How can we quantify, explain, and apply the uncertainty of complex soil maps predicted with neural networks?

Artificial neural networks (ANNs) have proven to be a useful tool for complex questions that involve large amounts of data. Our use case of predicting soil maps with ANNs is in high demand by government agencies, construction companies, or farmers, given cost and time intensive field work. However, there are two main challenges when applying ANNs. In their most common form, deep learning algorithms do not provide interpretable predictive uncertainty. This means that properties of an ANN such as the certainty and plausibility of the predicted variables, rely on the interpretation by experts rather than being quantified by evaluation metrics validating the ANNs. Further, these algorithms have shown a high confidence in their predictions in areas geographically distant from the training area or areas sparsely covered by training data. To tackle these challenges, we use the Bayesian deep learning approach "last-layer Laplace approximation", which is specifically designed to quantify uncertainty into deep networks, in our explorative study on soil classification. It corrects the overconfident areas without reducing the accuracy of the predictions, giving us a more realistic uncertainty expression of the model's prediction. In our study area in southern Germany, we subdivide the soils into soil regions and as a test case we explicitly exclude two soil regions in the training area but include these regions in the prediction. Our results emphasize the need for uncertainty measurement to obtain more reliable and interpretable results of ANNs, especially for regions far away from the training area. Moreover, the knowledge gained from this research addresses the problem of overconfidence of ANNs and provides valuable information on the predictability of soil types and the identification of knowledge gaps. By analyzing regions where the model has limited data support and, consequently, high uncertainty, stakeholders can recognize the areas that require more data collection efforts.

VIPurPCA: Visualizing and Propagating Uncertainty in Principal Component Analysis.

Variables obtained by experimental measurements or statistical inference typically carry uncertainties. When an algorithm uses such quantities as input variables, this uncertainty should propagate to the algorithm's output. Concretely, we consider the classic notion of principal component analysis (PCA): If it is applied to a finite data matrix containing imperfect (i.e., uncertain) multidimensional measurements, its output-a lower-dimensional representation-is itself subject to uncertainty. We demonstrate that this uncertainty can be approximated by appropriate linearization of the algorithm's nonlinear functionality, using automatic differentiation. By itself, however, this structured, uncertain output is difficult to interpret for users. We provide an animation method that effectively visualizes the uncertainty of the lower dimensional map. Implemented as an open-source software package, it allows researchers to assess the reliability of PCA embeddings.

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models.

Understanding neural computation on the mechanistic level requires models of neurons and neuronal networks. To analyze such models one typically has to solve coupled ordinary differential equations (ODEs), which describe the dynamics of the underlying neural system. These ODEs are solved numerically with deterministic ODE solvers that yield single solutions with either no, or only a global scalar error indicator on precision. It can therefore be challenging to estimate the effect of numerical uncertainty on quantities of interest, such as spike-times and the number of spikes. To overcome this problem, we propose to use recently developed sampling-based probabilistic solvers, which are able to quantify such numerical uncertainties. They neither require detailed insights into the kinetics of the models, nor are they difficult to implement. We show that numerical uncertainty can affect the outcome of typical neuroscience simulations, e.g. jittering spikes by milliseconds or even adding or removing individual spikes from simulations altogether, and demonstrate that probabilistic solvers reveal these numerical uncertainties with only moderate computational overhead.

Dismutation of Tricyanoboryllead Compounds: The Homoleptic Tetrakis(tricyanoboryl)plumbate Tetraanion.

