Hi,
Hi,
attached to MATH-784 (https://issues.apache.org/jira/browse/MATH-784)
is a Monte-Carlo simulation which might be used to explore this issue.
For the time being, it confirms that
* guessParametersErrors() indeed estimates the sd on the parameters,
* the sqrt of the diagonal coefficients of the covariance matrix also
provide a good estimate of these standard deviations, and also the 68%
confidence interval (as announced in Numerical Recipes, 15.6).
Although these values are very close, I do think they do not really
have the same mathematical meaning.
What remains to be explored
* use observations which are not normally distributed (e.g. Poisson?),
* use smaller sets of observations, which should emphasize the
difference between guessParametersErrors() and the sqrt of the
diagonal coefficients.
'Statistics in theory and practice' by R. Lupton (Princeton University
Press) might hold the solution in its page 90. The distinction is between
sigma and the half-width of some confidence interval. In Lupton's case,
he adopts a 99% confidence interval and obtains 2.57*sigma for the half
width of the confidence interval. A 68% confidence interval corresponds to
1*sigma as you obtain. With the formula presently in commons-math, the
confidence interval is based on a unit increase of the normalised chi
square: by how much the parameter can change without leading to an increase
of the normalised chi square by more than 1?
All these approaches make sense but should not be confused with each other.
I repeat what I stated in my previous post: unless otherwise stated, the
scientific community is used to publish sigma, i.e. just the square root
of the covariance matrix, i.e. the square root of the inverse of the
Fisher information matrix. Maybe renaming guessParametersErrors() into
getParametersConfidenceIntervalHalfWidth() would help avoiding the
confusion.
Regards,
Dim.
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Dimitri Pourbaix * Don't worry, be happy
Institut d'Astronomie et d'Astrophysique * and CARPE DIEM.
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