exponential regression for bacteria growth

Question

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this:

$$(0,32), (2,65),(4,132),(6,275)$$

(hours, qtty of bacteria)

This is a plot of exponential growth of bacterias. We know that its growth follows an exponential law, find a curve that best represent it and predict it for $t=10$ hours

I think I should pick a curve like $e^x$ but will it be with only $1$ coefficient?

Answer 1

In my opinion you can't use a transformation via the logarithmus. I think you need a funciton like $$ f(t) = a e^{bt}. $$ To use least squares on a nonlinear specification some characteristica from the linear regression must be dropped. But in general the asymptotic theory leads to more or less similar results (for the statistical properties).

With that function you need a algorithm for solving nonlinear equations like the Gauss Newton Method. In my opinion you can't solve it 'by hand'.

If you use R you can try the nls function:

t <- c(0, 2, 4, 6)

y <- c(32, 65, 132, 275)

(mod <- nls(y ~ a * exp(b * t), start = list(a = 1, b = 1)))

This leads you to the coefficients:

a = 31.2257

b = 0.3624

This parameters are now the least square estimators. The fit to the data looks like:

For a prediction with $t = 10$ you just have to compute $$ f(10) = 31.2257 \cdot e^{0.3624 \cdot 10} \approx 1170.565 $$

Answer 2

Better apply linear regression to the logarithms of the quantities. Then you have an approximation of the form $e^{ax+b}$ or equivalently $C.e^{ax}$

In a sense this is the 'best possible' approximation measured by the relative differences between the approximated quantities and the measured quantities.