compared with patients without diabetes. In summary, modeling may help to quantify disease progression and its predictors when only partially observable prospective data are available.