First-order Markov Process Research Articles

Wealth distribution and fiscal policy

SummaryThis paper examines the long-run equilibrium wealth distribution in the context of a general equilibrium model in which heterogeneous agents are born into dynasties. When agents are born, they are randomly assigned to be either altruists or egoists. An altruistic agent cares about his progeny and so consumes part of his income (wages plus bequests) and bequeaths the rest to his heirs. These bequests determine the capital stock next period. An egoistic agent consumes all his income (wages plus bequests) leaving nothing to his heirs. Assuming that the probability that an altruist (egoist) is born to an altruist or an egoist follows a first-order Markov process, and given initial conditions on the distribution of wealth and bequests, the equilibrium dynamics of the economy can be characterised, and a stationary equilibrium (numerically) solved. This done, two fiscal policy tax/intragenerational transfer regimes are evaluated in terms of individual welfare and the distribution of wealth.

Analytical solution of discrete colored noise ECA tracking filter

New analytical solutions of steady-state Kalman gains are presented for a discrete-time tracking filter with correlation in both the measurement noise and the target maneuver. The measurement noise model is a first-order discrete Markov process characterized by a correlation coefficient /spl rho/. The target motion is examined for an exponentially correlated acceleration maneuver type in which the vehicle oscillation such as wind-induced-bending is also considered. The present solution method is based on factorizing the observed spectral density matrix /spl Psi/(z) in frequency domain. The algorithm proposed here gives the Kalman gain matrix directly. For a case when the steady-state error covariance matrix is desired, such gains can be incorporated with the algebraic Riccati equation.

Distribution of Base Pair Repeats in Coding and Noncoding DNA Sequences

We analyze the histograms for the lengths of the 16 possible distinct repeats of identical dimers, known as dimeric tandem repeats, in DNA sequences. For coding regions, the probability of finding a repetitive sequence of , copies of a particular dimer decreases exponentially as , increases. For the noncoding regions, the distribution functions for most of the 16 dimers have long tails and can be approximated by power-law functions, while for coding DNA, they can be well fit by a firstorder Markov process. We propose a model, based on known biophysical processes, which leads to the observed probability distribution functions for noncoding DNA. We argue that this difference in the shape of the distribution functions between coding and noncoding DNA arises from the fact that noncoding DNA is more tolerant to evolutionary mutational alterations than coding DNA. [S0031-9007(97)04907-7]

Modeling the conditional distribution of interest rates as a regime-switching process

This paper develops a generalized regime-switching (GRS) model of the short-term interest rate. The model allows the short rate to exhibit both mean reversion and conditional heteroskedasticity and nests the popular generalized autoregressive conditional heteroskedasticity (GARCH) and square root process specifications. The conditional variance process accommodates volatility clustering and dependence on the level of the interest rate. A first-order Markov process with state-dependent transition probabilities governs the switching between regimes. The GRS model is compared with various existing models of the short rate in terms of (1) the statistical fit of short-term interest rate data and (2) out-of-sample forecasting performance.

Optimal cartel trigger strategies and the number of firms

Recent empirical papers have analyzed collusion in the Joint Executive Committee in an attempt to determine which of several theories of cartel behavior is supported by the behavior of this 19th century railroad cartel. Non-parametric tests of whether high and low profit regimes followed a first-order Markov process when one controls for the number of firms support the theory of optimal collusion given by Abreu, Pearce, and Stachetti. Results on whether transition probabilities depend on the number of firms are inconclusive.

Improved tracking of maneuvering targets: The use of turn-rate distributions for acceleration modeling

Tactically maneuvering targets are difficult to track since acceleration cannot be observed directly and the accelerations are induced by human control or an autonomous guidance system therefore they are not subject to deterministic models. A common tracking system is the two-state Kalman filter with a Singer maneuver model where the second-order statistics of acceleration is the same as a first-order Markov process. The Singer model assumes a uniform probability distribution on the targets acceleration which is independent of the x and y direction. In practice, it is expected that targets have constant forward speed and an acceleration vector normal to the velocity vector, a condition not present in the Singer model. The work of Singer is extended by presenting a maneuver model which assumes constant forward speed and a probability distribution on the targets turn-rate. Details of the model are presented along with sample simulation results.

