The model is based on perhaps overly simple reasoning and an overt mistake, but its motivation is clear.
In this paper, the authors investigated an hypothesized relationship between monetary damage $Y$ and flood depth $X$. Experience strongly suggests that certain monetary variables are better analyzed by re-expressing them as logarithms. Thus, the first step in the analysis is to assume that
- $\log(Y)$ should be inversely related to depth.
There is a natural bound to damages: they will not exist--be essentially $0$--when there is no flood; that is, when $X$ is below some threshold $x_0$ such as $x_0=0$. The associated logarithm is extremely large and negative. Therefore
- As $X$ approaches $0$ (through positive values), $\log(Y)$ should be extremely large and negative.
This rules out the mathematically simplest relationship, the linear one, where $\log(Y)$ would be directly proportional to $-X$: when $X=0$, the linear relationship will have a finite intercept predicting positive flood damages when no flood is present.
Now there are infinitely many possible nonlinear relationships, but at least the ones considered should all have the following property:
- When $X^\prime \gt X$, the damages associated with $X^\prime$ should be larger than the damages associated with $X$.
At this point a good way to proceed would be with an exploratory analysis of a relevant dataset and/or appeal to some underlying theory. We see no evidence of that in the paper. I surmise, then, that the authors contemplated a set of mathematical functions with which they were familiar, seeking the simplest among them that would meet the foregoing criteria. Many people would view a rational function--that is, a ratio of polynomials--as being pretty simple and tractable. (As a basis for developing a model with explanatory power this is very weak, but if all one wants is to develop some mathematical relationship that might have decent predictive power, there is nothing wrong with proceeding this way.) The simplest possible rational functions with these properties are those whose numerator and denominator are linear,
$$p(X) = \frac{\alpha + \beta X}{\gamma + \delta X}.$$
(The linear functions are a special case where $\delta=0$.)
In order for this to get very large in size as $X$ approaches $0$, it is necessary that $\gamma=0$, $\delta \ne 0$, and $\alpha \ne 0$. We can therefore divide both the numerator and denominator by $\delta$, yielding a function of the form
$$p(X) = \frac{\alpha/\delta + \left(\beta/\delta\right) X}{X} = \beta_0 + \beta_1\frac{1}{X}$$
where $\beta_0 = \beta/\delta$ and $\beta_1 = \alpha/\delta$. Moreover, because we need $p(X)$ to become very negative for small positive values of $X$, necessarily
$$\beta_1 \lt 0.$$
These considerations have suggested the possible relationship
$$\log(Y) = \beta_0 + \beta_1\frac{1}{X}.$$
Missing from this equation is any recognition that what is observed in response to flood events (the left hand side) can deviate from the value computed on the right based on the flood depths. The authors therefore introduce an additive error term $\varepsilon_2$ to accommodate such deviations, writing (in their equation (2))
$$\log(Y) = \beta_0 + \beta_1\frac{1}{X} + \varepsilon_2.$$
The values of $\varepsilon_2$ will be treated as random. Note that these random values will have a large spread because damages are not accurately predictable from flood depth alone. The same reasoning that originally suggested analyzing the logarithm of $Y$ also suggests that the distribution of $\varepsilon_2$ should be approximately symmetric around $0$.
By exponentiating both sides we obtain
$$Y = \exp\left(\beta_0 + \beta_1\frac{1}{X}\right)\exp(\varepsilon_2).$$
The authors equate this (in their equation (1)) with a model of the form
$$Y = \exp\left(\beta_0 + \beta_1\frac{1}{X}\right) + \varepsilon.$$
This is mathematically incorrect, but it could be a reasonable approximation were the errors all small and the damages almost unvarying. Neither of those will be the case. We therefore have to understand this last move not as an actual equation having any mathematical or statistical rigor, but only as motivation for the model.
In effect, this chain of reasoning amounts to proposing the use of reciprocal flood depth $1/X$, rather than the flood depth $X$ itself, as an appropriate way to represent flood impacts on monetary damages. This is the basis for a later extension of the model to include a covariate and its interaction with flood depth, which again enters in the form of $1/X$ rather than $X$. (There are ways to identify and test appropriate re-expressions of the independent variable in a regression, but they seem not to have been used here.)
A quick scan through the rest of paper--especially a glance at the residual plot (Figure 2)--indicates this model (a) has been overfitted (stepwise regression was used along with GWR, both of which tend to overfit) yet (b) still is not a good fit to the data. Understanding the reasoning that suggested this model--rather than the model itself--therefore has more lasting value than any results that might be claimed in the paper.
