Discrete-state Models Research Articles

Loop series for discrete statistical models on graphs

In this paper we present the derivation details, logic, and motivation for the three loopcalculus introduced in Chertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Generatingfunctions for each of the three interrelated discrete statistical models are expressed in termsof a finite series. The first term in the series corresponds to the Bethe–Peierlsbelief–propagation (BP) contribution; the other terms are labelled by loops on the factorgraph. All loop contributions are simple rational functions of spin correlation functionscalculated within the BP approach. We discuss two alternative derivations of the loopseries. One approach implements a set of local auxiliary integrations over continuous fieldswith the BP contribution corresponding to an integrand saddle-point value. Theintegrals are replaced by sums in the complementary approach, briefly explained inChertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Local gauge symmetrytransformations that clarify an important invariant feature of the BP solutionare revealed in both approaches. The individual terms change under the gaugetransformation while the partition function remains invariant. The requirement for allindividual terms to be nonzero only for closed loops in the factor graph (as opposed topaths with loose ends) is equivalent to fixing the first term in the series to beexactly equal to the BP contribution. Further applications of the loop calculus toproblems in statistical physics, computer and information sciences are discussed.

A Stochastic Model of Oscillatory Blood Testosterone Levels

A continuous-time, discrete-state stochastic model of testosterone secretion in men is considered. Blood levels of testosterone in men fluctuate periodically with a period of 2-3 h. The deterministic model, on which the stochastic model considered here is based, is well studied and has been shown to have a globally stable fixed point. Thus, no sustained oscillations are possible in the deterministic case. However, the stochastic model does observe periodic, pulsatile behavior. This demonstrates how oscillations can occur due to a switching behavior dependent on the random degradation of testosterone molecules in the system. The Gillespie algorithm is used to simulate the hormone secretion model. Important parameters of the model are discussed and results from the model are compared to experimental observations.

Nonlinear Terahertz Response ofn-Type GaAs

Excitation of an n-type GaAs layer by intense ultrashort terahertz pulses causes coherent emission at 2 THz. Phase-resolved nonlinear propagation experiments show a picosecond decay of the emitted field, despite the ultrafast carrier-carrier scattering at a sample temperature of 300 K. While the linear THz response is in agreement with the Drude response of free electrons, the nonlinear response is dominated by the super-radiant decay of optically inverted impurity transitions. A quantum mechanical discrete state model using the potential of the disordered impurities accounts for all experimental observations.

Timing and diversification: A state-dependent asset allocation approach

The influence of changing economic environment leads the distribution of stock market returns to be time-varying. A conditionally optimal investment hence requires a dynamic adjustment of asset allocation. In this context, this paper examines the improvement in portfolio performance by simulating portfolio strategies that are conditioned on the Markov regime switching behaviour of stock market returns. Including a memory effect eliminates the empirical shortcoming of discrete state models, namely that they produce a standard and an extreme state in stock returns. So far, this has prevented the regimes from being used as a valuable conditioning variable. Based on a discrete state indicator variable, is presented evidence of considerable performance improvement relative to the static model due to optimal shifting between aggressive and well diversified portfolio structures.

Continuous Distribution of Emission States from Single CdSe/ZnS Quantum Dots

The photoluminescence dynamics of colloidal CdSe/ZnS/streptavidin quantum dots were studied using time-resolved single-molecule spectroscopy. Statistical tests of the photon-counting data suggested that the simple "on/off" discrete state model is inconsistent with experimental results. Instead, a continuous emission state distribution model was found to be more appropriate. Autocorrelation analysis of lifetime and intensity fluctuations showed a nonlinear correlation between them. These results were consistent with the model that charged quantum dots were also emissive, and that time-dependent charge migration gave rise to the observed photoluminescence dynamics.

Long‐Time Protein Folding Dynamics from Short‐Time Molecular Dynamics Simulations

Protein folding involves physical timescales - microseconds to seconds - that are too long to be studied directly by straightforward molecular dynamics simulation, where the fundamental timestep is constrained to femtoseconds. Here we show how the long-time statistical dynamics of a simple solvated biomolecular system can be well described by a discrete-state Markov chain model constructed from trajectories that are an order of magnitude shorter than the longest relaxation times of the system. This suggests that such models, appropriately constructed from short molecular dynamics simulations, may have utility in the study of long-time conformational dynamics.

