ForestFit (version 0.4)

fitgrowth: Estimatinng the parametersof the fitted non-linear growth curve to the height-diameter(H-D) observations

Description

Estimates the parameters of the nine well-known three-parameter non-linear curves fitted to the height-diameter observations. These nine models are given by the following.

• Richards (Richards(1959)) $$H=1.3+\beta_1+\frac{\beta_2}{D+\beta_3},$$

• Gompertz (Winsor(1992)) $$H=1.3+\beta_1 e^{-\beta_2e^{-\beta_3 D}},$$

• Hossfeld IV (Zeide(1993)) $$H=1.3+\frac{\beta_1}{1+\frac{1}{\beta_2 D^{\beta_3}}},$$

• Korf (Flewelling and De Jong(1994)) $$H=1.3+\beta_1 e^{-\beta_2D^{-\beta_3}},$$

• logistic (Pearl and Reed (1920)) $$H=1.3+\frac{\beta_1}{1+\beta_2e^{-\beta_3D}},$$

• Prodan (Prodan(1968)) $$H=1.3+\frac{D^2}{\beta_1 D^2+\beta_2 D+\beta_3},$$

• Ratkowsky (Ratkowsky(1990)) $$H=1.3+\beta_1 e^{-\frac{\beta_2}{D+\beta_3}},$$

• Sibbesen (Huang et al. (1992)) $$H=1.3+\beta_1 D^{\beta_2 D^{-\beta_3}},$$

• Weibull (Yang et al. (1978)) $$H=1.3+\beta_1\Bigl(1-e^{-\beta_2 D^{\beta_3}}\Bigr),$$

Usage

fitgrowth(h,d,model,starts)

Arguments

h

Vector of height observations.

d

Vector of diameter observations.

model

The name of the fitted model including "chapman-richards", "gompertz", "hossfeldiv", "korf", "logistic", "prodan" , "ratkowsky", "Sibbesen", and "weibull".

starts

A list of starting values for the parameters \(\beta_1\), \(\beta_2\), and \(\beta_3\).

Value

A list of objects in four parts as

1. Estimated parameters and corresponding summaries including standard errors, computed \(t\)-statistics, and \(p\)-values.

2. Residuals.

3. Covariance matrix of the estimated model parameters (coefficients) \(\hat{\beta}_1\), \(\hat{\beta}_1\), and \(\hat{\beta}_1\).

4. Residual standard error, i.e., \(\hat{\sigma}\).

5. The hieght-diameter scatterplot superimposed by the fitted model.

References

J. W. Flewelling and R. De Jong. (1994). Considerations in simultaneous curve fitting for repeated height-diameter measurements, Canadian Journal of Forest Research, 24(7), 1408-1414.

S. Huang, S. J. Titus, and D. P. Wiens. 1992. Comparison of nonlinear height<U+00B1>diameter functions for major Alberta tree species. Canadian Journal of Forest Research, 22, 1297-1304.

R. Pearl and L. J. Reed. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proceedings of the National Academy of Sciences of the United States of America, 6(6), 275.

M. Prodan. 1968. The spatial distribution of trees in an area. Allg. Forst Jagdztg, 139, 214-217.

D. A. Ratkowsky. 1990. Handbook of nonlinear regression, New York, Marcel Dekker, Inc.

F. J. Richards. 1959. A flexible growth function for empirical use. Journal of Experimental Botany, 10, 290-300.

S. B. Winsor. 1992. The Gompertz curve as a growth curve. Proceedings of National Academic Science, USA, 18, 1-8.

R. C. Yang, A. Kozak, J. H. G. Smith. 1978. The potential of Weibull-type functions as a flexible growth curves. Canadian Journal of Forest Research, 8, 424-431.

B. Zeide. 1993. Analysis of growth equation. Forest Science, 39, 594-616.

Examples

# NOT RUN {
h<-c( 1.8,  2.7,  3.3,  2.4,  2.9,  3.4,  2.5,  3.2,  4.1,  2.7,
      2.7,  2.2,  4.1,  1.5,  2.6, 17.1,  3.0,  2.5,  3.8,  2.1,
      3.2,  2.5,  2.8,  2.2,  2.0,  2.4,  4.2,  2.6,  2.5,  3.7,
      2.2,  3.0,  3.2,  2.5,  3.1,  3.0,  9.7, 12.1,  2.0,  2.4,
      3.2,  2.4,  2.9,  2.4,  3.2,  2.0,  2.5, 12.8, 18.2)
d<-c(13.7, 16.8, 20.8, 13.5, 17.0, 16.5, 15.0, 40.9, 20.8, 18.0,
     16.0, 12.7, 26.4, 11.4, 16.8, 66.0, 18.3, 9.70, 19.8, 16.5,
     22.9, 15.2, 29.0, 22.4, 11.4, 22.9, 26.7, 19.3, 24.1, 22.4,
     13.5, 11.7, 19.3, 18.0, 19.6, 26.4, 72.1, 66.0, 11.7, 16.0,
     13.5, 15.2, 17.0, 12.4, 16.0, 11.4, 14.5, 63.0, 55.6)
starts<-c(18,0.0005,2)
fitgrowth(h,d,"weibull",starts=starts)
# }