Order Markov Chain Research Articles

On the Efficiency of the Maximum Likelihood and Maximum Quasi-Likelihood Estimators in the Second Order Markov Chains

On the Efficiency of the Maximum Likelihood and Maximum Quasi-Likelihood Estimators in the Second Order Markov Chains

The tensor splitting with application to solve multi-linear systems

In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong M -tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M -tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms.

Statistical analysis of high-order Markov dependencies

The paper deals with parsimonious models of integer valued time series. Such models are special cases of high-order Markov chain with a small number of parameters. Two new parsimonious models are presented. The first is Markov chain of order s with r partial connections, and the second model is called Markov chain of conditional order. Theoretical results on probabilistic properties and statistical inferences for these models are given.

Probabilistic Forecast of PV Power Generation Based on Higher Order Markov Chain

This paper presents a method to forecast the probability distribution function (PDF) of the generated power of PV systems based on the higher order Markov chain (HMC). Since the output power of the PV system is highly influenced by ambient temperature and solar irradiance, they are used as important features to classify different operating conditions of the PV system. The classification procedure is carried out by applying the pattern discovery method on the historical data of the mentioned variables. An HMC is developed based on the categorized historical data of PV power in each operating point. The 15-min ahead PDF of the PV output power is forecasted through the Gaussian mixture method (GMM) by combining several distribution functions and by using the coefficients defined based on parameters of the HMC-based model. In order to verify the proposed method, the genetic algorithm is applied to minimize a well-defined objective function to achieve the optimal GMM coefficients. Numerical tests using real data demonstrate that the forecast results follow the real probability distribution of the PV power well under different weather conditions.

Combining 3D digitizing data and architectural modelling to simulate shoot growth and geometry in apple

Various models have been developed to predict tree growth and architecture over years, but few of them are capable of simulating 3D geometry development and shoot typology from a pre-existing crown geometry. In this study, we developed an approach for simulating 3D apple tree structure by combining 3D digitizing data with topological rules adapted from MappleT, an existing architectural model. In this modelling approach, bud fates were simulated with a first and second order Markov chain for terminal successions and using a hidden semi-Markov chain for the branching pattern on the one-year-old shoots. In these Markov models, five bud/shoot types were considered (small, medium, long, floral and dormant). Parametric laws and allometric relationships for each shoot type were used to simulate shoot length and leaf area. Shoot spatial position was set according to branching angle and phyllotaxis of the simulated cultivar. One nine-year-old apple tree (Malus ×domestica 'Jubile®') grown in 110L pots was digitized in 2013 and 2014. The model was calibrated on data recorded in 2012 and 2013. The calibration allowed (i) deriving distribution parameters of shoot length according to their types; (ii) establishing allometric relationships between the internode number, leaf area and the total shoot length; and (iii) evaluating the parameters of the Markov and semi-hidden Markov chains. To validate the model, simulations were performed using the tree digitized in 2013 as inputs of the model. The results were compared to the same tree digitized in 2014, comparing within tree shoot demography and the shoot leaf areas according to their type and length. Using this approach we were able to simulate tree growth on year n+1 from observed architectures at the adult stage. This opens the way for the optimization of management practices on observed trees in orchards conditions.

Parsimonious Higher Order Markov Models for Rating Transitions

Summary We propose several parsimonious models for higher order Markov chains, applied to the study of municipal rating migrations in credit risk. In full parameterized Markov chain models, the number of parameters increases very rapidly as the order in the Markov chain grows and this can yield biased estimates when certain sequences of states are rare. For some processes, as in the case of credit ratings, this problem is accentuated because the transitions between distant states are unlikely (persistent transitions). We introduce the short and long persistence models and compare them with the full parameterized Markov chain, achieving a better fit with a lower number of parameters. Furthermore, downgrade momentum effects are found in the rating process, which are consistent with recent empirical findings.

