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${session.getAttribute("locale")}5Fitting the 4-parameter lineal basis model
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i, y_{i}} there is a corresponding "generating" point {X_{i}, Y_{i}}, which lies on the Lineal Basis. The difference vector between an {X_{i}, Y_{i}} and its corresponding {x_{i}, y_{i}} is modelled as a sample from a bivariate distribution which here is taken as a product of two independent Beta[α,β] distributions, using the notation Beta[αX, βX]xBeta[αY, βY]. Further, we will allow αX, βX, αY, and βY to vary with s, i.e., we have four functions αX[s, ...], βX[s, ...], αY[s, ...], and βY[s, ...]. Since the mean of the Beta[α, β] distribution is α / (α + β), the parameters of these four function must be such that for a given s, the joint Beta-Beta mean point {αX[s, ...] / (αX[s, ...] + βX[s, ...]), αY[s, ...] / (αY[s, ...] + βY[s, ...]) lies on the Lineal Basis. It turns out that simple linear functions suffice to fit many of the data sets typically encountered in these bounded spaces. Interesting computational issues arise when constructing the "mean prediction region" and the "single prediction region" for a Lineal Basis model, analogous to the "mean prediction bands" and the"single prediction bands" of simple linear regression. Concepts from computational geometry are employed, and in particular the logic of a key calculation is verified via a Manipulate.]]>Wed 11 Apr 2018 10:50:56 AEST]]>Prediction and validation of foliage projective cover from Landsat-5 TM and Landsat-7 ETM+ imagery
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