CDF(cumulative distribution function) starts not from 0

I have a dataset, then i want to know the distribution, i used exppdf(X,mu), but the result is not reasonable, please see the figure, why the CDF(cumulative distribution function) starts not from 0? And the PDF(probability density function) seems too small. How can i solve this problem?

[Xiaowei Wang]

The same problem also happens for chi2cdf(X,nu), the CDF starts not from 0?

[Greig]

The reason that the PDF and CDF don't start at zero probability is because the distributions that you specify exist outside of the X values you used.

These two functions generate the theoretical PDFs and CDFs for the specified distributions at the X values you input. They are not for data fitting.

See my answer below for more details.

Answer 1

[Greig]

It seems like your choice of parametric distribution is not appropriate for your data and it seems that you might be approaching the problem in the wrong way.

First things first... What are your data? What is their physical meaning? Do they have lower or upper bounds? What do you want to do with the distribution fit? Have previous analyses (by others) justified the use of a exponential or chi-squared distribution? All of these questions will help you to decide what type of distribution is appropriate to fit to your data.

Second things second, fitting... Do you have Statistics Toolbox? If so, then dfittool is a good place to start. In command window type

doc 'Model Data Using the Distribution Fitting App'

and choose the first option.

If you can provide us with more info about your data we can probably help some more.

Answer 2

[Xiaowei Wang]

Thanks for your nice reply. The data i used is uploaded, it is the simulation result, i could find the maximum and minimum (upper and lower)bounds, but no idea why doing this. My purpose is to know which distribution my data fits. Yes, i had Statistics Toolbox and tried with dfitool, there are lots of distributio functions, how to choose the most appropraie one? Is there some parameters as measurand, e.g, the value of the error?

[Xiaowei Wang]

Also the corresponding m file

[Greig]

The point I was trying to make above, is that you need to understand the physics behind what you are measuring to try and decide what distribution is appropriate. I don't know what your data are, so you are in the best position to know the properties of the data and the physical constraints that they are subject to. These properties and constraints tell you what distribution is appropriate.

At quick glance you are looking at directivity? Which, according to the Wikipedia page is the ratio of radiative intensity in one direction (U) to the average across all directions (P). Is that correct?

I know nothing about this, so the following might be wrong. But, I am guess that U >= 0, and assuming at least one direction has some power, P > 0. So this means that the smallest possible value is 0, and the largest is infinity. This range is known as the supported range of a distribution.

Another Wikipedia search gives a list probability distributions that are support in this range.

What are the units of your data? Have you transformed them in dB? Trusted Wikipedia says this is more common. If so, this will change the supported range of the distribution to [-infinity , infinity]. Other distributions found on the Wikipedia link above will then be more appropriate.

Maybe you can think of other properties of the data that will help to constrain what is a reasonable distribution. I guess both U and P will be subject to some noise or calculation, this may mean they themselves follow a distribution. In this case directivity may be the ratio of two distributions, which sometimes has a nice defined form (but not always).

[Xiaowei Wang]

Hello, Greig,that is correct, directivity is a quantity in electromagnetic theory, which is ratio of U over V. It has a range from 0 to infinity, where i use linear values, not in dB. Then i tried gamma distribution gampdf and gamcdf, still the problem, the values of CDF not starts from 0 (not appropraite). Then next i tried gammafit, but this seems only to obtain some parameters. Finally i used dfittool, and the normal distribution is better approached (in attachment). Although normal distribution is for the whole real line (-infinity, infinity) , nor sure i am correct. I will try the distribution of directivity itself, and also the distribution for U and P themselves, whether or not they are equivalent.