# A Spatial Concordance Correlation Coefficient with an Application to Image Analysis

## TL;DR

This paper introduces a spatial concordance coefficient for comparing spatial sequences along a fixed lag, explores its properties for specific covariance functions, and applies it to analyze digital images of forest canopies.

## Contribution

It defines a new spatial concordance coefficient, establishes its asymptotic normality, and develops a local estimation approach for large digital images.

## Key findings

- The coefficient was explored for Matérn and Wendland covariance functions.
- Asymptotic normality was proven for Wendland covariance in increasing domain sampling.
- The local approach revealed spatial decay of concordance in forest canopy images.

## Abstract

In this work we define a spatial concordance coefficient for second-order stationary processes. This problem has been widely addressed in a non-spatial context, but here we consider a coefficient that for a fixed spatial lag allows one to compare two spatial sequences along a 45-degree line. The proposed coefficient was explored for the bivariate Mat\'ern and Wendland covariance functions. The asymptotic normality of a sample version of the spatial concordance coefficient for an increasing domain sampling framework was established for the Wendland covariance function. To work with large digital images, we developed a local approach for estimating the concordance that uses local spatial models on non-overlapping windows. Monte Carlo simulations were used to gain additional insights into the asymptotic properties for finite sample sizes. As an illustrative example, we applied this methodology to two similar images of a deciduous forest canopy. The images were recorded with different cameras but similar fields-of-view and within minutes of each other. Our analysis showed that the local approach helped to explain a percentage of the non-spatial concordance and to provided additional information about its decay as a function of the spatial lag.