A series of unprecedently air-stable (tricyanoboryl)plumbate anions was obtained by the reaction of the boron-centered nucleophile B(CN)3 2- with triorganyllead halides. Salts of the anions [R3 PbB(CN)3 ]- (R=Ph, Et) were isolated and found to be stable in air at room temperature. In the case of Me3 PbHal (Hal=Cl, Br), a mixture of the anions [Me4-n Pb{B(CN)3 }n ]n- (n=1, 2) was obtained. The [Et3 PbB(CN)3 ]- ion undergoes stepwise dismutation in aqueous solution to yield the plumbate anions [Et4-n Pb{B(CN)3 }n ]n- (n=1-4) and PbEt4 as by-product. The reaction rate increases with decreasing pH value of the aqueous solution or by bubbling O2 through the reaction mixture. Adjustment of the conditions allowed the selective formation and isolation of salts of all anions of the series [Et4-n Pb{B(CN)3 }n ]n- (n=2-4) including the homoleptic tetraanion [Pb{B(CN)3 }4 ]4- .

Convergence rates of Gaussian ODE filters.

A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss-Markov process X , which is then iteratively conditioned on information about x ˙ . This article establishes worst-case local convergence rates of order q + 1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of q = 1 and an integrated Brownian motion prior, and analyses how inaccurate information on x ˙ coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to q ∈ { 2 , 3 , … } .

Analytical probabilistic modeling of dose-volume histograms.

PURPOSE: Radiotherapy, especially with charged particles, is sensitive to executional and preparational uncertainties that propagate to uncertainty in dose and plan quality indicators, for example, dose-volume histograms (DVHs). Current approaches to quantify and mitigate such uncertainties rely on explicitly computed error scenarios and are thus subject to statistical uncertainty and limitations regarding the underlying uncertainty model. Here we present an alternative, analytical method to approximate moments, in particular expectation value and (co)variance, of the probability distribution of DVH-points, and evaluate its accuracy on patient data. METHODS: We use Analytical Probabilistic Modeling (APM) to derive moments of the probability distribution over individual DVH-points based on the probability distribution over dose. By using the computed moments to parameterize distinct probability distributions over DVH-points (here normal or beta distributions), not only the moments but also percentiles, that is, α - DVHs, are computed. The model is subsequently evaluated on three patient cases (intracranial, paraspinal, prostate) in 30- and single-fraction scenarios by assuming the dose to follow a multivariate normal distribution, whose moments are computed in closed-form with APM. The results are compared to a benchmark based on discrete random sampling. RESULTS: The evaluation of the new probabilistic model on the three patient cases against a sampling benchmark proves its correctness under perfect assumptions as well as good agreement in realistic conditions. More precisely, ca. 90% of all computed expected DVH-points and their standard deviations agree within 1% volume with their empirical counterpart from sampling computations, for both fractionated and single fraction treatments. When computing α - DVH, the assumption of a beta distribution achieved better agreement with empirical percentiles than the assumption of a normal distribution: While in both cases probabilities locally showed large deviations (up to ±0.2), the respective - DVHs for α={0.05,0.5,0.95} only showed small deviations in respective volume (up to ±5% volume for a normal distribution, and up to 2% for a beta distribution). A previously published model from literature, which was included for comparison, exhibited substantially larger deviations. CONCLUSIONS: With APM we could derive a mathematically exact description of moments of probability distributions over DVH-points given a probability distribution over dose. The model generalizes previous attempts and performs well for both choices of probability distributions, that is, normal or beta distributions, over DVH-points.

Innovative Syntheses of Cyano(fluoro)borates: Catalytic Cyanation, Electrochemical and Electrophilic Fluorination.

Different types of high-yield, easily scalable syntheses for cyano(fluoro)borates Kt[BFn (CN)4-n ] (n=0-2) (Kt=cation), which are versatile building blocks for materials applications and chemical synthesis, have been developed. Tetrafluoroborates react with trimethylsilyl cyanide in the presence of metal-free Brønsted or Lewis acid catalysts under unprecedentedly mild conditions to give tricyanofluoroborates or tetracyanoborates. Analogously, pentafluoroethyltrifluoroborates are converted into pentafluoroethyltricyanoborates. Boron trifluoride etherate, alkali metal salts, and trimethylsilyl cyanide selectively yield dicyanodifluoroborates or tricyanofluoroborates. Fluorination of cyanohydridoborates is the third reaction type that includes direct fluorination with, for example, elemental fluorine, stepwise halogenation/fluorination reactions, and electrochemical fluorination (ECF) according to the Simons process. In addition, fluorination of [BH(CN)2 {OC(O)Et}]- to result in [BF(CN)2 {OC(O)Et}]- is described.

