Assessing the Competitiveness of Matrix-Free Block Likelihood Estimation in Spatial Models

Abstract

In geostatistics, block likelihood offers a balance between statistical accuracy and computational efficiency when estimating covariance functions. This balance is reached by dividing the sample into blocks and computing a weighted sum of (sub) log-likelihoods corresponding to pairs of blocks. Practitioners often choose block sizes ranging from hundreds to a few thousand observations, inherently involving matrix-based implementations. An alternative, residing at the opposite end of this methodological spectrum, treats each observation as a block, resulting in the matrix-free pairwise likelihood method. We propose an additional alternative within this broad methodological landscape, systematically constructing blocks of size two and merging pairs of blocks through conditioning. Importantly, our method strategically avoids large-sized blocks, facilitating explicit calculations that ultimately do not rely on matrix computations. Studies with both simulated and real data validate the effectiveness of our approach, on one hand demonstrating its superiority over pairwise likelihood, and on the other, challenging the intuitive notion that employing matrix-based versions universally lead to better statistical performance.

Figures (7)

• Figure 1: Spectrum of block likelihood estimation, categorized by the employed block sizes. The shaded box illustrates the utilization of blocks of size two, which is the basis for the matrix-free bi-conditional likelihood method introduced in Section \ref{['sec:bi']}.
• Figure 2: From left to right: $12$ spatial sites and two possible configurations with 6 bi-dimensional blocks. Pairs of blocks that are part of the objective function are symbolized by the solid line connecting them, while the dashed line represents a list of sites that do not interact simultaneously for that configuration.
• Figure 3: Simulated locations and convex hulls for varying block configurations with 16, 25, and 36 blocks (from left to right).
• Figure 4: Comparison of the relative root mean square error and relative global efficiency of the estimates with respect to the full likelihood, computed from 1000 independent replicates. The three color zones in each panel correspond to the exponential, Matérn (1.5) and Cauchy models (from left to right). BCL$_{16}$, BCL$_{25}$ and BCL$_{36}$ refer to block likelihood with 16, 25 and 36 blocks, respectively.
• Figure 5: Execution time (in seconds) for evaluating the bi-CL method and the Cholesky factorizations required for BCL$_8$ and BCL$_{16}$ methods.

Theorems & Definitions (1)

• Remark 3.1