[Halld-physics] correlations in neutral shower parameters

[Mark M. Ito]

Richard et al.,

A general comment: it seems some of these correlations are weak and in 
fact we may never have sensible values for them. If we are going for a 
compact format, we might think about initially dropping the off-diagonal 
elements that are only non-zero in principle.

I'm worried about keeping a significant amount of zero-valued data as a 
place-holder for future use and having that future never arrive.

As an extreme version of this, you can imagine a near-diagonal 
representation where none of the off-diagonal elements are kept. Don't 
know if this is actually possible. It likely would mean a 
parametrization aligned with detector geometry rather than an abstract 
global coordinate system.

For tracking, the off-diagonal elements arise naturally in the process 
of fitting and some of the correlations are certainly significant. So 
the values are both non-zero and meaningful; they should be kept.

   -- Mark

On 08/18/2011 06:19 PM, Richard Jones wrote:
> In laying out the dst format for MC studies, I claimed in an earlier 
> message that the error matrix used to store the errors on neutral hit 
> parameters only needs to contain 3 (bcal) or 2 (fcal) off-diagonal 
> elements.  Our argument follows below.  The starting point for this is 
> a single cluster reconstructed in the bcal or fcal.  No vertex 
> hypothesis has yet been applied to this hit, nor has a momentum vector 
> been assigned to it.  It is just a neutral cluster in a calorimeter.
>
>    1. *bcal* - the set of measured hit parameters which are most
>       loosely coupled in terms of the raw measurements are
>       (r,phi,z,E,t).  A transformation can be used to convert into any
>       other preferred set of hit parameters.  Correlations are (r,z) ,
>       (r,E) , and (z,E).
>           * phi is decoupled from all of the others.
>           * r is directly measured through the radial profile, but
>             couples to z and E implicitly.
>           * z is independently measured using timing, but is
>             implicitly correlated with r through fluctuations in the
>             energy deposition for oblique showers, and with E through
>             the position of the shower maximum, and explicitly with E
>             through the attenuation correction.
>           * E correlates implicitly to r,z through shower depth
>             variations, and explicitly to z through the attenuation
>             correction.
>           * the t of the shower maximum correlates with r,z (and
>             through them with E).  However this correlation is
>             trivial, and will be taken out when the time is projected
>             back from the shower maximum to a reference vertex.  Since
>             the speed of light is independent of E, this correlation
>             of t with r,z is trivial and need not be carried along
>             with the shower.
>    2. *fcal* - the set of measured hit parameters which are most
>       loosely coupled in terms of the raw measurements are (x,y,E,t). 
>       Note that the depth z of the shower maximum is not measured in
>       the fcal, so it does not belong in the list, being a pure
>       function of E.  Correlations are (x,y) , (x,E) , and (y,E).
>          1. x,y are implicitly correlated with each other by the shape
>             of shower deposition in the fcal, and with E through
>             shower depth.
>          2. as for the fcal, the t of the shower maximum is trivially
>             correlated with E,x,y.  I argue that this correlation is
>             trivial and need not be carried along with the shower
>             object for the same reasons as given above under the bcal
>             heading.
>
> Thus, for the bcal we must keep 3 off-diagonal error matrix elements 
> per shower.  For the fcal there are also 3 non-trivial off-diagonal 
> elements, but two of them are related to each other in a simple way:
>
>                x * sigma(y,E) = y * sigma(x,E)
>
> So strictly speaking there are only 2 off-diagonal error matrix 
> elements per shower that must be kept in the fcal.
>
> -Richard J.
>
>
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