Gaussian Process Method Research Articles

Geometrical deviation modeling and monitoring of 3D surface based on multi-output Gaussian process

Geometrical deviation is an important factor in determining the quality of a three-dimensional (3D) Surface. For 3D surfaces with complex shapes, the high-definition measurement (HDM) technology can provide detailed information on surface topography, which inspired new challenges in characterizing, modeling, and monitoring geometrical deviations. This paper proposes a spherical multi-output Gaussian process (S-MOGP) method to model and monitor 3D surfaces. Firstly, the surface in the 3D coordinate system is mapped to the spherical 2D parameter domain. Secondly, a state equation based on the multi-output Gaussian process is established to model the 3D surface. Finally, statistics are calculated and control charts are presented to monitor the geometrical deviation. The results of simulations and a case study show that the proposed method can effectively model 3D surfaces and monitor the geometrical deviations.

Scaled Vecchia Approximation for Fast Computer-Model Emulation

Many scientific phenomena are studied using computer experiments consisting of multiple runs of a computer model while varying the input settings. Gaussian processes (GPs) are a popular tool for the analysis of computer experiments, enabling interpolation between input settings, but direct GP inference is computationally infeasible for large datasets. We adapt and extend a powerful class of GP methods from spatial statistics to enable the scalable analysis and emulation of large computer experiments. Specifically, we apply Vecchia’s ordered conditional approximation in a transformed input space, with each input scaled according to how strongly it relates to the computer-model response. The scaling is learned from the data by estimating parameters in the GP covariance function using Fisher scoring. Our methods are highly scalable, enabling estimation, joint prediction, and simulation in near-linear time in the number of model runs. In several numerical examples, our approach substantially outperformed existing methods.

Transfer learning based on sparse Gaussian process for regression

Transfer learning is to use the knowledge obtained from the source domain to improve the learning efficiency when the target domain has insufficient labeled data. For regression problems, when the conditional distribution function and the marginal distribution function of the source domain and the target domain are different, how to effectively extract similar knowledge for transfer learning is still a problem. In this paper, we propose a transfer learning method for regression problem based on the sparse Gaussian process (GP). GP models are very popular in regression modeling, as they have the capability to produce uncertainty estimation, however, they cannot be used directly for transfer learning. We propose an adaptive neural kernel network (ANKN) to ensure that the GP model can effectively transfer knowledge. Additionally, although many sparse GP methods are proposed to solve the time consumption problem of the GP models in large datasets, they cannot maintain the transfer performance. We propose a transfer inducing point (TIP) algorithm for data selection in large datasets to maintain the transfer performance. The experiments with transfer regression problems on both real-world small datasets and large datasets indicate that the our method significantly increases prediction accuracy and effectiveness.

Constraints on Cosmographic Functions of Cosmic Chronometers Data Using Gaussian Processes

We study observational constraints on the cosmographic functions up to the fourth derivative of the scale factor with respect to cosmic time, i.e., the so-called snap function, using the non-parametric method of Gaussian Processes. As observational data we use the Hubble parameter data. Also we use mock data sets to estimate the future forecast and study the performance of this type of data to constrain cosmographic functions. The combination between a non-parametric method and the Hubble parameter data is investigated as a strategy to reconstruct cosmographic functions. In addition, our results are quite general because they are not restricted to a specific type of functional dependency of the Hubble parameter. We investigate some advantages of using cosmographic functions instead of cosmographic series, since the former are general definitions free of approximations. In general, our results do not deviate significantly from $\Lambda CDM$. We determine a transition redshift $z_{tr}=0.637^{+0.165}_{-0.175}$ and $H_{0}=69.45 \pm 4.34$. Also assuming priors for the Hubble constant we obtain $z_{tr}=0.670^{+0.210}_{-0.120}$ with $H_{0}=67.44$ (Planck) and $z_{tr}=0.710^{+0.159}_{-0.111}$ with $H_{0}=74.03$(SH0ES). Our main results are summarized in table 2.

Numerical methods for mean field games based on Gaussian processes and Fourier features

In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.

