Linear Regression Model based on Log-Cosh Loss Function in R - r

Question

I have read about loss functions theoretically and also how to build regression models based on them in R.
I can apply all of the regression models based on different loss functions in R programming except for Log-Cosh Loss Function.
For example, I would like to build a linear regression model on 5-folds subsets of the DATA, and then extract the coefficients and calculate the individuals and the aggregated variance as follows.
data = read.csv("train.csv") # "critical_temp" is the dependent variable.
data_nom_df=as.data.frame(scale(data))#Normalization
#Cross Validation
set.seed(12345)
k = 5
folds <- createFolds(data_nom_df$critical_temp, k = k, list = TRUE, returnTrain = TRUE)
## Ordinary Least Square regression
#block A
lm = list()
for (i in 1:k) {
lm[[i]] = lm(critical_temp~ .,
data = data_nom_df[folds[[i]],])
}
#block B
lm_coef = list()
lm_coef_var = list()
for(j in 1:(lm[[1]]$coefficients %>% length())){
for(i in 1:k){
lm_coef[[i]] = lm[[i]]$coefficients[j]
lm_coef_var[[j]] = lm_coef %>% unlist() %>% var()
}
}
#block C
lm_var = unlist(lm_coef_var)
lm_df = cbind(coefficients = lm[[1]]$coefficients %>% names() %>% as.data.frame()
, variance = lm_var %>% as.data.frame())
colnames(lm_df) = c("coefficients", "variance_lm")
lm_df
#block D
lm_var_sum = sum(lm_var)
lm_var_sum
The same for the rest of the regression models. However, I do not find any code or package to apply a regression model based on Log-Cosh Loss Function in R.
Could you please guide me to any source that would help me to solve this problem.

Answer 1

This can be done from first principles. Also note the existence of the logcosh function in the limma package which could be used in place of log(cosh(.)) if you have numeric difficulties.
f <- function(b) with(cars, sum(log(cosh(dist - b[1] - b[2] * speed))))
fm0 <- lm(dist ~ speed, cars)
res <- optim(coef(fm0), f, method = "BFGS")
str(res)
## List of 5
## $ par : Named num [1:2] -12.82 3.47
## ..- attr(*, "names")= chr [1:2] "(Intercept)" "speed"
## $ value : num 532
## $ counts : Named int [1:2] 28 10
## ..- attr(*, "names")= chr [1:2] "function" "gradient"
## $ convergence: int 0
## $ message : NULL
Graphics
# the black line is the ordinary least squares regression line and
# the red line is the log cosh regression line
plot(cars)
abline(fm0)
yfit <- res$par[1] + res$par[2] * cars$speed
lines(cars$speed, yfit, col = "red")
ADDED
Note that the optimization can also be written like this which may be useful if you have many independent variables.
fm0 <- lm(dist ~ speed, cars)
X <- model.matrix(fm0)
f <- function(b) with(cars, sum(log(cosh(dist - X %*% b))))
res <- optim(coef(fm0), f, method = "BFGS")
res
giving:
$par
(Intercept) speed
-12.816190 3.469536
$value
[1] 531.5872
$counts
function gradient
28 10
$convergence
[1] 0
$message
NULL