# Beyond Linear Modeling using R (part-2)

In a previous post we fitted a non linear model to some randomly generated data. Here we will a different type of non-linear function and evaluate the results.

**Modeling**

In this approach we will model the same data generated by the previous post but the relationship between x and y will be modeled with an exponential function:

With the following script we will directly fit an exponential model using again the nls() function.

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# exponential model fit > expmodel <- nls(y ~ I(exp(1)^(b + a * x)), data = ds, start = list(b = 0,a = 1), trace = F) # summary of coefficients > summary(expmodel)$coefficients Estimate Std. Error t value Pr(>|t|) b -4.114140 0.1802758 -22.82137 1.239745e-19 a 4.214029 0.1998630 21.08459 1.010778e-18 |

The model estimated that b = -4.114140 ± 0.1802758 and a = 4.214029 ± 0.1998630. Additionally, both of these variables are highly significant (p << 0.05).

**Plotting**

Plotting the fitted model and the cubic approximation we have the following graph:

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>plot(y ~ x, main = "Fitted exponential function") >s <- seq(0, 1, length = 100) >lines(s, s^3, lty = 2, col = "red") >lines(s, predict(expmodel, list(x = s)), lty = 1, col = "blue") >legend("topleft", legend=c("fitted exponential", "cubic"), col=c("blue", "red"), lty=1:2, cex=0.8, box.lty=1, box.lwd=1) |

We can see that the fitted model is close to the cubic model that we used initially.

*Goodness of fit*

To examine the goodness of fit of the model, we will calculate R squared, seen in a previous post for linear regression:

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# sum of squared error > SSE <- sum(residuals(expmodel)^2) # total sum of squares > SST <- sum((y - mean(y))^2) # Rsquared > Rsquared_exp <- 1-(SSE/SST) |

R squared is calculated at 0.9764 which is a very good result highlighting the overall good performance of the model. Also, it is slightly smaller than the respective measure of the power model used in the previous post.

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