Tercile Forecasts for Extending the Horizon of Skillful Hydrological Predictions

Abstract Medium to subseasonal hydrological forecasts contain more information relevant to water and environmental management tasks than climatological forecasts. However, extracting this information at the most appropriate level of accuracy and spatiotemporal resolution remains a difficulty. Many studies show that the skill of the extended range forecasts with daily resolution tends toward zero after 7–14 days for small mountainous catchments. Beyond that forecast horizon the application of highly sophisticated pre- and postprocessing methods generally produce limited gains. Consequently, current forecasting techniques cannot effectively represent forecast extremes at extended ranges such as anomalously high and low runoff or soil moisture. To tackle these deficiencies, this study analyzes the value of tercile forecasts for weekly aggregates of runoff and soil moisture forecasts available at a daily resolution for Switzerland. The forecasts are classified into three categories: below, above, and normal conditions, which are derived from long-term simulations and correspond approximately to climatological conditions. To achieve improved reliability and skill of the predicted tercile probabilities, a nonparametric probabilistic classification method has been tested. It is based on Gaussian process (GP), which is attractive in machine learning (ML) applications because of its ability to estimate the predictive uncertainty. The outcome of these postprocessed forecasts was compared to preprocessing methods where the meteorological predictions are statistically corrected before passing to the hydrological model. Our results indicate that tercile forecasts of weekly aggregates produce a suitable skill up to 3 weeks lead time using the preprocessed input and up to 4 weeks lead time using the GP method.

Bayesian dynamic linear model framework for structural health monitoring data forecasting and missing data imputation during typhoon events

A Bayesian dynamic linear model (BDLM) framework for data modeling and forecasting is proposed to evaluate the performance of an operational cable-stayed bridge, that is, Ting Kau Bridge in Hong Kong, by using SHM strain field data acquired. One of the major challenges in dealing with the existing in-service bridge under extreme typhoon loads is to forecast structural behavior using the typhoon response exhibiting non-stationarity, large data fluctuations and strong randomness. The first attempt for SHM data modeling during extreme events, that is, typhoons, using BDLM framework, was conducted in this study. The data from multiple sensors are analyzed for one-step, multi-step ahead forecasting and missing data imputation. The overall bridge behavior is incorporated into a forecasting model by superposition of forecasting results of trend (representing the structural baseline response), periodic component (response component evolving regularly over time), and autoregressive component (time-dependent error) through BDLM algorithm. The results demonstrate that the BDLM framework yielded more accurate calculations compared with Gaussian process and Variational Heteroscedasticity Gaussian Process methods with respect to one-step ahead forecasting for strain data under typhoons. Multi-step ahead forecasting was successfully carried out both for non-typhoon and typhoon responses within an acceptable precision range. The correlation between periodic component and temperature was also investigated. Regarding missing data imputation, BLDM algorithm can generate robust results due to making full use of the monitoring data both before and after the missing segments.

A new GW-based heteroscedastic Gaussian process method for online crack evaluation

Accurate evaluation of fatigue crack by using structural health monitoring (SHM) methods is very important to the life management of aircraft structures. However, fatigue crack propagation is always a complicated process for individual aircraft structures during their long-term service. And the monitoring has to be performed under different environmental and operational conditions. These factors result in the uncertain distribution of the damage index obtained by SHM. The distribution usually changes along with the service time, which can be defined as heteroscedasticity. The heteroscedasticity characteristic of the uncertain distribution of damage index has an important negative impact on the SHM based diagnostics algorithms if not considered during the evaluation. However, till now, few researchers have considered this important aspect. To improve the evaluation accuracy under different environmental and operational conditions for individual aircraft, this paper proposes a new guided wave-based heteroscedastic Gaussian process method. Gaussian process quantile regression is adopted to estimate the conditional quantiles of the damage index distribution during the service to deal with the heteroscedasticity. The method is validated on an attachment lug fatigue test, an important aircraft structural component. The experimental results demonstrate that the proposed method can quantify the heteroscedastic uncertainty associated and obviously improve the quality of crack evaluation. For the serious heteroscedastic specimen, the maximum evaluation is only 0.7 mm reduced from the original 7.4 mm, which is an order of magnitude.

