Curved Chaotic Map Time Series Models and Their Stochastic Reversals

This paper considers two types of chaotic map time series models, including the well-known tent, logistic and binary-shift maps as special cases; these are called curved tent and curved binary families. Deterministic behaviour is investigated by invariant distributions, Lyapunov exponents, and by serial dependency. Stochastic time reversal of the families is shown to produce models which have a broader range of stochastic and chaotic properties than their deterministic counterparts. The marginal distributions may have concentrations and restricted supports and are shown to be a non-standard class of invariant distribution. Dependenc y is generally weaker with the reversed stochastic models. The work gives a broad statistical account of deterministic and stochastically reversed map models, such as are emerging in random number generation, communica tion systems and cryptography     

Prediction of Chaotic Time Series in the Presence of Measurement Error: the Importance of Initial Conditions

In this paper we argue that even if a dynamic relationship can be well described by a deterministic system, retrieving this relationship from an empirical time series has to take into account some, although possibly very small measurement error in the observations. Therefore, measuring the initial conditions for prediction may become much more difficult since one now has a combination of deterministic and stochastic elements. We introduce a partial smoothing estimator for estimating the unobserved initial conditions. We will show that this estimator allows to reduce the effects of measurement error for predictions although the reduction may be small in the presence of strong chaotic dynamics. This will be illustrated using the logistic map.     

Chaos: But Not in Both Directions!

This paper will informally explore the reversal of some stochastic autoregressive processes, which lead to deterministically chaotic processes. Correspondingly, the stochastic reversal of map models is shown to lead to a new class of invariant distribution. Finally, some connections between congruential recursions and independence in discretized chaotic processes are illustrated.     

A Short Note on the Dependence Structure of a Stochastic Process Generated by the Asymmetric Cusp Map

Intermittency maps are well-known to be capable of generating stochastic processes with slowly decaying cross covariances. The present paper considers the asymmetric cusp map and derives an asymptotic lower bound of the autocovariance function of a stochastic process generated by this map. As a consequence, such a stochastic process is shown to belong to the class of long memory processes.     

A class of stochastic epidemic models and its deterministic counterpart

A general stochastic model for the spread of an epidemic developing in a closed population is introduced. Each model consisting of a discrete-time Markov chain involves a deterministic counterpart represented by an ordinary differential equation. Our framework involves various epidemic models such as a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. We prove the asymptotic consistency of the stochastic model regarding a deterministic model; this means that for a large population both modelings are similar. Moreover, a Central Limit Theorem for the fluctuations of the stochastic modeling regarding the deterministic model is also proved.     

Some statistical opportunities in the agricultural sciences

Quantitative plant-disease epidemiology gained its primary motivation with the publication of VanderPlank's (1963) mathematical treatise, since which a range of predominantly deterministic models has been proposed by plant pathologists to describe the temporal dynamic of plant diseases. In this article, a number of results are reviewed. In particular, a summary is made of the primary models developed in the literature together with diseases for which they were proposed; estimation of the infection rate is considered for the generalized logistic model; and a bivariate stochastic model for stem and leaf infection is developed and used to obtain measures of plant disease stress.     

Persistence and existence of stationary measures for a logistic growth model with predation

We consider a stochastic logistic growth model with a predation term, and a diffusive stochastic part with a power-type coefficient. We provide criteria for the persistence of the population and for the existence and uniqueness of a stationary measure. Furthermore, we perform a detailed study of the densities of the stationary measures resorting to the forward Kolmogorov equation. We compile our results in a stochastic bifurcation diagram, drawing comparisons with the corresponding deterministic model.     

