Characterizing prediction errors of a new tree height model for cut-to-length Pinus radiata stems through the Burr Type XII distribution

Abstract

Unlike height-diameter equations for standing trees commonly used in forest resources modelling, tree height models for cut-to-length (CTL) stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed. This feature was merely noticed in previous studies but never thoroughly investigated. This study characterized the prediction error distribution of a newly developed such tree height model for Pinus radiata (D. Don) through the three-parameter Burr Type XII (BXII) distribution. The model’s prediction errors (

\epsilon

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) exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent. Structured serial correlations were also present in the data. A total of 14 candidate weighting functions were compared to select the best two for weighting

\epsilon

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in order to reduce its conditional heteroskedasticity. The weighted prediction errors (

{\epsilon }_w

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) were shifted by a constant to the positive range supported by the BXII distribution. Then the distribution of weighted and shifted prediction errors (

{\epsilon }_{w+}

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) was characterized by the BXII distribution using maximum likelihood estimation through 1000 times of repeated random sampling, fitting and goodness-of-fit testing, each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences. The nonparametric two sample Kolmogorov–Smirnov (KS) goodness-of-fit test and its closely related Kuiper’s (KU) test showed the fitted BXII distributions provided a good fit to the highly leptokurtic and heavy-tailed distribution of

\epsilon

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. Random samples generated from the fitted BXII distributions of

{\epsilon }_{w+}

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derived from using the best two weighting functions, when back-shifted and unweighted, exhibited distributions that were, in about 97 and 95% of the 1000 cases respectively, not statistically different from the distribution of

\epsilon

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. Our results for cut-to-length P. radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution. The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P. radiata plantations, particularly when uncertainty assessments, statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.

Keywords

Conditional heteroskedasticity / Leptokurtic error distribution / Skedactic function / Nonlinear quantile regression / Weighted prediction errors / Serial correlation / Random sampling and fitting / Nonparametric goodness-of-fit tests