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Spatio-temporal Models
Published in Yu Ding, Data Science for Wind Energy, 2019
Another popular family of the covariance function is the Matérn covariance function, which has a smoothness parameter, υ, that can control the smoothness of sample functions more precisely. Specifically, the sample functions are almost surely continuously differentiable of order [υ] − 1, where ⌈·⌉ rounds up to the next integer. We choose to omit the presentation of the Matérn covariance function because we do not use it in this book. Interested readers can refer to [173] for more information.
Probabilistic nonlinear dimensionality reduction through gaussian process latent variable models: An overview
Published in Arun Kumar Sinha, John Pradeep Darsy, Computer-Aided Developments: Electronics and Communication, 2019
The element at ith row and jth column of K is given by the prior distribution (36). Thus, the marginal likelihood of dual probabilistic PCA is a product of d independent Gaussian processes. The covariance function of a Gaussian process describes the properties of functions, such as variability. Learning in Gaussian processes is to determine hyperparameters of a covariance function that is suitable for the problem being modelled.
Gaussian Process–Based Inverse Uncertainty Quantification for TRACE Physical Model Parameters Using Steady-State PSBT Benchmark
Published in Nuclear Science and Engineering, 2019
Chen Wang, Xu Wu, Tomasz Kozlowski
where the hyperparameters and define the proprieties of the covariance function such as magnitude, shape, and smoothness. They should be properly estimated to obtain the best possible prediction performance. MLE or cross validation can be used to estimate the hyperparameters. Details about parameter estimation and other types of covariance functions can be found in Ref. 26. The accuracy of the GP emulator needs to be assessed before use. We are interested in the predictive accuracy at untried points, which can be done by quantifying the predictive error at an additional set of validation data.
Dynamic sampling method for ship resistance performance optimisation based on approximated model
Published in Ships and Offshore Structures, 2021
Haichao Chang, Chengsheng Zhan, Zuyuan Liu, Xide Cheng, Baiwei Feng
To distribute sufficient samples nearby the samples with higher eLOO located in the CAMM regions, new samples are required. First, the scope of the CAMM regions should be determined before the new samples are selected. Subsequently, the new samples will be selected. The covariance function in statistics is the function used to reflect the correlation degree between the data. Herein, the scope of the CAMM regions is defined by it.
Layer-wise spatial modeling of porosity in additive manufacturing
Published in IISE Transactions, 2019
Jia (Peter) Liu, Chenang Liu, Yun Bai, Prahalada Rao, Christopher B. Williams, Zhenyu (James) Kong
ST-LGCP is defined as a hierarchical model, the first-level of which is a Gaussian Process (GP) that accommodates a nonparametric intensity function , where is an AM part layer and is a realization from the GP (Moller et al., 1998; Brix and Diggle, 2001): where mean , is the covariate of the realization , is the number of layers, is the parameter for the covariates. The covariance function is typically represented as a distance-based kernel function with an assumption that shorter distances result in higher correlations. The radial basis function is a popular choice, due to its compact form that involves only three parameters: namely, variance , spatial scale parameter and temporal scale parameter Accordingly, a separable spatiotemporal covariance function can be written as follows (Cressie and Wikle, 2011): where and are two locations within the region of interest on the layers and of XCT scan images, respectively, .