Functional Concurrent Linear Regression Model for Spatial Images

Abstract

Motivated by a problem in describing forest nitrogen cycling, in this paper we explore regression models for spatial images. Specifically, we present a functional concurrent linear model with varying coefficients for two-dimensional spatial images. To address overparameterization issues, the parameter surfaces in this model are transformed into the wavelet domain and a sparse representation is found by using a large-scale l 1 constrained least squares algorithm. Once the sparse representation is identified, an inverse wavelet transform is applied to obtain the estimated parameter surfaces. The optimal penalization term in the objective function is determined using the Bayesian Information Criterion (BIC) and we introduce measures of model quality. Our model is versatile and can be applied to both single and multiple replicate cases.

Missing Data and Regression Models for Spatial Images

“…P REVIOUSLY [12], we introduced a functional concurrent linear model (FCLM) for 2-D spatial images. The FCLM was proposed as a tool for modeling the relationship between two colocated spatial images, an example was given wherein it was desired to use a spatial image of elevation as a covariate to explain a colocated spatial image of gypsy moth defoliation.…”

“…In the example in [12], Y represented defoliation rates over a region in the Appalachian mountains from [1]; initially with one explanatory variable, X 1 represented elevation for the same region. The modeling showed that defoliation was indeed related to elevation, although it was useful to include a tree species classification as an additional covariate X 2 .…”

“…Pixel-wise, the model in (1) is ill-defined. To address this, [12] used wavelets to represent the matrices B 0 , B 1 , . .…”

“…To choose λ, [12] used Bayesian Information Criterion (BIC). More explicitly, if p = 1, then in terms of the wavelet bases, the FCLM looks like…”

Algorithmic Identification of Looted Archaeological Sites from Space

“…If each pixel or pixel neighborhood in the image is considered to be a data point, K-means clustering (Hartigan and Wong, 1979;Rekik et al, 2006) or a linear regression model (Galton, 1894;Zhang et al, 2011) can be used to separate out groups of changed and unchanged pixels, and principal component analysis can be effectively used to prepare the data (Celik, 2009a). Scientists have used the kernel trick (Camps-Valls and Bruzzone, 2009) to improve performance by creating non-linear classifiers.…”