Equation Fitting: Seeking Insight to Meet Requirements

[CRGreathouse]

I'm almost too embarrassed to post, but I thought someone might have insight that could help me here. I'm trying to fit an equation to data, but I'm just not sure what sort of equation I should use. I have some requirements on the form of the equation, and then I have points (yet to be determined, actually).

$$f'(x)>0\,\,\forall x>0$$ (cost is increasing)
$$f''(x)<0\,\,\forall x>0$$ (essentially, average cost is decreasing)
$$\lim_{x\rightarrow\infty} f(x)=k$$ (price is bounded above in this fashion)

It would make the most sense if the origin was included, but this can be one of the data points.

So before I even think about least-squares vs. minimal points on the curve, I wanted to consider different forms that could make sense. The first that comes to mind was the elementary

$$y=a-\frac{ac}{x+c}$$

but this has some instabilities and oddities. I guess I'm just looking for thoughts on widely-used models that fit my criteria or could be modified to fit them. In the perfect case I'd have a smooth equation that was nearly linear for small x and essentially hyperbolic for large x.

Any suggestions would be welcomed.

 

[SGT]

I'm almost too embarrassed to post, but I thought someone might have insight that could help me here. I'm trying to fit an equation to data, but I'm just not sure what sort of equation I should use. I have some requirements on the form of the equation, and then I have points (yet to be determined, actually).

$$f'(x)>0\,\,\forall x>0$$ (cost is increasing)
$$f''(x)<0\,\,\forall x>0$$ (essentially, average cost is decreasing)
$$\lim_{x\rightarrow\infty} f(x)=k$$ (price is bounded above in this fashion)

It would make the most sense if the origin was included, but this can be one of the data points.

So before I even think about least-squares vs. minimal points on the curve, I wanted to consider different forms that could make sense. The first that comes to mind was the elementary

$$y=a-\frac{ac}{x+c}$$

but this has some instabilities and oddities. I guess I'm just looking for thoughts on widely-used models that fit my criteria or could be modified to fit them. In the perfect case I'd have a smooth equation that was nearly linear for small x and essentially hyperbolic for large x.

Any suggestions would be welcomed.

One equation that fits your requirements is:
$$f(x) = k(1 - e^{-ax})$$ k > 0 a > 0
Then
$$f´(x) = ak e^{-ax} > 0$$
$$f"(x) = -a^2k e^{-ax} < 0$$

 

[CRGreathouse]

One equation that fits your requirements is:
$$f(x) = k(1 - e^{-ax})$$ k > 0 a > 0
Then
$$f´(x) = ak e^{-ax} > 0$$
$$f"(x) = -a^2k e^{-ax} < 0$$

Thanks. You know, I was just so out of it I didn't think to use this.

I'm going to play with this for a while and make it work. You gave me just what I needed to get my brain working again.