Nonstationary Transition Probabilities Research Articles

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.

Seasonal Variation of Power Distribution in Niger State of Nigeria using Markov Model with Non-Stationary Transition Probabilities

This paper presents the application of Markov chain model with non-stationary transition probabilities to study the monthly data of the power distribution in Niger state in the wet, Dry-Hot and Hamatten/Dry- Hot seasons. The result indicates an optimal power distribution of over 150,000MWwith probability 0.49 during the wet season, 0.25 during the hot-dry season and 0.19 in the hot-cold season respectively. The variation of power distribution directly affects the electricity consumers. Markov chain model could be used as a predictive tool for determining the power distribution pattern at different seasons in the Study area. These predictions might be used for the management of (NCC) for effective distribution of megawatts.Keywords: Markov Chain, Transition probability, Non-stationary, Power Distribution

A multistage decision-dependent stochastic bilevel programming approach for power generation investment expansion planning

ABSTRACTIn this article, we study the long-term power generation investment expansion planning problem under uncertainty. We propose a bilevel optimization model that includes an upper-level multistage stochastic expansion planning problem and a collection of lower-level economic dispatch problems. This model seeks for the optimal sizing and siting for both thermal and wind power units to be built to maximize the expected profit for a profit-oriented power generation investor. To address the future uncertainties in the decision-making process, this article employs a decision-dependent stochastic programming approach. In the scenario tree, we calculate the non-stationary transition probabilities based on discrete choice theory and the economies of scale theory in electricity systems. The model is further reformulated as a single-level optimization problem and solved by decomposition algorithms. The investment decisions, computation times, and optimality of the decision-dependent model are evaluated by case studies on IEEE reliability test systems. The results show that the proposed decision-dependent model provides effective investment plans for long-term power generation expansion planning.

Optimal size of a rental inventory with items available from a secondary source: a model with non-stationary probabilities

This article concerns operations of businesses that own inventories of rental items, and can hire additional items from secondary sources whenever they face a temporary exhaustion of their inventories. This set-up is relevant to many operations: the items may be tools, trucks, containers, communication channels, or individuals who provide services such as repairmen. A fundamental problem that emerges in the design of these operations is to determine the optimal size of the inventory of items the business should own. To solve this problem, this article takes the view of a finite horizon project and proposes an approach that chooses the inventory size that minimizes the expected present cost of the project. This approach models random times between consecutive item requests and random rental durations with corresponding expectations that may vary along the day. The expected present cost uses non-stationary transition probabilities that recent articles have computed resorting to stationary approximations. This article, in contrast, computes these probabilities faster solving a differential equation without resorting to such approximations. If the present cost is of interest, an analysis that plugs-in the optimal size into the present cost ignores the sampling variability that transfers from the traffic data to the optimal size. This article complements the analysis with simulations that provide the sampling distribution of the present cost.

Resource Availability Prediction in Distributed Systems: An Approach for Modeling Non-Stationary Transition Probabilities

Large scale distributed systems employ thousands of resources which inevitably suffer from the unavailability issue. Serious side effects like unexpected delay or failure in the application execution are probable in case of such an issue. The imposed outcome might then be catastrophic consequences for real time applications or penalties for the service providers. Better prediction of the resource unavailability helps diminishing the undesired outcomes. This paper proposes a resource availability prediction algorithm for the mentioned goal. The resource availability variation is modeled as a stochastic process. By analyzing the availability information of NDU resources and both physical and virtual machines of the PlantLab, we found that the transition probabilities among the availability levels are non-stationary. To cope with this characteristic, we introduce Availability Transition Patterns (ATPs); the ATPs are dynamically constructed and the transitions between them are modeled by a Markov chain. The future ATP is then predicted based on the constructed Markov chain, according to which the resource availability-level is predicted. Experimental results confirm the efficiency of the proposed prediction algorithm.

Determinants of transition in artificially discrete Markov chains using microdata

We describe an econometric procedure to model transitions in Markov chains whose state space is finite and classification stems from observed continuous variables. We show how stationary and non-stationary transition probabilities as well as the marginal effects of continuous and dichotomous variables determining transition can be estimated. The model resembles the ordered probit approach used in Epstein et al. (2006) but allows for the differences in the nature of the dependent variable and suggests some very important extensions pertaining to more meaningful representation of parameter estimates and the simultaneous construction of transition matrices.

Bayesian estimation of non-stationary Markov models combining micro and macro data

We develop a Bayesian framework for estimating non-stationary Markov models in situations where macro population data are available only on the proportion of individuals residing in each state, but micro-level sample data are available on observed transitions between states. Posterior distributions on non-stationary transition probabilities (TPs) are derived combining micro and macrodata using potentially asynchronous data observations, providing a new method for inferring TPs that merges previously disparate approaches. Monte Carlo simulations demonstrate how observed micro transitions can improve the precision of posterior information. We provide an empirical illustration in the context of farm structural change.

