Equation Fitting: Seeking Insight to Meet Requirements

  • Thread starter CRGreathouse
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In summary, the person is looking for help in finding an equation to fit their data. They have some requirements for this equation, such as increasing cost, decreasing average cost, and a bounded price. They mention trying a simple equation, but it has some issues. They are looking for suggestions on widely-used models that fit their criteria or can be modified to fit them. One suggested equation is f(x) = k(1 - e^{-ax}), which satisfies all the requirements. This equation was not initially considered by the person, but thanks to the suggestion, they are able to continue working on their problem.
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CRGreathouse
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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).

[tex]f'(x)>0\,\,\forall x>0[/tex] (cost is increasing)
[tex]f''(x)<0\,\,\forall x>0[/tex] (essentially, average cost is decreasing)
[tex]\lim_{x\rightarrow\infty} f(x)=k[/tex] (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

[tex]y=a-\frac{ac}{x+c}[/tex]

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.
 
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  • #2
CRGreathouse said:
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).

[tex]f'(x)>0\,\,\forall x>0[/tex] (cost is increasing)
[tex]f''(x)<0\,\,\forall x>0[/tex] (essentially, average cost is decreasing)
[tex]\lim_{x\rightarrow\infty} f(x)=k[/tex] (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

[tex]y=a-\frac{ac}{x+c}[/tex]

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:
[tex]f(x) = k(1 - e^{-ax}) [/tex] k > 0 a > 0
Then
[tex]f´(x) = ak e^{-ax} > 0[/tex]
[tex]f"(x) = -a^2k e^{-ax} < 0[/tex]
 
  • #3
SGT said:
One equation that fits your requirements is:
[tex]f(x) = k(1 - e^{-ax}) [/tex] k > 0 a > 0
Then
[tex]f´(x) = ak e^{-ax} > 0[/tex]
[tex]f"(x) = -a^2k e^{-ax} < 0[/tex]

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

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.
 

1. What is equation fitting?

Equation fitting is a statistical technique used to find the best mathematical representation of a set of data points. It involves finding a mathematical equation or model that closely approximates the relationship between the independent and dependent variables in the data set.

2. Why is equation fitting important?

Equation fitting is important because it allows us to better understand and make predictions about complex systems or processes. It can also help us identify patterns and relationships in data that may not be immediately apparent.

3. What are the steps involved in equation fitting?

The steps involved in equation fitting typically include selecting an appropriate mathematical model, estimating the model parameters, evaluating the model's fit to the data, and making any necessary adjustments or improvements to the model.

4. What are some common mathematical models used in equation fitting?

Some common mathematical models used in equation fitting include linear regression, polynomial regression, exponential growth/decay, and power law models. The choice of model depends on the type of data and the relationship between the variables being studied.

5. How do you assess the quality of a fitted equation?

The quality of a fitted equation can be assessed by looking at the coefficient of determination (R-squared), which measures the proportion of variation in the dependent variable that is explained by the independent variable(s). Additionally, residual analysis can be used to check for any patterns or trends in the errors of the fitted model.

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