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A Robbins-Monro type multidimensional stochastic approximation algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions.
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.
Ridge, Marquardt's generalized inverse, shrunken, and principal components estimators are discussed in terms of the objectives of point estimation of parameters, estimation of the predictive regression function, and hypothesis testing. It is found that as the normal equations approach singularity, more consideration must be given to estimable functions of the parameters as opposed to estimation of the full parameter vector; that biased estimators all introduce constraints on the parameter space; that adoption of mean squared error as a criterion of goodness should be independent of the degree of singularity; and that ordinary least-squares subset regression is the best overall method.
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.
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.
Functional variables are often used as predictors in regression problems. A commonly used parametric approach, called scalar-on-function regression, uses the $\mathbb L^2$ inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase–amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. In this paper, we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices.
Background: With the continuing increase in the use of definitive stereotactic radiosurgery (SRS) for patients with limited brain metastases (BM), clinicians need more specific prognostic tools. We investigated clinical predictors of outcomes in patients with limited breast cancer BM treated with SRS alone. Methods and Materials: We identified 136 patients with breast cancer and 1-3 BM who underwent definitive SRS for 186 BM between 2000 and 2012. The Kaplan-Meier method was used to assess overall survival (OS), regional failure (RF), and local failure (LF). Associations between clinical factors and outcomes were tested using Cox regression. A point scoring system was used to stratify patients based on OS, and the predictive power was tested with concordance probability estimate (CPE). Results: The median OS was 17.6 months. The 12-month RF and LF rates were 45% and 10%, respectively. On multivariate analysis, >1 lesion (hazard ratio [HR] = 1.6, P=.02), triple-negative (TN) disease (HR=2.0, P=.006), and active extracranial disease (ED) (HR=2.7, P<.0001) were significantly associated with worse OS. The point score system was defined using proportional simplification of the multivariate Cox proportional hazards regression function. The median OS for patients with 3.0-4.0 points (n=37), 4.5-5.5 points (n=28), 6.0-6.5 points (n=37), and 8-8.5 points (n=34) were 9.2, 15.6, 25.1, and 45.1 months, respectively (P<.0001, CPE = 0.72). Active ED (HR=2.4, P=.0007) was significantly associated with RF. Higher risk for LF was significantly associated with larger BM size (HR=3.1, P=.0001). Conclusion: Patients with >1 BM, active ED, and TN had the highest risk of death after SRS. Active ED is an important prognostic factor for OS and intracranial control.
An anoxic flow-through cell experiment was conducted to examine mechanisms controlling the real-time reduction of selenate (Se(VI)) by zerovalent iron (ZVI), which is commonly used in permeable reactive barriers to treat dissolved contaminants including Se(VI). Changes in selenium (Se) isotope composition were examined by increasing the influent Se concentration over time, thus changing the proportion of Se removed from solution. At the conclusion of the experiment, an anoxic Se-free solution was pumped through the cell to assess the stability of the reaction products. At all stages, X-ray absorption data were obtained from the solid phase and Se isotope data from the aqueous phase. Reduced Se in the form of adsorbed Se(IV), Fe 2 SeO 4 , Se(0), and iron selenides accumulated on the ZVI over time. A linear regression function was fit to the $δ$ 82/76 Se values of the effluent, yielding an isotopic separation of 9.6‰. A Rayleigh curve was fit to the isotope data from the effluent samples collected during the rinse stage with an effective fractionation of 2.4‰. The results from this experiment can be used to elucidate the effect of multiple concurrent mechanisms on Se isotope behavior.
