Non Linear Regression

#NON LINEAR REGRESSION

#data
x <- c(1,3,4,6,8,10,12,14,16,18)
y <- c(1.5,2.9,5.2,9.3,15.4,22.8,30.5,39.2,49.6,60.1)

#Fitting an exponential model and fitting the curve
MODEL <- nls(y ~ a * exp(b * x), start = list(a = 1, b = 0.2)) 
summary(MODEL)

## 
## Formula: y ~ a * exp(b * x)
## 
## Parameters:
##   Estimate Std. Error t value Pr(>|t|)    
## a  4.86321    0.85216   5.707 0.000451 ***
## b  0.14295    0.01109  12.886 1.24e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.515 on 8 degrees of freedom
## 
## Number of iterations to convergence: 8 
## Achieved convergence tolerance: 5.064e-06

plot(x, y, main = "Exponential Fit", pch = 16) 
x_pred <- seq(min(x), max(x), length.out = 100) 
y_pred <- predict(MODEL, newdata = list(x = x_pred)) 
lines(x_pred, y_pred, col = "red", lwd = 2)

#Fitting the logistic growth model

model_logistic <- nls(y ~ a / (1 + exp(-b * (x - c))), start = list(a = 30, b = 0.2, c = 5)) 
summary(model_logistic)

## 
## Formula: y ~ a/(1 + exp(-b * (x - c)))
## 
## Parameters:
##   Estimate Std. Error t value Pr(>|t|)    
## a  80.1465     6.2185   12.89 3.93e-06 ***
## b   0.2567     0.0180   14.26 1.98e-06 ***
## c  13.9412     0.7066   19.73 2.15e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.242 on 7 degrees of freedom
## 
## Number of iterations to convergence: 8 
## Achieved convergence tolerance: 1.549e-06

y_pred_logistic <- predict(model_logistic, newdata = list(x = x_pred)) 
plot(x, y, main = "Logistic Growth Model", pch = 16) 
lines(x_pred, y_pred_logistic, col = "green", lwd = 2)

#Comparing the residuals and goodness of fit

res_exp <- 3.515 
cat("The residual standard error value for exponential fit is:", res_exp, "and 8 degrees of freedom!\n")

## The residual standard error value for exponential fit is: 3.515 and 8 degrees of freedom!

res_log <- 1.242 
cat("The residual standard error value for logistic growth model is:", res_log, "and 7 degrees of freedom!\n") 

## The residual standard error value for logistic growth model is: 1.242 and 7 degrees of freedom!

cat("The good residual error value is:", min(res_exp, res_log), "\n")

## The good residual error value is: 1.242

#Predicting y values for new x values
#Now the new data set

model_log <- nls(y ~ a / (1 + exp(-b * (x - c))), start = list(a = 30, b = 0.2, c = 5)) 
new <- data.frame(x = c(11, 12, 13)) 
y_prediction <- predict(model_log, new) 
print(y_prediction)

## [1] 25.62582 30.29081 35.25587