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I am looking for a smooth (continuous differentiable) approximation of the following two three-phased functions with breakpoints at $B_1$ and $B_2$:$$ y_1(x, B_1, B_2, a, b) = \begin{cases} a; & x < B_1\\ a + b(x - B_1); & B_1 \leqslant x \leqslant B_2\\ a + b(B_2 - B_1); & x > B_2 \end{cases}, $$

and$$ y_2(x, B_1, B_2, a, b_1, b_2, b_3) = \begin{cases} a + b_1(x - B_1); & x < B_1\\ a + b_2(x - B_1); & B_1 \leqslant x \leqslant B_2\\ a + b_2(B_2 - B_1) + b3(x - B_2); & x > B_2 \end{cases}. $$

The derivative equals $b$ or $b2$ at $\dfrac{1}{2}(B_1+B_2)$ for functions $y_1$ and $y_2$, and equals $b1$ at $x\ll B_1$ and $b3$ at $x\gg B_2$ for function $y_2$.

I would like one additional parameter $s$ that would describe how closely the smooth function would approximate the piecewise linear versions. The function should extrapolate more or less linearly on both sides.

Any thoughts about what could be good functions for this? I want the functions to be smooth and continuously differentiable as that would help me to fit these parameters as I could then provide the analytical first-order derivative.

Tom Wenseleers
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2 Answers2

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Ha with the help of my brother found the answer myself in the end. If we define \begin{align*} f(x, b, s) &= \frac{1}{2}\sqrt{\smash[b]{b(4s+bx^2)}}\\ &\mathrel{\phantom{=}}{} \end{align*}

then my piecewise linear function $y_1(x, a, b, B1, B2, s)$ can be approximated as \begin{align*} y_1(x, a,b,B1,B2,s) &= a + f(x-B1,b,s)-f(x-B2,b,s) \end{align*} if the intercept is $a$ and slope of the middle part $b$.

Likewise, my piecewise linear function $y_2(x, a, b1, b2, b3, B1, B2, s)$ with slopes $b1$, $b2$ and $b3$ in the 3 linear parts can be approximated as \begin{align*} y_2(x, a,b1,b2,b3,B1,B2,s) &= a + b1.x + f(x-B1,b2-b1,s)-f(x-B2,b2-b3,s) \end{align*} These will approach the original piecewise model more as $s\to0$. The same system can be used for functions with any number of breakpoints. They are much easier to fit than the original piecewise model though, due to them being continuously differentiable.

Tom Wenseleers
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$\def\peq{\mathrel{\phantom{=}}{}}$Redefine $y_2(x)$ as$$ y_2(x) = \begin{cases} a_1 x + b_1; & x \leqslant x_1\\ a_2 x + b_2; & x_1 < x \leqslant x_2\\ a_3 x + b_3; & x > x_2 \end{cases}. $$ Assume that $a_2 > a_1$ and $a_2 > a_3$. Extending from your brother's construction, here is an approximation of two parameters $s$ and $n$:\begin{align*} y_2(x; s, n) &= \frac{1}{2} (a_1 + a_3)x + \frac{1}{2} (b_1 + b_3)\\ &\peq + \frac{1}{2} (a_2 - a_1) \left((x - x_1)^{2n} + (a_2 - a_3)^{2n} s^{2n}\right)^{\frac{1}{2n}}\\ &\peq + \frac{1}{2} (a_3 - a_2) \left((x - x_2)^{2n} + (a_2 - a_1)^{2n} s^{2n}\right)^{\frac{1}{2n}}, \end{align*} where $s > 0$, $n \in \mathbb{N}_+$. The smaller the $c$ or the larger the $n$, the better the approximation.

Ѕᴀᴀᴅ
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  • Thanks for that! Only downside is that this introduces two extra parameters s and n. I've added a solution with just a single extra parameter s in my own answer to approximated y1 and y2, which in principle should work for any number of breakpoints. – Tom Wenseleers Mar 19 '18 at 06:40
  • @TomWenseleers The parameter $n$ is just for an extra degree of choice, which can be taken as $1$ to reduce this model to yours :) Also, this model can be extended to finite-phased scenarios. – Ѕᴀᴀᴅ Mar 19 '18 at 07:00
  • @TomWenseleers For finite-phased models, $(a_2-a_3)^{2n}s^{2n}$ and $(a_2-a_1)^{2n}s^{2n}$ should be replaced by more general parameters, i.e. $d_1^{2n},\cdots,d_m^{2n}$, where $d_1,\cdots,d_m>0$. – Ѕᴀᴀᴅ Mar 19 '18 at 07:09
  • Ha thanks for the clarifaction - yes, I see now! Cool - thanks a lot for your help! – Tom Wenseleers Mar 19 '18 at 07:12