Allometric equations that are considered in the development of linear models.
eqns_infoA dataframe with 5 variables:
One of the six general types of allometric equations.
Basic form of the allometric equation.
The basic formula used to fit the model.
The weight argument in the model.
Character string used to represent the unique combinations of equations and weights.
Six allometric equations used to develop linear models for urban trees:
Linear: \(y_{i} = a + bx_{i} + \epsilon_{i}/\sqrt{w_{i}}\)
Quadratic: \(y_{i} = a + bx_{i} + cx^2_{i} + \epsilon_{i}/\sqrt{w_{i}}\)
Cubic: \(y_{i} = a + bx_{i} + cx^2_{i} + dx^3_{i} + \epsilon_{i}/\sqrt{w_{i}}\)
Quartic: \(y_{i} = a + bx_{i} + cx^2_{i} + dx^3_{i} + ex^4_{i} + \epsilon_{i}/\sqrt{w_{i}}\)
Log-log: \(\log{(y_{i})} = a + b(\log{(\log{(x_{i} + 1)})}) + \epsilon_{i}/\sqrt{w_{i}}\)
Exponential: \(\log{(y_{i})} = a + bx_{i} + 1 + \epsilon_{i}/\sqrt{w_{i}}\)
where
\(y_{i}\) = Response variable of individual tree \(i\), \(i = 1,2,3\)... \(n\), \(n\) = number of observations
\(x_{i}\) = Predictor variable
\(a,b,c,d,e\) = Parameters to be estimated
\(\epsilon_{i}\) = Normally distributed error term
\(w_{i}\) = Known weight that takes one of the four following forms: \(w_{i} = 1\), \(w_{i} = 1/\sqrt{x_{i}}\), \(w_{i} = 1/\sqrt{x_{i}}\), \(w_{i} = 1/x_{i}^2\).
McPherson E. G., van Doorn N. S. & Peper P. J. (2016) Urban Tree Database and Allometric Equations. General Technical Report PSW-GTR-253, USDA Forest Service, 86.
data(eqns_info)
head(eqns_info)
#> modeltype base_equation base_formula weights modelcode
#> 1 Linear y = a + bx y ~ x lin_w1
#> 2 Linear y = a + bx y ~ x I(1/sqrt(x)) lin_w2
#> 3 Linear y = a + bx y ~ x I(1/x) lin_w3
#> 4 Linear y = a + bx y ~ x I(1/x^2) lin_w4
#> 5 Quadratic y = a + bx + x^2 y ~ x + I(x^2) quad_w1
#> 6 Quadratic y = a + bx + x^2 y ~ x + I(x^2) I(1/sqrt(x)) quad_w2