How do we understand and visualize uncertainty?

Geophysicists are often concerned with reconstructing subsurface properties using observations collected at or near the surface. For example, in seismic migration, we attempt to reconstruct subsurface geometry from surface seismic recordings, and in potential field inversion, observations are used to map electrical conductivity or density variations in geologic layers. The procedure of inferring information from indirect observations is called an inverse problem by mathematicians, and such problems are common in many areas of the physical sciences. The inverse problem of inferring the subsurface using surface observations has a corresponding forward problem, which consists of determining the data that would be recorded for a given subsurface configuration. In the seismic case, forward modeling involves a method for calculating a synthetic seismogram, for gravity data it consists of a computer code to compute gravity fields from an assumed subsurface density model. Note that forward modeling often involves assumptions about the appropriate physical relationship between unknowns (at depth) and observations on the surface, and all attempts to solve the problem at hand are limited by the accuracy of those assumptions. In the broadest sense then, exploration geophysicists have been engaged in inversion since the dawn of the profession and indeed algorithms often applied in processing centers can all be viewed as procedures to invert geophysical data

Spatiospectral concentration on a sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200

The feedback-related negativity reflects the binary evaluation of good versus bad outcomes

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It's worse than you thought : the feedback negativity and violations of reward prediction in gambling tasks

The reinforcement learning theory suggests that the feedback negativity should be larger when feedback is unexpected. Two recent studies found, however, that the feedback negativity was unaffected by outcome probability. To further examine this issue, participants in the present studies made reward predictions on each trial of a gambling task where objective reward probability was indicated by a cue. In Study 1, participants made reward predictions following the cue, but prior to their gambling choice; in Study 2, predictions were made following their gambling choice. Predicted and unpredicted outcomes were associated with equivalent feedback negativities in Study 1. In Study 2, however, the feedback negativity was larger for unpredicted outcomes. These data suggest that the magnitude of the feedback negativity is sensitive to violations of reward prediction, but that this effect may depend on the close coupling of prediction and outcome

Dyson's Brownian Motion and Universal Dynamics of Quantum Systems

We establish a correspondence between the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random Gaussian perturbing matrix, and a Fokker-Planck equation postulated by Dyson. Within this model, we prove the equivalence conjectured by Altshuler et al between the space-time correlations of the Sutherland-Calogero-Moser system in the thermodynamic limit and a set of two-variable correlations for disordered quantum systems calculated by them. Multiple variable correlation functions are, however, shown to be inequivalent for the two cases.Comment: 10 pages, revte