Re: Additional ppmu correction concerns

[Steven Clayton <smclayto@npl.uiuc.edu>]

Tom,

I agree: the lambda_of and lambda_op corrections should be parametrized
together.  At one extreme, if lambda_of is zero, lambda_op has no effect.
Maybe we could write the molecular kinetics correction like this:
DeltaLambda_s(lambda_of, lambda_op) = F1(lambda_of)*(1 + F2(lamdba_op)),
for some functions F1 and F2, which could also be expressed in terms
of deviations from central values of lambda_of and lambda_op.
F1 and F2 are probably each linear to good approximation.

If we read off the corrections for lambda_of and lambda_op separately
from my full kinetics fits, then we are double-counting for lambda_op,
as you say, since I held one parameter to its central value while
sweeping the other.  However, on slide 9 of Peter's kinetics.pdf, the
correct contributions seem to be indicated at the bottom:
18 Hz due to lambda_of = 2.3/us (w/lambda_op = 0), then an additional
5 Hz from lambda_op = 0.08/us (w/lambda_of = 2.3/us).
The total correction from lambda_of & lambda_op can be read off the
plot on the left: 23.5 Hz (lambda_of = 2.3, lambda_op = 0.08,
phi = 0.0116).  I would guess, and Peter will have to confirm, that his
first moment calculations were done for lambda_of with lambda_op=0, then
the latter made non-zero to get the change due to lambda_op.

To be clear, the correction in Peter's results table in unblinding.pdf
of 19+5 Hz is close to what I get with the full kinetics fit, 23.5 Hz.
Here are the (fixed) parameters I used:
Lambda_t = 12.0 Hz
gamma_o = 1.009/2.  (mol. overlap factor)
gamma_p = 1.143/2.  (mol. overlap factor)
phi = 0.0116        (density)
lambda_pf = 0.0074/us
cZ = 0   (imp. concentration)
lambda_OP = 0.08/us
lambda_0 = (lambda from 3-par fit - 664 Hz)
Then the 23.5 Hz is the difference in the result for Lambda_s with
lambda_of fixed to 2.3/us from that with lambda_of fixed to 0.

For the publication we should fit with the full kinetics, with
lambda_of and lambda_op at their central values, since we "know"
these kinetics, with some error due to uncertainty in the parameters.
(Similarly, in my corrections table it doesn't really make sense to
have an entry for "seen mu-p scatters," since they should just be
removed.) Then the effect on Lambda_s from deviations of lambda_of
and lambda_op could be approximated:
DeltaLambda_s = A*(lambda_of - lambda_of_0)*(1 + B*(lambda_op - lambda_op_0)),
where lambda_of_0 and lambda_op_0 are the values we used in the fit,
A and B are constants to be calculated.

I don't understand why you see a slightly lower mol. correction of
20.1 Hz in your MC.  We should figure that one out.

Steven

On Wed, Oct 25, 2006 at 07:08:12PM -0700, Tom Banks wrote:
> 
> Dear all,
> 
> I recently sent out an e-mail describing my worries about the different
> ppmu corrections generated by first moment vs. best-fit methods.  I am
> still trying to understand this issue--in particular, why I observe a
> significant difference between the two methods, while Peter K. claims not
> to have observed any such difference in his simulations.  In the meantime,
> a *second* ppmu-related concern has arisen regarding the two components of
> the molecular correction, as I will explain below.
> 
> Peter K. yesterday described to me how he came by the 19 Hz "ppm
> correction" and the 5 Hz "op correction" which were used at the unblinding
> meeting to extract Lambda_S=730 Hz from our unblinded disappearance rate.
> Basically, the 19 Hz (which, incidentally, Peter tells me should have
> actually been 22 Hz) and 5 Hz numbers come from full-kinetics fits to the
> data performed by Steve, and they are presented on slide 9 in Peter's
> unblinding talk "Corrections and Uncertainties from Kinetics":
> 
>   http://www.npl.uiuc.edu/exp/mucapture/coll/unblind/kinetics.pdf
> 
> Steve performed full kinetics fit scans over a range of different
> lambda_of (~ lambda_ppmu) and different lambda_op values.  The
> corresponding variation in fitted Lambda_S gives us the magnitude of the
> molecular correction, and Steve's results appear to basically confirm
> Peter's MC results in slides 7 & 8 that the ppmu correction=45 ppm (this
> number actually needs some additional manipulations: multiplication by gas
> density adjustment 1.16, as well as multiplication by .455 to convert from
> ppm to Hz, to get 23.8 Hz).
> 
> What worries me is the separation of the molecular correction into a
> "ppmu" component and an "op" component, since Steve's study of the
> lambda_of (mislabeled as "lambda_pf" in the presentation) variation
> already *includes* a nonzero lambda_op component.  It seems to me that the
> 19/22 Hz correction therefore includes the 5 Hz effect which is tacked on
> as a separate "op" correction.
> 
> I ran some MC simulations to investigate the situation, and they appear to
> confirm my interpretation.  For example:
> 
>   Simulation scenario 1:
>     lambda_of = 0
>     lambda_op = 0
>     Mean(best-fit - lambda_0 - Lambda_S) = - 0.5 Hz
> 
>   Simulation scenario 2:
>     lambda_of = 2.3 E+6 Hz
>     lambda_op = 0
>     Mean(best-fit - lambda_0 - Lambda_S) = -14.8 Hz
> 
>   Simulation scenario 3:
>     lambda_of = 2.3 E+6 Hz
>     lambda_op = 8 E+4 Hz
>     Mean(best-fit - lambda_0 - Lambda_S) = -20.1 Hz
> 
> You can see here that the inclusion of lambda_op increases the correction
> from 15 Hz to 20 Hz, the latter of which is roughly the magnitude of
> Peter's proposed "lambda_ppmu" correction.
> 
> Perhaps I have simply misunderstood some aspect of Peter's and Steve's
> work, in which case hopefully someone can set me straight.  I want to
> emphasize again that I realize that we need not resolve this issue before
> Peter's DNP presentation, but I want to raise it now so that we have ample
> time for discussion before publication.
> 
> Regards,
> Tom
> 
> 
>