Applications of a Gaussian process framework for modelling of high-resolution exoplanet spectra

ABSTRACT Observations of exoplanet atmospheres in high resolution have the potential to resolve individual planetary absorption lines, despite the issues associated with ground-based observations. The removal of contaminating stellar and telluric absorption features is one of the most sensitive steps required to reveal the planetary spectrum and, while many different detrending methods exist, it remains difficult to directly compare the performance and efficacy of these methods. Additionally, though the standard cross-correlation method enables robust detection of specific atmospheric species, it only probes for features that are expected a priori. Here, we present a novel methodology using Gaussian process (GP) regression to directly model the components of high-resolution spectra, which partially addresses these issues. We use two archival CRyogenic Infra-Red Echelle Spectrograph (CRIRES)/Very Large Telescope (VLT) data sets as test cases, observations of the hot Jupiters HD 189733 b and 51 Pegasi b, recovering injected signals with average line contrast ratios of ∼4.37 × 10−3 and ∼1.39 × 10−3, and planet radial velocities ΔKp = 1.45 ± 1.53 $\mathrm{km\, s^{-1}}$ and ΔKp = 0.12 ± 0.12 $\mathrm{km\, s^{-1}}$ from the injection velocities, respectively. In addition, we demonstrate an application of the GP method to assess the impact of the detrending process on the planetary spectrum, by implementing injection-recovery tests. We show that standard detrending methods used in the literature negatively affect the amplitudes of absorption features in particular, which has the potential to render retrieval analyses inaccurate. Finally, we discuss possible limiting factors for the non-detections using this method, likely to be remedied by higher signal-to-noise data.

Spatiotemporal Geostatistical Analysis of Groundwater Level in Aquifer Systems of Complex Hydrogeology

AbstractDirect interpolation of groundwater levels often leads to contour maps which are hydrogeological inconsistent since numerical algorithms do not consider changes in flowline patterns caused by hydrogeological heterogeneities and aquifer boundary conditions. In the present work, this issue is assessed by conducting a geostatistical analysis based on Gaussian process method, using space‐time groundwater level observations, to generate reliable spatial maps of groundwater level variability and to identify groundwater level patterns over the island of Crete, Greece. Besides, two innovative tools are employed in the process: the Manhattan distance metric to obtain spatial correlation where faults are present in the aquifer and the spatiotemporal Spartan covariance function to obtain spatiotemporal interdependence. The results show significant prediction improvement over convectional geostatistical methods. Useful information is obtained from the delivered map notifying areas under high risk of groundwater resources shortage. The developed approach could be applied to other areas with analogous hydrogeological properties. It will be especially valuable in semi‐arid areas prone to droughts, where groundwater represents the main source of water.

Galaxy-scale Test of General Relativity with Strong Gravitational Lensing

Abstract Although general relativity (GR) has been precisely tested at the solar system scale, precise tests at a galactic or cosmological scale are still relatively insufficient. Here, in order to test GR at the galactic scale, we use the newly compiled galaxy-scale strong gravitational lensing (SGL) sample to constrain the parameter γ PPN in the parameterized post-Newtonian (PPN) formalism. We employ the Pantheon sample of Type Ia supernova observations to calibrate the distances in the SGL systems using the Gaussian Process method, which avoids the logical problem caused by assuming a cosmological model within GR to determine the distances in the SGL sample. Furthermore, we consider three typical lens models in this work to investigate the influences of the lens-mass distributions on the fitting results. We find that the choice of lens models has a significant impact on the constraints on the PPN parameter γ PPN. By using a minimum χ 2 comparison and the Bayesian information criterion as evaluation tools to make comparisons for the fitting results of the three lens models, we find that the most reliable lens model gives the result of γ PPN = 1.065 − 0.074 + 0.064 , which is in good agreement with the prediction of γ PPN = 1 by GR. As far as we know, our 6.4% constraint result is the best result so far among recent works using the SGL method.