A Non-Stationary Transition Probabilities for a Reservoir elevation of Hydro Electric Power Dam

This paper demonstrates the application of Markov chain model with non-stationary transition probabilities to study data of the Reservoir elevation of Shiroro dam in the dry and raining seasons. The result indicates an optimal of 40% and 49% transition probabilities at equilibrium for dry season and wet season respectively. The result confirms the reality on the ground that higher reservoir elevation is obtained more during the raining season. Conversely, the lower reservoir elevation is experienced largely in the dry season. The variation of the reservoir elevation directly affects the hydro electric power generation and the availability of the other dam resources. Markov chain model could be used as a predictive device for studying reservoir elevation of Shiroro dam. These predictions might be used for the management of the dam resources. 1 1 1 n 1 n n 1 n

Model reduction for a class of nonstationary Markov jump linear systems

This paper concerns the problem of model reduction for a class of Markov jump linear system (MJLS) with nonstationary transition probabilities (TPs) in discrete-time domain. The nonstationary character of TPs is considered as finite piecewise stationary and the variations in the finite set are considered as two types: arbitrary variation and stochastic variation, respectively. The latter means that the variation is subject to a higher-level transition probability matrix. Invoking the idea in the recent studies of partially unknown TPs for the traditional MJLS with stationary TPs, a generalized framework covering the two kinds of variation is proposed. The model reduction results for the underlying systems are obtained in H∞ sense. A numerical example is presented to illustrate the effectiveness and potential of the developed theoretical results.

Stationarity of the transition probabilities in the Markov chain formulation of owner payments on projects

Stationarity of the transition probabilities in the Markov chain formulation of owner payments on projects

Credit risk rating migration and unobserved borrower heterogeneity

Some past studies of credit risk ratings migration have found trend reversals and evidence that the data‐generating process is nonstationary. Using a sample of Farm Credit System mortgages, we find no compelling statistical evidence of either phenomenon. We do find evidence that our sample of loans may be characterized by two types of borrowers – namely, movers and stayers. This type of borrower heterogeneity is unobserved because movers who do not migrate are indistinguishable from stayers who never migrate. We report on the development of a flexible nonparametric model for estimating transition probabilities. The model can also be used to estimate nonstationary transition probabilities and an example is provided.

Credit Risk Rating Migration and Unobserved Borrower Heterogeneity

Some past studies of credit risk ratings migration have found trend reversals and evidence that the data generating process is non-stationary. Using a sample of FCS mortgages, we find no compelling statistical evidence of either phenomenon. We do find evidence that our sample of loans may be characterized by two types of borrowers, namely, movers and stayers. This type of borrower heterogeneity is unobserved because movers that do not migrate are indistinguishable from stayers who never migrate. We report on the development of a flexible nonparametric model for estimating transition probabilities. The model can also be used to estimate non-stationary transition probabilities and an example is provided.

Food consumption dynamics: a nonparametric approach using maximum entropy estimation

AbstractWe model dynamic consumer choice in a stochastic optimal control framework and show conditions under which observable market share data possess the Markov property. Using 30 years of annual aggregate milk consumption data differentiated by fat content, maximum entropy is used to estimate nonstationary transition probabilities showing how consumer tastes and preferences have changed over time. The maximum entropy approach allows for the estimation of a 4 × 4 transition probability matrix for each year of the sample. Results suggest that skim milk was an absorbing state over most of the sample but that the trend toward skim milk has decelerated and possibly reversed itself since 1998. Our approach provides a useful complement to existing parametric approaches to demand analysis when data are limited or the problem is ill‐posed.

Forecasting Herd Structure and Milk Production for Production Risk Management

Substantial increases in milk price volatility have resulted from changes in federal dairy policies. For a dairy farm, however, monthly gross milk receipts are a function of unit price and quantity produced. Both can vary substantially over time. Therefore, to be effective, risk management strategies must address milk and input price volatility (price risk management) and fluctuations in milk production per cow and cow numbers (production risk management). Herd milk production through time can be modeled as a discrete stochastic process using finite Markov chains. Cows at time t = 0 are assigned to homogeneous production cells in four-dimensional arrays with coordinates determined by parity (1,2,3), week in milk (1,…,104), pregnancy status (0,1), and week pregnant (1,…,40). The processes of aging, pregnancy, involuntary cull, voluntary cull, abortion, dry-off, and freshening from week i-1 to week i are accounted for, using nonstationary transition probabilities. Bayesian estimates of transition probabilities are derived from historical herd data, assuming that individual outcomes are from Bernoulli distributions. The values of parameters θi for the Bernoulli distributions are unknown but have prior distributions that follow beta distributions with parameters αi and βi estimated from historical data. Herd observations are then used to generate posterior distributions of θi, also from beta distributions. Projecting from one week to the next is accomplished by moving virtual animals from one production cell to the next based on the transition probability assigned to that path. Summing production estimates and variances of all independent cells provides for an expected herd production with an associated variance. As expected, the forecast variance increases with time, reflecting increased uncertainty of distant projections. Model validation presents an interesting problem because future observations used for validation are under human control and are not independent of the forecast.

