Comparing two spatial variables with the probability of agreement

TL;DR

This work extends the probability of agreement (PA) from scalar comparisons to spatial and spatiotemporal settings by defining PA as a function of spatial lag (and temporal lag in the spatiotemporal case). It derives closed-form PA expressions under Gaussian assumptions and common covariance families (e.g., Matérn, Wendland), and proves monotonic decay of PA with lag under suitable conditions. Estimation uses plug-in parameters with the delta method to obtain asymptotic uncertainty, enabling confidence intervals and hypothesis tests, demonstrated through simulations and a PhenoCam-based forest imagery example. The results show PA robustly captures practical agreement patterns across space and time, offering a tool to quantify and test spatial-temographic similarity in ecological and remote-sensing applications.

Abstract

Computing the agreement between two continuous sequences is of great interest in statistics when comparing two instruments or one instrument with a gold standard. The probability of agreement (PA) quantifies the similarity between two variables of interest, and it is useful for accounting what constitutes a practically important difference. In this article we introduce a generalization of the PA for the treatment of spatial variables. Our proposal makes the PA dependent on the spatial lag. As a consequence, for isotropic stationary and nonstationary spatial processes, the conditions for which the PA decays as a function of the distance lag are established. Estimation is addressed through a first-order approximation that guarantees the asymptotic normality of the sample version of the PA. The sensitivity of the PA is studied for finite sample size, with respect to the covariance parameters. The new method is described and illustrated with real data involving autumnal changes in the green chromatic coordinate (Gcc), an index of "greenness" that captures the phenological stage of tree leaves, is associated with carbon flux from ecosystems, and is estimated from repeated images of forest canopies.

Figures (16)

• Figure 1: The effect of variation in $\rho_{XY}$ (top left), $\mu_D$ (top right), $\phi_{XY}$ (bottom left), and $\sigma_Y$ (bottom right), on the behavior of PA as a function of lag $\|h\|$ in a bivariate Gaussian random field with a Matérn covariance structure. Note differences in range limits of the y-axis among the four panels.
• Figure 1: $\psi_c(h)$ versus $h \in \{0,1, \ldots,15\}$. (a) $c=1.5$; (b) $c=2$; (c) $c=2.5$.
• Figure 2: Estimates of the probability of agreement as a function of $\|\bm h\|$, $u$ and $c$ for a spatiotemporal Gaussian process with a positive linear trend and a non-separable Iacocesare covariance structure with fixed parameters given in Table \ref{['tab:spatiotemp']}, and for $N_s=50$. Note differences in range limits of the y-axis among the nine panels.
• Figure 2: Simulated realizations of a spatiotemporal process defined by a Gaussian random field with an exponential separable covariance function for $N_S=50$ and $N_T =6$.
• Figure 3: A. The PhenoCam image taken by a stationary camera from the EMS tower at Harvard Forest, Massachusetts, USA on 15 October 2008 at 13:31 (UTC $-4$). A rectangular section ($570 \times 660$ pixels) of the image was clipped (B; black outline in A) and then down-scaled (to $44 \times 58)$ pixels (C) for estimation of the spatial PA. Note that our rectangular image is different from the clipped "region of interest" (ROI; unmasked area in D) analyzed by the PhenoCam network.

Theorems & Definitions (8)

• Example 1
• Theorem 1
• Theorem 2
• Lemma 1
• Theorem 3
• Definition 1
• Lemma 2
• Theorem 4