Sparse on-line Gaussian processes

We develop an approach for sparse representations of Gaussian Process (GP) models (which are Bayesian types of kernel machines) in order to overcome their limitations for large data sets. The method is based on a combination of a Bayesian online algorithm together with a sequential construction of a relevant subsample of the data which fully specifies the prediction of the GP model. By using an appealing parametrisation and projection techniques that use the RKHS norm, recursions for the effective parameters and a sparse Gaussian approximation of the posterior process are obtained. This allows both for a propagation of predictions as well as of Bayesian error measures. The significance and robustness of our approach is demonstrated on a variety of experiments.

Institutional antecedents and performance consequences of influence strategies in export channels to transitional economies

This study examines how the institutional environment of transitional economies impacts institutional arrangements in the form of influence strategies employed by Western exporters in managing relationships with local firms. Reflecting environmental components, a Western firm’s understanding of Eastern Europe’s regulatory volatility, foreignness, and partner’s control locus is posited to impact economic performance by affecting key coercive and non-coercive influence strategies. A model specifying the effects of the institutional environment on economic outcomes is developed and tested on data from US exporters to Eastern Europe. A structural equation analysis indicates institutional components have a differential impact on the influence strategies employed by these Western firms and on export performance. In particular, use of coercive legalistic pleas is increased by regulatory volatility but reduced by perceived foreignness while use of non-coercive recommendations is increased by the partner’s external locus of control but not by perceived foreignness. Importantly, the institutional environment’s impact on economic performance is shown to be direct as well as indirect through the influence strategies Western firms employ in Eastern Europe. The study concludes with a discussion of implications for managers and researchers.

Two sides to attitudinal commitment:the effect of calculative and loyalty commitment on enforcement mechanism in distribution channels

Previous conceptualizations of attitudinal commitment are extended by considering two very different components of a manufacturer’s attachment to an independent channel intermediary. Relying on commitment theory, a model is developed that describes how attitudinal commitment may reside in either the instrumental or the social strain of a manufacturer’s relationship with its distributor. For each strain, the developmental role played by key facets of the channel setting—relative dependence, pledges, and trust—are shown. Furthermore, the nature of the attachment bond is posited to motivate very different governance mechanisms as the distribution agreement is enforced by either social or contractual means. Empirical support for the model demonstrates that an expanded view of attitudinal commitment is important in understanding the complex nature of attachment in channel relationships.

Bayesian classification with Gaussian processes

We consider the problem of assigning an input vector to one of m classes by predicting P(c|x) for c=1,...,m. For a two-class problem, the probability of class one given x is estimated by s(y(x)), where s(y)=1/(1+e-y). A Gaussian process prior is placed on y(x), and is combined with the training data to obtain predictions for new x points. We provide a Bayesian treatment, integrating over uncertainty in y and in the parameters that control the Gaussian process prior the necessary integration over y is carried out using Laplace's approximation. The method is generalized to multiclass problems (m>2) using the softmax function. We demonstrate the effectiveness of the method on a number of datasets.

Seeing light vs dark lines: psychophysical performance is based on separate channels, limited by noise and uncertainty

Visual detection performance (d') is usually an accelerating function of stimulus contrast, which could imply a smooth, threshold-like nonlinearity in the sensory response. Alternatively, Pelli (1985 Journal of the Optical Society of America A 2 1508 - 1532) developed the 'uncertainty model' in which responses were linear with contrast, but the observer was uncertain about which of many noisy channels contained the signal. Such internal uncertainty effectively adds noise to weak signals, and predicts the nonlinear psychometric function. We re-examined these ideas by plotting psychometric functions (as z-scores) for two observers (SAW, PRM) with high precision. The task was to detect a single, vertical, blurred line at the fixation point, or identify its polarity (light vs dark). Detection of a known polarity was nearly linear for SAW but very nonlinear for PRM. Randomly interleaving light and dark trials reduced performance and rendered it non-linear for SAW, but had little effect for PRM. This occurred for both single-interval and 2AFC procedures. The whole pattern of results was well predicted by our Monte Carlo simulation of Pelli's model, with only two free parameters. SAW (highly practised) had very low uncertainty. PRM (with little prior practice) had much greater uncertainty, resulting in lower contrast sensitivity, nonlinear performance, and no effect of external (polarity) uncertainty. For SAW, identification was about v2 better than detection, implying statistically independent channels for stimuli of opposite polarity, rather than an opponent (light - dark) channel. These findings strongly suggest that noise and uncertainty, rather than sensory nonlinearity, limit visual detection.

