Learning on dynamic statistical manifolds.

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback-Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

Anti-sulfatide/galactocerebroside antibodies in immunoglobulin M paraproteinemic neuropathies.

BACKGROUND AND PURPOSE: Anti-sulfatide antibodies have been observed in heterogeneous neuropathies and their clinical relevance is still controversial. Whether the combination of sulfatide with galactocerebroside would increase sensitivity or specificity of enzyme-linked immunosorbent assay testing compared to sulfatide alone was assessed. METHODS: Immunoglobulin M (IgM) antibodies to sulfatides, galactocerebroside and combined sulfatide and galactocerebroside (Sulf/GalC) were measured in 229 neuropathy patients, including 73 with IgM paraproteinemic neuropathy [62 with anti-myelin-associated glycoprotein (anti-MAG) antibody] and 156 with other neuropathies. Results from 27 patients with IgM monoclonal gammopathy without neuropathy and 28 healthy subjects served as control. RESULTS: Thirty-three patients showed increased titers of anti-sulfatide antibodies, 28 of whom had an IgM paraproteinemic neuropathy (P < 0.0001). When evaluating the reactivity for the combination Sulf/GalC, 57/229 patients were found to be positive, including 36/73 (49%) with IgM paraproteinemic neuropathy (P < 0.0001). Patients with known anti-sulfatide antibodies also showed anti-Sulf/GalC reactivity, with increased titers in 48.5% of the cases. Testing for anti-Sulf/GalC antibodies allowed 24 additional patients to be detected (eight with IgM paraproteinemic neuropathies), who had no reactivity to the individual glycolipids. Amongst the 11 subjects with IgM paraproteinemic neuropathy who were negative for anti-MAG antibodies, only two were reactive to sulfatide, whilst six (55%) were found to be positive when tested against the combination of sulfatide and galactocerebroside. CONCLUSIONS: Testing for both sulfatide and galactocerebroside in IgM paraproteinemic neuropathies seems to increase the sensitivity compared to anti-sulfatide antibodies alone (49% and 39%, respectively, with a slightly reduced specificity, from 97% to 87%), helping the characterization of otherwise undefined neuropathy that could benefit from immunomodulatory therapy.

A theoretical framework for modeling dilution enhancement of non-reactive solutes in heterogeneous porous media.

Spatial heterogeneity of the hydraulic properties of geological porous formations leads to erratically shaped solute clouds, thus increasing the edge area of the solute body and augmenting the dilution rate. In this study, we provide a theoretical framework to quantify dilution of a non-reactive solute within a steady state flow as affected by the spatial variability of the hydraulic conductivity. Embracing the Lagrangian concentration framework, we obtain explicit semi-analytical expressions for the dilution index as a function of the structural parameters of the random hydraulic conductivity field, under the assumptions of uniform-in-the-average flow, small injection source and weak-to-mild heterogeneity. Results show how the dilution enhancement of the solute cloud is strongly dependent on both the statistical anisotropy ratio and the heterogeneity level of the porous medium. The explicit semi-analytical solution also captures the temporal evolution of the dilution rate; for the early- and late-time limits, the proposed solution recovers previous results from the literature, while at intermediate times it reflects the increasing interplay between large-scale advection and local-scale dispersion. The performance of the theoretical framework is verified with high resolution numerical results and successfully tested against the Cape Cod field data.

Cumulative distribution function solutions of advection-reaction equations with uncertain parameters.

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddy-diffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.

Classical and new proving methodology: provings of Plumbum metallicum and Piper methysticum and comparison with a classical proving of Plumbum metallicum.

OBJECTIVE: To study the reliability of a proving methodology and the reproducibility of proving symptoms. METHODS: Two homeopathic medicines and placebo were given, in a double-blind randomized design, to 31 healthy volunteers (13 Piper methysticum 30C, 11 placebo and 7 Plumbum metallicum 30C), 5 drops 4 times daily, until the onset of unbearable symptoms, or at most for 1 week. The primary outcome measure was the number of phrases containing unusual or new symptoms selected by supervisors (SEL) from unstructured diaries and the number of these symptoms (SYM) present in SELs. The secondary outcome measures were the number of symptoms with modalities of both verum groups concordant with symptoms reported in a previous proving of Plumbum 12C. Other parameters evaluated were repeated and crossed symptoms in SELs. RESULTS: Both medicines showed qualitative and quantitative differences from placebo. Piper: 146 SELs (median: 5), Plumbum: 118 SELs (16), placebo: 48 SELs (2), containing 260 (8), 199 (29) and 58 (2) SYMs, respectively. There was a significant difference from placebo in Plumbum but not in Piper SELs and SYMs (P < 0.05). 31, 24 and 4 'repeated' and 18, 22 and 2 'crossed' symptoms were found in Piper, Plumbum and placebo. 8 and 30 symptoms concordant with the classical proving of Plumbum were found for Piper and Plumbum, corresponding to about 10% and 45% of their total SELs. CONCLUSIONS: Open diaries, supervision and double-blind placebo are useful methods in homeopathic pathogenetic trials. Estimates of concordance should be introduced in proving methodology.