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Results for “functional regression”
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Taking PDE Solutions from Low-Fidelity to High-Fidelity Using Function-on-Function Regression.
Abstract not provided.
Taking PDE Solutions from Low-fidelity to High-Fidelity Using Function-on-Function Regression.
Abstract not provided.
Gradient-based Optimization for Regression in the Functional Tensor-Train Format
We consider the task of low-multilinear-rank functional regression, i.e., learning a low-rank parametric representation of functions from scattered real-valued data. Our first contribution is the development and analysis of gradient-based optimization procedures, including stochastic gradient descent and quasi-Newton methods, for learning the parameters of a functional tensor-train (FT) that yields improved accuracy over standard alternating least squares methods. The functional tensor-train uses the tensor-train (TT) representation of low-rank arrays as an ansatz for a class of low-multilinear-rank functions. The FT is represented by a set of matrix-valued functions that contain a set of univariate functions, and the regression task is to learn the parameters of these univariate functions. Our second contribution demonstrates that using nonlinearly parameterized univariate functions, e.g., symmetric kernels with moving centers, within each core can outperform the standard approach of using a linear expansion of basis functions. Our final contributions are new rank adaptation and group-sparsity regularization procedures to minimize overfitting. We use several benchmark problems to demonstrate at least an order of magnitude lower accuracy with gradient-based optimization methods than ALS in the low-sample number regime. We also demonstrate an order of magnitude reduction in accuracy on a test problem resulting from nonlinear parameterizations over linear parameterizations. Finally we compare regression performance with 22 other nonparametric and parametric regression methods on 10 real-world data sets. We achieve top-five accuracy for seven of the data sets and best accuracy for two of the data sets. These rankings are the best amongst parametric models and competetive with the best non-parametric methods.
Elastic functional principal component regression
We study regression using functional predictors in situations where these functions contains both phase and amplitude variability. In other words, the functions are misaligned due to errors in time measurements, and these errors can significantly degrade both model estimation and prediction performance. The current techniques either ignore the phase variability, or handle it via preprocessing, that is, use an off–the–shelf technique for functional alignment and phase removal. We develop a functional principal component regression model which has a comprehensive approach in handling phase and amplitude variability. The model utilizes a mathematical representation of the data known as the square–root slope function. These functions preserve the L 2 norm under warping and are ideally suited for simultaneous estimation of regression and warping parameters. Furthermore, using both simulated and real–world data sets, we demonstrate our approach and evaluate its prediction performance relative to current models. In addition, we propose an extension to functional logistic and multinomial logistic regression.
Predicting fatigue from heart rate signatures using functional logistic regression
Physical fatigue can have adverse effects on humans in extreme environments. Therefore, being able to predict fatigue using easy to measure metrics such as heart rate (HR) signatures has potential to have an impact in real-life scenarios. We apply a functional logistic regression model that uses HR signatures to predict physical fatigue, where physical fatigue is defined in a data-driven manner. Data were collected using commercially available wearable devices on 47 participants hiking the 20.7-mile Grand Canyon rim-to-rim trail in a single day. Fitted model provides good predictions and interpretable parameters for real-life application.
Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations
Here, this work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The proposed statistical model comprises the sum of a deterministic regression-function model and a stochastic noise model. The method constructs the regression-function model by applying regression techniques from machine learning (e.g., support vector regression, artificial neural networks) to map features (i.e., error indicators such as sampled elements of the residual) to a prediction of the approximate-solution error. The method constructs the noise model as a mean-zero Gaussian random variable whose variance is computed as the sample variance of the approximate-solution error on a test set; this variance can be interpreted as the epistemic uncertainty introduced by the approximate solution. This work considers a wide range of feature-engineering methods, data-set-construction techniques, and regression techniques that aim to ensure that (1) the features are cheaply computable, (2) the noise model exhibits low variance (i.e., low epistemic uncertainty introduced), and (3) the regression model generalizes to independent test data. Finally, numerical experiments performed on several computational-mechanics problems and types of approximate solutions demonstrate the ability of the method to generate statistical models of the error that satisfy these criteria and significantly outperform more commonly adopted approaches for error modeling.
From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas
Functional data analysis (FDA), which is a branch of statistics on modeling infinite dimensional random vectors resided in functional spaces, has become a major research area for Journal of Multivariate Analysis. We review some fundamental concepts of FDA, their origins and connections from multivariate analysis, and some of its recent developments, including multi-level functional data analysis, high-dimensional functional regression, and dependent functional data analysis. Here, we also discuss the impact of these new methodology developments on genetics, plant science, wearable device data analysis, image data analysis, and business analytics. Two real data examples are provided to motivate our discussions.
Inverse prediction of PuO2 processing conditions using Bayesian seemingly unrelated regression with functional data
Over the past decade, a variety of innovative methodologies have been developed to better characterize the relationships between processing conditions and the physical, morphological, and chemical features of special nuclear material (SNM). Different processing conditions generate SNM products with different features, which are known as “signatures” because they are indicative of the processing conditions used to produce the material. These signatures can potentially allow a forensic analyst to determine which processes were used to produce the SNM and make inferences about where the material originated. This article investigates a statistical technique for relating processing conditions to the morphological features of PuO 2 particles. We develop a Bayesian implementation of seemingly unrelated regression (SUR) to inverse-predict unknown PuO 2 processing conditions from known PuO 2 features. Model results from simulated data demonstrate the usefulness of the technique. Applied to empirical data from a bench-scale experiment specifically designed with inverse prediction in mind, our model successfully predicts nitric acid concentration, while results for Pu concentration and precipitation temperature were equivalent to a simple mean model. Our technique compliments other recent methodologies developed for forensic analysis of nuclear material and can be generalized across the field of chemometrics for application to other materials.