An application of Gaussian process modeling for high-order accurate adaptive mesh refinement prolongation

We present a new polynomial-free prolongation scheme for Adaptive Mesh Refinement (AMR) simulations of compressible and incompressible computational fluid dynamics. The new method is constructed using a multi-dimensional kernel-based Gaussian Process (GP) prolongation model. The formulation for this scheme was inspired by the GP methods introduced by A. Reyes et al. (A New Class of High-Order Methods for Fluid Dynamics Simulation using Gaussian Process Modeling, Journal of Scientific Computing, 76 (2017), 443-480; A variable high-order shock-capturing finite difference method with GP-WENO, Journal of Computational Physics, 381 (2019), 189-217). In this paper, we extend the previous GP interpolations and reconstructions to a new GP-based AMR prolongation method that delivers a high-order accurate prolongation of data from coarse to fine grids on AMR grid hierarchies. In compressible flow simulations special care is necessary to handle shocks and discontinuities in a stable manner. To meet this, we utilize the shock handling strategy using the GP-based smoothness indicators developed in the previous GP work by A. Reyes et al. We demonstrate the efficacy of the GP-AMR method in a series of testsuite problems using the AMReX library, in which the GP-AMR method has been implemented.

Agbots 3.0: Adaptive Weed Growth Prediction for Mechanical Weeding Agbots

This work presents advances in predictive modeling of weed growth, as well as an improved planning index to be used in conjunction with these techniques, for the purpose of improving the performance of coordinated weeding algorithms being developed for industrial agriculture. We demonstrate that the evolving Gaussian process (E-GP) method applied to measurements from the agents can predict the evolution of the field within the realistic simulation environment, Weed World. This method also provides physical insight into the seed bank distribution of the field. In this work, we extend the E-GP model in two important ways. First, we have developed a model that has a bias term, and we show how it is connected to the seed bank distribution. Second, we show that one may decouple the component of the model representing weed growth from the component, which varies with the seed bank distribution, and adapt the latter online. We compare this predictive approach with one that relies on known properties of the weed growth model and show that the E-GP method can drive down the total weed biomass for fields with high seed bank densities using less agents, without assuming this model information. We use an improved planning index, the Whittle index, which allows a balanced tradeoff between exploiting a row or allowing it to accrue reward and conforms to what we show is the theoretical limit for the fewest number of agents, which can be used in this domain.

PairGP: Gaussian process modeling of longitudinal data from paired multi-condition studies

High-throughput technologies produce gene expression time-series data that need fast and specialized algorithms to be processed. While current methods already deal with different aspects, such as the non-stationarity of the process and the temporal correlation, they often fail to take into account the pairing among replicates.We propose PairGP, a non-stationary Gaussian process method to compare gene expression time-series across several conditions that can account for paired longitudinal study designs and can identify groups of conditions that have different gene expression dynamics. We demonstrate the method on both simulated data and previously unpublished RNA sequencing (RNA-seq) time-series with five conditions. The results show the advantage of modeling the pairing effect to better identify groups of conditions with different dynamics. The pairing effect model displays good capabilities of selecting the most probable grouping of conditions even in the presence of a high number of conditions.The developed method is of general application and can be applied to any gene expression time series dataset. The model can identify common replicate effects among the samples coming from the same biological replicates and model those as separate components. Learning the pairing effect as a separate component, not only allows us to exclude it from the model to get better estimates of the condition effects, but also to improve the precision of the model selection process. The pairing effect that was accounted before as noise, is now identified as a separate component, resulting in more accurate and explanatory models of the data.

Structural performance assessment considering both observed and latent environmental and operational conditions: A Gaussian process and probability principal component analysis method

Structural faults like damage and degradations will cause changes in structure response data. Performance assessment can be conducted by investigating such changes. In real implementations however, structural responses are affected by environmental and operational variations (EOVs) as well. Such variation should be well captured by the assessment model when detecting structural changes. It should be noted that not all EOVs can be measured by the monitoring system. When both observed and latent EOVs have significant effects on the monitored structural responses, these two effects should be considered properly. Furthermore, uncertainties can be significant for the monitoring data since loads and EOVs cannot be directly controlled under working conditions. To address these problems, this work proposes a performance assessment method considering both observed and latent EOVs. A Gaussian process is used to model the functional behaviour between structural response and observed EOVs whilst principal component analysis is used to eliminate the effect of latent EOVs. These two methods are combined using a Bayesian formulation and the effect of both observed and latent EOVs are modelled. The associated model parameters are inferred through probability density functions to account for the uncertainties. A synthetic data example is presented to validate the proposed method. It is also applied to the monitoring data of a long-span cable-stayed bridge with different damage scenarios considered, illustrating its capability of real implementations in structural health monitoring.

