Modelling of non-stationary processes using radial basis function networks

This paper reports preliminary progress on a principled approach to modelling nonstationary phenomena using neural networks. We are concerned with both parameter and model order complexity estimation. The basic methodology assumes a Bayesian foundation. However to allow the construction of pragmatic models, successive approximations have to be made to permit computational tractibility. The lowest order corresponds to the (Extended) Kalman filter approach to parameter estimation which has already been applied to neural networks. We illustrate some of the deficiencies of the existing approaches and discuss our preliminary generalisations, by considering the application to nonstationary time series.

Dynamics of on-line learning in radial basis function networks

On-line learning is examined for the radial basis function network, an important and practical type of neural network. The evolution of generalization error is calculated within a framework which allows the phenomena of the learning process, such as the specialization of the hidden units, to be analyzed. The distinct stages of training are elucidated, and the role of the learning rate described. The three most important stages of training, the symmetric phase, the symmetry-breaking phase, and the convergence phase, are analyzed in detail; the convergence phase analysis allows derivation of maximal and optimal learning rates. As well as finding the evolution of the mean system parameters, the variances of these parameters are derived and shown to be typically small. Finally, the analytic results are strongly confirmed by simulations.

On-line learning in radial basis functions networks

An analytic investigation of the average case learning and generalization properties of Radial Basis Function Networks (RBFs) is presented, utilising on-line gradient descent as the learning rule. The analytic method employed allows both the calculation of generalization error and the examination of the internal dynamics of the network. The generalization error and internal dynamics are then used to examine the role of the learning rate and the specialization of the hidden units, which gives insight into decreasing the time required for training. The realizable and over-realizable cases are studied in detail; the phase of learning in which the hidden units are unspecialized (symmetric phase) and the phase in which asymptotic convergence occurs are analyzed, and their typical properties found. Finally, simulations are performed which strongly confirm the analytic results.

Monthly fluctuations in radial growth of individual lobes of the lichen Parmelia conspersa (Erhr. ex Ach.) Ach.

The radial growth (RG) of 120 lobes from 35 thalli of the foliose lichen Parmelia conspersa (Ehrh. ex Ach.) Ach. was studied monthly over 22 months in south Gwynedd, Wales, UK. Autocorrelation analysis of each lobe identified three patterns of fluctuation: 1) random fluctuations (58% of lobes), 2) a cyclic pattern of growth (23% of lobes), and 3) fluctuating growth interrupted by longer periods of very low or zero growth (19% of lobes). In 80% of thalli, two or three patterns of fluctuation were present within the same thallus. Growth fluctuations were correlated with climatic variables in 31% of lobes, most commonly with either total rainfall or number of rain days per month. Lobes correlated with climate were not associated with a particular type of growth fluctuation. RG of a lobe was positively correlated with the degree of bifurcation of the lobe tip. It is hypothesised that lobes of P. conspersa exhibit a cyclic pattern of growth due in part to lobe division. The effects of climate, periods of zero growth, and microvariations in the environment of a lobe are superimposed on this cyclic pattern resulting in the random growth of many lobes. Random growth fluctuations may contribute to the maintenance of thallus symmetry in P. conspersa.

Radial growth of Rhizocarpon section Rhizocarpon lichen thalli over six years at snoqualmie pass in the Cascade Range, Washington State

This article describes a 6-yr study of the radial growth rates (RGR, mm yr-1) of Rhizocarpon section Rhizocarpon thalli on a talus slope at Snoqualmie Pass in the Cascade Range, Washington State, United States (47°27'N; 121°26'W). At the end of the growth period, 32 of a total of 39 thalli had exhibited a positive RGR, and 7 of a total of 39 thalli showed no measurable growth. Mean RGR of all thalli was 0.07 mm yr-1 (range, 0-0.19 mm, SD = 0.06). Analysis of variance suggested no significant variation in RGR in successive growth periods, but significant differences were present both within and between thalli. The slope of a boulder facet did not influence RGR, but growth was affected by aspect, the least growth being observed on north-northwest facets. A plot of RGR against thallus diameter revealed a wide scatter of data points with little evidence for a significant change in growth with thallus size. Hence, the study showed that the RGR of Rhizocarpon thalli at Snoqualmie is extremely slow and highly variable and significantly less than estimates based on lichenometry. To determine the growth curve of a yellow-green Rhizocarpon by direct measurement at such a site would require a large sample of thalli and careful standardization of the species studied, the aspect conditions under which the thalli were measured, and the initial hypothallus width of the thalli. © 2005 Regents of the University of Colorado.