Reactor Power Level Estimation by Fusing Multi-Modal Sensor Measurements
Estimates of the power level of a nuclear reactor based on measurements from an independent monitoring sensor system can help in the compliance verification of its declared operations. We present a three-level fusion method to estimate the power level of a nuclear reactor using features derived from infrared, electromagnetic, and acoustic sensor measurements collected in proximity to the reactor. Based on a simplified analytical model of the secondary coolant system of the reactor, we identify partial regression functions of the power level in terms of the temperature difference between inlet and outlet of coolant pipes, and the activity levels of four fans and four pumps, which are estimated as features from the sensor measurements. The power level estimator employs a combination of aggregate and complementary fusion steps at three levels to incorporate the multi-modal features in a structure that reflects the secondary cooling system and its partial regression functions. Using the measurements from a test campaign at an operational reactor, we show that this estimator achieves 3.47% or lower root mean square error under 5-fold cross validation. More generally, these results illustrate a progressive reduction in estimation error as additional modalities are appropriately incorporated, and that the fuser outperforms single modality features and their sub-combinations.
Regression-Based Analytics for Response Dynamics of SDN Solutions and Components
We consider Software-Defined Network (SDN) solutions that dynamically setup and teardown connections, and also their response dynamics of connectivity and TCP throughput. There is a wide variety of choices for controllers, switches, andsoftware modules, and objective analytics of measurements are needed to compare the response dynamics of solutions based on their combinations. Throughput and ping measurements during connection changes contain significant statistical variations due to the complexities of controllers, northbound scripts, and network devices. We propose analytical methods to estimate response regression functions, which provide a rigorous and objective comparison of different SDN solutions and their components, namely, controllers, switches, and OpenFlow versions. We apply this approach in two scenarios: (a) switching of long-haul connections emulated in hardware by ANUE devices over a testbed consisting of HP and Cisco switches, and (b) multi-site science infrastructure emulated using VMs and Mininet. The results provide useful practical insights including: (i) the dpctl method responds seconds faster than the OpenDayLight (ODL) method on average under OpenFlow 1.0, but performs similar to ODL under OpenFlow 1.3, and (ii) the ping response of dpctl is about 0.6 sec faster than ODL for the second scenario.
Elastic Partial Functional PrincipalComponent Regression with UnmatchedBoundaries
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Physics Model Calibration via Functional Kernel Regression and Multilevel Modeling
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Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics
Here, in this paper, we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. Cozad and Sahinidis propose a nonconvex mixed-integer nonlinear program (MINLP), in which binary variables are used to assign operators and nonlinear expressions are used to propagate data values through nonlinear operators, such as square, square root, and exponential. We extend this formulation by adding new cuts that improve the solution of this challenging MINLP. We also propose a heuristic that iteratively builds an expression tree by solving a restricted MINLP. We perform computational experiments and compare our approach with a mixed-integer program–based method and a neural network–based method from the literature.
Deep transfer operator learning for partial differential equations under conditional shift
Transfer learning enables the transfer of knowledge gained while learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labelling, potential computational power limitations and dataset distribution mismatches. Here, we propose a new transfer learning framework for task-specific learning (functional regression in partial differential equations) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of the target data. Inspired by conditional embedding operator theory, we minimize the statistical distance between labelled target data and the surrogate prediction on unlabelled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various transfer learning scenarios involving nonlinear partial differential equations under diverse conditions due to shifts in the geometric domain and model dynamics. Our transfer learning framework enables fast and efficient learning of heterogeneous tasks despite considerable differences between the source and target domains.
Robust analysis of trends in noisy tokamak confinement data using geodesic least squares regression
Regression analysis is a very common activity in fusion science for unveiling trends and parametric dependencies, but it can be a difficult matter. We have recently developed the method of geodesic least squares (GLS) regression that is able to handle errors in all variables, is robust against data outliers and uncertainty in the regression model, and can be used with arbitrary distribution models and regression functions. We here report on first results of application of GLS to estimation of the multi-machine scaling law for the energy confinement time in tokamaks, demonstrating improved consistency of the GLS results compared to standard least squares.
A new surrogate modeling technique combining Kriging and polynomial chaos expansions – Application to uncertainty analysis in computational dosimetry
In numerical dosimetry, the recent advances in high performance computing led to a strong reduction of the required computational time to assess the specific absorption rate (SAR) characterizing the human exposure to electromagnetic waves. However, this procedure remains time-consuming and a single simulation can request several hours. As a consequence, the influence of uncertain input parameters on the SAR cannot be analyzed using crude Monte Carlo simulation. The solution presented here to perform such an analysis is surrogate modeling. This paper proposes a novel approach to build such a surrogate model from a design of experiments. Considering a sparse representation of the polynomial chaos expansions using least-angle regression as a selection algorithm to retain the most influential polynomials, this paper proposes to use the selected polynomials as regression functions for the universal Kriging model. The leave-one-out cross validation is used to select the optimal number of polynomials in the deterministic part of the Kriging model. The proposed approach, called LARS-Kriging-PC modeling, is applied to three benchmark examples and then to a full-scale metamodeling problem involving the exposure of a numerical fetus model to a femtocell device. The performances of the LARS-Kriging-PC are compared to an ordinary Kriging model and to a classical sparse polynomial chaos expansion. The LARS-Kriging-PC appears to have better performances than the two other approaches. A significant accuracy improvement is observed compared to the ordinary Kriging or to the sparse polynomial chaos depending on the studied case. This approach seems to be an optimal solution between the two other classical approaches. A global sensitivity analysis is finally performed on the LARS-Kriging-PC model of the fetus exposure problem.