Regression techniques for capped independent variables Suppose you are trying to predict height ($y$) based on age ($x$). A straight linear regression won’t work well, since humans stop growing at a certain age. This suggests instead trying to model height as something like:
$y = w\cdot \mathrm{squash}(x, a, b) + w_0$
where 
$\mathrm{squash}(x,a,b) = \max(a, \min(x,b))$
In general, when trying to predict some dependent variable $y$ from dependent variables $x_1, x_2, ..., x_n$, some of which you expect to display "squash"-characteristics like this, it seems a good idea to try to model $y$ by something like:
$y = w_0 + \sum_{i=1}^n w_i\cdot\mathrm{squash(x_i, a_i, b_i)}$
I'm wondering what regularized techniques exist to estimate $\vec{w}, \vec{a}, \vec{b}$.
 A: It's a little bit tough, because of the kinks in your squash function. A related idea would be to use something like $\tan^{-1}\{(x-a)/b\}$ in place of squash, and use nonlinear regression to fit the model. This would be a similar idea, but smoothes out the transitions between affecting the response and not affecting it. Other S-shaped curves could be used in place of the arc tangent, e.g., logistic, cumulative normal, or even something not symmetrical around its inflection point.
Addendum
If it really is height versus age, you probably don't want an S-shaped curve. Something more like $w_0+w_1(1-e^{-ax})$, with one less parameter, provides for rapid initial growth, slowing down later. 
A: Realistic height-age curves are ... well, curved, rather than like your linear-with-a-barrier function.
As you see, the population curves show several slowdowns and speedups and are not likely to be well-fitted by an overly-simple functional form.
Individual growth curves (when taken at reasonably high frequency, say monthly) tend to be a little more complex, with a number of growth spurts and slow patches.
Your multiple-regression situation looks like you have an additive model with natural linear splines in each variable, and also estimate the knot-positions. This can be treated in a number of ways, for example as a nonlinear/partially linear least squares problem.
But if your motivation is growth curves, it doesn't strike me as a close match.