Dense connectomic reconstruction in layer 4 of the somatosensory cortex.

The dense circuit structure of mammalian cerebral cortex is still unknown. With developments in three-dimensional electron microscopy, the imaging of sizable volumes of neuropil has become possible, but dense reconstruction of connectomes is the limiting step. We reconstructed a volume of ~500,000 cubic micrometers from layer 4 of mouse barrel cortex, ~300 times larger than previous dense reconstructions from the mammalian cerebral cortex. The connectomic data allowed the extraction of inhibitory and excitatory neuron subtypes that were not predictable from geometric information. We quantified connectomic imprints consistent with Hebbian synaptic weight adaptation, which yielded upper bounds for the fraction of the circuit consistent with saturated long-term potentiation. These data establish an approach for the locally dense connectomic phenotyping of neuronal circuitry in the mammalian cortex.

Carba-closo-dodecaboranylethynyl ligands facilitating luminescent reversed charge-transfer excited states in gold/silver complexes.

A set of mono- and dinuclear AuI and AgI alkynyl complexes bearing the carba-closo-dodecaboranylethynyl ligand show intense room temperature phosphorescence. The {closo-1-CB11} cage participates in an unprecedented way as an electron donating moiety, changing the direction of the charge-transfer excited state.

Cyanohydridoborate Anions: Synthesis, Salts, and Low-Viscosity Ionic Liquids.

High-yield syntheses up to molar scales for salts of [BH(CN)3 ]- (2) and [BH2 (CN)2 ]- (3) starting from commercially available Na[BH4 ] (Na5), Na[BH3 (CN)] (Na4), BCl3 , (CH3 )3 SiCN, and KCN were developed. Direct conversion of Na5 into K2 was accomplished with (CH3 )3 SiCN and (CH3 )3 SiCl as a catalyst in an autoclave. Alternatively, Na5 is converted into Na[BH{OC(O)R}3 ] (R=alkyl) that is more reactive towards (CH3 )3 SiCN and thus provides an easy access to salts of 2. Some reaction intermediates were identified, for example, Na[BH(CN){OC(O)Et}2 ] (Na7 b) and Na[BH(CN)2 {OC(O)Et}] (Na8 b). A third entry to 2 and 3 uses ether adducts of BHCl2 or BH2 Cl such as the commercial 1,4-dioxane adducts that react with KCN and (CH3 )3 SiCN. Alkali metal salts of 2 and 3 are convenient starting materials for organic salts, especially for low viscosity ionic liquids (ILs). [EMIm]3 has the lowest viscosity and highest conductivity with 10.2 mPa s and 32.6 mS cm-1 at 20 °C known for non-protic ILs. The ILs are thermally, chemically, and electrochemically robust. These properties are crucial for applications in electrochemical devices, for example, dye-sensitized solar cells (Grätzel cells).

Analytical incorporation of fractionation effects in probabilistic treatment planning for intensity-modulated proton therapy.

