Weighted Least Squares Method and Construction of Volume Tables

Abstract

Abstract Constructing tree volume tables by the method of least squares has one main drawback, namely that the variance of the tree volume is not homogeneous. In the first part of the paper "General Theory" two theorems are given. Theorem 1 shows that if (1) [formula] are independent random variables, (2) [formula] and (3) [formula] where [formula] may be an unknown quantity but the wj are known positive constants then the best linear estimate of y is [formula] where the b's are obtained by minimizing the weighted sum of squares [Equation] with respect to bi. Formulae for calculating the standard errors of the regression estimates are also given. Theorem 2 shows that if (3) from Theorem 1 is changed to (3a) [formula] where [formula] is any known function of the additional known variable Zj and K² may be any unknown quantity, then the weighted least squares estimates of the regression of y on [formula] is equivalent to the least squares estimate of the regression of [formula] on the independent variables [formula]. Consequently, standard least squares techniques for calculating regression coefficients, their standard errors and tests of significance are fully applicable. Part 2 analyses the variance of the tree volume and shows from actual data that it can vary from .02 (cu ft)² to more than 400 (cu ft)². Furthermore, the conditions of Theorem 2 are for all practical purposes fulfilled. Part 3 gives several examples of the application of the Theorem 1 and Theorem 2 to actual data. It also compares the least squares and the weighted least squares estimates with the experimental data and shows by visual inspection how much better the weighted least squares method of regression can be.