The variable-interval arithmetic coding using asymptotic deterministic randomness for data compression and encryption

Different from the traditional way which compresses the data first and then encrypts the compressed bit-stream later, a novel compression and encryption scheme using variable model arithmetic coding and coupled chaotic system can encrypt and compress the input plaintext synchronously. However, there will be a compromise between the amount of compression achieved and the amount of security incorporated, because the compression efficiency is determined by the key bit-stream. In this paper, an improved scheme using variable-interval arithmetic coding and asymptotic deterministic randomness has been proposed. The improved scheme is secure because the key bit-stream generated by the asymptotic deterministic randomness can resist previous attacks against chaotic encryption. In addition, the compression efficiency will not change with the key bit-stream, because the statistical model will no longer be changed. The results show that the new scheme can achieve high security and compression efficiency.     

On positive harris recurrence of stochastic fluid networks

The stochastic fluid networks we consider here are open multiclass fluid networks with Markov-modulated inflow rate. They are typically used as models for manufacturing and telecommunication systems. The main aim of this paper is to investigate the question of positive Harris recurrence of the joint process of buffer content and inflow rate. It will turn out that a Markovian server allocation exists under which the process is positive Harris recurrent if the usual traffic conditions are satisfied on average. Moreover, for special networks like single-server networks, re-entrant lines and Jackson networks it is possible to show that certain service disciplines induce positive Harris recurrent state processes. As a by-product we show that these stochastic fluid networks converge under appropriately defined scaling to solutions of deterministic fluid models.     

TWO-SPECIES SYSTEMS IN RANDOM ENVIRONMENT

An exactly solvable model for two interacting species with stochastic parameters is investigated. The parameter describing the interaction is assumed to be characterized by a dichotomic Markov process. It is remarkable that in contrast to the usual oscillatory behaviour in the deterministic case the expected value of the logarithm of each population size, as a consequence of parametric stochastic perturbations, attains a stable value as time increases. check that in their absence the damping term vanishes, leading to the deterministic model of Gomatam.     

Estimation of finite population kurtosis under two-phase sampling for nonresponse

In this paper an estimator of finite population kurtosis computed under the two-phase sampling for nonresponse is proposed. The formulas characterizing its asymptotic properties are derived using Taylor linearization technique for the general situation of arbitrary sampling designs in both phases and stochastic nonresponse represented by arbitrary response distribution. An important special case of simple random sampling without replacement and deterministic nonresponse is also considered.     

在经济数据中寻找混沌

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Improved bridge constructs for stochastic differential equations

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Itô stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler–Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.     

Simple diagnostic tools for inverstigating linear trends in time series

This paper presents a simple diagnostic tool for time series. Based on a coefficient α that veries between 1 and 0, the tool measures the approximation of a time series to an arithmetic progression (i.e., a linear function of time). The proposed α is based on the ratio of the average squared second difference to the average squared first difference of the ginven series. As such, α reduces to the Von Neumann ratio η of the series of first differences, namely, α = 1-η/4. For an arithmetic progression α = 1, and deviations therefrom cause it to decrease. Unlike the correlation coefficient (between the entries and the indics), α is sensitive to local, or piecewise, linearity. Here α is evaluated for an assortment of simple time series models such as random walk, AR(1) and MA(1). Large-sample distribution yields a number of commonly used stochastic models including non-normal process. For most standard deterministic and stochastic models, α stabilizes as n approaches infinity, and provides a statistic that is capable of distinguishing between many different standard random and deterministic models. A further measure τ, which together with α distinguisches between random walks and deterministic trend plus i.i.d., is also suggested. Some examples based on empirical data are also studied.     

A new characterization of the jump rate for piecewise-deterministic Markov processes with discrete transitions

Piecewise-deterministic Markov processes form a general class of non diffusion stochastic models that involve both deterministic trajectories and random jumps at random times. In this paper, we state a new characterization of the jump rate of such a process with discrete transitions. We deduce from this result a non parametric technique for estimating this feature of interest. We state the uniform convergence in probability of the estimator. The methodology is illustrated on a numerical example.     

A consistency result in general censoring models

In this paper we prove a consistency result for sieved maximum likelihood estimators of the density in general random censoring models with covariates. The proof is based on the method of functional estimation. The estimation error is decomposed in a deterministic approximation error and the stochastic estimation error. The main part of the proof is to establish a uniform law of large numbers for the conditional log-likelihood functional, by using results and techniques from empirical process theory.     