Extracting Information from Spot Interest Rates and Credit Ratings using Double Higher-Order Hidden Markov Models

Estimating and forecasting the unobservable states of an economy are important and practically relevant topics in economics. Central bankers and regulators can use information about the market expectations on the hidden states of the economy as a reference for decision and policy makings, for instance, deciding monetary policies. Spot interest rates and credit ratings of bonds contain important information about the hidden sequence of the states of the economy. In this paper, we develop double higher-order hidden Markov chain models (DHHMMs) for extracting information about the hidden sequence of the states of an economy from the spot interest rates and credit ratings of bonds. We consider a discrete-state model described by DHHMMs and focus on the qualitative aspect of the unobservable states of the economy. The observable spot interest rates and credit ratings of bonds depend on the hidden states of the economy which are modelled by DHHMMs. The DHHMMs can incorporate the persistent phenomena of the time series of spot interest rates and the credit ratings. We employ the maximum likelihood method and the EM algorithm, namely Viterbi's algorithm, to uncover the optimal hidden sequence of the states of the economy which can be interpreted the “best” estimate of the sequence of the underlying economic states generating the spot interest rates and credit ratings of the bonds. Then, we develop an efficient maximum likelihood estimation method to estimate the unknown parameters in our model. Numerical experiment will be conducted to illustrate the implementation of the model.

Kinesin crouches to sprint but resists pushing

Recent optical trap experiments have applied resisting, assisting, and sideways loads to conventional kinesin moving on microtubules at fixed [ATP]. To gain insight into intermediate motions when the motor protein takes its 8.2-nm steps, the velocity and randomness data have been analyzed by using discrete-state stochastic models with a three-dimensional "energy landscape." The bead size and tether angle play a crucial role. The analysis implies that on binding ATP the motor "crouches," the point of attachment of the tether at the necklinker junction moving downward toward the microtubule by 0.5-0.7 nm, while inching forward by only 0.1-0.2 nm, before completing the step from a transition state by a unitary "sprint" of approximately 7.8 nm. These inferences accord with high-resolution observations that exclude a previously predicted substep of 1.8-2.1 nm. Assisting and leftward loads are opposed in that the perpendicular component of the tension in the tether is enhanced by approximately 2 pN, which reduces the velocity, but sideways lurching is not supported.

Long-Time Conformational Transitions of Alanine Dipeptide in Aqueous Solution: Continuous and Discrete-State Kinetic Models

We present an analysis of the thermodynamics, conformational dynamics, and kinetics of the solvated alanine dipeptide molecule. Solvation was treated in the framework of the OPLS/analytic generalized born with nonpolar interactions effective potential model. The effective free energy map was generated in a series of multiwindow umbrella sampling all-atom simulations using the weighted histogram analysis method. A Brownian dynamics approach was used to examine the room-temperature dynamics of the dipeptide. To emulate the realistic dynamics of conformational interconversions of the alanine dipeptide in water, the parameters that govern the time evolution of the reduced model were chosen to mimic the intermediate-time dynamics of a model which treats both the solute and the solvent explicitly. The characteristic time (mean first passage time) for the “fast” C7eq → αR transition was found to be around 249 ps in good agreement with the results of previously reported studies. From an ensemble of microsecond Brownian dynamics trajectories we were able to numerically estimate with high statistical precision the mean transition time for the “slow” transition αR → C7ax, which was found to be approximately 11 ns. Conformational kinetics of the dipeptide was further analyzed by introducing a discrete-state kinetic model consisting of four states (αR, β/C5/C7eq, αL, and C7ax) and 12 rate constants, which reproduces the long-time behavior of the Brownian dynamics simulations. The simulations described here complement and help to motivate single-molecule spectroscopic studies of alanine dipeptide and other peptides in aqueous solution. The close correspondence established between the continuous dynamics and a discrete-state kinetic model provides a possible route for studying the long-time kinetic behavior of larger polypeptides using detailed all-atom effective potentials.