Application of ADI Iterative Methods to the Restoration of Noisy Images

The restoration of two-dimensional images in the presence of noise by Wiener's minimum mean square error filter requires the solution of large linear systems of equations. When the noise is white and Gaussian, and under suitable assumptions on the image, these equations can be written as a Sylvester's equation \[ T_1^{-1}\hat{F}+\hat{F}T_2=C \] for the matrix $\hat{F}$ representing the restored image. The matrices $T_1$ and $T_2$ are symmetric positive definite Toeplitz matrices. We show that the ADI iterative method is well suited for the solution of these Sylvester's equations, and illustrate this with computed examples for the case when the image is described by a separable first-order Markov process. We also consider generalizations of the ADI iterative method, propose new algorithms for the generation of iteration parameters, and illustrate the competitiveness of these schemes.

Hidden Markov Random Fields

A noninvertible function of a first-order Markov process or of a nearest-neighbor Markov random field is called a hidden Markov model. Hidden Markov models are generally not Markovian. In fact, they may have complex and long range interactions, which is largely the reason for their utility. Applications include signal and image processing, speech recognition and biological modeling. We show that hidden Markov models are dense among essentially all finite-state discrete-time stationary processes and finite-state lattice-based stationary random fields. This leads to a nearly universal parameterization of stationary processes and stationary random fields, and to a consistent nonparametric estimator. We show the results of attempts to fit simple speech and texture patterns.

Rate-aided multisignal time delay detection and tracking

Time delay detection and tracking problems have been investigated in previous papers. The approach used is applicable to many problems. In those papers, a first-order Markov process was assumed for the signal parameter (time delay) dynamic model and a constant signal parameter value was assumed within a block, or processing unit of time. A performance degradation is incurred when the signal parameter value is nearly constant-rate. Also, crossing signal parameter values from different signals cannot be tracked. The detector-trackers will switch signals. To combat these difficulties, the signal parameter dynamic model is assumed to be a second-order Markov process, and signal parameter rate of change information is computed within a block. A rate-aided multisignal detection and tracking system is developed with these assumptions. >

Influence of the BWB mechanism on some performance figures of a DQDB subnetwork

This paper is based on a methodology which evaluates the parameters of a discrete-time first-order Markov process that models the slot-occupancy-pattern process on the forward bus of a DQDB network (i.e., the pattern of the slots (busy vs. empty) traveling on that bus). The methodology is an extension of the one proposed in [9] and, although applied in this paper to analyze DQDB with the Bandwidth Balancing mechanism (BWB) and Poisson arrival processes, it can be used for dual-bus networks whenever the arrival process to each node is a superposition of compound Poisson processes. The results show that the interdependence between slots is significant and, in some configurations, this interdependence is affected significantly by the BWB mechanism. To analyze how the BWB mechanism influences certain delay performance figures, we investigated the performance of a gateway that concentrates several DQDB subnetworks into a slotted trunk and we determined the access delay of the rightmost DQDB node. The accuracy of our results is verified via a simulative analysis.

Slow asymptotic convergence of LMS acoustic echo cancelers

In most acoustic echo canceler (AEC) applications, an adaptive finite impulse response (FIR) filter is employed with coefficients that are computed using the LMS algorithm. The paper establishes a theoretical basis for the slow asymptotic convergence that is often noted in practice for such applications. The analytical approach expresses the mean-square error trajectory in terms of eigenmodes and then applies the asymptotic theory of Toeplitz matrices to obtain a solution that is based on a general characterization of the actual room impulse response. The method leads to good approximations even for a moderate number of taps (N>16) and applies to both full-band and subband designs. Explicit mathematical expressions of the mean-square error convergence are derived for bandlimited white noise, a first-order Markov process, and, more generally, pth-order rational spectra and a direct power-law model, which relates to lowpass FIR filters. These expressions show that the asymptotic convergence is generally slow, being at best of order 1/t for bandlimited white noise. It is argued that input filter design cannot do much to improve slow convergence. However, the theory suggests postfiltering as a remedy that would be useful for the full-band LMS AEC and may also be applicable to subband designs.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A new transform with symmetrical coding performance for Markov (1) signals

