Computing Functions of Random Variables via Reproducing Kernel Hilbert Space Representations

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as {\em kernel probabilistic programming}. We illustrate it on synthetic data, and show how it can be used for nonparametric structural equation models, with an application to causal inference

Solving Satisfiability Problems with Genetic Algorithms

We show how to solve hard 3-SAT problems using genetic algorithms. Furthermore, we explore other genetic operators that may be useful to tackle 3-SAT problems, and discuss their pros and cons

Validation of nonlinear PCA

Linear principal component analysis (PCA) can be extended to a nonlinear PCA by using artificial neural networks. But the benefit of curved components requires a careful control of the model complexity. Moreover, standard techniques for model selection, including cross-validation and more generally the use of an independent test set, fail when applied to nonlinear PCA because of its inherent unsupervised characteristics. This paper presents a new approach for validating the complexity of nonlinear PCA models by using the error in missing data estimation as a criterion for model selection. It is motivated by the idea that only the model of optimal complexity is able to predict missing values with the highest accuracy. While standard test set validation usually favours over-fitted nonlinear PCA models, the proposed model validation approach correctly selects the optimal model complexity.Comment: 12 pages, 5 figure

Image analysis for cosmology: results from the GREAT10 Galaxy Challenge

In this paper, we present results from the weak-lensing shape measurement GRavitational lEnsing Accuracy Testing 2010 (GREAT10) Galaxy Challenge. This marks an order of magnitude step change in the level of scrutiny employed in weak-lensing shape measurement analysis. We provide descriptions of each method tested and include 10 evaluation metrics over 24 simulation branches. GREAT10 was the first shape measurement challenge to include variable fields; both the shear field and the point spread function (PSF) vary across the images in a realistic manner. The variable fields enable a variety of metrics that are inaccessible to constant shear simulations, including a direct measure of the impact of shape measurement inaccuracies, and the impact of PSF size and ellipticity, on the shear power spectrum. To assess the impact of shape measurement bias for cosmic shear, we present a general pseudo-Cℓ formalism that propagates spatially varying systematics in cosmic shear through to power spectrum estimates. We also show how one-point estimators of bias can be extracted from variable shear simulations. The GREAT10 Galaxy Challenge received 95 submissions and saw a factor of 3 improvement in the accuracy achieved by other shape measurement methods. The best methods achieve sub-per cent average biases. We find a strong dependence on accuracy as a function of signal-to-noise ratio, and indications of a weak dependence on galaxy type and size. Some requirements for the most ambitious cosmic shear experiments are met above a signal-to-noise ratio of 20. These results have the caveat that the simulated PSF was a ground-based PSF. Our results are a snapshot of the accuracy of current shape measurement methods and are a benchmark upon which improvement can be brought. This provides a foundation for a better understanding of the strengths and limitations of shape measurement method

Extreme events in gross primary production: a characterization across continents

Climate extremes can affect the functioning of terrestrial ecosystems, for instance via a reduction of the photosynthetic capacity or alterations of respiratory processes. Yet the dominant regional and seasonal effects of hydrometeorological extremes are still not well documented and in the focus of this paper. Specifically, we quantify and characterize the role of large spatiotemporal extreme events in gross primary production (GPP) as triggers of continental anomalies. We also investigate seasonal dynamics of extreme impacts on continental GPP anomalies. We find that the 50 largest positive extremes (i.e., statistically unusual increases in carbon uptake rates) and negative extremes (i.e., statistically unusual decreases in carbon uptake rates) on each continent can explain most of the continental variation in GPP, which is in line with previous results obtained at the global scale. We show that negative extremes are larger than positive ones and demonstrate that this asymmetry is particularly strong in South America and Europe. Our analysis indicates that the overall impacts and the spatial extents of GPP extremes are power-law distributed with exponents that vary little across continents. Moreover, we show that on all continents and for all data sets the spatial extents play a more important role for the overall impact of GPP extremes compared to the durations or maximal GPP. An analysis of possible causes across continents indicates that most negative extremes in GPP can be attributed clearly to water scarcity, whereas extreme temperatures play a secondary role. However, for Europe, South America and Oceania we also identify fire as an important driver. Our findings are consistent with remote sensing products. An independent validation against a literature survey on specific extreme events supports our results to a large extent

Machine-learning-based prediction of respiratory flow and lung volume from real-time cardiac MRI using MR-compatible spirometry

Background:Cardiac real-time MRI (RT-MRI) in combination with MR-compatible spirometry (MRcS) offers unique opportunities to study heart-lung interactions. In contrast to other techniques that monitor respiration during MRI, MRcS provides quantitative respiratory data. Though MRcS is well tolerated, shortening of the scanning time with MRcS would be desirable, especially in young and sick patients.Purpose:The aim of the study was to predict airflow and lung volume based on RT-MR images after a short learning phase of combined RT-MRI and MRcS to provide respiratory data for subsequent short axis stack-based volumetries.Methods:Cardiac RT-MRI (1.5 T; short axis; 30 frames/s) was acquired during free breathing in combination with MRcS in adult healthy subjects (n = 10). MR images with MRcS were recorded during a learning phase to collect training data. The iterative Lucas-Kanade method was applied to estimate optical flow from the captured MR images. A ridge regression model was fitted to predict airflow and thus also the lung volume from the estimated optical flow. Hyperparameters were estimated using leave-one-out cross validation and the performance was assessed on a held-out test dataset. Different durations and compositions of the learning phase were investigated to develop the most efficient measurement protocol. Coefficient of determination (R2), relative mean squared error (rMSE), Bland-Altman analysis on absolute tidal volume difference (aTVD), and absolute maximal airflow difference (aMFD) were used to validate the predictions on held-out test data.Results:MRI combined with MRcS can train a machine learning algorithm to provide excellent predictive quantitative respiratory volume and flow for the remaining study. The optimal trade-off between predictive power and time necessary for training was reached with a shortened cardiac volumetry protocol covering only about two breaths per slice and every second slice (airflow: mean R2: 0.984, mean rMSE: 0.015, Bias aMFD: -0.01 L/s with +0.084/-0.1 95% CI and volume: mean R2: 0.990, mean rMSE: 0.003, Bias aTVD: 4.27 mL with +33/-24 95% CI) at a total duration of 100 s. Shorter protocols or application of the algorithm to subsequent studies in the same subject or even in different subjects still provided useful qualitative data.Conclusion:Machine-learning-based prediction of respiratory flow and lung volume from cardiac RT-MR images after a short training phase with MRcS is feasible and can help to shorten the time with MRcS while providing accurate respiratory data during RT-MRI.<br