PURPOSE: We show that it is possible to explicitly incorporate fractionation effects into closed-form probabilistic treatment plan analysis and optimization for intensity-modulated proton therapy with analytical probabilistic modeling (APM). We study the impact of different fractionation schemes on the dosimetric uncertainty induced by random and systematic sources of range and setup uncertainty for treatment plans that were optimized with and without consideration of the number of treatment fractions. METHODS: The APM framework is capable of handling arbitrarily correlated uncertainty models including systematic and random errors in the context of fractionation. On this basis, we construct an analytical dose variance computation pipeline that explicitly considers the number of treatment fractions for uncertainty quantitation and minimization during treatment planning. We evaluate the variance computation model in comparison to random sampling of 100 treatments for conventional and probabilistic treatment plans under different fractionation schemes (1, 5, 30 fractions) for an intracranial, a paraspinal and a prostate case. The impact of neglecting the fractionation scheme during treatment planning is investigated by applying treatment plans that were generated with probabilistic optimization for 1 fraction in a higher number of fractions and comparing them to the probabilistic plans optimized under explicit consideration of the number of fractions. RESULTS: APM enables the construction of an analytical variance computation model for dose uncertainty considering fractionation at negligible computational overhead. It is computationally feasible (a) to simultaneously perform a robustness analysis for all possible fraction numbers and (b) to perform a probabilistic treatment plan optimization for a specific fraction number. The incorporation of fractionation assumptions for robustness analysis exposes a dose to uncertainty trade-off, i.e., the dose in the organs at risk is increased for a reduced fraction number and/or for more robust treatment plans. By explicit consideration of fractionation effects during planning, we demonstrate that it is possible to exploit this trade-off during optimization. APM optimization considering the fraction number reduced the dose in organs at risk compared to conventional probabilistic optimization neglecting the fraction number. CONCLUSION: APM enables computationally efficient incorporation of fractionation effects in probabilistic uncertainty analysis and probabilistic treatment plan optimization. The consideration of the fractionation scheme in probabilistic treatment planning reveals the trade-off between number of fractions, nominal dose, and treatment plan robustness.

Perfluoroalkyltricyanoborate and Perfluoroalkylcyanofluoroborate Anions: Building Blocks for Low-Viscosity Ionic Liquids.

The potassium perfluoroalkyltricyanoborates K[Cn F2 n+1 B(CN)3 ] [n=1 (1 d), 2 (2 d)] and the potassium mono(perfluoroalkyl)cyanofluoroborates K[Cn F2 n+1 BF(CN)2 ] [n=1 (1 c), 2 (2 c)] and [Cn F2 n+1 BF2 (CN)]- [n=1 (1 b), 2 (2 b), 3 (3 b), 4 (4 b)] are accessible with perfect selectivities on multi-gram scales starting from K[Cn F2 n+1 BF3 ] and Me3 SiCN. The K+ salts are starting materials for the preparation of salts with organic cations, for example, [EMIm]+ (EMIm=1-ethyl-3-methylimidazolium). These [EMIm]+ salts are hydrophobic room-temperature ionic liquids (RTILs) that are thermally, chemically and electrochemically very robust, offering electrochemical windows up to 5.8 V. The RTILs described herein, exhibit very low viscosities with a minimum of 14.0 mPa s at 20 °C for [EMIm]1 c, low melting points down to -57 °C for [EMIm]2 b and extraordinary high conductivities up to 17.6 mS cm-1 at 20 °C for [EMIm]1 c. The combination of these properties makes these ILs promising materials for electrochemical devices as exemplified by the application of selected RTILs as component of electrolytes in dye-sensitised solar cells (DSSCs, Grätzel cells). The efficiency of the DSSCs was found to increase with a decreasing viscosity of the neat ionic liquid. In addition to the spectroscopic characterisation, single crystals of the potassium salts of the anions 1 b-d, 2 d, 3 b and 4 c as well as of [nBu4 N]2 c have been studied by X-ray diffraction.

Borylation of fluorinated arenes using the boron-centred nucleophile B(CN)32- - a unique entry to aryltricyanoborates.