A Monte Carlo Sampling Scheme for the Ising Model

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete-state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data. We show how to reconstruct the entropy S of the model, from which, e.g., the free energy can be obtained. Furthermore we discuss how this scheme allows us to more or less completely remove the effects of critical fluctuations near the critical temperature and likewise how it reduces critical slowing down. This makes it possible to use simple state generation methods like the Metropolis algorithm also for large lattices.

Stochastic Resonance in a Non-Markovian Discrete State Model for Excitable Systems

We study a non-Markovian three state model, subjected to an external periodic signal. This model is intended to describe an excitable system with periodical driving. In the limit of a small amplitude of the external signal we derive expressions for the spectral power amplification and the signal to noise ratio as well as for the interspike interval distribution.

Bayesian approaches to multiple sources of evidence and uncertainty in complex cost-effectiveness modelling.

Increasingly complex models are being used to evaluate the cost-effectiveness of medical interventions. We describe the multiple sources of uncertainty that are relevant to such models, and their relation to either probabilistic or deterministic sensitivity analysis. A Bayesian approach appears natural in this context. We explore how sensitivity analysis to patient heterogeneity and parameter uncertainty can be simultaneously investigated, and illustrate the necessary computation when expected costs and benefits can be calculated in closed form, such as in discrete-time discrete-state Markov models. Information about parameters can either be expressed as a prior distribution, or derived as a posterior distribution given a generalized synthesis of available data in which multiple sources of evidence can be differentially weighted according to their assumed quality. The resulting joint posterior distributions on costs and benefits can then provide inferences on incremental cost-effectiveness, best presented as posterior distributions over net-benefit and cost-effectiveness acceptability curves. These ideas are illustrated with a detailed running example concerning the cost-effectiveness of hip prostheses in different age-sex subgroups. All computations are carried out using freely available software for conducting Markov chain Monte Carlo analysis.

Computation of Infrastructure Transition Probabilities Using Stochastic Duration Models

Sound infrastructure deterioration models are essential for accurately predicting future conditions that, in turn, are key inputs to effective maintenance and rehabilitation decision making. The challenge central to developing accurate deterioration models is that condition is often measured on a discrete scale, such as inspectors’ ratings. Furthermore, deterioration is a stochastic process that varies widely with several factors, many of which are generally not captured by available data. Consequently, probabilistic discrete-state models are often used to characterize deterioration. Such models are based on transition probabilities that capture the nature of the evolution of condition states from one discrete time point to the next. However, current methods for determining such probabilities suffer from several serious limitations. An alternative approach addressing these limitations is presented. A probabilistic model of the time spent in a state (referred to as duration) is developed, and the approach used for estimating its parameters is described. Furthermore, the method for determining the corresponding state transition probabilities from the estimated duration models is derived. The testing for the Markovian property is also discussed, and incorporating the effects of history dependence, if found present, directly in the developed duration model is described. Finally, the overall methodology is demonstrated using a data set of reinforced concrete bridge deck observations.

Algebraic Bethe ansatz for a discrete-state BCS pairing model

We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe Ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities, H_i, from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the H_i (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe Ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the H_i as Hamiltonian, more general exactly solvable models can be constructed. -- To make the paper accessible to readers that are not Bethe Ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.

On the form of ROCs constructed from confidence ratings.

A classical question for memory researchers is whether memories vary in an all-or-nothing, discrete manner (e.g., stored vs. not stored, recalled vs. not recalled), or whether they vary along a continuous dimension (e.g., strength, similarity, or familiarity). For yes-no classification tasks, continuous- and discrete-state models predict nonlinear and linear receiver operating characteristics (ROCs), respectively (D. M. Green & J. A. Swets, 1966; N. A. Macmillan & C. D. Creelman, 1991). Recently, several authors have assumed that these predictions are generalizable to confidence ratings tasks (J. Qin, C. L. Raye, M. K. Johnson, & K. J. Mitchell, 2001; S. D. Slotnick, S. A. Klein, C. S. Dodson, & A. P. Shimamura, 2000, and A. P. Yonelinas, 1999). This assumption is shown to be unwarranted by showing that discrete-state ratings models predict both linear and nonlinear ROCs. The critical factor determining the form of the discrete-state ROC is the response strategy adopted by the classifier.

Computing transient gating charge movement of voltage-dependent ion channels.

