Effective Sample Size for Functional Spatial Data

TL;DR

This work extends the scalar notion of effective sample size (ESS) to functional spatial data by defining a functional ESS via the trace-covariogram, capturing dependence across space in a Hilbert-space setting. The authors derive the fundamental properties, provide estimation procedures based on empirical trace-covariograms, and illustrate the method with a FAR(1) functional model and a real oceanographic dataset on geometric vertical velocities, showing substantial information reduction can be achieved (e.g., maintaining key structure with roughly 17% of the data). They also demonstrate practical subsampling diagnostics through functional boxplots and resampling. The framework offers a principled tool for redundancy quantification and data-subsampling decisions in functional spatial analyses, with avenues for extending to multivariate, anisotropic settings and uncertainty quantification.

Abstract

The effective sample size quantifies the amount of independent information contained in a dataset, accounting for redundancy due to correlation between observations. While widely used in geostatistics for scalar data, its extension to functional spatial data has remained largely unexplored. In this work, we introduce a novel definition of the effective sample size for functional geostatistical data, employing the trace-covariogram as a measure of correlation, and show that it retains the intuitive properties of the classical scalar ESS. We illustrate the behavior of this measure using a functional autoregressive process, demonstrating how serial dependence and the allocation of variability across eigen-directions influence the resulting functional ESS. Finally, the approach is applied to a real meteorological dataset of geometric vertical velocities over a portion of the Earth, showing how the method can quantify redundancy and determine the effective number of independent curves in functional spatial datasets.

Figures (5)

• Figure 3.1: Functional ESS under the FAR(1) model. Left: dependence on $\lambda_0 \in (0,1)$ with $\lambda_k = \lambda_0^k$ and fixed $\eta_k = (1/2)^k$. Right: dependence on $\eta_0 \in (0,1)$ with $\eta_k = \eta_0^k$ and fixed $\lambda_k = (1/2)^k$. Results are shown for $n = 30, 60,$ and $120$.
• Figure 4.1: Geometric vertical velocity (in m$\cdot$s$^{-1}$) over a region of the Pacific Ocean for January 2024. Left: spatial maps at three depth levels (20 m, 10 m, and 1 m, from top to bottom). Right: vertical profiles of velocity as a function of depth (in m) for each of the 600 spatial locations.
• Figure 4.2: Empirical (circles) and fitted (curves) (semi) trace-variograms for the geometric vertical velocity dataset.
• Figure 4.3: Functional boxplots of the full dataset and of three independent subsamples of size 106, corresponding to the maximum functional ESS value obtained across all models.
• Figure 4.4: Functional boxplot of the full sample, with an additional dashed band around the functional median. The band extends by $\pm 2.11\times 10^{-7}$, corresponding to the mean absolute discrepancy between the functional median of the full sample and the medians computed across 1,000 subsamples.

Theorems & Definitions (9)

• Definition 2.1: Griffith:2005
• Definition 2.2: Weak Stationarity
• Definition 2.3: Isotropy
• Definition 3.1
• Proposition 3.1
• Proof
• Proposition 3.2
• Proposition 3.3
• Proof