Functional linear regression on Namibian and South African data - Mzimela, Nosipho (Department of Statistical Sciences, 2017)
Indigenous to Southern Africa, the Aloe Dichotoma, most commonly known as the Quiver tree, are species of Aloe found mostly in the southern parts of Namibia and the Northern Cape Province in South Africa. Researchers noticed that Quiver trees assumed very different shapes depending on their geographical location. This project aims to model the observed differences in structural form of the trees between geographically spate populations with functional regression analysis using climate variables at each location. A number of statistical challenges present themselves such as the multivariate nature of the data. Functional data analysis was used in this project to display the data so as to highlight various characteristics while allowing us to study important sources of pattern and variation among the data. Functional data analysis can be best summarised as approximating discrete data with a function by assuming the existence of a function giving rise to the observed data. The underlying function is assumed to be smooth such that a pair of adjacent data values are necessarily linked together and unlikely to be too different from each other. There are a number of smoothing methods used to fit a function to the discrete data. In this project we use Roughness Penalty Smoothing methods which are based on optimising a fitting criterion that defines what a smooth of the data is trying to achieve. The meaning of smooth is explicitly expressed at the level of the criterion being optimised, rather than implicitly in terms of the number of basis functions used. Once the continuous functions for the climate variables have been fitted, these are used as predictors in a functional regression model with the structural variables as responses. This allows for the estimation of regression coefficients to describe the effect of the climate variables on each structural variable. The functional models suggest that maximum temperature has an effect on the structural form of Aloe Dichotoma. Further, the structural form of Aloe Dichotoma does differ in geographically spate locations. Trees found in the warmer Northern regions are more likely to have taller trees. The results did not necessarily prove the hypothesis that the number of branches found on trees in the North is fewer than those in the South, but these trees are more likely to have more dichotomous branches which may be translated to more branches.