Curve Fitting Tool - Weibull distribution

Hi all,

I am trying to do a fitting of a graph, using the curve fitting Tool and, in particular, using the Weibull option that use the formula:

a*b*x^(b-1)*exp(-a*x^b)

Despite the fact that the shape of the Weibull distribution seems to be the same of the one of my graph, the height of the Weibull distribution is lower. I have tried to calculate the integral of the Weibull function of the curve fitting tool of some data and the result is always 1. I think that this is due to the fact that it is a density function. With the above mentioned formula, it is impossible to fit properly the dataset because the Weibull distribution is always lower.

I used the approach shown here https://it.mathworks.com/help/curvefit/weibull.html, using the formula:

c*a*b*x^(b-1)*exp(-a*x^b)

Using this approach I have seen that the parameter c helps to scale the data. Is this approach correct? The second formula that I used should not substitute the first one that is used as default in the curve fitting tool?

if I want to obtain a Weibull distribution or a Lognormal distribution is it sufficient to apply a multiplier coefficient (e.g. the c parameter, which is necessary in the formula above to allow the curve's height to adjust to data) to a density function?

[dpb]

doc curvefit

Weibull Distributions

About Weibull Distribution Models

...

Curve Fitting Toolbox™ does not fit Weibull probability distributions to a sample of data. Instead, it fits curves to response and predictor data such that the curve has the same shape as a Weibull distribution.

So, the answer to that supposition would seem to be "No."

W/o data, not much we could do to see why...

[SYML2nd]

Ok, using the 3 parameter Weibull suggested here https://it.mathworks.com/help/curvefit/weibull.html (at the end of the post) the curve seems to fit much better. The formula that I used is:

f(x)=c*a*b*x^(b-1)*exp(-a*x^b)

Why these two results should be different?

[dpb]

For the reason given at the link...his Weibull model is defined with three parameters: the first scales the curve along the horizontal axis, the second defines the shape of the curve, and the third scales the curve along the vertical axis. Notice that while this curve has almost the same form as the Weibull probability density function, it is not a density because it includes the parameter c, which is necessary to allow the curve's height to adjust to data.

The third parameter allows for the vertical scaling not present in the packaged model.

[SYML2nd]

In my opinion there is an error in the Weibull fitting of the Curve Fitting Tool. I have tried to calculate the integral of the Weibull function of the curve fitting tool of some data and the result is always 1.

So, my initial supposition that the Weibull option of the Curve fitting tool is a density function, in my opinion was true. I agree with the fact that the third parameter is essential to scale the function, but in my opinion it should have been already placed in the default Weibull function of the curve fitting tool, otherwise it is not a real fitting. isnt'it? Maybe I am wrong.

Now, I wonder if it is sufficient a third scaling parameter (like the c in the above mentioned equation) to adapt a density distribution to a function. I have noticed that the Gaussian fitting option in the curve fitting tool has also 3 parameter (one more than the density function).

[dpb]

I've never used the fitting tool appliction; I always just use fit at the command line so I know what I'm getting (plus, I think it is at least as simple if not simpler than going through the gui interface). Hence, I just quoted the doc which implies the default isn't a pdf; however, I agree your experiments indicate the doc and the routine are not in synch with each other.

Which behavior is considered to be the design would be up to TMW to clarify and correct whichever is incorrect in order to remove the inconsistency.

As to the comment above

"The second formula that I used should not substitute the first one that is used as default in the curve fitting tool?"

As you observed, "No.". That behavior I agree is as designed and documented; the tool uses what it has set as default for each type you can select; if you want something different inside it you must define the fit explicitly. I presume you could then save that and re-import that definition to avoid having to recreate it over every time, but that's also supposition on having not used the tool enough to be very familiar with it at all.

[SYML2nd]

Ok. What it is not clear to me now is: if I want to obtain a Weibull distribution or a Lognormal distribution is it sufficient to apply a multiplier coefficient (e.g. the c parameter, which is necessary in the formula above to allow the curve's height to adjust to data) to a density function?