From filters to features:Scale-space analysis of edge and blur coding in human vision

To make vision possible, the visual nervous system must represent the most informative features in the light pattern captured by the eye. Here we use Gaussian scale-space theory to derive a multiscale model for edge analysis and we test it in perceptual experiments. At all scales there are two stages of spatial filtering. An odd-symmetric, Gaussian first derivative filter provides the input to a Gaussian second derivative filter. Crucially, the output at each stage is half-wave rectified before feeding forward to the next. This creates nonlinear channels selectively responsive to one edge polarity while suppressing spurious or "phantom" edges. The two stages have properties analogous to simple and complex cells in the visual cortex. Edges are found as peaks in a scale-space response map that is the output of the second stage. The position and scale of the peak response identify the location and blur of the edge. The model predicts remarkably accurately our results on human perception of edge location and blur for a wide range of luminance profiles, including the surprising finding that blurred edges look sharper when their length is made shorter. The model enhances our understanding of early vision by integrating computational, physiological, and psychophysical approaches. © ARVO.

Binocular combination at threshold: temporal filtering and summation of signals in separate ON and OFF channels

How does the brain combine spatio-temporal signals from the two eyes? We quantified binocular summation as the improvement in 2AFC contrast sensitivity for flickering gratings seen by two eyes compared with one. Binocular gratings in-phase showed sensitivity up to 1.8 times higher, suggesting nearly linear summation of contrasts. The binocular advantage decreased to 1.4 at lower spatial and higher temporal frequencies (0.25 cycle deg-1, 30 Hz). Dichoptic, antiphase gratings showed only a small binocular advantage, by a factor of 1.1 to 1.2, but no evidence of cancellation. We present a signal-processing model to account for the contrast-sensitivity functions and the pattern of binocular summation. It has linear sustained and transient temporal filters, nonlinear transduction, and half-wave rectification that creates ON and OFF channels. Binocular summation occurs separately within ON and OFF channels, thus explaining the phase-specific binocular advantage. The model also accounts for earlier findings on detection of brief antiphase flashes and the surprising finding that dichoptic antiphase flicker is seen as frequency-doubled (Cavonius et al, 1992 Ophthalmic and Physiological Optics 12 153 - 156). [Supported by EPSRC project GR/S74515/01].

Perceiving edge blur: Gaussian-derivative filtering and a rectifying nonlinearity

We studied the visual mechanisms that encode edge blur in images. Our previous work suggested that the visual system spatially differentiates the luminance profile twice to create the `signature' of the edge, and then evaluates the spatial scale of this signature profile by applying Gaussian derivative templates of different sizes. The scale of the best-fitting template indicates the blur of the edge. In blur-matching experiments, a staircase procedure was used to adjust the blur of a comparison edge (40% contrast, 0.3 s duration) until it appeared to match the blur of test edges at different contrasts (5% - 40%) and blurs (6 - 32 min of arc). Results showed that lower-contrast edges looked progressively sharper. We also added a linear luminance gradient to blurred test edges. When the added gradient was of opposite polarity to the edge gradient, it made the edge look progressively sharper. Both effects can be explained quantitatively by the action of a half-wave rectifying nonlinearity that sits between the first and second (linear) differentiating stages. This rectifier was introduced to account for a range of other effects on perceived blur (Barbieri-Hesse and Georgeson, 2002 Perception 31 Supplement, 54), but it readily predicts the influence of the negative ramp. The effect of contrast arises because the rectifier has a threshold: it not only suppresses negative values but also small positive values. At low contrasts, more of the gradient profile falls below threshold and its effective spatial scale shrinks in size, leading to perceived sharpening.

