[Paper Review] Bivariate DeepKriging for Large-scale Spatial Interpolation of Wind Fields

Pratik Nag, Ying Sun+1 more|arXiv (Cornell University)|Jul 16, 2023

Energy Load and Power Forecasting4 citations

TL;DR

This paper proposes Bivariate DeepKriging, a spatially adaptive deep neural network with radial basis function embeddings for large-scale bivariate wind field interpolation. It outperforms traditional cokriging in accuracy and reduces computation time by 20×, achieving superior prediction and uncertainty quantification on 506,771 wind data points across the Middle East region.

ABSTRACT

High spatial resolution wind data are essential for a wide range of applications in climate, oceanographic and meteorological studies. Large-scale spatial interpolation or downscaling of bivariate wind fields having velocity in two dimensions is a challenging task because wind data tend to be non-Gaussian with high spatial variability and heterogeneity. In spatial statistics, cokriging is commonly used for predicting bivariate spatial fields. However, the cokriging predictor is not optimal except for Gaussian processes. Additionally, cokriging is computationally prohibitive for large datasets. In this paper, we propose a method, called bivariate DeepKriging, which is a spatially dependent deep neural network (DNN) with an embedding layer constructed by spatial radial basis functions for bivariate spatial data prediction. We then develop a distribution-free uncertainty quantification method based on bootstrap and ensemble DNN. Our proposed approach outperforms the traditional cokriging predictor with commonly used covariance functions, such as the linear model of co-regionalization and flexible bivariate Matérn covariance. We demonstrate the computational efficiency and scalability of the proposed DNN model, with computations that are, on average, 20 times faster than those of conventional techniques. We apply the bivariate DeepKriging method to the wind data over the Middle East region at 506,771 locations. The prediction performance of the proposed method is superior over the cokriging predictors and dramatically reduces computation time.

Motivation & Objective

• Address the challenge of large-scale, non-Gaussian, non-stationary bivariate wind field interpolation in climate and renewable energy applications.
• Overcome the limitations of traditional cokriging, which assumes Gaussianity and suffers from O(N³) computational complexity on large datasets.
• Develop a deep learning-based spatial prediction model that handles non-Gaussianity, non-stationarity, and high spatial variability without parametric distributional assumptions.
• Enable efficient uncertainty quantification via a bootstrap-based ensemble method for deep neural networks in spatial statistics.
• Demonstrate scalability and performance on a real-world dataset of 506,771 wind observation points across the Middle East, including high-resolution downscaling for NEOM region.

Proposed method

• Propose a deep neural network (DNN) with spatially dependent feature embeddings using radial basis functions (RBFs) to model bivariate wind components (U and V) as inputs.
• Integrate spatial coordinates into the DNN via RBFs to encode spatial dependence, enabling the network to learn complex, non-stationary spatial patterns.
• Use a weighted mean squared error (MSE) loss function with variance-proportional weights to improve prediction accuracy for both wind components.
• Implement a distribution-free uncertainty quantification method using bootstrap resampling and ensemble DNNs to generate prediction intervals.
• Train the model on full datasets (up to 506,771 locations), while cokriging is restricted to subsamples (147,000 locations) due to computational constraints.
• Compare performance using RMSPE, PICP (prediction interval coverage probability), and MPIW (mean prediction interval width) metrics.

Experimental results

Research questions

• RQ1 Can a deep learning model outperform classical cokriging in predicting large-scale, non-Gaussian bivariate wind fields?
• RQ2 How does the proposed Bivariate DeepKriging method handle non-stationarity and non-Gaussianity in spatial wind data compared to parametric models?
• RQ3 To what extent does the model reduce computational time while maintaining or improving prediction accuracy on massive spatial datasets?
• RQ4 Can a nonparametric, ensemble-based uncertainty quantification method provide reliable prediction intervals without assuming symmetric t-distributions?
• RQ5 How effective is the RBF-based spatial embedding in capturing complex spatial dependence in high-resolution wind field interpolation?

Key findings

• Bivariate DeepKriging reduced RMSPE to 0.394 (with 450,000 training points), significantly outperforming cokriging (RMSPE = 0.882).
• The model achieved 97.1% and 95.0% PICP for U and V components, respectively, compared to 60.1% and 73.4% for cokriging, indicating far more reliable prediction intervals.
• Bivariate DeepKriging computed point predictions in 16.81 minutes (1,009 seconds), while cokriging required 2.18 days (188,352 seconds), achieving a 20× speedup.
• The model’s interval prediction took 55.61 minutes (3,337 seconds), still substantially faster than cokriging, demonstrating scalability.
• The method enabled high-resolution spatial downscaling from 5km×5km to 1km×1km over the NEOM region, providing detailed wind field maps for renewable energy planning.
• The model’s performance was robust even when trained on 450,000 points, showing strong generalization and scalability on large spatial datasets.