An introduction to radial basis functions for system identification: a comparison with other neural network methods

A look is taken at the use of radial basis functions (RBFs), for nonlinear system identification. RBFs are firstly considered in detail themselves and are subsequently compared with a multi-layered perceptron (MLP), in terms of performance and usage.

Application of cylindrical distribution functions to wide-angle X-ray scattering from oriented polymers

A systematic approach is presented for obtaining cylindrical distribution functions (CDF's) of noncrystalline polymers which have been oriented by extension. The scattering patterns and CDF's are also sharpened by the method proposed by Deas and by Ruland. Data from atactic poly(methyl methacrylate) and polystyrene are analysed by these techniques. The methods could also be usefully applied to liquid crystals.

From ensemble forecasts to predictive distribution functions

The translation of an ensemble of model runs into a probability distribution is a common task in model-based prediction. Common methods for such ensemble interpretations proceed as if verification and ensemble were draws from the same underlying distribution, an assumption not viable for most, if any, real world ensembles. An alternative is to consider an ensemble as merely a source of information rather than the possible scenarios of reality. This approach, which looks for maps between ensembles and probabilistic distributions, is investigated and extended. Common methods are revisited, and an improvement to standard kernel dressing, called ‘affine kernel dressing’ (AKD), is introduced. AKD assumes an affine mapping between ensemble and verification, typically not acting on individual ensemble members but on the entire ensemble as a whole, the parameters of this mapping are determined in parallel with the other dressing parameters, including a weight assigned to the unconditioned (climatological) distribution. These amendments to standard kernel dressing, albeit simple, can improve performance significantly and are shown to be appropriate for both overdispersive and underdispersive ensembles, unlike standard kernel dressing which exacerbates over dispersion. Studies are presented using operational numerical weather predictions for two locations and data from the Lorenz63 system, demonstrating both effectiveness given operational constraints and statistical significance given a large sample.

C-H...O hydrogen bonding in 4-phenyl-benzaldehyde: a comprehensive crystallographic, spectroscopic and computational study

The crystal structure of 4-phenyl-benzaldehyde reveals the presence of a dimer linked by the C=O and C( 9)-H groups of adjacent molecules. In the liquid phase, the presence of C-(HO)-O-... bonded forms is revealed by both vibrational and NMR spectroscopy. A Delta H value of - 8.2 +/- 0.5 kJ mol(-1) for the dimerisation equilibrium is established from the temperature-dependent intensities of the bands assigned to the carbonyl-stretching modes. The NMR data suggest the preferential engagement of the C(2,6)-H and C(10/12)/C(11)-H groups as hydrogen bond donors, instead of the C(9)-H group. While ab initio calculations for the isolated dimers are unable to corroborate these NMR results, the radial distribution functions obtained from molecular dynamics simulations show a preference for C(2,6)-H and C(10/12)/C(11)-(HO)-O-... contacts relative to the C(9)-(HO)-O-... ones.

Complex formation in solutions of oppositely charged polyelectrolytes at different polyion compositions and salt content

The formation of complexes in solutions containing positively charged polyions (polycations) and a variable amount of negatively charged polyions (polyanions) has been investigated by Monte Carlo simulations. The polyions were described as flexible chains of charged hard spheres interacting through a screened Coulomb potential. The systems were analyzed in terms of cluster compositions, structure factors, and radial distribution functions. At 50% charge equivalence or less, complexes involving two polycations and one polyanion were frequent, while closer to charge equivalence, larger clusters were formed. Small and neutral complexes dominated the solution at charge equivalence in a monodisperse system, while larger clusters again dominated the solution when the polyions were made polydisperse. The cluster composition and solution structure were also examined as functions of added salt by varying the electrostatic screening length. The observed formation of clusters could be rationalized by a few simple rules.

A Monte Carlo study of solutions of oppositely charged polyelectrolytes

The formation of complexes appearing in solutions containing oppositely charged polyelectrolytes has been investigated by Monte Carlo simulations using two different models. The polyions are described as flexible chains of 20 connected charged hard spheres immersed in a homogenous dielectric background representing water. The small ions are either explicitly included or their effect described by using a screened Coulomb potential. The simulated solutions contained 10 positively charged polyions with 0, 2, or 5 negatively charged polyions and the respective counterions. Two different linear charge densities were considered, and structure factors, radial distribution functions, and polyion extensions were determined. A redistribution of positively charged polyions involving strong complexes formed between the oppositely charged polyions appeared as the number of negatively charged polyions was increased. The nature of the complexes was found to depend on the linear charge density of the chains. The simplified model involving the screened Coulomb potential gave qualitatively similar results as the model with explicit small ions. Finally, owing to the complex formation, the sampling in configurational space is nontrivial, and the efficiency of different trial moves was examined.