Learning Dynamics, Lock-In, and Equilibrium Selection in Experimental Coordination Games

This paper compares the leading theoretical approaches to equilibrium selection, both traditional and adaptive, in the light of recent experiments by Van Huyck, Battalio, and Beil (henceforth "VHBB") in which subjects repeatedly played coordination games, uncertain only about each other's strategy choices. The large strategy spaces of VHBB's designs and the variety of interaction patterns they considered yielded rich dynamics, with systematic differences in limiting outcomes across treatments. Explaining these differences promises to shed considerable light on equilibrium selection and coordination more generally, in the field as well as the laboratory. Following earlier analyses by Bruno Broseta and myself, I propose a model that gives a flexible characterization of individual behavior and allows for strategic uncertainty, in the form of idiosyncratic random shocks to players' adjustments. The model includes representatives of the leading approaches to equilibrium selection, which are distinguished by different values of behavioral parameters, including variances that represent the level of strategic uncertainty. The model provides a framework within which to estimate the parameters econometrically, using data from the experiments. The estimates suggest that VHBB's treatments evoked high initial levels of strategic uncertainty, declining steadily to zero as subjects learned to predict each other's responses. The resulting model has nonstationary transition probabilities and history-dependent dynamics that lock in on an equilibrium of the stage game, whose prior probability distribution is nondegenerate due to persistent effects of strategic uncertainty. The analysis shows that even when strategic uncertainty is eventually eliminated by learning, it imparts a drift to the learning dynamics, whose magnitude and direction depend on the environment and the behavioral parameters. This drift makes the distribution of the limiting outcome vary across treatments much as its empirical frequency distribution varied in the experiments. In this sense, taking the persistent effects of strategic uncertainty into account allows a simple, unified explanation of VHBB's results.

A Comprehensive Model for Managing Credit Risk on Home Mortgage Portfolios

Managing credit risk in financial institutions requires the ability to forecast aggregate losses on existing loans, predict the length of time that loans will be on the books before prepayment or default, analyze the expected performance of particular segments in the existing portfolio, and project payment patterns of new loans. Described in this paper are tools created for these functions in a large California financial institution. A forecasting model with Markovian structure and nonstationary transition probabilities is used to model the life of a mortgage. Logistic and regression models are used to estimate severity of losses. These models are integrated into a system that allows analysts and managers to depict the expected performance of individual loans and portfolio segments under different economic scenarios. With this information, analysts and managers can establish appropriate loss reserves, suggest pricing differentials to compensate for risk, and make strategic lending decisions.

Forecasting losses on a liquidating long-term loan portfolio

Assessing the condition of financial institutions and valuation of loan portfolios in secondary markets require the estimation of exposure to losses on existing portfolios of long-term loans. Data from a major U.S. financial institution were used to construct a forecasting model with Markovian structure and nonstationary transition probabilities. The model proved to be effective in representing changes in probability of default that occur as individual loans mature and accurate in forecasting aggregate defaults and losses on a nationwide portfolio of long-term loans.

Technical change and the structure of production: A non-stationary Markov analysis

Elasticities of farm structure and of the probabilities of movement from one size of farm to another are derived and calculated from a Markov model. They are used to examine the impact of technical change on the size distribution of farms over time. Imperfectly observed aggregate data are used to estimate non-stationary transition probabilities which characterise movement in farm size. The model examines the structure of small (one to 29 cows), medium (30 to 49 cows), large (50 to 99 cows), and very large (100 or more cows) Wisconsin dairy farms during the 1980s and early 1990s. Findings confirm the non-stationary nature of the evolution in farm structure. Copyright 1995 by Oxford University Press.

Statistical theory of X-ray scattering from crystals of finite size with pure displacement disorder in one dimension

The statistical theory of X-ray scattering from a crystal that is disordered in one dimension is discussed. The state of disorder is characterized by probabilities that different types of scattering elements occupy specified positions. The effect this disorder has on the corresponding ensemble average axial reflections is determined. Finite size is explicitly accounted for. Motivated by recent experiments on water intercalation into thin, lyotropic multilayers, special consideration is given to systems whose components differ with respect to size, but not with respect to their scattering factors. Focusing on this case of pure displacement disorder in a binary mixture, a representation of the scattering function in closed form is derived. The case of a random binary mixture leads to the results of Hendricks & Teller [J. Chem. Phys. (1942), 10, 147- 167], and Méring [Acta Cryst. (1949), 2, 371-377], while in the presence of nearest-neighbor correlations, a connection is established with the theory of Kakinoki & Komura [J. Phys. Soc. Jpn (1952), 7, 30-35]. In extension of their treatment, systems with non-stationary transition probabilities are investigated. The effects on the scattering function of constraining composition fluctuations are also studied. Particular characteristics of the scattering function are displayed and discussed.

A Markov Chain Analysis of Structural Changes in the Texas High Plains Cotton Ginning Industry

AbstractMarkov chain analysis of changes in the number and size of cotton gin firms in West Texas was conducted assuming stationary and non-stationary transition probabilities. Projections of industry structure were made to 1999 with stationary probability assumptions and six sets of assumed conditions for labor and energy costs and technological change in the non-stationary transition model. Results indicate a continued decline in number of firms, but labor, energy, and technology conditions alter the configuration of the structural changes.