Error in the intercept

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Adam Parry
Adam Parry 2012-7-3
I think this might be a dumb question, but how do you find the error of the intercept for a straigh line graph that I have fittded using polyfit? and polyval. I got the error for the slope by doing something like this.
[p2,s2] = polyfit(A2,B2,1);
[f2,delta] = polyval(p2,x,s2);
deltaf2=s2.normr/sqrt(s2.df);
C2=deltaf2^2*inv(s2.R)*inv(s2.R)';
deltap2=sqrt(diag(C2));
ok
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Richard Brown
Richard Brown 2012-7-3
typo: inv(A) * B should be A \ b :)
Ganessen Moothooveeren
Ganessen Moothooveeren 2013-3-14
you used this to find error in slope but which variable is the error in slope??..is it deltaf2?? [p2,s2] = polyfit(A2,B2,1); [f2,delta] = polyval(p2,x,s2); deltaf2=s2.normr/sqrt(s2.df); C2=deltaf2^2*inv(s2.R)*inv(s2.R)'; deltap2=sqrt(diag(C2));

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Adam Parry
Adam Parry 2012-7-4
I think I have cracked it if anyone is interested.
Fairly simple if you use nlinfit. (I would be interested to know if you can get the same results from lsqcurvefit though if anyone has the answer).
So what you so is
function F = myfun(p,X)
F = (p(1)*X)+p(2); end
!sepearte file!
X = [1:4]; Y = [5:8]; % or something like this
[beta,resid,J,COVB,mse] = nlinfit(X,Y,@myfun2,[1;1]) ; [ci se] = nlparci(beta,resid,'covar',COVB);
%beta(1) is coefficient p(1)(the gradient) and beta(2) is the intercept. % se(1) is the error on the gradient and se(2) is the error of the intercept.
Easy, i feel a little ashamed..
oh yeah, you also had to alter nlparci
by going
open nlparci in matlab
then changing on the first line
function ci = nlparci(beta,resid,varargin)
to
function [ci se] = nlparci(beta,resid,varargin)
and thats it...
Thanks
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Star Strider
Star Strider 2012-7-4
I believe both 'nlinfit' and 'lsqcurvefit' can give you the information you need to give to 'nlparci' to calculate the parameter confidence intervals. The principal difference between 'nlinfit' and 'lsqcurvefit' is that 'lsqcurvefit' allows parameter constraints. Both will give you either the covariance matrix or the jacobian as well as the other results that 'nlparci' can use as arguments.
You don't have to alter 'nlparci' to get the standard errors, since 'nlinfit' gives you the ability to calculate those from the 'COVB' matrix it returns. You already calculated the standard errors as 'deltap2' from the covariance matrix you calculated as 'C2' in your original code. I used your results to calculate the 'CI95' matrix.

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Star Strider
Star Strider 2012-7-3
Not dumb at all. The problem is that if you want confidence limits on the estimated parameters, the 'polyfit' and 'polyval' functions won't get you there.
If you have the Statistics or Optimization Toolboxes, you can fit your model with 'lsqcurvefit' or 'nlinfit' respectively, then use 'nlparci' to get the confidence limits on the parameters. (Use 'nlpredci' to get confidence limits on the fitted data.)
If you don't have access the these, 'lscov' will likely give you what you need to calculate the confidence intervals yourself.
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Star Strider
Star Strider 2012-7-3
编辑：Star Strider 2012-7-3
You didn't do anything wrong that I can see. When I ran 'lscov' on it (with simulated data), it produced the same covariance matrix you calculated. If anything, you didn't go far enough. The 95% confidence limits are ±1.96*SE, so with respect to your code they would be:
CI95 = [p2-1.96*deltap2 p2+1.96*deltap2];
and of course unless the 'CI95' interval for a parameter included zero, the parameter belongs in the model. Use 'norminv' to get critical values for other confidence intervals.
Other than that, using 'inv' is generally frowned upon because of condition concerns. The '\' operator avoids these because it does the division directly.
It took a bit of experimenting, but an alternate way of calculating C2 that uses '\' and avoids 'inv' is:
C2 = deltaf2^2 * (s2.R'*s2.R)\eye(2);
That's the only improvement I can think of.
Adam Parry
Adam Parry 2012-7-4
I was just looking in to finding errors using lsqcurvefit, but it ssems that you have to use nlinfit. Is this right? It also seems to suggest that you have to change some of the code in nparci in order to get the actual error on the parameters?
Can you help with that at all.
Thanks very much for the help so far by the way

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