Perturbation Free-Energy Toolkit: Automated Alchemical Topology Builder and Optimized Simulation Update Scheme

Free-energy calculations play an important role in the application of computational chemistry to a range of fields, including protein biochemistry, rational drug design or material science. Importantly, the free energy difference is directly related to experimentally measurable quantities such as partition and adsorption coefficients, water activity and binding affinities. Among several techniques aimed at predicting the free-energy differences, perturbation approaches, involving alchemical transformation of one molecule into another through intermediate states, stand out as rigorous methods based on statistical mechanics. However, despite the importance of efficient and accurate free energy predictions, applicability of the perturbation approaches is still largely impeded by a number of challenges. This study aims at addressing two of them: 1) the definition of the perturbation path, i.e., alchemical changes leading to the transformation of one molecule to the other, and 2) determining the amount of sampling along the path to reach desired convergence. In particular, an automatic perturbation builder based on a graph matching algorithm is developed, that is able to identify the maximum common substructure of two molecules and provide the perturbation topologies suitable for free-energy calculations using GROMOS and GROMACS simulation packages. Moreover, it was used to calculate the changes in free energy of a set of post-translational modifications and analyze their convergence behavior. Different methods were tested, which showed that MBAR and extended thermodynamic integration (TI) in combination with MBAR show better performance as compared to BAR, extended TI with linear interpolation and plain TI. Also, a number of error estimators were explored and how they relate to the true error, estimated as the difference in free energy from an extensive set of simulation data. This analysis shows that most of the estimators provide only a qualitative agreement to the true error, with little quantitative predictive power. This notwithstanding, the preformed analyses provided insight into the convergence of free-energy calculations, which allowed for development of an iterative update scheme for perturbation simulations that aims at minimizing the simulation time to reach the convergence, i.e., optimizing the efficiency. Importantly, this toolkit is made available online as an open-source python package (https://github.com/drazen-petrov/SMArt).

Perturbation Free-Energy Toolkit: Automated Alchemical Topology Builder and Optimized Simulation Update Scheme

Free-energy calculations play an important role in the application of computational chemistry to a range of fields, including protein biochemistry, rational drug design or material science. Importantly, the free energy difference is directly related to experimentally measurable quantities such as partition and adsorption coefficients, water activity and binding affinities. Among several techniques aimed at predicting the free-energy differences, perturbation approaches, involving alchemical transformation of one molecule into another through intermediate states, stand out as rigorous methods based on statistical mechanics. However, despite the importance of efficient and accurate free energy predictions, applicability of the perturbation approaches is still largely impeded by a number of challenges. This study aims at addressing two of them: 1) the definition of the perturbation path, i.e., alchemical changes leading to the transformation of one molecule to the other, and 2) determining the amount of sampling along the path to reach desired convergence. In particular, an automatic perturbation builder based on a graph matching algorithm is developed, that is able to identify the maximum common substructure of two molecules and provide the perturbation topologies suitable for free-energy calculations using GROMOS and GROMACS simulation packages. Moreover, it was used to calculate the changes in free energy of a set of post-translational modifications and analyze their convergence behavior. Different methods were tested, which showed that MBAR and extended thermodynamic integration (TI) in combination with MBAR show better performance as compared to BAR, extended TI with linear interpolation and plain TI. Also, a number of error estimators were explored and how they relate to the true error, estimated as the difference in free energy from an extensive set of simulation data. This analysis shows that most of the estimators provide only a qualitative agreement to the true error, with little quantitative predictive power. This notwithstanding, the preformed analyses provided insight into the convergence of free-energy calculations, which allowed for development of an iterative update scheme for perturbation simulations that aims at minimizing the simulation time to reach the convergence, i.e., optimizing the efficiency. Importantly, this toolkit is made available online as an open-source python package (https://github.com/drazen-petrov/SMArt).

Empirical determination of the covariance of forecast errors: an empirical justification and reformulation of Hybrid covariance models

<p>The replacement of climatological background error covariance models with Hybrid error covariance models that linearly combine a localized ensemble covariance matrix and a climatological error covariance matrix has led to significant forecast improvements at several forecasting centres. To deepen understanding of why the Hybrid’s superficially ad-hoc mix of ensemble based covariances and climatological covariances yielded such significant improvements, we derive the linear state estimation equations that minimize analysis error variance given an imperfect ensemble covariance. For high dimensional models, the computational cost of the very large sample sizes required to empirically estimate the terms in these equations is prohibitive. However, a reasonable and computationally feasible approximation to these equations can be obtained from empirical estimates of the true error covariance between two model variables given an imperfect ensemble covariance between the same two variables.   Here, using a Data Assimilation (DA) system featuring a simplified Global Circulation Model (SPEEDY), pseudo-observations of known error variance and an ensemble data assimilation scheme (LETKF),  we quantitatively demonstrate that the traditional Hybrid used by many operational centres is a much better approximation to the true covariance given the ensemble covariance than either the static climatological covariance or the localized ensemble covariance. These quantitative findings help explain why operational centres have found such large forecast improvements when switching from a static error covariance model to a Hybrid forecast error covariance model. Another fascinating finding of our empirical study is that the form of current Hybrid error covariance models is fundamentally incorrect in that the weight given to the static covariance matrix is independent of the separation distance of model variables. Our results show that this weight should be an increasing function of variable separation distance.  It is found that for ensemble covariances significantly different to zero, the true error covariance of spatially separated variables is an approximately linear function of the corresponding ensemble covariance, However, for small ensemble sizes and ensemble covariances near zero, the true covariance is an increasing function of the magnitude of the ensemble covariance and reaches a local minimum at the precise point where the ensemble covariance is equal to zero. It is hypothesized that this behaviour is a consequence of small ensemble size and, specifically, associated spurious fluctuations of the ensemble covariances and variances. Consistent with this hypothesis, this local minimum is almost eliminated by quadrupling the ensemble size.</p>

