Efficient algorithms for Bayesian Nearest Neighbor Gaussian Processes

Abstract

We consider alternate formulations of recently proposed hierarchical Nearest Neighbor Gaussian Process (NNGP) models (Datta et al., 2016a) for improved convergence, faster computing time, and more robust and reproducible Bayesian inference. Algorithms are defined that improve CPU memory management and exploit existing high-performance numerical linear algebra libraries. Computational and inferential benefits are assessed for alternate NNGP specifications using simulated datasets and remotely sensed light detection and ranging (LiDAR) data collected over the US Forest Service Tanana Inventory Unit (TIU) in a remote portion of Interior Alaska. The resulting data product is the first statistically robust map of forest canopy for the TIU.

Figure 1:

Structure of the factors making up the sparse ${\tilde{C}}^{-1}$ matrix for n=200 and m=10.

Figure 2:

(a) Run time required for one sampler iteration using n=5 × 10⁴ by number of CPUs (y-axis is on the log scale). (b) Run time required for one sampler iteration by number of locations.

Figure 3:

Conjugate model cross-validation results for selection of α and ϕ using the simulated dataset. Parameter combination with minimum scoring rule indicated with open circle symbol ◦ and true combination used to generate the data indicated with a plus symbol +.

Figure 4:

TIU, Alaska, study region. (a) G-LiHT flight lines where canopy height was measured at 5 × 10⁶ locations and percent tree cover predictor variable. (b) Occurrence of forest fire within the past 20 years predictor variable and two example areas for prediction illustration.

Figure 5:

95th LiDAR percentile height posterior predictive distribution summary at a 30 m pixel resolution for the two example areas identified in Figure 4(b).