Shading and texture:Separate information channels with a common adaptation mechanism?

We outline a scheme for the way in which early vision may handle information about shading (luminance modulation, LM) and texture (contrast modulation, CM). Previous work on the detection of gratings has found no sub-threshold summation, and no cross-adaptation, between LM and CM patterns. This strongly implied separate channels for the detection of LM and CM structure. However, we now report experiments in which adapting to LM (or CM) gratings creates tilt aftereffects of similar magnitude on both LM and CM test gratings, and reduces the perceived strength (modulation depth) of LM and CM gratings to a similar extent. This transfer of aftereffects between LM and CM might suggest a second stage of processing at which LM and CM information is integrated. The nature of this integration, however, is unclear and several simple predictions are not fulfilled. Firstly, one might expect the integration stage to lose identity information about whether the pattern was LM or CM. We show instead that the identity of barely detectable LM and CM patterns is not lost. Secondly, when LM and CM gratings are combined in-phase or out-of-phase we find no evidence for cancellation, nor for 'phase-blindness'. These results suggest that information about LM and CM is not pooled or merged - shading is not confused with texture variation. We suggest that LM and CM signals are carried by separate channels, but they share a common adaptation mechanism that accounts for the almost complete transfer of perceptual aftereffects.

The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering

Marr's work offered guidelines on how to investigate vision (the theory - algorithm - implementation distinction), as well as specific proposals on how vision is done. Many of the latter have inevitably been superseded, but the approach was inspirational and remains so. Marr saw the computational study of vision as tightly linked to psychophysics and neurophysiology, but the last twenty years have seen some weakening of that integration. Because feature detection is a key stage in early human vision, we have returned to basic questions about representation of edges at coarse and fine scales. We describe an explicit model in the spirit of the primal sketch, but tightly constrained by psychophysical data. Results from two tasks (location-marking and blur-matching) point strongly to the central role played by second-derivative operators, as proposed by Marr and Hildreth. Edge location and blur are evaluated by finding the location and scale of the Gaussian-derivative `template' that best matches the second-derivative profile (`signature') of the edge. The system is scale-invariant, and accurately predicts blur-matching data for a wide variety of 1-D and 2-D images. By finding the best-fitting scale, it implements a form of local scale selection and circumvents the knotty problem of integrating filter outputs across scales. [Supported by BBSRC and the Wellcome Trust]

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.

Functional presynaptic HCN channels in the rat globus pallidus

Hyperpolarization-activated, cyclic nucleotide-gated cation (HCN) channels are expressed postsynaptically in the rodent globus pallidus (GP), where they play several important roles in controlling GP neuronal activity. To further elucidate the role of HCN channels in the GP, immunocytochemical and electrophysiological approaches were used to test the hypothesis that HCN channels are also expressed presynaptically on the local axon collaterals of GP neurons. At the electron microscopic level, immunoperoxidase labelling for HCN1 and HCN2 was localized in GP somata and dendritic processes, myelinated and unmyelinated axons, and axon terminals. One population of labelled terminals formed symmetric synapses with somata and proximal dendrites and were immunoreactive for parvalbumin, consistent with the axon collaterals of GABAergic GP projection neurons. In addition, labelling for HCN2 and, to a lesser degree, HCN1 was observed in axon terminals that formed asymmetric synapses and were immunoreactive for the vesicular glutamate transporter 2. Immunogold labelling demonstrated that HCN1 and HCN2 were located predominantly at extrasynaptic sites along the plasma membrane of both types of terminal. To determine the function of presynaptic HCN channels in the GP, we performed whole-cell recordings from GP neurons in vitro. Bath application of the HCN channel blocker ZD7288 resulted in an increase in the frequency of mIPSCs but had no effect on their amplitude, implying that HCN channels tonically regulate the release of GABA. Their presence, and predicted role in modulating transmitter release, represents a hitherto unidentified mechanism whereby HCN channels influence the activity of GP neurons. © The Authors (2007).

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.