Conditional mean functions of non-linear models of US output

We compare a number of models of post War US output growth in terms of the degree and pattern of non-linearity they impart to the conditional mean, where we condition on either the previous period's growth rate, or the previous two periods' growth rates. The conditional means are estimated non-parametrically using a nearest-neighbour technique on data simulated from the models. In this way, we condense the complex, dynamic, responses that may be present in to graphical displays of the implied conditional mean.

A comparison of tests of non-linear cointegration with an application to the predictability of US interest rates using the term structure

We test whether there are nonlinearities in the response of short- and long-term interest rates to the spread in interest rates, and assess the out-of-sample predictability of interest rates using linear and nonlinear models. We find strong evidence of nonlinearities in the response of interest rates to the spread. Nonlinearities are shown to result in more accurate short-horizon forecasts, especially of the spread.

Forecasting economic and financial time series with non-linear models

In this paper we discuss the current state-of-the-art in estimating, evaluating, and selecting among non-linear forecasting models for economic and financial time series. We review theoretical and empirical issues, including predictive density, interval and point evaluation and model selection, loss functions, data-mining, and aggregation. In addition, we argue that although the evidence in favor of constructing forecasts using non-linear models is rather sparse, there is reason to be optimistic. However, much remains to be done. Finally, we outline a variety of topics for future research, and discuss a number of areas which have received considerable attention in the recent literature, but where many questions remain.

Forecasting US output growth with non-linear models in the presence of data uncertainty

We consider the impact of data revisions on the forecast performance of a SETAR regime-switching model of U.S. output growth. The impact of data uncertainty in real-time forecasting will affect a model's forecast performance via the effect on the model parameter estimates as well as via the forecast being conditioned on data measured with error. We find that benchmark revisions do affect the performance of the non-linear model of the growth rate, and that the performance relative to a linear comparator deteriorates in real-time compared to a pseudo out-of-sample forecasting exercise.

Linear and non-linear transmission of equity return volatility: evidence from the US, Japan and Australia

This paper models the transmission of shocks between the US, Japanese and Australian equity markets. Tests for the existence of linear and non-linear transmission of volatility across the markets are performed using parametric and non-parametric techniques. In particular the size and sign of return innovations are important factors in determining the degree of spillovers in volatility. It is found that a multivariate asymmetric GARCH formulation can explain almost all of the non-linear causality between markets. These results have important implications for the construction of models and forecasts of international equity returns.

Linear and non-linear (non-)forecastability of high-frequency exchange rates

This paper forecasts Daily Sterling exchange rate returns using various naive, linear and non-linear univariate time-series models. The accuracy of the forecasts is evaluated using mean squared error and sign prediction criteria. These show only a very modest improvement over forecasts generated by a random walk model. The Pesaran–Timmerman test and a comparison with forecasts generated artificially shows that even the best models have no evidence of market timing ability.

A new adaptive multiple modelling approach for non-linear and non-stationary systems

This paper proposes a novel adaptive multiple modelling algorithm for non-linear and non-stationary systems. This simple modelling paradigm comprises K candidate sub-models which are all linear. With data available in an online fashion, the performance of all candidate sub-models are monitored based on the most recent data window, and M best sub-models are selected from the K candidates. The weight coefficients of the selected sub-model are adapted via the recursive least square (RLS) algorithm, while the coefficients of the remaining sub-models are unchanged. These M model predictions are then optimally combined to produce the multi-model output. We propose to minimise the mean square error based on a recent data window, and apply the sum to one constraint to the combination parameters, leading to a closed-form solution, so that maximal computational efficiency can be achieved. In addition, at each time step, the model prediction is chosen from either the resultant multiple model or the best sub-model, whichever is the best. Simulation results are given in comparison with some typical alternatives, including the linear RLS algorithm and a number of online non-linear approaches, in terms of modelling performance and time consumption.

Torque in mooring chain. Part 1: Background and theory

Chain in both its forms - common (or stud-less) and stud-link - has many engineering applications. It is widely used as a component in the moorings of offshore floating systems, where its ruggedness and corrosion resistance make it an attractive choice. Chain exhibits some interesting behaviour in that when straight and subject to an axial load it does not twist or generate any torque, but if twisted or loaded when in a twisted condition it behaves in a highly non-linear manner, with the torque dependent upon the level of twist and axial load. Clearly an understanding of the way in which chains may behave and interact with other mooring components (such as wire rope, which also exhibits coupling between axial load and generated torque) when they are in service is essential. However, the sizes of chain that are in use in offshore moorings (typical bar diameters are 75 mm and greater) are too large to allow easy testing. This paper, which is in two parts, aims to address the issues and considerations relevant to torque in mooring chain. The first part introduces a frictionless theory that predicts the resultant torques and 'lift' in the links as non-dimensionalized functions of the angle of twist. Fortran code is presented in an Appendix, which allows the reader to make use of the analysis. The second part of the paper presents results from experimental work on both stud-less (41 mm) and stud-link (20.5 and 56 mm) chains. Torsional data are presented in both 'constant twist' and 'constant load' forms, as well as considering the lift between the links.