Based on the DCT and DST-II, we present a new transform that provides a symmetrical performance for positively and negatively correlated Markov (1) signals (first-order stationary Markov process). A fast computational algorithm is presented. A sufficient condition of symmetrical transforms is also provided, and the idea of constructing the discrete cosine-sine transform (DCST) is then generalized to the Karhunen-Loeve transform (KLT) of any Markov (1) signal. >

An analytical comparative study of a class of discrete linear basis transforms

For suitability to carry out a comparative study of discrete linear basis transforms, a parametric class of these transforms is considered. It is shown that under certain conditions on the parameters, every member of this class is superior to the Walsh-Hadamard transform for first-order stationary Markov process. For high correlation of this process, the class is proved to be inferior to the slant transform. A lower bound of the performance of this class is also found.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the definition and persistence of blocking

In this paper, we have investigated under what conditions positive geopotential height anomalies can be interpreted as blocking structures. Furthermore, we have studied the persistence of blocking. The data used here are the observed daily 500 hPa geopotential heights for 10 winter seasons from 1982/83 to 1991/92. The objective blocking criteria of Dole (1978) have been compared with the subjective blocking criteria established by Rex (1950a, b). Our study indicates that if we want to relate positive geopotential height anomalies to blocking as defined by Rex, not only the amplitude, but also the latitudinal position of the anomalies is important. Positive anomalies centered at about 45°N represent a northward extension of the subtropical anticyclone. Large positive anomalies centered at about 60°N are accompanied with negative anomalies at low latitudes and blocking anticyclones centered at about 50°N. We found that blocking and strong zonal flow have the same anomaly pattern but with opposite sign. The day-to-day changes in geopotential height associated with blocking are computed and the relevance of these day-to-day changes to the persistence of the blocking pattern is examined. We find that the day-to-day changes are smaller than normal over the blocking area, but larger than normal north of it. The anomalous day-to-day changes are mainly caused by the northward shift of the jet stream and the corresponding shift in the track of the transient weather disturbances. The strength of the day-to-day changes depends very much on the exact latitudinal location where the changes are computed. These results indicate that the day-to-day changes in geopotential height do not contain information about the persistence of the blocking pattern itself, but merely about the shift of the transient eddies caused by the blocking pattern. Hence, we can not simply interpret the persistence of blocking in terms of the day-to-day changes in positive anomalies. The persistence of blocking and strong zonal flow has been examined. As blocking and strong zonal flow are associated with approximately the same anomaly pattern but with opposite sign, the persistence of blocking and strong zonal flow can be expressed in terms of the duration of the corresponding anomaly patterns. We have shown that the average duration of the blocking and strong zonal flow patterns is approximately the same. This leads us to conclude that although the daily changes in geopotential height are small near the center of the blocking high, the blocking regime itself is not more persistent than the strong zonal flow regime. The persistence of blocking and strong zonal flow has also been compared with a first-order Markov process. Generally speaking, we find no significant differences in persistence between the blocking and strong zonal flow regimes and the first-order Markov process. This leads us to conclude that blocking is not more persistent than other atmospheric anomalies of the same geographical scale. This is in agreement with Lindzen (1986). DOI: 10.1034/j.1600-0870.1994.t01-2-00004.x

An evaluation of sexual behaviour change using statistical and cognitive models

We evaluate sexual behaviour change among homosexual men enrolled in the cohorts in four AIDS Community Demonstration Projects. Behaviour change is classified following the stages of behaviour change model and described using a Markov model. Predictors of behaviour change are identified and evaluated using logit models for correlated data. Sexual behaviour change within the cohort could be modelled as a first-order Markov process. In addition, predictors suggested by models of health behaviour were correlated with particular patterns of sexual behaviour change. Our evaluation revealed a variety of patterns of sexual behaviour change in the cohorts and suggests multi-faceted interventions for promotion of behaviour change.

