- Title
- Fitting the 4-parameter lineal basis model
- Creator
- Stokes, Barrie J.; Moffiet, Trevor N.
- Publisher Link
- http://dx.doi.org/10th International Mathematica Symposium. Proceedings of the 10th International Mathematica Symposium (Beijing, China 16-18 July 2010)
- Relation
- http://www.internationalmathematicasymposium.org
- Publisher
- Wolfram Research
- Resource Type
- conference paper
- Date
- 2010
- Description
- This Mathematica 7 Notebook develops a novel regression method applied to bivariate data on a bounded space (we will work in the canonical {0, 1 }x{O, 1} space). This approach was developed by [Moffiet, 2008] in the context of analysis of satellite remote sensing data. A distinguishing feature of the development presented here is that the X and Y variables are treated with complete symmetry; neither variable takes dependent or independent roles. The fitted "regression line", which we call a Lineal Basis, is described parametrically, i.e., as {X= X[s, ...], y = y{s, ...}}, with 0 ≤ s ≤ 1 and the constraints {X[0, ...], Y[0, ...]} = {0, 0} and {X[1, ...], Y[1, ...]} = {1, 1}, rather than conventionally as Y = f[x, ...] with f[0] = 0 and f[1] = 1 (ellipsis ... denotes parameters). This Lineal Basis is fitted to the data according to a model in which for each observed data point {xi, yi} there is a corresponding "generating" point {Xi, Yi}, which lies on the Lineal Basis. The difference vector between an {Xi, Yi} and its corresponding {xi, yi} 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.
- Subject
- Mathematica 7 Notebook; bivariate data; variables; parameters; functions
- Identifier
- http://hdl.handle.net/1959.13/935284
- Identifier
- uon:12024
- Language
- eng
- Full Text
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