Type-I intermittency from Markov binary block visibility graph perspective

In this work, the type-I intermittency is studied from the optimized Markov binary visibility graphs perspective. We consider a local Poincaré map such as the logistic map that is a simple model for exhibiting this type of intermittency. To consider the acceptance gate as G≪0.01, we show that the transition between laminar and non-laminar zones in type-I intermittency takes distinct phases and regions. According to their behavioral characteristics, we call them as pure, switching, threshold, trapping, and transforming phases for the laminar zone and initial, terminal reinjection, and chaotic burst regions for non-laminar zone. We investigate their properties based on statistical tools such as the maximum and the mean length of the laminar zone and also length distributions of the laminar zone. For further investigation, we study degree distribution of the complex network generated by type-I intermittency time series and finally, predict various behaviors of phases and regions by proposed theoretical degree distributions.KEYWORDS: Type-I intermittency, binary block design, Markov binary visibility graph, chaos

We study the effect of the acceptance gate on the type-I intermittency derived from a local Poincaré map such as the logistic map. Then, based on system behaviors during the transition between laminar and non-laminar zones, we consider them as dynamic states such as pure, switching, threshold, trapping, and transforming phases for laminar zone and initial, terminal reinjection, and chaotic burst regions for non-laminar zone. We define an optimized Markov binary visibility graphs using the binary block design in order to describe the type-I intermittency from the complex network perspective. Statistical results proved this claim and in order to further illustrate this claim, we use from network properties of the optimized Markov binary visibility graphs.     

New concepts in deterministic growth curve forecasting

Attempts to better the Performance of classical growth curve functions have met with limited success. Construction industry projects highlighted the need to improve deterministic models rather than the stochastic methodologies which are nearly always based on the former. New concepts (changed for the first time since 1825) are formulated and used to generate multi-component deterministic models. Six highly diverse case studies, of which three are presented, were used to test one model and its autocorrelation form. Trial forecast standard errors showed a drop of 50% when compared to classical and stochastic models. Among the by-products of this work are uses of normalisation, scaling and a simple statistical procedure to estimate linear constants. A different consequence of the new concepts thew light on the problem of predicting a consumption process in marketing. The major implications of this research show the import of the new concepts and diversification of the fields of study on deterministic modelling; and also the need to reappraise the functional interface with many of the underlying processes of growth.     

SEMIFAR forecasts, with applications to foreign exchange rates

SEMIFAR models introduced in Beran (1997. Estimating trends, long-range dependence and nonstationarity, preprint) provide a semiparametric modelling framework that enables the data analyst to separate deterministic and stochastic trends as well as short- and long-memory components in an observed time series. A correct distinction between these components, and in particular, the decision which of the components may be present in the data have an important impact on forecasts. In this paper, forecasts and forecast intervals for SEMIFAR models are obtained. The forecasts are based on an extrapolation of the nonparametric trend function and optimal forecasts of the stochastic component. In the data analytical part of the paper, the proposed method is applied to foreign exchange rates from Europe and Asia.     

Bridges between deterministic and probabilistic models for binary data

For the analysis of binary data, various deterministic models have been proposed, which are generally simpler to fit and easier to understand than probabilistic models. We claim that corresponding to any deterministic model is an implicit stochastic model in which the deterministic model fits imperfectly, with errors occurring at random. In the context of binary data, we consider a model in which the probability of error depends on the model prediction. We show how to fit this model using a stochastic modification of deterministic optimization schemes.The advantages of fitting the stochastic model explicitly (rather than implicitly, by simply fitting a deterministic model and accepting the occurrence of errors) include quantification of uncertainty in the deterministic model’s parameter estimates, better estimation of the true model error rate, and the ability to check the fit of the model nontrivially. We illustrate this with a simple theoretical example of item response data and with empirical examples from archeology and the psychology of choice.