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Precision & Accuracy: SUMMARY
*To*: mathgroup at christensen.cybernetics.net
*Subject*: [mg743] Precision & Accuracy: SUMMARY
*From*: Paul E Howland <PEHOWLAND at taz.dra.hmg.gb>
*Date*: Wed, 12 Apr 1995 15:43:49 +0200 (IST)
Mathgroup,
I posted a question yesterday regarding the difference between Accuracy[]
and Precision[] in Mathematica. I had two different but interesting
replies, which I summarise below. To solve my problem, though, I
eventually delved into the package NonlinearFit.m to see how the Pro's did
it: I'll also mention my findings about this here.
The first answer provided an analogy with playing darts. Accuracy
implies that the average position of the darts is near the bull's eye,
although their variance can be quite high, whereas Precision implies that
all the darts hit the same place, but this may be nowhere near the bull's
eye. That is, Accuracy=Unbiased answer, Precision=Low Variance answer.
The second answer provided a useful example of where the Accuracy and
Precision of a number vary:
Accuracy[1.00000000000000000001 * 10^-5] = 25
Precision[1.00000000000000000001 * 10^-5] = 20
I was still left wondering how this could be applied when optimising a
function. What is the difference between the accuracy of my optimisation
and its precision? I had a delve into the code of NonlinearFit.m, and
found the answer. When trying to minimise a cost function (NonlinearFit
uses the Chi squared function), the function satisfies the accuracy
constraint if:
0 <= chisq-newchi < (10^-AccuracyGoal)
but satisfies the precision constraint if:
0 <= chisq-newchi < newchi*(10^-PrecisionGoal)
where chisq is the value of the cost function at the last iteration, and
newchi is the value of the cost function at the current iteration.
Thus AccuracyGoal specifies that the algorithm be declared converged when
the _absolute_ difference in the cost function is less than a certain
threshold, whereas PrecisionGoal specifies that the algorithm be declared
converged when the _proportional_ change in the cost function is less
than a specified threshold.
*** TWO WARNINGS ABOUT NonlinearFit.m ***
=========================================
1. Whilst looking at the code for NonlinearFit, I noticed that the
default values for AccuracyGoal and PrecisionGoal are:
AccuracyGoal -> 1
PrecisionGoal -> 3
rather than
AccuracyGoal -> $MachinePrecision-10
PrecisionGoal -> $MachinePrecision-10
as stated in the "Guide to Standard Mathematica Packages V2.2". Thus
results may be considerably less precise than you expect!
2. Be wary of the message "NonlinearFit::lmpnocon" which warns that the
values of the parameters do not appear to have converged. The algorithm
produces this warning if the difference between the previous value of a
parameter and its new value is greater than 1 (for any of the parameters),
at the end of the calculation. It will thus be _highly_ dependent on the
units of the parameters you are fitting! Be warned!
Paul E Howland
Long Range Ground Radar Systems Section tel. +44 (0)1684 895767
CSS2 Division, Room BY209 fax. +44 (0)1684 896315
Defence Research Agency email: PEHOWLAND at DRA.HMG.GB
Malvern, Worcs, WR14 3PS, UK.
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