Mean-tracking clustering algorithm for radial basis function centre selection

Radial basis functions can be combined into a network structure that has several advantages over conventional neural network solutions. However, to operate effectively the number and positions of the basis function centres must be carefully selected. Although no rigorous algorithm exists for this purpose, several heuristic methods have been suggested. In this paper a new method is proposed in which radial basis function centres are selected by the mean-tracking clustering algorithm. The mean-tracking algorithm is compared with k means clustering and it is shown that it achieves significantly better results in terms of radial basis function performance. As well as being computationally simpler, the mean-tracking algorithm in general selects better centre positions, thus providing the radial basis functions with better modelling accuracy

The distribution of the ring current: Cluster observations

Extending previous studies, a full-circle investigation of the ring current has been made using Cluster 4-spacecraft observations near perigee, at times when the Cluster array had relatively small separations and nearly regular tetrahedral configurations, and when the Dst index was greater than −30 nT (non-storm conditions). These observations result in direct estimations of the near equatorial current density at all magnetic local times (MLT) for the first time and with sufficient accuracy, for the following observations. The results confirm that the ring current flows westward and show that the in situ average measured current density (sampled in the radial range accessed by Cluster 4–4.5RE) is asymmetric in MLT, ranging from 9 to 27 nAm−2. The direction of current is shown to be very well ordered for the whole range of MLT. Both of these results are in line with previous studies on partial ring extent. The magnitude of the current density, however, reveals a distinct asymmetry: growing from 10 to 27 nAm−2 as azimuth reduces from about 12:00MLT to 03:00 and falling from 20 to 10 nAm−2 less steadily as azimuth reduces from 24:00 to 12:00MLT. This result has not been reported before and we suggest it could reflect a number of effects. Firstly, we argue it is consistent with the operation of region-2 field aligned-currents (FACs), which are expected to flow upward into the ring current around 09:00MLT and downward out of the ring current around 14:00MLT. Secondly, we note that it is also consistent with a possible asymmetry in the radial distribution profile of current density (resulting in higher peak at 4– 4.5RE). We note that part of the enhanced current could reflect an increase in the mean AE activity (during the periods in which Cluster samples those MLT).

Lagrangian validation of numerical drifter trajectories using drifting buoys: Application to the Agulhas system

The skill of numerical Lagrangian drifter trajectories in three numerical models is assessed by comparing these numerically obtained paths to the trajectories of drifting buoys in the real ocean. The skill assessment is performed using the two-sample Kolmogorov–Smirnov statistical test. To demonstrate the assessment procedure, it is applied to three different models of the Agulhas region. The test can either be performed using crossing positions of one-dimensional sections in order to test model performance in specific locations, or using the total two-dimensional data set of trajectories. The test yields four quantities: a binary decision of model skill, a confidence level which can be used as a measure of goodness-of-fit of the model, a test statistic which can be used to determine the sensitivity of the confidence level, and cumulative distribution functions that aid in the qualitative analysis. The ordering of models by their confidence levels is the same as the ordering based on the qualitative analysis, which suggests that the method is suited for model validation. Only one of the three models, a 1/10° two-way nested regional ocean model, might have skill in the Agulhas region. The other two models, a 1/2° global model and a 1/8° assimilative model, might have skill only on some sections in the region

Orientational ordering in the nematic phase of a polyethylene glycol-peptide conjugate in aqueous solution

The orientational ordering of the nematic phase of a polyethylene glycol (PEG)-peptide block copolymer in aqueous solution is probed by small-angle neutron scattering (SANS), with the sample subjected to steady shear in a Couette cell. The PEG-peptide conjugate forms fibrils that behave as semiflexible rodlike chains. The orientational order parameters (P) over bar (2) and (P) over bar (4) are obtained by modeling the data using a series expansion approach to the form factor of uniform cylinders. The method used is independent of assumptions on the form of the singlet orientational distribution function. Good agreement with the anisotropic two-dimensional SANS patterns is obtained. The results show shear alignment starting at very low shear rates, and the orientational order parameters reach a plateau at higher shear rates with a pseudologarithmic dependence on shear rate. The most probable distribution functions correspond to fibrils parallel to the flow direction under shear, but a sample at rest shows a bimodal distribution with some of the rodlike peptide fibrils oriented perpendicular to the flow direction.

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.