International Trade Potential of Indian Onion: An Analysis of Competitive Advantage and Direction of Trade

Agriculture being the backbone of Indian economy supports the country to be in international trade domain with a great position. This is observed by looking at the share of food grains and some horticultural produce especially onion in international trade. This paper made an attempt to analyze the international trade potential of Indian onion with an emphasis on examination of the drivers of change in onion production through Hazell's decomposition analysis. It also focused on identifying the direction of onion exports by using stationary form of the first order Markov chain model and suggested a diversification of trade markets. In addition to finding out the direction of trade, the study also informs about whether India has a comparative advantage in onion exports or not with the price competitiveness by using Revealed comparative advantage (RCA) index and Nominal protection coefficient (NPC) respectively.

A text steganography method based on Markov chains

A new method of text steganography based on Markov chains of different orders that allows the introduction of hidden information in texts is presented together with test results of a software solution which generate texts with a good approximation to the natural language model.

Analyzing the profit-loss sharing contracts with Markov model

The purpose of this paper is to examine how to use first order Markov chain to build a reliable monitoring system for the profit-loss sharing based contracts (PLS) as the mode of financing contracts in Islamic bank with censored continuous-time observations. The paper adopts the longitudinal analysis with the first order Markov chain framework. Laplace transform was used with homogenous continuous time assumption, from discretized generator matrix, to generate the transition matrix. Various metrics, i.e.: eigenvalue and eigenvector were used to test the first order Markov chain assumption. Cox semi parametric model was used also to analyze the momentum and waiting time effect as non-Markov behavior. The result shows that first order Markov chain is powerful as a monitoring tool for Islamic banks. We find that waiting time negatively affected present rating downgrade (upgrade) significantly. Likewise, momentum covariate showed negative effect. Finally, the result confirms that different origin rating have different movement behavior. The paper explores the potential of Markov chain framework as a risk management tool for Islamic banks. It provides valuable insight and integrative model for banks to manage their borrower accounts. This model can be developed to be a powerful early warning system to identify which borrower needs to be monitored intensively. Ultimately, this model could potentially increase the efficiency, productivity and competitiveness of Islamic banks in Indonesia. The analysis used only rating data. Further study should be able to give additional information about the determinant factors of rating movement of the borrowers by incorporating various factors such as contract-related factors, bank-related factors, borrower-related factors and macroeconomic factors.

Bayesian Nonparametric Modeling of Higher Order Markov Chains

ABSTRACTWe consider the problem of flexible modeling of higher order Markov chains when an upper bound on the order of the chain is known but the true order and nature of the serial dependence are unknown. We propose Bayesian nonparametric methodology based on conditional tensor factorizations, which can characterize any transition probability with a specified maximal order. The methodology selects the important lags and captures higher order interactions among the lags, while also facilitating calculation of Bayes factors for a variety of hypotheses of interest. We design efficient Markov chain Monte Carlo algorithms for posterior computation, allowing for uncertainty in the set of important lags to be included and in the nature and order of the serial dependence. The methods are illustrated using simulation experiments and real world applications. Supplementary materials for this article are available online.

International Real Estate Review

Metro market real estate cycles for office, industrial, retail, apartment, and hotel properties may be specified as first order Markov chains, which allow analysts to use a well-developed application, ¡§staying time¡¨. Anticipations for time spent at each cycle point are consistent with the perception of analysts that these cycle changes speed up, slow down, and pause over time. We find that these five different property types in U.S. markets appear to have different first order Markov chain specifications, with different staying time characteristics. Each of the five property types have their longest mean staying time at the troughs of recessions. Moreover, industrial and office markets have much longer mean staying times in very poor trough conditions. Most of the shortest mean staying times are in hyper supply and recession phases, with the range across property types being narrow in these cycle points. Analysts and investors should be able to use this research to better estimate future occupancy and rent estimates in their discounted cash flow (DCF) models.

On the length and the position of the minimum sequence containing all runs of ones in a Markovian binary sequence

Let a sequence of binary (zero–one) trials which forms a nonhomogeneous/homogeneous Markov chain of first order with given initial probability distribution and one-step transition probability matrix. The trials are assumed to be ordered on a line. The statistics denoting the length and the position (starting, ending) of the shortest segment of the sequence containing all runs of ones, are considered. The study concerns with conditional probability distributions of the statistics given that there are at least two runs of ones in the sequence. Two application case studies related to learning experiments and DNA sequences are discussed. Illustrative numerics are presented.

