The relative order and inverses of recurrent networks

Differential geometry is used to investigate the structure of neural-network-based control systems. The key aspect is relative order—an invariant property of dynamic systems. Finite relative order allows the specification of a minimal architecture for a recurrent network. Any system with finite relative order has a left inverse. It is shown that a recurrent network with finite relative order has a local inverse that is also a recurrent network with the same weights. The results have implications for the use of recurrent networks in the inverse-model-based control of nonlinear systems.

Dynamic recurrent neural network for system identification and control

A dynamic recurrent neural network (DRNN) that can be viewed as a generalisation of the Hopfield neural network is proposed to identify and control a class of control affine systems. In this approach, the identified network is used in the context of the differential geometric control to synthesise a state feedback that cancels the nonlinear terms of the plant yielding a linear plant which can then be controlled using a standard PID controller.

Testing for spatial autocorrelation: the regressors that make the power disappear

We show that for any sample size, any size of the test, and any weights matrix outside a small class of exceptions, there exists a positive measure set of regression spaces such that the power of the Cli-Ord test vanishes as the autocorrelation increases in a spatial error model. This result extends to the tests that dene the Gaussian power envelope of all invariant tests for residual spatial autocorrelation. In most cases, the regression spaces such that the problem occurs depend on the size of the test, but there also exist regression spaces such that the power vanishes regardless of the size. A characterization of such particularly hostile regression spaces is provided.

On the use of geometric moments to examine the continuum of sudden stratospheric warmings

The polar winter stratospheric vortex is a coherent structure that undergoes different types of deformation that can be revealed by the geometric invariant moments. Three moments are used—the aspect ratio, the centroid latitude, and the area of the vortex based on stratospheric data from the 40-yr ECMWF Re-Analysis (ERA-40) project—to study sudden stratospheric warmings. Hierarchical clustering combined with data image visualization techniques is used as well. Using the gap statistic, three optimal clusters are obtained based on the three geometric moments considered here. The 850-K potential vorticity field, as well as the vertical profiles of polar temperature and zonal wind, provides evidence that the clusters represent, respectively, the undisturbed (U), displaced (D), and split (S) states of the polar vortex. This systematic method for identifying and characterizing the state of the polar vortex using objective methods is useful as a tool for analyzing observations and as a test for climate models to simulate the observations. The method correctly identifies all previously identified major warmings and also identifies significant minor warmings where the atmosphere is substantially disturbed but does not quite meet the criteria to qualify as a major stratospheric warming.

Resource allocation analysis of the stochastic diffusion search

The Stochastic Diffusion Search (SDS) was developed as a solution to the best-fit search problem. Thus, as a special case it is capable of solving the transform invariant pattern recognition problem. SDS is efficient and, although inherently probabilistic, produces very reliable solutions in widely ranging search conditions. However, to date a systematic formal investigation of its properties has not been carried out. This thesis addresses this problem. The thesis reports results pertaining to the global convergence of SDS as well as characterising its time complexity. However, the main emphasis of the work, reports on the resource allocation aspect of the Stochastic Diffusion Search operations. The thesis introduces a novel model of the algorithm, generalising an Ehrenfest Urn Model from statistical physics. This approach makes it possible to obtain a thorough characterisation of the response of the algorithm in terms of the parameters describing the search conditions in case of a unique best-fit pattern in the search space. This model is further generalised in order to account for different search conditions: two solutions in the search space and search for a unique solution in a noisy search space. Also an approximate solution in the case of two alternative solutions is proposed and compared with predictions of the extended Ehrenfest Urn model. The analysis performed enabled a quantitative characterisation of the Stochastic Diffusion Search in terms of exploration and exploitation of the search space. It appeared that SDS is biased towards the latter mode of operation. This novel perspective on the Stochastic Diffusion Search lead to an investigation of extensions of the standard SDS, which would strike a different balance between these two modes of search space processing. Thus, two novel algorithms were derived from the standard Stochastic Diffusion Search, ‘context-free’ and ‘context-sensitive’ SDS, and their properties were analysed with respect to resource allocation. It appeared that they shared some of the desired features of their predecessor but also possessed some properties not present in the classic SDS. The theory developed in the thesis was illustrated throughout with carefully chosen simulations of a best-fit search for a string pattern, a simple but representative domain, enabling careful control of search conditions.

