Statistical work in hydrology on the topic of riverflow time series is discussed. The theme is the "data-generation method" which in the present situation aims to produce simulations corresponding to a chosen historical riverflow series; they are required to resemble the historical data in respect of hydrologically important properties, such as runs of low flows. In practice the simulations are then used as input to a projected water resources system, but this aspect is not treated here. Capture of statistical resemblence entails a thorough statistical analysis of the historical series, and the paper describes the distinctive non-Gaussian and periodic features of typical riverflow series; it emphasizes the hydrological importance of seasonality, marginal distribution, dependence and crossing properties. The paper then describes hydrological adaptions and use of some of the traditional time series and shot noise models. It discusses the hydrologically motivated long memory fractional noise and broken line models; here the aim is to bring them to the attention of statisticians for appraisal while at the same time making their theory more accessible to hydrologists. Places where the presently used methodology needs consolidating are mentioned, as are areas where further statistical and mathematical work would be of value.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. International Statistical Institute (ISI) is collaborating with JSTOR to digitize, preserve and extend access to International Statistical Review / Revue Internationale de Statistique. SummaryThis paper gives a review and development of what is currently known about the directionality (irreversibility) of time series models, together with briefer coverage of the still limited statistical methodology. Reversibility is shown to imply stationarity; Weiss's result concerning the reversibility of linear Gaussian processes is stressed, and contrasted to the directional nature of much time series data. Reversed ARMA models are explored, and non-linear examples given; the stationarity and invertibility conditions of ARMA models are shown to be implicitly directional, and a consequence of the future-independent nature of such models. Invertibility is extended to the two-sided futuredependent generalised linear model, and applied to reversible moving average models. The directional and reversible implications of autoregressive roots are covered. Work applying directional-sensitive methods of statistical analysis to reversed data series is mentioned; possible dangers in transforming directional series to Gaussian marginal distributions are noted. The directional nature of most non-linear models is invoked to emphasise the current importance of the area.

A new time series model for exponential variables having first-order autoregressive structure is presented. Unlike the recently studied standard autoregressive model in exponential variables (ear(1)), runs of constantly scaled values are avoidable, and the two parameter structure allows some adjustment of directional effects in sample path behaviour. The model is further developed by the use of cross-coupling and antithetic ideas to allow negative dependency. Joint distributions and autocorrelations are investigated. A transformed version of the model has a uniform marginal distribution and its correlation and regression structures are also obtained. Estimation aspects of the models are briefly considered.

The paper explores the calculation of exact bit errror rates (BERs) for some single-user chaotic-shift-keying (CSK) communications systems, in contrast to approximate Gaussian-based approximations in current use. The conventional signal-to-noiseratio approach is shown to give only lower bounds on the BERs. An analytical Gaussian approach based on exact mean and variance of the decoder function gives inexact results. Exact BERs are given here for several CSK systems with spreading sequences from different types of chaotic map. They achieve exactness from fully exploiting the dynamical and statistical features of the systems and the results correspond theoretically to impractically large Monte Carlo simulations. A further aspect of the paper is the derivation of likelihood optimal bit decoders which can be superior to correlation decoders. The nonapplicability of Gaussian assumptions is viewed through some exact distributional results for one system.

SUMMARY An approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented; the approach is illustrated by an analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution. The stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients, called the NEAR(2) model; it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random. The exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable; other applications of this structure are briefly considered. Dependency in the NEAR(2) process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure; this involves defining linear uncorrelated residuals which are dependent, and then assessing this higher order dependence by standard time series computations. The application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling.