How to compare two data sets with statistics?

[elegysix]

I have two questions:

I have a set of data, a measured spectrum. When I model the spectrum with a function, I calculate r²=1-($\sum$(y-y_model)²/$\sum$(y-y_avg)²).

Q1) However, I have reference data now, which is what the spectrum should be. So is it right to use the same calculation on it for r², but instead of using y_model, using y_reference?

Q2) The model function I was fitting to the data is
S_λ = 2πhc²/λ⁵(e^hc/λkT-1)
Is it correct to calculate goodness of fit in that way for such a distribution?


Here is a plot of my two data sets

thanks!

 

[Simon Bridge]

Q1> what does it mean: "what the spectrum should be"
There is what the spectrum is and what the model predicts - surely it "should be" whatever it actually is.

Q2> To decide what to do you need, first, to define the problem.
What is it you are trying to find out?

If you want to see if the model is a good fit to the data, then a goodness fit is probably warranted.
Make sure that the approach you use answers the questions you are asking.

What I am reading above is that you have not asked a clear enough question to know how to proceed.

Suspect you may need these:
http://home.comcast.net/~szemengtan/
... "Inverse Problems" towards the bottom of the page.

Those data plots are seriously cool btw.

 

[elegysix]

Thanks, sorry for being unclear.
Forget that I mentioned a "model"

"what the spectrum should be" is the ASTMG173.
We captured the solar spectrum and want to compare it with a reference spectrum (the ASTMG173) to show that our measurements are accurate.

the question is - how can I properly use statistics to say how well these two data sets match?

Is it appropriate to use this calculation: $r^{2} = 1 - \frac{\sum(y_{r} - y_{s})^{2} }{\sum(y_{r} - \bar{y_{r}})^{2} }$

where $y_{r}$ is the reference y data, and $y_{s}$ is our measured y data, and $\bar{y_{r}}$ is the mean of the reference y data.

thanks

 

[Simon Bridge]

So you are testing the measuring method, to show that it is sound?

You want to use the coefficient of determination test?
I think you have the roles of the data-sets reversed.

There are other goodness of fit tests - i.e. chi-squared - what lead you to choose this one?

 

[elegysix]

So you are testing the measuring method, to show that it is sound?

yes.

You want to use the coefficient of determination test?

Not necessarily. I want to use whatever test is appropriate for this.

There are other goodness of fit tests - i.e. chi-squared - what lead you to choose this one?

I am not familiar with the others, that is why I made this thread. Which test should I use? what would you use?

thanks

 

[Simon Bridge]

I see ... I cannot see anything immediately ruling out a CoD test.
I would use Chi-squared... but that's me.

Really you are comparing two data-sets and asking if they are close enough to come from the same forward function rather than checking a data set against a theoretical model of a forward function.

The inverse problems papers I linked you to (post #2) gives a lot of detail on different rationales for goodness of fit in different circumstances.