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! 



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

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> How many ways we know to join two line segments with a smooth transitional function?
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We can do this in a lot of ways. If I understand your question, some of the more common ways are mentioned on Wikipedia.


<http://en.wikipedia.org/wiki/Heaviside_step_function#Analytic_approximations>