Sparse Gaussian Processes for Solving Nonlinear Pdes

In this article, we propose a numerical method based on sparse Gaussian processes (SGPs) to solve nonlinear partial differential equations (PDEs). The SGP algorithm is based on a Gaussian process (GP) method, which approximates the solution of a PDE with the maximum a posteriori probability estimator of a GP conditioned on the PDE evaluated at a finite number of sample points. The main bottleneck of the GP method lies in the inversion of a covariance matrix, whose cost grows cubically with respect to the size of samples. To improve the scalability of the GP method while retaining desirable accuracy, we draw inspiration from SGP approximations, where inducing points are introduced to summarize the information of samples. More precisely, our SGP method uses a Gaussian prior associated with a low-rank kernel generated by inducing points randomly selected from samples. In the SGP method, the size of the matrix to be inverted is proportional to the number of inducing points, which is much less than the size of the samples. The numerical experiments show that the SGP method using less than half of the uniform samples as inducing points achieves comparable accuracy to the GP method using the same number of uniform samples, which significantly reduces the computational cost. We give the existence proof for the approximation to the solution of a PDE and provide rigorous error analysis.

Remote Sensing Monitoring of Wheat Stripe Rust Based on Fractional Order Differential and Gaussian Process Methods

Remote Sensing Monitoring of Wheat Stripe Rust Based on Fractional Order Differential and Gaussian Process Methods

Probabilistic Wind Power Forecasting for Newly-Built Wind Farm Based on Multi-Task Gaussian Process Method

Probabilistic Wind Power Forecasting for Newly-Built Wind Farm Based on Multi-Task Gaussian Process Method

Optimization and comparison of models for core temperature prediction of mother rabbits using infrared thermography

Core temperature is a measurement used to evaluate and diagnose the health status of mother rabbits. The preferred method is to use an indwelling probe to measure the rectal temperature. This procedure can be stressful to rabbits, are time and labor-consuming. The objective of this paper was to develop the fastest, accurate and non-invasive method to measure the rectal temperature of mother rabbits using Infrared thermography (IRT) and machine learning (ML). Two hundred lactating mother rabbits (LR) and two hundred non-lactating mother rabbits (NLR) of the YPLU breed were used in the experiment and IRT data, rectal temperature and ambient temperature were recorded. The results showed that there was a significant correlation between the maximum infrared temperature of eyes (EMIT) and ears (RMIT) and rectal temperature (p < 0.05) and a significant correlation between the EMIT, RMIT and the ambient temperature (p < 0.01) in two kinds of mother rabbits. Least squares support vector machine (LS-SVM), Gaussian process (GP), and partial least squares (PLS) machine learning methods were used to establish models of mother rabbits’ core temperature. Using the over-sampling method to process the training set, three models of LS-SVM, GP, and PLS with improved generalization performance were obtained. The RMSE for the test set by LS-SVM, GP, and PLS methods were 0.13℃, 0.12℃ and 0.07℃ and the R2 were 0.94, 0.96 and 0.98, respectively. Compared with the results obtained from two other regressions, LS-SVM and GP, the PLS method exhibited the best performance. The prediction error of the test set is <0.2℃, which can better predict the core temperature of the mother rabbits.

A Gaussian Process Method with Uncertainty Quantification for Air Quality Monitoring

The monitoring and forecasting of particulate matter (e.g., PM2.5) and gaseous pollutants (e.g., NO, NO2, and SO2) is of significant importance, as they have adverse impacts on human health. However, model performance can easily degrade due to data noises, environmental and other factors. This paper proposes a general solution to analyse how the noise level of measurements and hyperparameters of a Gaussian process model affect the prediction accuracy and uncertainty, with a comparative case study of atmospheric pollutant concentrations prediction in Sheffield, UK, and Peshawar, Pakistan. The Neumann series is exploited to approximate the matrix inverse involved in the Gaussian process approach. This enables us to derive a theoretical relationship between any independent variable (e.g., measurement noise level, hyperparameters of Gaussian process methods), and the uncertainty and accuracy prediction. In addition, it helps us to discover insights on how these independent variables affect the algorithm evidence lower bound. The theoretical results are verified by applying a Gaussian processes approach and its sparse variants to air quality data forecasting.