Locally varying geostatistical machine learning for spatial prediction

Machine learning methods dealing with the spatial auto-correlation of the response variable have garnered significant attention in the context of spatial prediction.Nonetheless,under these methods,the relationship between the response variable and explanatory variables is assumed to be homogeneous throughout the entire study area.This assumption,known as spatial stationarity,is very questionable in real-world situations due to the influence of contextual factors.Therefore,allowing the relationship between the target variable and predictor variables to vary spatially within the study region is more reasonable.However,existing machine learning techniques accounting for the spatially varying relationship between the dependent variable and the predictor variables do not capture the spatial auto-correlation of the dependent variable itself.Moreover,under these techniques,local machine learning models are effectively built using only fewer observations,which can lead to well-known issues such as over-fitting and the curse of dimensionality.This paper introduces a novel geostatistical machine learning approach where both the spatial auto-correlation of the response variable and the spatial non-stationarity of the regression relationship between the response and predictor variables are explicitly considered.The basic idea consists of relying on the local stationarity assumption to build a collection of local machine learning models while leveraging on the local spatial auto-correlation of the response variable to locally augment the training dataset.The proposed method’s effectiveness is showcased via experiments conducted on synthetic spatial data with known characteristics as well as real-world spatial data.In the synthetic(resp.real)case study,the proposed method’s predictive accuracy,as indicated by the Root Mean Square Error(RMSE)on the test set,is 17%(resp.7%)better than that of popular machine learning methods dealing with the response variable’s spatial auto-correlation.Additionally,this method is not only valuable for spatial prediction but also offers a deeper understanding of how the relationship between the target and predictor variables varies across space,and it can even be used to investigate the local significance of predictor variables.

An interval finite element method for electromagnetic problems with spatially uncertain parameters

During the manufacturing process of dielectric materials used in electromagnetic engineering, the electromagnetic parameters are often spatially uncertain due to the processing technology, environmental temperature, personal operations, etc. Traditionally,the random field model can be used to measure the spatial uncertainties, but its construction requires a large number of samples.On the contrary, the interval field model only needs the upper and lower bounds of the spatially uncertain parameters, which requires much less samples and furthermore is easy to understand and use for engineers. Therefore, in this paper, the interval field model is introduced to describe the spatial uncertainties of dielectric materials, and then an interval finite element method(IFEM) is proposed to calculate the upper and lower bounds of electromagnetic responses. Firstly, the interval field of the dielectric material is represented by the interval K-L expansion and inserted into the scalar Helmholtz wave equations, and thus the interval equilibrium equations are constructed according to the node-based finite element method. Secondly, a perturbation interval finite element method is developed for calculating the upper and lower bounds of electromagnetic responses such as the electric strength and magnetic strength. Finally, the effectiveness of the proposed method is verified by three numerical examples.

Uncertain eigenvalue analysis of the dielectric-filled waveguide by an interval vector finite element method

Eigenvalues of the dielectric-filled waveguide are of great importance to its transmission characteristic analysis and optimization design, which could be easily affected by spatially uncertain dielectric parameters. For the sake of quantifying their influence on eigenvalues of the dielectric-filled waveguide and overcoming the limitation of less samples, an interval vector finite element method(IVFEM) is proposed to acquire the lower and upper bounds of the eigenvalues with spatial uncertainty of the medium parameters. Firstly, the uncertain dielectric material properties are described by the interval field model and the corresponding interval Karhunen-Loève(K-L) approximate method. Secondly, by inserting the interval uncertainties into the constitutive relationship of the standard generalized eigenvalue equations of the dielectric-filled waveguide, an interval standard generalized eigenvalue equation is then formulated. At last, the lower and upper bounds of the eigenvalues are calculated according to the first-order perturbation method, which can be used to estimate the transmission properties of the waveguide efficiently. Three kinds of the dielectric-filled waveguides are analyzed by the proposed IVFEM and verified by Monte Carlo simulation method.