# Fitting linear and non-linear equations by group" In forestmangr: Forest Mensuration and Management

knitr::opts_chunk$set(collapse = T, comment = "#>") knitr::opts_chunk$set(fig.width=7, fig.height=5)
options(tibble.print_min = 6L, tibble.print_max = 6L)
library(forestmangr)
library(dplyr)
library(tidyr)

We'll fit some linear and non-linear models for dominant height, and compare them. We'll use the first 10 strata of the exemple dataset exfm16.

library(forestmangr)
library(dplyr)
library(tidyr)

data(exfm14)
data_ex <- exfm14 %>% filter(strata%in%1:10)
data_ex

In order to fit Schumacher's dominant height model, we can use lm_table. Thanks to the log and inv functions, there is no need to create new variables:

mod1 <- lm_table(data_ex, log(dh) ~ inv(age))
mod1

To fit a non-linear model, like Chapman-Richards' we can use the nls_table function. This function uses Levenberg-Marquardt's algorithm by default, in order to assure a good fit. Since this is a non-linear fit, we have to input initial values for all coefficients:

mod2 <- nls_table(data_ex, dh ~ b0 * (1 - exp( -b1 * age )  )^b2,
mod_start = c( b0=23, b1=0.03, b2 = 1.3  ) )
mod2

If we wanted to fit one model of each stratum, we can use the .groups argument:

mod1 <- lm_table(data_ex, log(dh) ~ inv(age), .groups = "strata")
mod1
mod2 <- nls_table(data_ex, dh ~ b0 * (1 - exp( -b1 * age )  )^b2,
mod_start = c( b0=23, b1=0.03, b2 = 1.3  ),
.groups = "strata" )
mod2

If the fit is not ideal, it's possible to use a dataframe with starting values for each stratum, and use it as an input for mod_start:

tab_start <- data.frame(strata = c(1:10),
rbind(
data.frame(b0=rep(23, 5),b1=rep(0.03,5),b2=rep(1.3,5) ),
data.frame(b0=rep(23, 5),b1=rep(0.03,5),b2=rep(.5,5) )))
tab_start
mod2 <- nls_table(data_ex, dh ~ b0 * (1 - exp( -b1 * age )  )^b2,
mod_start = tab_start,
.groups = "strata" )
mod2

Now we're going to fit some other models. These are:

Schumacher: $$Ln(DH) = \beta_0 + \beta_1 * \frac{1}{age}$$

Chapman-Richards: $$DH = \beta_0 * (1 - exp^{-\beta_1 * age})^{\beta_2}$$

Bayley-Clutter: $$Ln(DH) = \beta_0 + \beta_1 * \begin{pmatrix} \frac{1}{age} \end{pmatrix} ^{\beta_2}$$

Curtis: $$DH = \beta_0 + \beta_1 * \frac{1}{age}$$

We'll fit these models and add their estimated values to the original data using the merge_est output and naming each estimated variable with the est_name argument:

data_ex_est <- data_ex %>%
lm_table(log(dh) ~ inv(age), .groups = "strata",
output = "merge_est", est.name = "Schumacher") %>%

nls_table(dh ~ b0 * (1 - exp( -b1 * age )  )^b2,
mod_start = c( b0=23, b1=0.03, b2 = 1.3  ),.groups="strata",
output ="merge_est",est.name="Chapman-Richards") %>%

nls_table(log(dh) ~ b0 + b1 * ( inv(age)^b2 ) ,
mod_start = c( b0=3, b1=-130, b2 = 1.5),.groups = "strata",
output ="merge_est",est.name = "Bailey-Clutter") %>%

lm_table(dh ~ inv(age), .groups = "strata",
output = "merge_est", est.name = "Curtis")

Ps: The lm_table function checks if the model has log in the y variable, and if it does, it removes it automatically when estimating variables. Because of that, there's no need to calculate the exponential for the estimated variables.

In order to compare these models, we'll calculate the root mean square error and bias for all models. To do this, we'll gather all estimated variables in a single column using tidyr::gather, group by model, and use the rmse_per and bias_per functions:

data_ex_est %>%
gather(Model, Value,
Schumacher, Chapman-Richards, Bailey-Clutter, Curtis) %>%
group_by(Model) %>%
summarise(
RMSE = rmse_per(y = dh, yhat = Value),
BIAS = bias_per(y = dh, yhat = Value) )

Another way of comparing and evaluating these models is using residual graphical analysis. The function resid_plot can help us with that:

resid_plot(data_ex_est, "dh", "Schumacher", "Chapman-Richards", "Bailey-Clutter", "Curtis")

There are other types of plots avaiable, such as histogram:

resid_plot(data_ex_est, "dh", "Schumacher","Chapman-Richards","Bailey-Clutter", "Curtis",
type = "histogram_curve")

And versus:

resid_plot(data_ex_est, "dh", "Schumacher", "Chapman-Richards", "Bailey-Clutter", "Curtis",
type = "versus")

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forestmangr documentation built on Aug. 16, 2021, 5:08 p.m.