Author: tlinnet
Date: Thu Aug 28 12:34:25 2014
New Revision: 25376
URL: http://svn.gna.org/viewcvs/relax?rev=25376&view=rev
Log:
Yet another try to make the API documentation working.
task #7822(https://gna.org/task/index.php?7822): Implement user function to
estimate R2eff and associated errors for exponential curve fitting.
Modified:
trunk/specific_analyses/relax_disp/estimate_r2eff.py
Modified: trunk/specific_analyses/relax_disp/estimate_r2eff.py
URL:
http://svn.gna.org/viewcvs/relax/trunk/specific_analyses/relax_disp/estimate_r2eff.py?rev=25376&r1=25375&r2=25376&view=diff
==============================================================================
--- trunk/specific_analyses/relax_disp/estimate_r2eff.py (original)
+++ trunk/specific_analyses/relax_disp/estimate_r2eff.py Thu Aug 28
12:34:25 2014
@@ -184,39 +184,39 @@
The parameter 'epsrel' is used to remove linear-dependent columns when J
is rank deficient.
- The covariance matrix is given by,
-
- covar = (J^T J)^{-1}
-
- and is computed by QR decomposition of J with column-pivoting. Any
columns of R which satisfy
-
- |R_{kk}| <= epsrel |R_{11}|
-
- are considered linearly-dependent and are excluded from the covariance
matrix
- (the corresponding rows and columns of the covariance matrix are set to
zero).
-
- If the minimisation uses the weighted least-squares function:
-
- f_i = (Y(x, t_i) - y_i) / sigma_i
+ The covariance matrix is given by::
+
+ covar = (J^T J)^{-1} ,
+
+ and is computed by QR decomposition of J with column-pivoting. Any
columns of R which satisfy::
+
+ |R_{kk}| <= epsrel |R_{11}| ,
+
+ are considered linearly-dependent and are excluded from the covariance
matrix (the corresponding rows and columns of the covariance matrix are set
to zero). If the minimisation uses the weighted least-squares function::
+
+ f_i = (Y(x, t_i) - y_i) / sigma_i ,
then the covariance matrix above gives the statistical error on the
best-fit parameters resulting from the Gaussian errors 'sigma_i' on the
underlying data 'y_i'.
- This can be verified from the relation 'd_f = J d_c' and the fact that
the fluctuations in 'f from the data 'y_i' are normalised by 'sigma_i'
- and so satisfy <d_f d_f^T> = I.
-
- For an unweighted least-squares function f_i = (Y(x, t_i) - y_i) the
covariance matrix above should be multiplied by
- the variance of the residuals about the best-fit
-
- sigma^2 = sum ( (y_i - Y(x, t_i))^2 / (n-p) )
-
- to give the variance-covariance matrix sigma^2 C.
- This estimates the statistical error on the best-fit parameters from the
scatter of the underlying data.
-
- See:
- U{GSL - GNU Scientific Library<http://www.gnu.org/software/gsl/>}
- U{Manual:
Overview<http://www.gnu.org/software/gsl/manual/gsl-ref_37.html#SEC510>}
- U{Manual: Computing the covariance matrix of best fit
parameters<http://www.gnu.org/software/gsl/manual/gsl-ref_38.html#SEC528>}
- U{Other reference<http://www.orbitals.com/self/least/least.htm>}
+ This can be verified from the relation 'd_f = J d_c' and the fact that
the fluctuations in 'f' from the data 'y_i' are normalised by 'sigma_i' and
so satisfy::
+
+ <d_f d_f^T> = I. ,
+
+ For an unweighted least-squares function f_i = (Y(x, t_i) - y_i) the
covariance matrix above should be multiplied by the variance of the residuals
about the best-fit::
+
+ sigma^2 = sum ( (y_i - Y(x, t_i))^2 / (n-p) ) ,
+
+ to give the variance-covariance matrix sigma^2 C. This estimates the
statistical error on the best-fit parameters from the scatter of the
underlying data.
+
+ Links
+ =====
+
+ More information ca be found here:
+
+ - U{GSL - GNU Scientific Library<http://www.gnu.org/software/gsl/>}
+ - U{Manual:
Overview<http://www.gnu.org/software/gsl/manual/gsl-ref_37.html#SEC510>}
+ - U{Manual: Computing the covariance matrix of best fit
parameters<http://www.gnu.org/software/gsl/manual/gsl-ref_38.html#SEC528>}
+ - U{Other reference<http://www.orbitals.com/self/least/least.htm>}
@param J: The Jacobian matrix.
@type J: numpy array
r25376 - /trunk/specific_analyses/relax_disp/estimate_r2eff.py