Optimal piecewise locally linear modeling

Associative memory networks such as Radial Basis Functions, Neurofuzzy and Fuzzy Logic used for modelling nonlinear processes suffer from the curse of dimensionality (COD), in that as the input dimension increases the parameterization, computation cost, training data requirements, etc. increase exponentially. Here a new algorithm is introduced for the construction of a Delaunay input space partitioned optimal piecewise locally linear models to overcome the COD as well as generate locally linear models directly amenable to linear control and estimation algorithms. The training of the model is configured as a new mixture of experts network with a new fast decision rule derived using convex set theory. A very fast simulated reannealing (VFSR) algorithm is utilized to search a global optimal solution of the Delaunay input space partition. A benchmark non-linear time series is used to demonstrate the new approach.

Torque in mooring chain. Part 2: Experimental investigation

Chain is a commonly used component in offshore moorings where its ruggedness and corrosion resistance make it an attractive choice. Another attractive property is that a straight chain is inherently torque balanced. Having said this, if a chain is loaded in a twisted condition, or twisted when under load, it exhibits highly non-linear torsional behaviour. The consequences of this behaviour can cause handling difficulties or may compromise the integrity of the mooring system, and care must be taken to avoid problems for both the chain and any components to which it is connected. Even with knowledge of the potential problems, there will always be occasions where, despite the utmost care, twist is unavoidable. Thus it is important for the engineer to be able to determine the effects. A frictionless theory has been developed in Part 1 of the paper that may be used to predict the resultant torques and movement or 'lift' in the links as non-dimensional functions of the angle of twist. The present part of the paper describes a series of experiments undertaken on both studless and stud-link chain to allow comparison of this theoretical model with experimental data. Results are presented for the torsional response and link lift for 'constant twist' and 'constant load' type tests on chains of three different link sizes.

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

Linear-wavelet models applied to the identification of a two-link manipulator

This paper shows that a wavelet network and a linear term can be advantageously combined for the purpose of non linear system identification. The theoretical foundation of this approach is laid by proving that radial wavelets are orthogonal to linear functions. A constructive procedure for building such nonlinear regression structures, termed linear-wavelet models, is described. For illustration, sim ulation data are used to identify a model for a two-link robotic manipulator. The results show that the introduction of wavelets does improve the prediction ability of a linear model.

Aerosol-induced correlation between visibility and atmospheric electricity.

Atmospheric aerosol acts to both reduce the background concentration of natural cluster ions, and to attenuate optical propagation. Hence, the presence of aerosol has two consequences, the reduction of the air’s electrical conductivity and the visual range. Ion-aerosol theory and Koschmieder’s visibility theory are combined here to derive the related non-linear variation of the atmospheric electric potential gradient with visual range. A substantial sensitivity is found under poor visual range conditions, but, for good visual range conditions the sensitivity diminishes and little influence of local aerosol on the fair weather potential gradient occurs. This allows visual range measurements, made simply and routinely at many meteorological sites, to provide inference about the local air’s electrical properties.

Measures of observation impact in non-Gaussian data assimilation

ABSTRACT Non-Gaussian/non-linear data assimilation is becoming an increasingly important area of research in the Geosciences as the resolution and non-linearity of models are increased and more and more non-linear observation operators are being used. In this study, we look at the effect of relaxing the assumption of a Gaussian prior on the impact of observations within the data assimilation system. Three different measures of observation impact are studied: the sensitivity of the posterior mean to the observations, mutual information and relative entropy. The sensitivity of the posterior mean is derived analytically when the prior is modelled by a simplified Gaussian mixture and the observation errors are Gaussian. It is found that the sensitivity is a strong function of the value of the observation and proportional to the posterior variance. Similarly, relative entropy is found to be a strong function of the value of the observation. However, the errors in estimating these two measures using a Gaussian approximation to the prior can differ significantly. This hampers conclusions about the effect of the non-Gaussian prior on observation impact. Mutual information does not depend on the value of the observation and is seen to be close to its Gaussian approximation. These findings are illustrated with the particle filter applied to the Lorenz ’63 system. This article is concluded with a discussion of the appropriateness of these measures of observation impact for different situations.