A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping

Bootstrapping time series is one of the most acknowledged tools to study the statistical properties of an evolutive phenomenon. An important class of bootstrapping methods is based on the assumption that the sampled phenomenon evolves according to a Markov chain. This assumption does not apply when the process takes values in a continuous set, as it frequently happens with time series related to economic and financial phenomena. In this paper we apply the Markov chain theory for bootstrapping continuous-valued processes, starting from a suitable discretization of the support that provides the state space of a Markov chain of order \(k \ge 1\). Even for small k, the number of rows of the transition probability matrix is generally too large and, in many practical cases, it may incorporate much more information than it is really required to replicate the phenomenon satisfactorily. The paper aims to study the problem of compressing the transition probability matrix while preserving the “law” characterising the process that generates the observed time series, in order to obtain bootstrapped series that maintain the typical features of the observed time series. For this purpose, we formulate a partitioning problem of the set of rows of such a matrix and propose a mixed integer linear program specifically tailored for this particular problem. We also provide an empirical analysis by applying our model to the time series of Spanish and German electricity prices, and we show that, in these medium size real-life instances, bootstrapped time series reproduce the typical features of the ones under observation.

Cryptanalysis of an improved fragile watermarking scheme

This paper presents two attacks on Teng et al.'s fragile watermarking algorithm. Both attacks allow the attacker to apply valid watermarks on tampered images, therefore rendering the watermarking scheme useless. The first attack uses the watermarked version of two chosen images, and the second attack, a generalization of the first, uses a number of arbitrary watermarked images. The paper also models the cryptanalysis process for the second attack using Markov chains in order to demonstrate that the necessary number of images is relatively small for a high probability of successful attack. All the results that are presented in this paper have been confirmed by a practical implementation.

Stochastic analysis of rainfall and its application in appropriate planning and management for Eastern India agriculture

Rainfall analysis is essential for agricultural crop planning and water resources management, especially under water scarcity conditions. The 36 years (1977–2013) of rainfall data for Kharagpur were analyzed for characterization of different seasonal events. The Weibull's formula predicted the probability of mean onset on 23rd standard meteorological weeks (SMW) (3rd–9th June) and withdrawal on 43rd SMW (21st–27th October). There was 80–83% probability of a wet week [P(W)] occurring within 25th–35th SMW. The first order Markov chain process shows the conditional probability of one wet week preceded by another wet week [P(W/W)] varied between 0 and 86%; whereas, dry week preceded by another dry week [P(D/D)] varied in the range of 70 to 100%. The stochastic analysis of successive wet or dry weeks formulates the adaptation strategies to avoid the possible effect of wet or dry spell during cropping seasons. The wet spell analysis suggests rainwater harvesting to control soil erosion and maximization of water use efficiency. The probability of getting consecutive wet [P(W/W)] and dry days [P(D/D)] were varied in the range of 40%–70% and 50%–90%, respectively. The probability of getting different magnitude of rainfall (10 to 40 mm) during the monsoon weeks (25th–39th week) were found to be more than 50% probability level, which suggest for harvesting of excess runoff water for future supplemental irrigation.

Industrial Real Estate Cycles: Markov Chain Applications

Executive SummaryAdding a stochastic element to a well-understood real estate cycle model offers opportunities like those seen in earlier such syntheses of real estate analysis and statistics. The discrete real estate cycle points in the model require a discrete probability model, here a first order Markov chain. Many statistical applications flow from the combined model. Three Markov chain count variables have obvious real estate cycle appeal. Staying time, first recurrence time, and first passage time already exist in the Markov chain literature but only staying time is in the real estate cycle literature. The most fundamental innovation is in probabilistic forecasting. Being able to describe real estate cycle risk, cycle point by cycle point many quarters ahead, could improve evaluation of prospects for property disposal. It is also a simple spreadsheet application to describe real estate cycle risks that influence cash flows from operations across four-quarter spans.