The potassium salt of the boron-centred nucleophile B(CN)32- (1) readily reacts with perfluorinated arenes, such as hexafluorobenzene, decafluorobiphenyl, octafluoronaphthalene and pentafluoropyridine, which results in KF and the K+ salts of the respective borate anions with one {B(CN)3} unit bonded to the (hetero)arene. An excess of K21 leads to the successive reaction of two or, in the case of perfluoropyridine, even three C-F moieties and the formation of di- and trianions, respectively. Moreover, all of the 11 partially fluorinated benzene derivatives, C6F6-n H n (n = 1-5), generally react with K21 to give new tricyano(phenyl)borate anions with high chemo- and regioselectivity. A decreasing number of fluorine substituents on benzene results in a decrease in the reaction rate. In the cases of partially fluorinated benzenes, the addition of LiCl is advantageous or even necessary to facilitate the reaction. Also, pentafluorobenzenes R-C6F5 (R = -CN, -OMe, -Me, or -CF3) react via C-F/C-B exchange that mostly occurs in the para position and to a lesser extent in the meta or ortho positions. Most of the reactions proceed via an SNAr mechanism. The reaction of 1,4-F2C6H4 with K21 shows that an aryne mechanism has to be considered in some cases as well. In summary, a wealth of new stable tricyano(aryl)borates have been synthesised and fully characterized using multi-NMR spectroscopy and most of them were characterised using single-crystal X-ray diffraction.

Probabilistic numerics and uncertainty in computations.

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation with limited time or hardware, are important for much contemporary science and industry. Within applications such as climate science and astrophysics, the need to make decisions on the basis of computations with large and complex data have led to a renewed focus on the management of numerical uncertainty. We describe how several seminal classic numerical methods can be interpreted naturally as probabilistic inference. We then show that the probabilistic view suggests new algorithms that can flexibly be adapted to suit application specifics, while delivering improved empirical performance. We provide concrete illustrations of the benefits of probabilistic numeric algorithms on real scientific problems from astrometry and astronomical imaging, while highlighting open problems with these new algorithms. Finally, we describe how probabilistic numerical methods provide a coherent framework for identifying the uncertainty in calculations performed with a combination of numerical algorithms (e.g. both numerical optimizers and differential equation solvers), potentially allowing the diagnosis (and control) of error sources in computations.

Probabilistic shortest path tractography in DTI using Gaussian Process ODE solvers.

Tractography in diffusion tensor imaging estimates connectivity in the brain through observations of local diffusivity. These observations are noisy and of low resolution and, as a consequence, connections cannot be found with high precision. We use probabilistic numerics to estimate connectivity between regions of interest and contribute a Gaussian Process tractography algorithm which allows for both quantification and visualization of its posterior uncertainty. We use the uncertainty both in visualization of individual tracts as well as in heat maps of tract locations. Finally, we provide a quantitative evaluation of different metrics and algorithms showing that the adjoint metric (8] combined with our algorithm produces paths which agree most often with experts.

Analytical probabilistic modeling for radiation therapy treatment planning.

This paper introduces the concept of analytical probabilistic modeling (APM) to quantify uncertainties in quality indicators of radiation therapy treatment plans. Assuming Gaussian probability densities over the input parameters of the treatment plan quality indicators, APM enables the calculation of the moments of the induced probability density over the treatment plan quality indicators by analytical integration. This paper focuses on analytical probabilistic dose calculation algorithms and the implications of APM regarding treatment planning. We derive closed-form expressions for the expectation value and the (co)variance of (1) intensity-modulated photon and proton dose distributions based on a pencil beam algorithm and (2) the standard quadratic objective function used in inverse planning. Complex correlation models of high dimensional uncertain input parameters and the different nature of random and systematic uncertainties in fractionated radiation therapy are explicitly incorporated into APM. APM variance calculations on phantom data sets show that the correlation assumptions and the difference of random and systematic uncertainties of the input parameters have a crucial impact on the uncertainty of the resulting dose. The derivations regarding the quadratic objective function show that APM has the potential to enable robust planning at almost the same computational cost like conventional inverse planning after a single probabilistic dose calculation. Beneficial applications of APM in the context of radiation therapy treatment planning are feasible.