The binary bootstrap: inference with autocorrelated binary data

We introduce the binary bootstrap for inference with autoconelated binary data. Weempirically evaluate the standard eirors and confidence intervals created with the binary bootstrap using four stochastic process with known results: Bernoulli trials, first-order Markov processes, and long customer delays in M/M/l and D/M/10 queues. The binary bootstrap has certain advantagesover the conventional batch means method for creating confidence intervals on the probability oflong delay using only one run of a discrete-event simulation

On the approximation of the slot occupancy pattern in a DQDB network

This paper presents a novel approach to DQDB modeling in underload conditions; the innovative aspect lies in the representation of the slot-occupancy-pattern process on the forward bus. In the literature this process has been often modeled as a Bernoulli process, implying that the status (empty/busy) of consecutive slots are independent. Results presented in this paper, however, show that even at light loads the interdependence between slots is significant. Our model captures some of this interdependence by using a discrete-time first-order Markov process for the slot-occupancy-pattern process on the forward bus. In the paper we identify a simplified DQDB network for which we derive a closed formula for the joint probability density function for the slot occupancy pattern of two consecutive slots. The accuracy of our results is investigated via simulation. Finally, an application of our first-order Markov model is shown through an investigation of the performance of a gateway concentrating several DQDB networks in a slotted trunk.

Persistence and Predictability in a Perfect Model

The relationship between predictability and persistence is examined using a realistic two-level general circulation model (GCM). Predictability is measured by the average divergence of ensembles of solutions starting from perturbed initial conditions. Persistence is defined in terms of the autocorrelation function based on a single long-term model integration. The average skill of the dynamical forecasts is compared with the skill of simple persistence-based statistical forecasts. For initial errors comparable in magnitude to present-day analysis errors, the statistical forecast loses all skill after about one week, reflecting the lifetime of the lowest frequency fluctuations in the model. On the other hand, large ensemble mean dynamical forecasts would be expected to remain skillful for about three weeks. The disparity between the skill of the statistical and dynamical forecasts is greatest for the higher frequency modes, which have little memory beyond 1 day, yet remain predictable for about two weeks. For small ensembles, the error of the untempered dynamical forecasts must exceed that of the statistical forecasts for sufficiently long predictions. It is noteworthy, however, that for the low-frequency modes this is found to occur at a time when the GCM error is significantly less than would be obtained by forecasting climatology. These results are analyzed in terms of two characteristic time scales. A dynamical time scale (Td) is defined as the limiting decay time of a pseudoanomaly correlation and is taken as the measure of predictability. This is compared to the usual statistical time scale (Ts), which is the integrated autocorrelation function and measures the typical time scale of fluctuations. For the dominant low-frequency (Ts ≥ 10 days) modes of fluctuation, the dynamical time scale is between two and three times the statistical time scale. For shorter time scales, the ratio Td/Ts is even greater, reaching six for the shortest time scales considered. This is in contrast to a class of first-order Markov processes with forced-dissipative dynamics for which the two time scales, as defined, are identical.

Detection probabilities for correlated Rayleigh fading signals

Both the method of saddlepoint integration and its associated saddlepoint approximation are applied to calculating the probability of detecting correlated Rayleigh-fading signals in Gaussian noise by means of a detector that integrates M samples of the output of a quadratic rectifier. The quadrature components of the signal samples are modeled as an autoregressive moving-average process, and specific results are exhibited for a first-order Markov process. By these methods the fluctuation loss can be computed for much larger values of M and for larger values of the detection probability than previously. Values calculated by the saddlepoint approximation prove to be close enough to the exact values to be useful over a broad range of signal parameters. >

Probabilistic modelling of crossing in small samples and application of runs to hydrology

The first part of this paper presents exact probability distribution functions (PDF) of upcrossings in stationary first-order Markov processes for finite sample lengths. In the derivation of these PDF, the enumeration technique is used and the correctness of the results obtained is checked with the total probability of all the possible upcrossings, which has to equal unity for any sample length. Numerical calculations of these PDF on digital computers led to the condition that the expected number of crossings is linearly proportional to the truncation level probability. Hence, the proportionality factor is a function of the first-order autorun coefficient, which is intimately related to the first-order autocorrelation coefficient. The PDF are valid for stationary and dependent processes irrespective of their underlying probability distribution functions. In the second part of the paper, various analytical expressions concerning run properties, such as the mean positive and negative run lengths, run sums, and number of crossings, are presented. Their applications to a set of serially dependent annual flow sequences are performed. Good agreements are obtained between the derived expressions for calculating run properties and their counterparts from observed sequences.