Multi-fidelity sparse polynomial chaos expansion based on Gaussian process regression and least angle regression

Polynomial chaos (PC) expansion meta-model has been wildly employed and investigated in the field of uncertainty quantification (UQ) and sensitivity analysis (SA). However, the majority of the multi-fidelity polynomial chaos expansion (MF-PC) models in the literature are still focused on using high-fidelity (HF) PC model to correct low fidelity (LH) model directly, without cross-correlation between PC models of different fidelities. To address this shortcoming, a multi-fidelity sparse polynomial chaos expansion (MF-sPC) model is proposed based on least angle regression (LAR) and recursive Gaussian process regression (GPR) in this paper. From low to high degree of fidelity, the autoregressive scheme in MF GPR is employed to construct MF-sPC model, in which the sparse polynomial chaos (sPC) model of each fidelity is built iteratively coupling with GPR, LAR and cross validation (CV), as gradually expanding the design of experiment (DoE) to reach a given CV error. This recursive scheme finally yields a MF-sPC model with highest fidelity which takes advantage of all sPC models of the lower fidelities. And the proposed MF-sPC model is validated by a test example in detail, and the results reveal that this MF meta-model performs outstanding both in convergence speed and model accuracy

Three-Dimensional Extended Object Tracking and Shape Learning Using Gaussian Processes

In this study, we investigate the problem of tracking objects with unknown shapes using three-dimensional (3D) point cloud data. We propose a Gaussian process-based model to jointly estimate object kinematics, including position, orientation and velocities, together with the shape of the object for online and offline applications. We describe the unknown shape by a radial function in 3D, and induce a correlation structure via a Gaussian process. Furthermore, we propose an efficient algorithm to reduce the computational complexity of working with 3D data. This is accomplished by casting the tracking problem into projection planes which are attached to the object's local frame. The resulting algorithms can process 3D point cloud data and accomplish tracking of a dynamic object. Furthermore, they provide analytical expressions for the representation of the object shape in 3D, together with confidence intervals. The confidence intervals, which quantify the uncertainty in the shape estimate, can later be used for solving the gating and association problems inherent in object tracking. The performance of the methods is demonstrated both on simulated and real data. The results are compared with an existing random matrix model, which is commonly used for extended object tracking in the literature

Nonlinear Gaussian Filtering : Theory, Algorithms, and Applications

By restricting to Gaussian distributions, the optimal Bayesian filtering problem can be transformed into an algebraically simple form, which allows for computationally efficient algorithms. Three problem settings are discussed in this thesis: (1) filtering with Gaussians only, (2) Gaussian mixture filtering for strong nonlinearities, (3) Gaussian process filtering for purely data-driven scenarios. For each setting, efficient algorithms are derived and applied to real-world problems

Horizon-based autonomous navigation and mapping for small body missions

This report expands upon a previously developed approach to simultaneously estimate asteroid physical characteristics and relative spacecraft state with limited prior knowledge using optical observations of the illuminated horizon from resolved imagery. The approach is intended for eventual autonomous use onboard a spacecraft. The asteroid surface is represented as a star-convex shape where the radial extent is a function of the input spherical coordinates. This unknown radial extent function is modeled as a Gaussian Process, which is formulated as a state space model that is well-suited to sequential Bayesian inference methods, namely, the Extended Kalman Filter. Early versions of this algorithm solidly demonstrated proof-of-concept, but this works aims to adjust and refine the filter equations to increase robustness to larger nonlinearities. Efficacy of the reworked estimator is demonstrated through Monte Carlo simulations.Aerospace Engineerin

Data and hybrid models of dynamical systems

Tato práce prezentuje hybridní přístup k modelování dynamiky pomocí kombinace modelování prvních principů a modelování založeného na datech. Využita je unikátní vlastnost RGP, a to že je schopen přizpůsobit svojí dynamiku online, bez nutnosti předsběru dat během trénování, na časově proměnné aerodynamické síly. Navrhujeme metodu RGPMPC, která používá hybridní model v MPC regulátoru, přičemž mění datově založenou složku hybridního modelu tak, aby zohledňovala rozdíly mezi mod- elem a reálným systémem. Metoda je demonstrována na modelu quadrotoru v simulaci, pomocí simulátoru Gazebo. RGPMPC je schopen sledovat požadovanou trajektorii a přizpůsobit se měnícím se aerodynamickým silám. Tento simulační experiment posky- tuje důkaz, že metoda RGPMPC je schopna zlepšit výkon MPC regulátoru v přítom- nosti neznámých rozdílů mezi modelem a reálným systémem.ObhájenoThis thesis presents a hybrid approach for modeling and of dynamics by combining first principles modeling and data-driven modeling. An unique property of the RGP is exploited, namely that it is able to fit the dynamics online, without the need for a training run, to fit time-varying aerodynamics. We propose a method RGPMPC, which uses the hybrid model in a MPC controller, while changing the data-driven component of the hybrid model to account for model discrepancies. We demonstrate our method on a model of a quadrotor in simulation, using the Gazebo simulator. The RGPMPC is able to track the desired trajectory and adapt to the changing drag forces present. This simulation experiment provides a proof of concept that the RGPMPC method is able to improve the performance of the MPC controller in the presence of unknown discrepancies in the model

COVID-19 mortality rate prediction for India using statistical neural networks and gaussian process regression model

The primary purpose of this research is to identify the best COVID-19 mortality model for India using regression models and is to estimate the future COVID-19 mortality rate for India. Specifically, Statistical Neural Networks ( Radial Basis Function Neural Network (RBFNN), Generalized Regression Neural Network (GRNN)), and Gaussian Process Regression (GPR) are applied to develop the COVID-19 Mortality Rate Prediction (MRP) model for India. For that purpose, there are two types of dataset used in this study: One is COVID-19 Death cases, a Time Series Data and the other is COVID-19 Confirmed Case and Death Cases where Death case is dependent variable and the Confirmed case is an independent variable. Hyperparameter optimization or tuning is used in these regression models, which is the process of identifying a set of optimal hyperparameters for any learning process with minimal error. Here, sigma (\u3c3) is a hyperparameter whose value is used to constrain the learning process of the above models with minimum Root Mean Squared Error (RMSE). The performance of the models is evaluated using the RMSE and 'R2 values, which shows that the GRP model performs better than the GRNN and RBFNN