# Vertically translated depreciation curve: Update the exponential regression coefficient

I have an exponential regression equation that I use to predict the condition of roads.

The equation can be found on page 53 of the original master's thesis: Development of a Flexible Framework for Deterioration Modelling in Infrastructure Asset Management

From page 53:

The equation has some IF statements which deal with specific roads-related scenarios. For the purpose of this question, I'll boil it down to it's simplest form:

condition = 21 - EXP(0.072*age)

## Question:

The equation was developed for my organization back in 2012. At that time, the roads had only been inspected once. Now, fast-forward to 2017, and we have a lot more data to work with.

So, I think it would be wise to update the equation. If we were able to redo the analysis with the new data, I suspect we'd be able to predict the future condition of roads more accurately.

However, as non-stats guy, I'm finding this to be rather difficult.

How can I update the coefficient in the exponential regression equation?

Related:

Tune an exponential regression estimate using calculus

• Do you need it to be of the form $y=a+be^{cx}$ necessarily? Is there any reason for that? Dec 13, 2017 at 4:38

If you want to follow the same curve form, you set the y-intercept to 21 like the original author did. Then, the equation you need to estimate is $$y=21-e^{ax},$$

which is equivalent to

$$21-y=e^{ax}.$$

If you take logarithms both sides (you can do it because $y<21$), then $$log(21-y)=ax.$$

Renaming $log(21-y)=z$, this is of the form $$z=ax,$$

which is a linear regression with no intercept that can be estimated with many standard software packages.

For the lower bound you would, instead of the 21, use the 21-1.282-8.277 (the worst-case condition in the conditionals) as a y-intercept, and apply the same estimation method.

• So at age 0, $e^{ax}=e^0=1$, and we want to find an intercept such that at age $x=0$ we have $y=20$. If the intercept was $20$, then, at age 0, $y=20-e^{0}=20-1=19$, which is not the state of new roads. Dec 13, 2017 at 5:09
• A more general approach to estimate the y-intercept, if it is not as explicitly determined by the context as in this case, would be to compute the average of the $y$ conditional on $x=0$ (so the average condition for all roads of age=0 but only those) and add 1 for the same reason we add 1 to make the 20-score be an intercept of 21. Or, if you want upper-lower bounds like in the example, the average for the best-case roads at $x=0$, and of the worst-case roads at $x=0$ will give you two the two intercepts. Dec 13, 2017 at 5:28