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dominik.mierzwa

Poland
118 Posts
Posted - 10/16/2018 :  3:51:35 PM  Show Profile  Edit Topic  Reply with Quote  View user's IP address  Delete Topic
Origin Ver. and Service Release (Select Help-->About Origin):
Operating System:

Hi there,
In this topic:

https://my.originlab.com/forum/topic.asp?TOPIC_ID=44293

I asked about the meaning of reduced chi-square. Well I'm still really confused about this parameter especially that Origin documentation is inconsistent in this topic. Moreover, I found this topic https://my.originlab.com/forum/topic.asp?TOPIC_ID=4058 which is also related to the meaning of this parameter. In my opinion it was not fully explained

Generally in all source reduced chi-square is defined as follows:


source Bevington Data reduction and error analysis in physical sciences pp 194-195.

Same definition may be find here:
https://www.originlab.com/doc/Quick-Help/Bad-ReducedChiSqr

In this case the reduced chi-square is the ratio of estimated variance and the parent variance so it should be close to the 1 (for good fit)

But here we have got another definition:
https://www.originlab.com/doc/Origin-Help/NLFit-Theory#Scale_Error_with_sqrt.28Reduced_Chi-Sqr.29

which is different, but same as variance s^2 in Bevington (without weighting):


According to this site:
https://www.originlab.com/doc/Origin-Help/Interpret-Regression-Result#Scale_Error_with_sqrt.28Reduced_Chi-Sqr.29
the reduced chi-squared = RSS/DOF should also be close to 1 can you explain me why? If it is a variance (as s^2 in Bevington) the smaller RSS/DOF the better fit is and it should not be equal 1 (especially if DOF is very big)

Can someone explain me this discrepancies, I'm really confused but I know that in statistics details are key-meaning.
Thank you

Kind regards,
Dominik
Edited by - dominik.mierzwa on 10/18/2018 02:37:12 AM

Shirley_GZ

China
Posts
Posted - 10/17/2018 :  04:54:56 AM  Show Profile  Edit Reply  Reply with Quote  View user's IP address  Delete Reply
Hi Dominik,

For the equation in the page(https://www.originlab.com/doc/Origin-Help/NLFit-Theory#Scale_Error_with_sqrt.28Reduced_Chi-Sqr.29), when the weighting being Instrumental,

Then, it is same to the "general" definition after you plug the RRS to the equation, right?

About the doubt about the value of Reduced Chi-Sqr at the end of your post, please read the description in this FAQ,
https://www.originlab.com/doc/Quick-Help/Bad-ReducedChiSqr

Thanks,
Shirley
OriginLab

Originlab Technical Service Team
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dominik.mierzwa

Poland
118 Posts
Posted - 10/17/2018 :  06:55:47 AM  Show Profile  Edit Reply  Reply with Quote  View user's IP address  Delete Reply
Hi Shirley,

Please see the s^2 in above post. It is exactly the 'reduced chi-square' defined in Origin documentation:

reduced chi-square = RSS/DOF

where
RSS = sum[wi*(yi-yf)^2]
DOF = N-m
wi - weight (in instrumental: 1/<sigma(i)>^2>
yi - i-th experimental value of y
yf - i-th fitted value of y
N - number of experimental points (yi)
m - number of parametrs in model

So, it is not the reduced chi-square as defined in Bevington:

reduced chi-square = chi/DOF = s^2/<sigma(i)^2>

Thus, why should it be close to 1 (as mentioned in Origin documentation)? This is a variance (s^2) so it should be as small as possible.

The reduced chi-square defined as in Bevington should be close to 1 because it is the ratio of estimated variance and the parent variance.


Kind regards,
Dominik
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dominik.mierzwa

Poland
118 Posts
Posted - 10/18/2018 :  03:12:27 AM  Show Profile  Edit Reply  Reply with Quote  View user's IP address  Delete Reply
Hi Shirley,

I checked once again the notation and you are absolutely right. In the case of instrumental weighing, where
RSS = sum[wi*(yi-yf)^2]
wi=1/sigma(i)^2
the equation is coherent with the general one.

But if another weighting method is chosen or there is no weighing and:
RSS = sum(yi-yf)^2

Should reduced chi-square be still close to 1?



Kind regards,
Dominik
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Shirley_GZ

China
Posts
Posted - 10/22/2018 :  06:03:53 AM  Show Profile  Edit Reply  Reply with Quote  View user's IP address  Delete Reply
Hi Dominik,

I think the description in this FAQ is clearly enough,
https://www.originlab.com/doc/Quick-Help/Bad-ReducedChiSqr

quote:
If a weight is included in the fitting process and the Reduced Chi-Sqr is very different from 1, please examine if an improper weighting method is chosen. If the Reduced Chi-Sqr value is much smaller than 1, it may indicate a too large weight. Vice versa.


Thanks,
Shirley

Originlab Technical Service Team
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dominik.mierzwa

Poland
118 Posts
Posted - 10/22/2018 :  08:28:02 AM  Show Profile  Edit Reply  Reply with Quote  View user's IP address  Delete Reply
Thank you!

Kind regards,
Dominik
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