Small-world effects in lattice stochastic diffusion search

Stochastic Diffusion Search is an efficient probabilistic bestfit search technique, capable of transformation invariant pattern matching. Although inherently parallel in operation it is difficult to implement efficiently in hardware as it requires full inter-agent connectivity. This paper describes a lattice implementation, which, while qualitatively retaining the properties of the original algorithm, restricts connectivity, enabling simpler implementation on parallel hardware. Diffusion times are examined for different network topologies, ranging from ordered lattices, over small-world networks to random graphs.

Blind equalization of nonminimum phase channels: higher order cumulant based algorithm

Higher order cumulant analysis is applied to the blind equalization of linear time-invariant (LTI) nonminimum-phase channels. The channel model is moving-average based. To identify the moving average parameters of channels, a higher-order cumulant fitting approach is adopted in which a novel relay algorithm is proposed to obtain the global solution. In addition, the technique incorporates model order determination. The transmitted data are considered as independently identically distributed random variables over some discrete finite set (e.g., set {±1, ±3}). A transformation scheme is suggested so that third-order cumulant analysis can be applied to this type of data. Simulation examples verify the feasibility and potential of the algorithm. Performance is compared with that of the noncumulant-based Sato scheme in terms of the steady state MSE and convergence rate.

Blind equalisation of multilevel PAM data for nonminimum phase channels via second- and fourth-order cumulants

This paper introduces a new blind equalisation algorithm for the pulse amplitude modulation (PAM) data transmitted through nonminimum phase (NMP) channels. The algorithm itself is based on a noncausal AR model of communication channels and the second- and fourth-order cumulants of the received data series, where only the diagonal slices of cumulants are used. The AR parameters are adjusted at each sample by using a successive over-relaxation (SOR) scheme, a variety of the ordinary LMS scheme, but with a faster convergence rate and a greater robustness to the selection of the ‘step-size’ in iterations. Computer simulations are implemented for both linear time-invariant (LTI) and linear time-variant (LTV) NMP channels, and the results show that the algorithm proposed in this paper has a fast convergence rate and a potential capability to track the LTV NMP channels.

Representing geometrical knowledge

This paper introduces perspex algebra which is being developed as a common representation of geometrical knowledge. A perspex can currently be interpreted in one of four ways. First, the algebraic perspex is a generalization of matrices, it provides the most general representation for all of the interpretations of a perspex. The algebraic perspex can be used to describe arbitrary sets of coordinates. The remaining three interpretations of the perspex are all related to square matrices and operate in a Euclidean model of projective space-time, called perspex space. Perspex space differs from the usual Euclidean model of projective space in that it contains the point at nullity. It is argued that the point at nullity is necessary for a consistent account of perspective in top-down vision. Second, the geometric perspex is a simplex in perspex space. It can be used as a primitive building block for shapes, or as a way of recording landmarks on shapes. Third, the transformational perspex describes linear transformations in perspex space that provide the affine and perspective transformations in space-time. It can be used to match a prototype shape to an image, even in so called 'accidental' views where the depth of an object disappears from view, or an object stays in the same place across time. Fourth, the parametric perspex describes the geometric and transformational perspexes in terms of parameters that are related to everyday English descriptions. The parametric perspex can be used to obtain both continuous and categorical perception of objects. The paper ends with a discussion of issues related to using a perspex to describe logic.

On-line handwritten segmentation in linear drawings

This paper addresses the problem of tracking line segments corresponding to on-line handwritten obtained through a digitizer tablet. The approach is based on Kalman filtering to model linear portions of on-line handwritten, particularly, handwritten numerals, and to detect abrupt changes in handwritten direction underlying a model change. This approach uses a Kalman filter framework constrained by a normalized line equation, where quadratic terms are linearized through a first-order Taylor expansion. The modeling is then carried out under the assumption that the state is deterministic and time-invariant, while the detection relies on double thresholding mechanism which tests for a violation of this assumption. The first threshold is based on an approach of layout kinetics. The second one takes into account the jump in angle between the past observed direction of layout and its current direction. The method proposed enables real-time processing. To illustrate the methodology proposed, some results obtained from handwritten numerals are presented.

