Confidence Intervals and Regions for Proportions under Various Three-Endmember Linear Mixture Models

Many studies in recent years have been devoted to estimating the per-pixel proportions of three broad classes of materials (e.g., photosynthetic vegetation, non-photosynthetic vegetation and bare soil) using data from multispectral sensors. Oftentimes, the estimated proportions are used to monitor environmental change in both urban and non-urban environments. Many of these papers use proportion estimation methods based on the linear mixture model. Very few of these papers assess the accuracy of their estimators. This paper shows how to produce confidence intervals (CIs) and joint confidence regions (JCRs) for the proportions associated with various linear mixture models. There are two main models, both of which assume that the coefficients in the model are non-negative. The first model assumes that the coefficients sum to 1. The second does not, but uses rescaling of the estimated coefficients to produce estimated proportions. Three variants of these two models are also analysed. JCRs are shown to be particularly informative, because they are typically better at localising the information than CIs are. The methodology is illustrated using examples from Landsat Thematic Mapper data at 1169 locations across Australia, each of which has associated field observations. There is also a discussion about the extent to which the methodology can be extended to hyperspectral data.