The opening of voltage-gated sodium, potassium, and calcium ion channels has a steep relationship with voltage. In response to changes in the transmembrane voltage, structural movements of an ion channel that precede channel opening generate a capacitative gating current. The net gating charge displacement due to membrane depolarization is an index of the voltage sensitivity of the ion channel activation process. Understanding the molecular basis of voltage-dependent gating of ion channels requires the measurement and computation of the gating charge, Q. We derive a simple and accurate semianalytic approach to computing the voltage dependence of transient gating charge movement (Q-V relationship) of discrete Markov state models of ion channels using matrix methods. This approach allows rapid computation of Q-V curves for finite and infinite length step depolarizations and is consistent with experimentally measured transient gating charge. This computational approach was applied to Shaker potassium channel gating, including the impact of inactivating particles on potassium channel gating currents.

Using neural network function approximation for optimal design of continuous-state parallel–series systems

This paper presents a novel continuous-state system model for optimal design of parallel–series systems when both cost and reliability are considered. The advantage of a continuous-state system model is that it represents realities more accurately than discrete-state system models. However, using conventional optimization algorithms to solve the optimal design problem for continuous-state systems becomes very complex. Under general cases, it is impossible to obtain an explicit expression of the objective function to be optimized. In this paper, we propose a neural network (NN) approach to approximate the objective function. Once the approximate optimization model is obtained with the NN approach, the subsequent optimization methods and procedures are the same and straightforward. A 2-stage example is given to compare the analytical approach with the proposed NN approach. A complicated 4-stage example is given to illustrate that it is easy to use the NN approach while it is too difficult to solve the problem analytically. Scope and purpose The classical reliability theory assumes that the system and each component may only be in one of two possible states: working or failed. Thus, it is also referred to as binary reliability theory. A well-known reliability design problem under the binary reliability theory involves the determination of the number of redundancies in a parallel–series system which consists of N subsystems connected in series whereas each subsystem consists of a few components connected in parallel. In this paper, we consider the optimal design problem of a multi-state parallel–series system wherein both the system and its components may assume more than two levels of performance. Specifically, we assume that the state of each component and the system may be represented by a continuous random variable that may take values in the closed interval [0,1]. An optimization model is formulated for the determination of the number of redundancies in order to maximize the system's expected utility function. Because of the complexity of the optimization problem, we propose a neural network (NN) approach to approximate the objective function. The resulting optimization model is much easier to solve. Examples are given to illustrate the proposed approach.

Lattice effects observed in chaotic dynamics of experimental populations.

Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.

Measurement-induced decoherence in a simple quantum system

Measurement-induced decoherence and its consequent effect on the time evolution of the measured system is investigated via the Wigner representation W. Using a discrete-state computer model put forward by Wallace [D. Wallace, Phys. Rev. A 63, 022109 (2001)], we simulate a free particle moving in a periodic space, which is subjected to repeated projection-valued measurements of position. The measurements are made at rapid intervals, such as to induce a Zeno effect in the motion of the particle. We construct a discrete form of the Wigner distribution, which is used to represent the system ensemble at various times during its evolution. In addition, we calculate the entropy of the system ensemble and compare its time development with the Wigner distribution of the system state. We conclude that, in accordance with the findings of other similar investigations, the attenuation of negative W is a measure of the increasing decoherence in the system density operator, and is characteristic of the transition from the quantum to the classical descriptions of the system.

Modeling Stochastic Dynamical Systems for Interactive Simulation

We present techniques for constructing approximate stochastic models of complicated dynamical systems for applications in interactive computer graphics. The models are designed to produce realistic interaction at low cost. We describe two kinds of stochastic models: continuous state (ARX) models and discrete state (Markov chains) models. System identi cation techniques are used for learning the input-output dynamics automatically, from either measurements of a real system or from an accurate simulation. The synthesis of behavior in this manner is several orders of magnitude faster than physical simulation.We demonstrate the techniques with two examples: (1) the dynamics of candle ame in the wind, modeled using data from a real candle and (2) the motion of a falling leaf, modeled using data from a complex simulation. We have implemented an interactive Java program which demonstrates real-time interaction with a realistically behaving simulation of a cartoon candle ame. The user makes the ame animation icker by blowing into a microphone.