Interpreting GAM Coefficients I could use some advice interpreting GAM (Generalized Additive Model) coefficients.
I get the following results when I call coef on my model:
(Intercept)     s(TM).1     s(TM).2     s(TM).3
   817.9501    383.7981   2358.5184   -545.4162

Does that mean:
y = 817.9501 + 383.7981*TM + 2358.5184*TM^2 - 545.4162*TM^3

 A: From your R syntax, it seems like you are fitting a regression model which postulates that the predictor variable TM has a potentially nonlinear effect on the response (or outcome) variable Rev. (If you had reasons to believe the effect is linear, you would use the lm() function instead of the gam() function.) 
I am presuming the reason you are trying to fit a gam model is because you plotted Rev against TM and noticed some kind of nonlinearity in that plot. (Rev should be a continuous variable.) 
When the gam() function is invoked for your data, it will essentially try to fit the following model to the data: 
Rev = intercept + f(TM) + error

where f() is an unknown potentially linear function whose shape must be identified from the data and error is a random error term assumed to come from a normal distribution with mean 0 and unknown standard deviation sigma.
To estimate the function f(), gam will approximate f() by a linear combination of known basis functions and estimate the unknown weights these functions will receive in that combination. 
What you see in your output is that gam identified that three such basis functions were sufficient to recover the underlying shape of f(): s(TM).1, s(TM).2 and s(TM).3.  (All of these are known functions which depend on TM.) 
R estimated the weights one should give to each of these functions in the linear combination that will approximate f(). So the function f() can be estimated as follows:
Estimated value of f(TM) ~ 383.7981*s(TM).1 + 2358.5184*s(TM).2 - 545.4162*s(TM).3

This means that the expected/average value of Rev can be estimated as: 
Expected Rev = 817.9501 + 383.7981*s(TM).1 + 2358.5184*s(TM).2 - 545.4162*s(TM).3

In this context, you wouldn't want to interpret the effects of s(TM).1, s(TM).2 and s(TM).3 on the expected value of Rev. Rather, you would simply want to see how these three basis functions combine together via the estimated weights to produce a possibly nonlinear function which represents the nonlinear effect of TM on Rev. That is why it is important to simply plot 383.7981*s(TM).1 + 2358.5184*s(TM).2 - 545.4162*s(TM).3 against TM and interpret qualitatively the shape of that plot. For example, does the plot show that the expected value of Rev increases up to a point and then levels off? 
This link will give you more detailed explanations on gam models and what they do under their hood: http://environmentalcomputing.net/intro-to-gams/. It will also give you clues on how to plot the possibly nonlinear effect f() you are trying to estimate.
A: Not at all. For one thing, GAM's need a link function, and also smoothers, the $g$ and the $f$'s, respectively, in the general definition:
$$\mathbb{E}(y)=g^{-1}\left(\alpha+f_1(x_1)+f_2(x_2) + \dots + f_n(x_n) \right)$$
For this reason you cannot follow the exact same usual marginal contribution interpretation done with simple linear regression; the link function and the smoothing terms get in the way.
One usual approach for GAM's is plotting the partial effects and inspect the relationships between $y$ and $x$'s visually. This article has a really nice intro to GAM's with a short discussion on interpretation in the end.
