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Weighted Least Squares Method and Construction of Volume Tables

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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.

Document Type: Journal Article

Affiliations: Staff of the Canadian International Paper Co., Montreal, Quebec.

Publication date: June 1, 1964

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  • Forest Science is a peer-reviewed journal publishing fundamental and applied research that explores all aspects of natural and social sciences as they apply to the function and management of the forested ecosystems of the world. Topics include silviculture, forest management, biometrics, economics, entomology & pathology, fire & fuels management, forest ecology, genetics & tree improvement, geospatial technologies, harvesting & utilization, landscape ecology, operations research, forest policy, physiology, recreation, social sciences, soils & hydrology, and wildlife management.
    Forest Science is published bimonthly in February, April, June, August, October, and December.

    Also published by SAF:
    Journal of Forestry
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