Stochastic volatility jump-diffusions for European equity index dynamics

Major research on equity index dynamics has investigated only US indices (usually the S&P 500) and has provided contradictory results. In this paper a clarification and extension of that previous research is given. We find that European equity indices have quite different dynamics from the S&P 500. Each of the European indices considered may be satisfactorily modelled using either an affine model with price and volatility jumps or a GARCH volatility process without jumps. The S&P 500 dynamics are much more difficult to capture in a jump-diffusion framework.

Dynamic input/output linearization using recurrent neural networks

A dynamic recurrent neural network (DRNN) is used to input/output linearize a control affine system in the globally linearizing control (GLC) structure. The network is trained as a part of a closed loop that involves a PI controller, the goal is to use the network, as a dynamic feedback, to cancel the nonlinear terms of the plant. The stability of the configuration is guarantee if the network and the plant are asymptotically stable and the linearizing input is bounded.

Stable linearization using multilayer neural networks

The main limitation of linearization theory that prevents its application in practical problems is the need for an exact knowledge of the plant. This requirement is eliminated and it is shown that a multilayer network can synthesise the state feedback coefficients that linearize a nonlinear control affine plant. The stability of the linearizing closed loop can be guaranteed if the autonomous plant is asymptotically stable and the state feedback is bounded.

The relationship between land–ccean surface temperature contrast and radiative forcing

Previous work has demonstrated that observed and modeled climates show a near-time-invariant ratio of mean land to mean ocean surface temperature change under transient and equilibrium global warming. This study confirms this in a range of atmospheric models coupled to perturbed sea surface temperatures (SSTs), slab (thermodynamics only) oceans, and a fully coupled ocean. Away from equilibrium, it is found that the atmospheric processes that maintain the ratio cause a land-to-ocean heat transport anomaly that can be approximated using a two-box energy balance model. When climate is forced by increasing atmospheric CO2 concentration, the heat transport anomaly moves heat from land to ocean, constraining the land to warm in step with the ocean surface, despite the small heat capacity of the land. The heat transport anomaly is strongly related to the top-of-atmosphere radiative flux imbalance, and hence it tends to a small value as equilibrium is approached. In contrast, when climate is forced by prescribing changes in SSTs, the heat transport anomaly replaces ‘‘missing’’ radiative forcing over land by moving heat from ocean to land, warming the land surface. The heat transport anomaly remains substantial in steady state. These results are consistent with earlier studies that found that both land and ocean surface temperature changes may be approximated as local responses to global mean radiative forcing. The modeled heat transport anomaly has large impacts on surface heat fluxes but small impacts on precipitation, circulation, and cloud radiative forcing compared with the impacts of surface temperature change. No substantial nonlinearities are found in these atmospheric variables when the effects of forcing and surface temperature change are added.

Development of orientation during electrospinning of fibres of poly(ε-caprolactone)

We explore the influence of a rotating collector on the internal structure of poly(ε-caprolactone) fibres electrospun from a solution in dichloroethane. We find that above a threshold collector speed, the mean fibre diameter reduces as the speed increases and the fibres are further extended. Small-angle and wide-angle X-ray scattering techniques show a preferred orientation of the lamellar crystals normal to the fibre axis which increases with collector speed to a maximum and then reduces. We have separated out the processes of fibre alignment on the collector and the orientation of crystals within the fibres. There are several stages to this behaviour which correspond to the situations (a) where the collector speed is slower than the fibre spinning rate, (b) the fibre is mechanically extended by the rotating collector and (c) where the deformation leads to fibre fracture. The mechanical deformation leads to a development of preferred orientation with extension which is similar to the prediction of the pseudo-affine deformation model and suggests that the deformation takes place during the spinning process after the crystals have formed.