# How many ways we know to join two line segments with a smooth transitional function?

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!

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