Purpose: To develop an automated pulmonary image analysis framework for infectious lung diseases in small animal models. Methods: The authors describe a novel pathological lung and airway segmentation method for small animals. The proposed framework includes identification of abnormal imaging patterns pertaining to infectious lung diseases. First, the authors’ system estimates an expected lung volume by utilizing a regression function between total lung capacity and approximated rib cage volume. A significant difference between the expected lung volume and the initial lung segmentation indicates the presence of severe pathology, and invokes a machine learning based abnormal imaging pattern detection system next. The final stage of the proposed framework is the automatic extraction of airway tree for which new affinity relationships within the fuzzy connectedness image segmentation framework are proposed by combining Hessian and gray-scale morphological reconstruction filters. Results: 133 CT scans were collected from four different studies encompassing a wide spectrum of pulmonary abnormalities pertaining to two commonly used small animal models (ferret and rabbit). Sensitivity and specificity were greater than 90% for pathological lung segmentation (average dice similarity coefficient > 0.9). While qualitative visual assessments of airway tree extraction were performed by the participating expert radiologists, for quantitative evaluation the authors validated the proposed airway extraction method by using publicly available EXACT’09 data set. Conclusions: The authors developed a comprehensive computer-aided pulmonary image analysis framework for preclinical research applications. The proposed framework consists of automatic pathological lung segmentation and accurate airway tree extraction. The framework has high sensitivity and specificity; therefore, it can contribute advances in preclinical research in pulmonary diseases.
Airborne particulates are collected at U.S. Department of Energy sites that exhibit radiological contamination on the soil surface to help assess the potential for wind to transport radionuclides from the contamination sites. Collecting these samples was originally accomplished by drawing air through a cellulose-fiber filter. These filters were replaced with glass-fiber filters in March 2011. Airborne particulates were collected side by side on the two filter materials between May 2013 and May 2014. Comparisons of the sample mass and the radioactivity determinations for the side-by-side samples were undertaken to determine if the change in the filter medium produced significant results. The differences in the results obtained using the two filter types were assessed visually by evaluating the time series and correlation plots and statistically by conducting a nonparametric matched-pair sign test. Generally, the glass-fiber filters collect larger samples of particulates and produce higher radioactivity values for the gross alpha, gross beta, and gamma spectroscopy analyses. However, the correlation between the radioanalytical results for the glass-fiber filters and the cellulose-fiber filters was not strong enough to generate a linear regression function to estimate the glass-fiber filter sample results from the cellulose-fiber filter sample results.
With support from the U.S. Department of Energy-Office of Nuclear Energy’s (DOE-NE’s) Nuclear Energy Advanced Modeling and Simulation (NEAMS) program, an effort has been pursued to support high-temperature gas-cooled reactor (HTGR) technology development and its modeling and simulation needs. There is a particular need for advanced modeling and simulation tools to predict thermal-fluid behavior in the nuclear reactor primary system, especially in the core and the lower and upper plena, during safety-related transients. In this report, two main such activities are presented relevant to the HTGRs: (1) three-dimensional (3D) computational fluid dynamics (CFD) validation using benchmark data from the upper plenum of Texas A&M University’s 1/16th scaled very-high-temperature gas-cooled reactor (VHTR), and (2) development of wall heat-transfer correlation for laminar flow in a wall-heated pipe. The CFD tool validation exercises can be helpful to choose the models and CFD tools to simulate and design specific components of the HTRGs such as upper plenum where jet mixing is a complex phenomenon. In a loss of forced circulation event, the laminar flow can be observed during the development of natural circulation flow. This work includes the development and validation of heat transfer correlations for laminar flow using the Nek5000 CFD code due to limited available experimental data for laminar flow conditions to guide low-order models (1D). In this report, the flow characteristics of a single isothermal jet discharging into the upper plenum was investigated using the Nek5000 Large-Eddy Simulation (LES) CFD tool. Several numerical simulations were performed for various jet-discharged Reynolds numbers ranging from 3,413 to 12,819. A grid-independent study was performed. The numerical results of mean velocity, root-mean-square fluctuating velocity, and Reynolds stress were compared against the benchmark data. Good agreement was obtained between simulated and measured data for axial mean velocities, except near the upper plenum hemisphere. The maximum predicted errors for axial mean velocities at various normalized coolant channel diameter heights of 1, 5, and 10 are 1.56%, 1.88%, and 3.82%, respectively. In addition, the predicted root-mean-square fluctuating velocity and Reynolds stress are qualitatively in agreement with the experimental data. The Nek5000 code was used to develop wall-heat transfer correlation for laminar flow in a cylindrical tube. Several simulations were performed for various Reynolds flow and wall-heat fluxes. A new heat transfer correlation was developed using data from Nek5000 simulation results and regression functions in Matlab. The developed heat transfer correlation is valid for various Reynolds flows from 200 to 2000. The predicted R² value for model fit was 0.875, which ensures that 87.5% of the model data lies on the Nek5000 data. Moreover, a machine learning (ML) tool was used to train and test the Nek5000 data. A good fit of the ML-based model was observed with the test data.
Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a nonparametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications.
The kriging approach, a parametric regression method used by hydrologists and mining engineers, among others also provides an error estimate the integral of the regression function. The kriging method is explored and some of its statistical characteristics are described. The Watson method and theory are extended so that the kriging features are displayed. Theoretical and computational comparisons of the kriging and Watson approaches are offered.
For 435 accelerator produced antipions jets of 20 GeV/c and 300 GeV, in nuclear emulsion, eta(theta)'s have been individually calculated for each jet, where eta(theta) is a kinematic parameter introduced in order to approximate the LS (laboratory system) rapidity, eta = arctan h (beta cos theta). By taking further averages by dividing the samples into groupings of the LS energy E sub pi = m cos h eta sub pi N sub h, the number of heavy prongs with LS velocity beta 0.7, and n , the number of charged shower particles with LS velocity beta 0.7, much less than eta (theta) much greater than are obtained. By use of the KNO (Koba-Nielsen-Olesen) scaling variable, xi = n sub s/,n sub s. good fit is found of data to regression function.
Procedures that were developed to optimize the information content of multitemporal thematic mapper (TM) data sets for forest cover disturbance monitoring in Minnesota are described. TM imagery from three different years was calibrated to exoatmospheric reflectance. An atmospheric correction routine was applied combining two major components, atmospheric normalization over time and transformation to ground reflectance. Atmospheric conditions were modeled over time using regression functions derived from five ground features known to be unchanged over the time interval of interest and spanning the entire image reflectance range. The correlation between digital data and the forest cover was subsequently maximized and irrelevant information content was reduced by converting the band-specific reflectances into seven vegetation indices that were assumed to carry unique information. The application of two change detection algorithms to these seven indices ultimately resulted in 14 change features for each time interval of interest. Results show that the preprocessing sequence is vital to forest cover monitoring methodology.
We consider a scenario of two sites connected over a dedicated, long-haul connection that must quickly fail-over in response to degradations in host-to-host application performance. The traditional layer-2/3 hot stand-by fail-over solutions do not adequately address the variety of application degradations, and more recent single controller Software Defined Networks (SDN) solutions are not effective for long-haul connections. We present two methods for such a path fail-over using OpenFlow enabled switches: (a) a light-weight method that utilizes host scripts to monitor application performance and dpctl API for switching, and (b) a generic method that uses two OpenDaylight (ODL) controllers and REST interfaces. For both methods, the restoration dynamics of applications contain significant statistical variations due to the complexities of controllers, north bound interfaces and switches; they, together with the wide variety of vendor implementations, complicate the choice among such solutions. We develop the impulse-response method based on regression functions of performance parameters to provide a rigorous and objective comparison of different solutions. We describe testing results of the two proposed methods, using TCP throughput and connection rtt as main parameters, over a testbed consisting of HP and Cisco switches connected over longhaul connections emulated in hardware by ANUE devices. Lastly, the combination of analytical and experimental results demonstrate that the dpctl method responds seconds faster than the ODL method on average, even though both methods eventually restore original TCP throughput.
Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a nonparametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications