Theory Estimating the size distribution Applications Modelling the Size of Forest Trees Using Statistical Distributions Lauri Meht¨ atalo University of Helsinki Guest lecture at SLU, February 19, 2009 Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Estimating the size distribution Applications Outline of the presentation Theory Distribution function and density Transformation Weighting Estimating the size distribution Applications Scaling, height distribution and stand characteristics Overlapping crowns PRM for ALS PRM for traditional data Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Tree size ◮ Let X be a random variable characterizing the size of a tree in a forest stand. ◮ Randomness may be due to that ◮ a tree has been selected randomly from the stand, or ◮ the tree is regarded as a realization of an underlying, stochastic model of the stand. ◮ Most commonly diameter is used as tree size. ◮ Other alternatives are tree height, crown diameter, crown area, basal area, tree volume, etc. Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution function ◮ The variability in tree size within the stand is accounted for through tree size distribution. ◮ The size distribution is defined as F ( x ) = P ( X ≤ x ) ◮ Two alternative interpretations for the distribution are ◮ The probability for the size of a randomly selected tree to be below x ◮ The proportion of trees with size below x Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution function ◮ For a function F ( x ) to be a cdf, the following conditions need to hold: 1. F ( x ) is defined for −∞ < x < ∞ , has minimum of 0 and maximum of 1 (i.e., lim x →−∞ F ( x ) = 0 and lim x →∞ F ( x ) = 1). 2. F ( x ) is a nondecreasing function of x . 3. F ( x ) is (right) continuous, i.e., for any x 0 , lim x → x 0 F ( x ) = F ( x 0 ) ◮ Examples of a distribution function Weibull Uniform Percentile−based 0.8 0.8 0.8 F(x) F(x) F(x) 0.4 0.4 0.4 0.0 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 30 x x x Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting The density The density corresponding to distribution function F ( x ) is f ( x ) = F ′ ( x ) Examples Weibull Uniform Percentile−based 0.8 0.8 0.8 F(x) F(x) F(x) 0.4 0.4 0.4 0.0 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 30 x x x 0.12 0.08 0.04 0.06 0.04 f(x) f(x) f(x) 0.02 0.00 0.00 0.00 0 5 10 15 20 25 0 5 10 15 20 25 30 0 5 10 15 20 25 30 x x x Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, � ∞ ◮ for computing expected values E [ g ( X )] = −∞ g ( x ) f ( x ) d x , Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, � ∞ ◮ for computing expected values E [ g ( X )] = −∞ g ( x ) f ( x ) d x , ◮ for computing the likelihood L ( x ) = � n i =1 f ( x ) in ML-estimation, Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, � ∞ ◮ for computing expected values E [ g ( X )] = −∞ g ( x ) f ( x ) d x , ◮ for computing the likelihood L ( x ) = � n i =1 f ( x ) in ML-estimation, ◮ for computing weighted distributions, Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, � ∞ ◮ for computing expected values E [ g ( X )] = −∞ g ( x ) f ( x ) d x , ◮ for computing the likelihood L ( x ) = � n i =1 f ( x ) in ML-estimation, ◮ for computing weighted distributions, ◮ The distribution can be used e.g., ◮ for computing the proportion of trees between x 1 and x 2 as F ( x 2 ) − F ( x 1 ), Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Distribution or density? ◮ The density is most commonly used ◮ for illustration, as it is related to the commonly used histogram or stand table, � ∞ ◮ for computing expected values E [ g ( X )] = −∞ g ( x ) f ( x ) d x , ◮ for computing the likelihood L ( x ) = � n i =1 f ( x ) in ML-estimation, ◮ for computing weighted distributions, ◮ The distribution can be used e.g., ◮ for computing the proportion of trees between x 1 and x 2 as F ( x 2 ) − F ( x 1 ), ◮ for computing distributions of random variables that are related to X , e.g., that of transformed random variables. Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Transformation ◮ A transformed random variable is random variable Y that is obtained from X using transformation Y = g ( X ). ◮ Useful examples are volume, height, basal area, or crown area ( Y ) as a function of tree diameter ( X ). Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Transformation ◮ A transformed random variable is random variable Y that is obtained from X using transformation Y = g ( X ). ◮ Useful examples are volume, height, basal area, or crown area ( Y ) as a function of tree diameter ( X ). ◮ The distribution of Y = g ( X ) is F Y ( y ) = F X ( g − 1 ( y )) if g is increasing F Y ( y ) = 1 − F X ( g − 1 ( y )) if g is decreasing , where g − 1 ( y ) is the inverse transformation ◮ If X is tree diameter, then g − 1 ( y ) is tree diameter as a function of volume, height, basal area or crown area. Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Transformation ◮ A transformed random variable is random variable Y that is obtained from X using transformation Y = g ( X ). ◮ Useful examples are volume, height, basal area, or crown area ( Y ) as a function of tree diameter ( X ). ◮ The distribution of Y = g ( X ) is F Y ( y ) = F X ( g − 1 ( y )) if g is increasing F Y ( y ) = 1 − F X ( g − 1 ( y )) if g is decreasing , where g − 1 ( y ) is the inverse transformation ◮ If X is tree diameter, then g − 1 ( y ) is tree diameter as a function of volume, height, basal area or crown area. ◮ Applications ◮ Formulating different distributions based on allometric relationships of trees Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
Theory Distribution function and density Estimating the size distribution Transformation Applications Weighting Weighted distribution ◮ Frequencies are often proportional to the number of stems. ◮ It may be more convenient to have them proportional to some other characteristics, such as basal area or volume ◮ The density of a weighted diameter distribution is w ( x ) f X ( x ) f w X ( x ) = � ∞ w ( u ) f X ( u ) d u . 0 ◮ The nominator is the mean of w ( x ), e.g., mean basal area ( G / N ), or mean volume ( V / N ). Lauri Meht¨ atalo Modelling the Size of Forest Trees Using Statistical Distributions
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