Markov Chain Modelling for Short-Term NDVI Time Series Forecasting

In this paper, the NDVI time series forecasting model has been developed based on the use of discrete time, continuous state Markov chain of suitable order. The normalised difference vegetation index (NDVI) is an indicator that describes the amount of chlorophyll (the green mass) and shows the relative density and health of vegetation; therefore, it is an important variable for vegetation forecasting. A Markov chain is a stochastic process that consists of a state space. This stochastic process undergoes transitions from one state to another in the state space with some probabilities. A Markov chain forecast model is flexible in accommodating various forecast assumptions and structures. The present paper discusses the considerations and techniques in building a Markov chain forecast model at each step. Continuous state Markov chain model is analytically described. Finally, the application of the proposed Markov chain model is illustrated with reference to a set of NDVI time series data.

Stability Analysis of India's Raw Cotton Exports

India is one among the major players in raw cotton exports and its share in global cotton trade is expected to increase by 20 percent in 2023. The present study attempts to quantify the changing structure of Indian raw cotton exports from 2004–05 to 2014–15 using first order Markov Chain Approach. The results highlighted that China, Bangladesh, Vietnam, and Other countries group including Hongkong, Thailand, Malaysia were the stable markets for Indian raw cotton exports while Pakistan, Turkey, Taiwan and Indonesia were the unstable markets. As China's raw cotton imports from India is declining specially from 2012–13 onwards, India should not rely much on China, instead, it has to concentrate on Bangladesh, Vietnam and Taiwan whose export share is expected to increase in future. In addition, India need to establish world class manufacturing units along the cotton value chain to produce finished products which absorbs surplus domestic cotton and in turn provide additional employment opportunities.

On Preventing Location Attacks for Urban Vehicular Networks

The prevalence of global positioning system (GPS) equipped in vehicular networks exposes users’ location information to the location-based services. We argue that such data contains rich informative cues on drivers’ private behaviors and preferences, which will lead to the location privacy attacks. In this paper, we proposed a sophisticated prediction model to predict driver’s next location by using ak-order Markov chain-based third-rank tensor representing the partially observed transfer information of vehicles. Then Bayesian Personalized Ranking (BPR) is used to learn the unobserved transitions within the tensor for transition predication. Experimental results manifest the efficacy of the proposed model in terms of location predication accuracy, compared with several state-of-the-art predication methods. We also point out that the precision achieved by such advanced predication model is restricted to the order of the Markov chaink. Accordingly, we propose a schema to decrease the risks of such attacks by preventing the conformation of higher order Markov chain. Experimental results obtained based on the real-world vehicular network data demonstrated the effectiveness of our proposed schema.

Markov chain order estimation with parametric significance tests of conditional mutual information

Besides the different approaches suggested in the literature, accurate estimation of the order of a Markov chain from a given symbol sequence is an open issue, especially when the order is moderately large. Here, parametric significance tests of conditional mutual information (CMI) of increasing order m, Ic(m), on a symbol sequence are conducted for increasing orders m in order to estimate the true order L of the underlying Markov chain. CMI of order m is the mutual information of two variables in the Markov chain being m time steps apart, conditioning on the intermediate variables of the chain. The null distribution of CMI is approximated with a normal and gamma distribution deriving analytic expressions of their parameters, and a gamma distribution deriving its parameters from the mean and variance of the normal distribution. The accuracy of order estimation is assessed with the three parametric tests, and the parametric tests are compared to the randomization significance test and other known order estimation criteria using Monte Carlo simulations of Markov chains with different order L, length of symbol sequence N and number of symbols K. The parametric test using the gamma distribution (with directly defined parameters) is consistently better than the other two parametric tests and matches well the performance of the randomization test. The tests are applied to genes and intergenic regions of DNA sequences, and the estimated orders are interpreted in view of the results from the simulation study. The application shows the usefulness of the parametric gamma test for long symbol sequences where the randomization test becomes prohibitively slow to compute.