The evaluation of CloudSat and CALIPSO ice microphysical products using ground-based cloud radar and lidar observations

In this paper, the statistical properties of tropical ice clouds (ice water content, visible extinction, effective radius, and total number concentration) derived from 3 yr of ground-based radar–lidar retrievals from the U.S. Department of Energy Atmospheric Radiation Measurement Climate Research Facility in Darwin, Australia, are compared with the same properties derived using the official CloudSat microphysical retrieval methods and from a simpler statistical method using radar reflectivity and air temperature. It is shown that the two official CloudSat microphysical products (2B-CWC-RO and 2B-CWC-RVOD) are statistically virtually identical. The comparison with the ground-based radar–lidar retrievals shows that all satellite methods produce ice water contents and extinctions in a much narrower range than the ground-based method and overestimate the mean vertical profiles of microphysical parameters below 10-km height by over a factor of 2. Better agreements are obtained above 10-km height. Ways to improve these estimates are suggested in this study. Effective radii retrievals from the standard CloudSat algorithms are characterized by a large positive bias of 8–12 μm. A sensitivity test shows that in response to such a bias the cloud longwave forcing is increased from 44.6 to 46.9 W m−2 (implying an error of about 5%), whereas the negative cloud shortwave forcing is increased from −81.6 to −82.8 W m−2. Further analysis reveals that these modest effects (although not insignificant) can be much larger for optically thick clouds. The statistical method using CloudSat reflectivities and air temperature was found to produce inaccurate mean vertical profiles and probability distribution functions of effective radius. This study also shows that the retrieval of the total number concentration needs to be improved in the official CloudSat microphysical methods prior to a quantitative use for the characterization of tropical ice clouds. Finally, the statistical relationship used to produce ice water content from extinction and air temperature obtained by the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite is evaluated for tropical ice clouds. It is suggested that the CALIPSO ice water content retrieval is robust for tropical ice clouds, but that the temperature dependence of the statistical relationship used should be slightly refined to better reproduce the radar–lidar retrievals.

Evaluating raw ensembles with the continuous ranked probability score

The continuous ranked probability score (CRPS) is a frequently used scoring rule. In contrast with many other scoring rules, the CRPS evaluates cumulative distribution functions. An ensemble of forecasts can easily be converted into a piecewise constant cumulative distribution function with steps at the ensemble members. This renders the CRPS a convenient scoring rule for the evaluation of ‘raw’ ensembles, obviating the need for sophisticated ensemble model output statistics or dressing methods prior to evaluation. In this article, a relation between the CRPS score and the quantile score is established. The evaluation of ‘raw’ ensembles using the CRPS is discussed in this light. It is shown that latent in this evaluation is an interpretation of the ensemble as quantiles but with non-uniform levels. This needs to be taken into account if the ensemble is evaluated further, for example with rank histograms.

Electron trapping in shear Alfvén waves that power the aurora

Results from 1D Vlasov drift-kinetic plasma simulations reveal how and where auroral electrons are accelerated along Earth’s geomagnetic field. In the warm plasma sheet, electrons become trapped in shear Alfven waves, preventing immediate wave damping. As waves move to regions with larger vTe=vA, their parallel electric field decreases, and the trapped electrons escape their influence. The resulting electron distribution functions compare favorably with in situ observations, demonstrating for the first time a self-consistent link between Alfven waves and electrons that form aurora.

Seasonal cycles and variability of O3 and H2O in the UT/LMS during SPURT

Airborne high resolution in situ measurements of a large set of trace gases including ozone (O3) and total water (H2O) in the upper troposphere and the lowermost stratosphere (UT/LMS) have been performed above Europe within the SPURT project. SPURT provides an extensive data coverage of the UT/LMS in each season within the time period between November 2001 and July 2003. In the LMS a distinct spring maximum and autumn minimum is observed in O3, whereas its annual cycle in the UT is shifted by 2–3 months later towards the end of the year. The more variable H2O measurements reveal a maximum during summer and a minimum during autumn/winter with no phase shift between the two atmospheric compartments. For a comprehensive insight into trace gas composition and variability in the UT/LMS several statistical methods are applied using chemical, thermal and dynamical vertical coordinates. In particular, 2-dimensional probability distribution functions serve as a tool to transform localised aircraft data to a more comprehensive view of the probed atmospheric region. It appears that both trace gases, O3 and H2O, reveal the most compact arrangement and are best correlated in the view of potential vorticity (PV) and distance to the local tropopause, indicating an advanced mixing state on these surfaces. Thus, strong gradients of PV seem to act as a transport barrier both in the vertical and the horizontal direction. The alignment of trace gas isopleths reflects the existence of a year-round extra-tropical tropopause transition layer. The SPURT measurements reveal that this layer is mainly affected by stratospheric air during winter/spring and by tropospheric air during autumn/summer. Normalised mixing entropy values for O3 and H2O in the LMS appear to be maximal during spring and summer, respectively, indicating highest variability of these trace gases during the respective seasons.