Compact Schemes for the Spatial Discretization of Linear Elliptic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative 2 2 1 f x   and 2 2 2 f x   of Linear Elliptic Partial Differential Equations (LEPDE) ,on the non-boundary nodes, located along a particular horizontal grid line for 2 2 1 f x   and along a particular vertical grid line for 2 2 2 f x   of a two-dimensional structured Cartesian uniform grid. The aim of the numerical experiment is to demonstrate the higher order spatial accuracy and better rate of convergence of the solution, produced using the developed compact scheme. Further, these solutions are compared with the same, produced using the conventional 2 nd order scheme. The comparison is made, in terms of the discrete l l 2 &  norms, of the true error. The true error is defined as, the difference between the computed numerical and the available exact solution, of the chosen test problems. It is computed on every non-boundary node bounded in the computational domain.

Recommendations for spectral fitting of SO₂ from MAX-DOAS measurements

Fitting SO2 dSCDs from MAX-DOAS measurements of scattered sunlight is challenging because actinic light intensity is low in wavelength regions where the SO2 absorption features are strongest. SO2 dSCDs were fit with different wavelength windows (λlow to λhigh) from ambient measurements with calibration cells of 2.2 × 1017 and 2.2 × 1016 molec cm−2 inserted in the light path at different viewing elevation angles. SO2 dSCDs were the least accurate and fit errors were the largest for fitting windows with λlow 312 nm. The SO2 dSCDs also exhibited an inverse relationship with the SO2 absorption cross-section for fitting windows with λlow 400 nm. Deviation of the SO2 dSCD from the true value depended on the SO2 concentration for some fitting windows rather than exhibiting a consistent bias. Uncertainties of the SO2 dSCD reported by the fit algorithm were significantly less than the true error for many windows, particularly for the measurements without the filter or offset function. For retrievals with the filter or offset function, increasing λhigh > 320 nm tended to decrease the reported fit uncertainty but did not increase the accuracy. Based on the results of this study, a short-pass filter and a fitting window of 307.5

Certifying the True Error: Machine Learning in Coq with Verified Generalization Guarantees

We present MLCERT, a novel system for doing practical mechanized proof of the generalization of learning procedures, bounding expected error in terms of training or test error. MLCERT is mechanized in that we prove generalization bounds inside the theorem prover Coq; thus the bounds are machine checked by Coq’s proof checker. MLCERT is practical in that we extract learning procedures defined in Coq to executable code; thus procedures with proved generalization bounds can be trained and deployed in real systems. MLCERT is well documented and open source; thus we expect it to be usable even by those without Coq expertise. To validate MLCERT, which is compatible with external tools such as TensorFlow, we use it to prove generalization bounds on neural networks trained using TensorFlow on the extended MNIST data set.

Observation-Informed Generalized Hybrid Error Covariance Models

Because of imperfections in ensemble data assimilation schemes, one cannot assume that the ensemble-derived covariance matrix is equal to the true error covariance matrix. Here, we describe a simple and intuitively compelling method to fit calibration functions of the ensemble sample variance to the mean of the distribution of true error variances, given an ensemble estimate. We demonstrate that the use of such calibration functions is consistent with theory showing that, when sampling error in the prior variance estimate is considered, the gain that minimizes the posterior error variance uses the expected true prior variance, given an ensemble sample variance. Once the calibration function has been fitted, it can be combined with ensemble-based and climatologically based error correlation information to obtain a generalized hybrid error covariance model. When the calibration function is chosen to be a linear function of the ensemble variance, the generalized hybrid error covariance model is the widely used linear hybrid consisting of a weighted sum of a climatological and an ensemble-based forecast error covariance matrix. However, when the calibration function is chosen to be, say, a cubic function of the ensemble sample variance, the generalized hybrid error covariance model is a nonlinear function of the ensemble estimate. We consider idealized univariate data assimilation and multivariate cycling ensemble data assimilation to demonstrate that the generalized hybrid error covariance model closely approximates the optimal weights found through computationally expensive tuning in the linear case and, in the nonlinear case, outperforms any plausible linear model.

Comparative Analysis of Selected Methods for Estimating the Prediction Error of Classifie

Classification is an algorithm, which assigns studied companies, taking into consideration their attributes, to specific population. An essential part of it is classifier. Its measure of quality is especially predictability, measured by true error rate. The value of this error, due to lack of sufficiently large and independent test set, must be estimated on the basis of available learning set.The aim of this article is to make a review and compare selected methods for estimating the prediction error of classifier, constructed with linear discriminant analysis. It was examined if the results of the analysis depends on the sample size and the method of selecting variables for a model. Empirical research was made on example of problem of bankruptcy prediction of join-stock companies in Poland.