Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

(2003). Risk Management: Value at Risk and Beyond. Journal of the American Statistical Association: Vol. 98, No. 462, pp. 494-494.

Risk Management: Value at Risk and Beyond

Computer Methods in Applied Mechanics and Engineering

https://www.cell.com/joule/pdf/S2542-4351(19)30585-9.pdf

Battery Lifetime Prognostics

Using mathematical models for a quantitative description of dynamical systems requires the identification of uncertain parameters by minimising the difference between simulation and measurement. Owing to the measurement noise also, the estimated parameters possess an uncertainty expressed by their variances. To obtain highly predictive models, very precise parameters are needed. The optimal experimental design (OED) as a numerical optimisation method is used to reduce the parameter uncertainty by minimising the parameter variances iteratively. A frequently applied method to define a cost function for OED is based on the inverse of the Fisher information matrix. The application of this traditional method has at least two shortcomings for models that are nonlinear in their parameters: (i) it gives only a lower bound of the parameter variances and (ii) the bias of the estimator is neglected. Here, the authors show that by applying the sigma point (SP) method a better approximation of characteristic values of the parameter statistics can be obtained, which has a direct benefit on OED. An additional advantage of the SP method is that it can also be used to investigate the influence of the parameter uncertainties on the simulation results. The SP method is demonstrated for the example of a widely used biological model.

Optimal experimental design with the sigma point method

Currently applied methods for characterization and optimization of bio-pharmaceutical processes are still strongly empirical. This often involves Design of Experiment (DoE) methods that require a large number of time-consuming experiments and can hardly fulfill the requirements of ‘Quality by Design’ (QbD). Linking mathematical models with experimental methods is seen as an efficient strategy that significantly reduces development time and costs. Furthermore, if combined with advanced Process Analytical Technology (PAT), higher automation and more efficient workflows can be established along the product lifecycle. This contribution presents the state of the art of model-based tools for experimental design and gives an outlook on future trends in the field of bio-process engineering.

Model-based tools for optimal experiments in bioprocess engineering

Efficient Global Sensitivity Analysis of 3D Multiphysics Model for Li-Ion Batteries

In the field of chemical engineering, mathematical models have been proven to be an indispensable tool for process analysis, process design, and condition monitoring. To gain the most benefit from model-based approaches, the implemented mathematical models have to be based on sound principles, and they need to be calibrated to the process under study with suitable model parameter estimates. Often, the model parameters identified by experimental data, however, pose severe uncertainties leading to incorrect or biased inferences. This applies in particular in the field of pharmaceutical manufacturing, where usually the measurement data are limited in quantity and quality when analyzing novel active pharmaceutical ingredients. Optimally designed experiments, in turn, aim to increase the quality of the gathered data in the most efficient way. Any improvement in data quality results in more precise parameter estimates and more reliable model candidates. The applied methods for parameter sensitivity analyses and design criteria are crucial for the effectiveness of the optimal experimental design. In this work, different design measures based on global parameter sensitivities are critically compared with state-of-the-art concepts that follow simplifying linearization principles. The efficient implementation of the proposed sensitivity measures is explicitly addressed to be applicable to complex chemical engineering problems of practical relevance. As a case study, the homogeneous synthesis of 3,4-dihydro-1H-1-benzazepine-2,5-dione, a scaffold for the preparation of various protein kinase inhibitors, is analyzed followed by a more complex model of biochemical reactions. In both studies, the model-based optimal experimental design benefits from global parameter sensitivities combined with proper design measures.

/pdf/the-impact-of-global-sensitivities-and-design-measures-in-l63g4jlla2.pdf

The Impact of Global Sensitivities and Design Measures in Model-Based Optimal Experimental Design

In this study, we show an effective data-driven identification of the State-of-Health of Lithium-ion batteries by Nonlinear Frequency Response Analysis. A degradation model based on support vector regression is derived from highly informative Nonlinear Frequency Response Analysis data sets. First, an ageing test of a Lithium-ion battery at 25 °C is presented and the impact of relevant ageing mechanisms on the nonlinear dynamics of the cells is analysed. A correlation measure is used to identify the most sensitive frequency range for ageing tests. Here, the mid-frequency range from 1 Hz to 100 Hz shows the strongest correlation to Lithium-ion battery degradation. The focus on the mid-frequency range leads to a dramatic reduction in measurement time of up to 92% compared to standard measurement protocols. Next, informative features are extracted and used to parametrise the support vector regression model for the State of Health degradation. The performance of the degradation model is validated with additional cells and validation data sets, respectively. We show that the degradation model accurately predicts the State of Health values. Validation data demonstrate the usefulness of the Nonlinear Frequency Response Analysis as an effective and fast State of Health identification method and as a versatile tool in the diagnosis of ageing of Lithium-ion batteries in general.

René Schenkendorf

Papers

Optimal experimental design with the sigma point method

Model-based tools for optimal experiments in bioprocess engineering

Efficient Global Sensitivity Analysis of 3D Multiphysics Model for Li-Ion Batteries

The Impact of Global Sensitivities and Design Measures in Model-Based Optimal Experimental Design

State-of-Health Identification of Lithium-Ion Batteries Based on Nonlinear Frequency Response Analysis: First Steps with Machine Learning