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I have a small dataset with 5 points where x values are log(time) and y = obs. It is known that the behavior is an exponential increase for the first and second (x,y) values and then decreases linearly. The interpolant fit to the data points looks something like shown, but I want to create something like the red curve, which is more meaning full to my data, as it decays slowly and then should become constant. I have tried the double exponential decay function, but it doesn't give me good results, as the first peak becomes too high, close to 170% on the Y scale, which has no physical meaning in my case. The fitted equation: $y = a*exp(b*x)+c*exp(d*x)$

enter image description here

I'm not a Maths expert, but I would really appreciate it if you could suggest to me some equations or an improvement in my fit to help me resolve this problem.

The data looks something like this:

X = [0 3.3 4.02 4.71 5.41] Y = [100 111.38 78.64 71.89 66.75]

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With only five points the curve to fit is very poorly defined.

For example one can choose a parabolic equation for the three first points. But this is rather subjective.

A linear equation appears more convenient for the three last points.

The result appears on the next figure. This way the model equation is a piecewise function. Equivalently the model equation can be written thanks to the Heaviside's H function.

enter image description here

Of course the whole curve isn't smooth. If one want a smooth equation one can replace the Heaviside function by an approximate. The smoothness is controles by the parameter lambda.

enter image description here

Also you might play with the parameter xm and/or change the parabola equation to modify more strongly the shape of the curve and approach the expected shape. But this has few mathematical signifiance.

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