Suggesting a form of the equation of the given curve

Question

I am really struggling with this problem for a long time.

I think it is easy to understand, but difficult to solve;

We have a curve that has a lower limit as $x \rightarrow -\infty$, and it has another lower limit as $x \rightarrow \infty$. This curve increases only before the peak and decreases only after the peak. The rate at which this curve increases may not be the same rate at which this curve decreases (apart from the sign). What I mean is: the function may increases rapidly but decreases slowly, or vice versa.

Also, it is not a piecewise function. Moreover, it should be integerable at any domain.

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If the form of that function contains something like this: $ae^{-x^2}$ (Gaussian Distribution), then it will not be integerable.

I was trying to combine two expression (piecewise) to make it (not piecewise), but no luck: like combining

$y=a-be^{cx}$ for $-\infty <x \le j$, and $y=A-Be^{Cx}$ for $j < x < \infty$; where $a,b,c,A,B,C,j$ are constants.

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The following curve is plotted (using piecewise function) just to illustrate the problem.

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So the goal is to find the form of the equation of the curve with the following criteria:

• has a lower limit before the peak.

• has another lower limit after the peak.

• The rate of increasing may be different from the rate of decreasing (apart from the sign).

• Must be integerable in any domain.

• Must not be a piecewise function.

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Your help would be appreciated. THANKS!

Answer 1

There are many functions which satisfy your requirements. Perhaps there are more requirements that you have not stated? One solution to the problem as stated is

$$y=a+b\frac{\frac{x}{c}}{\sqrt{(\frac{x}{c})^2+1}}+d^2\frac{1}{(\frac{x}{c})^2+1}$$ It has a lower limit both before and after the peak, and it is smooth, both integrable, and not "piecewise".

You may play around with the parameterized function at https://www.desmos.com/calculator/o0ja1e0c0o

Answer 2

The piecewise function : $$y(x)=\begin{cases} a+b\:e^{c\:x}\qquad x<\chi \\ A+B\:e^{C\:x}\qquad x>\chi \end{cases}$$ is defined on an other form : $$y(x)=\left(a+b\:e^{c\:x}\right)H(\chi-x)+\left(A+B\:e^{C\:x}\right)H(x-\chi)$$ in which $H$ is the Heaviside step function.

If one want to avoid the picewise form one can replace the Heaviside function by a continuous function which approximates the Heaviside function. They are several possibilities. For example a function of tbe logistic kind can be used : $$H(X)\simeq\frac{1}{1+e^{-K\:X}}$$ $K$ is an arbitrary positive large number. Doesn't mater the value of $K$ insofar it is very large. For example :

In order to fit the equation to data one have to adjust the parameters $a,b,c,A,B,C,\chi$ using a regression method.

Since there is no scale on the graph published by the OP, arbitrary nomalized scales are added on the above graph. The values of parameters were adjusted in order to fit the blue curve to the given red points.

NOTE :

If one want a smoother transition between the two exponentials, a smaller value of $K$ has to be adjusted. So, one parameter more for the regression.

Even more general, one can introduce more paramerers such as slightly different $K_1,K_2$ and slightly different $\chi_1,\chi_2$ in order to have a wide range of possible shapes of the curve. $$y(x)=(a+be^{cx})\frac{1}{1+e^{K_1(x-\chi_1)}} +(A+Be^{Cx})\frac{1}{1+e^{K_2(\chi_2-x)}}$$ On the other hand, a so large number of parameters $a,b,c,\chi_1,K_1,A,B,C,\chi_2,K_2$ would increases a lot the difficulty for a robust regression process. With so many parameres I doubt that the available softwares will converge to reliable result.