A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA

Data Envelopment Analysis (DEA) is a nonparametric method for measuring the efficiency of a set of decision making units such as firms or public sector agencies, first introduced into the operational research and management science literature by Charnes, Cooper, and Rhodes (CCR) [Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the efficiency of decision making units. European Journal of Operational Research 2, 429–444]. The original DEA models were applicable only to technologies characterized by positive inputs/outputs. In subsequent literature there have been various approaches to enable DEA to deal with negative data. In this paper, we propose a semi-oriented radial measure, which permits the presence of variables which can take both negative and positive values. The model is applied to data on a notional effluent processing system to compare the results with those yielded by two alternative methods for dealing with negative data in DEA: The modified slacks-based model suggested by Sharp et al. [Sharp, J.A., Liu, W.B., Meng, W., 2006. A modified slacks-based measure model for data envelopment analysis with ‘natural’ negative outputs and inputs. Journal of Operational Research Society 57 (11) 1–6] and the range directional model developed by Portela et al. [Portela, M.C.A.S., Thanassoulis, E., Simpson, G., 2004. A directional distance approach to deal with negative data in DEA: An application to bank branches. Journal of Operational Research Society 55 (10) 1111–1121]. A further example explores the advantages of using the new model.

Faster radial strain relaxation in InAs-GaAs core-shell heterowires

The structure of wurtzite and zinc blende InAs-GaAs (001) core-shell nanowires grown by molecular beam epitaxy on GaAs (001) substrates has been investigated by transmission electron microscopy. Heterowires with InAs core radii exceeding 11 nm, strain relax through the generation of misfit dislocations, given a GaAs shell thickness greater than 2.5 nm. Strain relaxation is larger in radial directions than axial, particularly for shell thicknesses greater than 5.0 nm, consistent with molecular statics calculations that predict a large shear stress concentration at each interface corner. © 2012 American Institute of Physics.

Does radial growth of the lichen Parmelia conspersa depend exclusively on growth processes at the lobe tip?

The objective of this study was to test the hypothesis that the radial growth of lobes of the lichen Parmelia conspersa depends largely on growth processes which occur at the lobe tip. First, individual lobes were removed from thalli and portions of the lobe removed to within various distances from the tip. Radial growth of the lobe was unaffected until less than 2 mm of the lobe tip remained. Second, the surfaces of individual lobes were painted with acrylic paint leaving different portions of the lobe exposed. Painting lobes to within 0.5 mm and 1 mm of the tip substantially reduced radial growth. Third, the levels of ribitol, arabitol and mannitol were measured in different regions behind the lobe tip on four occasions during 1994. The concentration of the three carbohydrates was greatest at the lobe tip and the levels declined linearly with distance from the tip. Fourth, painting one vertical half of the lobe tip did not affect radial growth but artificially bisecting the lobe tip with a scalpel reduced radial growth. Although transport of carbohydrate from other regions of the lobe cannot be ruled out, the results support the hypothesis that radial growth in P. conspersa depends largely on processes within a region approximately 2 mm behind the lobe tip.

Do the lichens Xanthoparmelia conspersa (Ach.) Hale and Rhizocarpon Ram. Em. Th. Fr. subgenus Rhizocarpon utilise exogenous carbohydrates for radial growth?

To field test the hypothesis that lichen thalli can use environmental sources of carbon, solutions of ribitol, arabitol and mannitol were added to intact thalli of Xanthoparmelia conspersa (Ach.) Hale and a yellow species of Rhizocarpon (Rhizocarpon Ram. Em. Th. Fr. subgenus Rhizocarpon). In addition, ribitol and an arabitol/mannitol mixture were added to the marginal hypothalli of Rhizocarpon thalli after removal of the areolae. Carbohydrates were added at the beginning of 2- or 3-month growth periods for up to 15 months at concentrations approximately three times the levels estimated to be in the thalli. Addition of carbohydrates to intact thalli of both species had no effect on total radial growth but addition of mannitol significantly increased growth of X. conspersa thalli in the September/October growth period in one experiment. However, this effect was not repeated in a subsequent experiment in which different concentrations of mannitol were added to intact thalli. Addition of ribitol to hypothalli of Rhizocarpon resulted in significantly increased growth in the first few months of the experiment, growth then declining to levels below that of untreated thalli. The data suggest that although hypothalli of Rhizocarpon may have the ability to utilise exogenous carbohydrates for growth, there was little evidence that intact thalli of either species utilise environmental sources of carbon in the field.

Radial growth and carbohydrate levels in the lichen Parmelia conspersa on north and south facing rock surfaces

Parmelia conspersa (Ehrh. Ex Ach.)Ach. is a foliose lichen found more frequently on south facing compared with north facing rock surfaces in South Gwynedd, Wales, UK. The radial growth of thalli of P. conspersa from a north and a south facing rock surface was measured in situ at intervals of two months for 1 yr during 1990/1991. Mean annual radial growth rates were greater on the south compared with the north facing rock surface. In addition, the pattern of radial growth varied during the year with maximum growth recorded in the Feb/Mar. period especially on the south facing rock surface. The levels of ribitol, arabitol and mannitol were measured in individual lobes of P. conspersa collected from the same rock surfaces on 4 days (2 Jun; 7 July and 30 Nov. 1990 and 29 Mar. 1991) during 1990/1991. The total of the three carbohydrates varied between days; the levels of arbitol and ribitol being significantly lower in the 7 July sample on both north and south facing rock surfaces. In addition, the levels ribitol, arabitol and mannitol were higher on the south facing rock surface especially in the summer samples. The ratio of arabitol plus mannitol to ribitol and the mannitol/arabitol ratio varied more between days sampled than between north and south facing rock surfaces. The level of ribitol in individual thalli was positively correlated with arabitol on the north facing and with mannitol on the south facing slope. These results suggest that differences in the radial growth of P. conspersa thalli with aspect are more likely to reflect higher rates of photosynthesis on the south facing rock surface rather than large difference in the way carbohydrates were partitioned on the different surfaces. Lower radial growth rates may place P. conspersa at a competitive disadvantage on north facing rock surfaces.

