Mirion Technologies GEM-5 - Background Variance Ratio

Background Rate
Gem-5 User’s Manual 277
To optimize the variance calculation, the standard equation for variance can be
rewritten as:
Equation 20 Optimized Variance
1n
))R(n())R((
2
B
n
1i
2
i
2
B
−
×−
=
∑
−
σ
(20)
...where
i
i
i
T
C
R =
.
Using this formula, an algorithm for incrementally computing the variance after the
first data point is dropped and a new one added can be developed. This algorithm
assumes that
∑
2
i
R
has been cached (note that other values may be cached, such as,
which is used in the background variance test).
Background average starts with one update. This running total accumulates until
∑
i
T
is greater than or equal to the “Background Average Period”.
Step change in background (K Delta test) or clear all faults will restart average with
one update. New updates are added until
∑
i
T
reaches “Background Average
Period”. The oldest point is then subtracted out before adding a new one. The oldest
point is also subtracted out if it is about to be overwritten in background history
buffer which holds up to 1024 updates.
Background Variance Ratio
Each time the background rate is calculated a variance ratio is also calculated so the
background rate value can be compared to its theoretical value:
Equation 21 Variance Ratio
B
B
2
B
R
Rt
tioVarianceRa
−×
=
σ
(21)
…where
t
is the average actual data collection interval,
2
B
σ
is the calculated
variance in the background rate and
B
R
is the background rate. If
B
R
is zero, then
the ratio is explicitly set to zero.
Note on derivation: In a Poisson distribution, the average number of counts per
background interval,
B
, should equal the variance in
B
; i.e.,
B
2
B
= σ
. Since it is
the rates that are being calculated, the original terms can be expressed as rates, i.e.,
tRB ×=
B
and
22
B
2
B
t×= σσ
.