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

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

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  • $\begingroup$ These are approximations of the step function and clearly are helpful. I am intrigued with some approximations that are good, but in which the limit (i) don't holds - the Cauchy, for example. Is the limit (i) actually necessary? The piecewise models can be constructed by using another discontinuous operator, such as the maximum or module function. Then others families of approximation functions can be applied. This ways are very similar. Beyond these methods, do we know other form to model smooth transitions? $\endgroup$
    – Emm Ferreira
    Commented Apr 8, 2013 at 2:39

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