The influence of bird droppings and uric acid on the radial growth of five species of saxicolous lichens

Bird droppings were applied over 1 year as a thick paste and as a suspension in deionized water to five species of lichens with different distributions on and off bird perching stones. The paste and suspension increased the radial growth of Parmelia conspersa while the paste increased the growth of Xanthoria parietina and reduced the growth and caused loss of colour in Parmelia glabratula ssp. fuliginosa. There were no statistically significant effects of paste or suspension on the growth of Physcia grisea or Parmelia saxatilis. In P. conspersa and X. parietina the growth responses were similar through the year but in P. glabratula the inhibitory effect of the paste was significant after 8 months growth. Application of a suspension of uric acid over 1 year had no statistically significant effects on the growth of P. conspersa, P. glabratula or X. parietina and was unlikely to be responsible for the effects of bird droppings on growth. The growth responses of the five species agreed well with their distributions in the field.

The influence of the frequency of wetting and drying on the radial growth of three saxicolous lichens in the field

Three lichen species were wetted with distilled water at different frequencies during August 1973 to July 1974. The radial growth rates of Parmelia glabratula ssp. fuliginosa and Physcia orbicularis thalli declined with increased wetting while the radial growth rate of Parmelia conspersa thalli increased with wetting frequency until ten experimental wettings per month but at fifteen wettings per month fell to a value near to the control. In the summer months, wetting resulted in a decline in the radial growth of P. glabratula ssp fuliginosa compared with the control but had little influence on the growth of P. conspersa and Physcia orbicularis. In the winter months, wetting had no significant influence on the radial growth of Parmelia glabratula ssp. fuliginosa, while the radial growth of P. conspersa increased and Physcia orbicularis declined compared with controls. These results are interpreted physiologically and in relation to the aspect distribution of the three lichens on rock surfaces.

Large diameter optic nerve axon fibres in Alzheimer's disease

A variety of visual symptoms have been associated with Alzheimer's disease (AD). These include delays in flash visual evoked potentials which indicate a disruption of the integrity of the visual pathway. Examination of the visual cortex has revealed the presence of both senile plaques and neurofibrillary tangles. The purpose of this study was to determine whether there were differences in the number and/or size of optic nerve axons between AD patients and non-demented age-matched controls. Five optic nerves from AD patients and five from age-matched controls were embedded in epon resin and 1 micron sections prepared on a Reichert ultramicrotome. The sections were then stained in toluidine blue and examined at x400 magnification. The numbers of axons were counted in photographs of three fields taken at random from each section. To evaluate the axon diameters, 70 axons were chosen at random from each patient and measured using a calibrated eyepiece graticule. The total axon counts revealed no significant differences between the AD optic nerves and the age-matched controls. However, the frequency distribution of axon diameters was significantly different in the two groups. In particular, there were fewer larger diameter axons in patients with AD as previously reported. Degeneration of the large diameter axons suggests involvement of the magnocellular as opposed to the parvocellular pathways. Hence, there could be differences in visual performance of AD patients compared with normals which could be important in clinical diagnosis.

The incidence of corpora amylacea (CA) in the optic nerve of patients with Alzheimer's disease

Corpora amylacea (CA) are spherical or ovoid bodies 50-50 microns in diameter. They have been described in normal elderly brain as well as in a number of neurodegenerative disorders. In this study, the incidence of CA in the optic nerves of Alzheimer's disease (AD) patients was compared with normal elderly controls. Samples of optic nerves (MRC Brain Bank, Institute of Psychiatry) were taken from 12 AD patients (age range 69-94 years) and 18 controls (43-82 years). Optic nerves were fixed in 2% buffered glutaraldehyde, post-fixed in osmium tetroxide, embedded in epoxy resin and then sectioned to a thickness of 2 microns. Sections were stained with toluidine blue. CA were present in all of the optic nerves examined. In addition, a number of similarly stained but more irregularly shaped bodies were present. Fewer CA were found in the optic nerves of AD patients compared with controls. By contrast, the number or irregularly shaped bodies was increased in AD. In AD, there may be a preferential decline in the large diameter fibres which may mediate the M-cell pathway. Hence, the decline in the incidence of CA in AD may be associated with a reduction in these fibres. It is also possible that the irregualrly shaped bodies are a degeneration product of the CA.

A new variational radial basis function approximation for inference in multivariate diffusions

In this paper we present a radial basis function based extension to a recently proposed variational algorithm for approximate inference for diffusion processes. Inference, for state and in particular (hyper-) parameters, in diffusion processes is a challenging and crucial task. We show that the new radial basis function approximation based algorithm converges to the original algorithm and has beneficial characteristics when estimating (hyper-)parameters. We validate our new approach on a nonlinear double well potential dynamical system.