This topic was created to discuss how many ways we know to create piecewise linear functions with smooth transitions between the phases. An alternative is presents by Bacon & Watts (1971): the idea is build the model by using the signal operator and then replace it by a smooth approximation. Let

$y(x_i) = \theta_0 + \theta_1 (x_i-\tau) + \theta_2 (x_i-\tau) sgn(x_i-\tau) + \epsilon_i$,

in which $\theta_0$ is the $y$ value in the change point, $\theta_1$ is the mean of the line inclinations, $\theta_2$ is the half difference of the inclinations, and $\tau$ is the abscissa of the change point.

Griffiths & Miller (1973) define a family of smooth transitional functions to replace the signal. A transitional function must obeys three conditions:

(i) $\displaystyle{\lim_{s \rightarrow \pm \infty}} [s~trn(s) - |s|] = 0$,

(ii) $\displaystyle{\lim_{\gamma \rightarrow 0}} trn(s,\gamma)= sgn(s)$,

(iii) $trn(0) = 0$.

How many different ways could we use to obtain similar results?

thank you!