ludlum measurements, inc
DESCRIPTION
Ludlum Measurements, Inc. User Group Meeting June 22-23, 2009 San Antonio, TX. Counting Statistics. James K. Hesch Santa Fe, NM. Binary Processes. Success vs. Failure Go or No Go Hot or Not Yes or No Win vs. Lose 1 or 0 Disintegrate or not Count a nuclear event or not. Uncertainty. - PowerPoint PPT PresentationTRANSCRIPT
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Ludlum Measurements, Inc.Ludlum Measurements, Inc.
User Group MeetingUser Group Meeting
June 22-23, 2009June 22-23, 2009
San Antonio, TXSan Antonio, TX
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Counting StatisticsCounting Statistics
James K. HeschJames K. Hesch
Santa Fe, NMSanta Fe, NM
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Binary ProcessesBinary Processes
Success vs. FailureSuccess vs. Failure Go or No GoGo or No Go Hot or NotHot or Not Yes or NoYes or No Win vs. LoseWin vs. Lose 1 or 01 or 0 Disintegrate or notDisintegrate or not Count a nuclear event or notCount a nuclear event or not
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UncertaintyUncertainty
Shades of gray – neither black nor whiteShades of gray – neither black nor white How gray is gray?How gray is gray? More black than white, or more white than More black than white, or more white than
black?black?
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Some Familiar Real World ApplicationsSome Familiar Real World Applications
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What is the probability of drawing a What is the probability of drawing a Royal Flush in five cards drawn Royal Flush in five cards drawn
randomly from a deck of 52 cards?randomly from a deck of 52 cards?
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The first card must be a member of The first card must be a member of the set [10, J, Q, K, A] in any of the the set [10, J, Q, K, A] in any of the four suites. Thus it can be any one four suites. Thus it can be any one
of 20 cards.of 20 cards.
3846.052
20p
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The set of valid cards diminishes to The set of valid cards diminishes to four for the second card out of the four for the second card out of the
remaining 51 cards, etc.remaining 51 cards, etc.
48
1
49
2
50
3
51
4
52
20p
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Probability 1 : 649740Probability 1 : 649740
000001359.0!52
)!47)(!4(20p
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Plato’s Real vs. Ideal WorldsPlato’s Real vs. Ideal Worlds
Observed vs. ExpectedObserved vs. Expected Predicting with uncertaintyPredicting with uncertainty Science is inexactScience is inexact Stating the precisionStating the precision “ “+/- 2% at the 95% confidence level”+/- 2% at the 95% confidence level”
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Toss of One DieToss of One Die
Single Die Results Distribution
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
1 2 3 4 5 6
Value
Fre
qu
ency
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Toss of Two DiceToss of Two Dice
Two Dice Results Distribution
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
18.00%
2 3 4 5 6 7 8 9 10 11 12
Value (Sum)
Fre
qu
ency
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Four Tosses of a Pair of DiceFour Tosses of a Pair of Dice
33 1010 55 22 Total = 20Total = 20 Average (Mean) = 20/4 = 5Average (Mean) = 20/4 = 5 Compute the average value by which each Compute the average value by which each
toss in this sample VARIES from the mean.toss in this sample VARIES from the mean.
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Variance = Variance = σσ²²
1
)(2
n
Xx
1
)(1
2
2
n
Xxn
ii
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Toss of Three DiceToss of Three Dice
Three Dice Results Distribution
0
5
10
15
20
25
30
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Value (Sum)
Fre
qu
ency
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Toss of Four DiceToss of Four Dice
Four Dice Results Distribution
0%
2%
4%
6%
8%
10%
12%
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Value (Sum)
Fre
qu
ency
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Probability Distribution FunctionsProbability Distribution Functions
BinomialBinomial PoissonPoisson Gaussian or Normal (the famous bell curve)Gaussian or Normal (the famous bell curve)
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Binomial Distribution FunctionBinomial Distribution Function
kNkp pp
kNk
N
k
NP
)1(
)!(!
!
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Poisson Distribution FunctionPoisson Distribution Function
!)(
x
exp
x
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Sample ExerciseSample Exercise
In a counting exercise where the average In a counting exercise where the average number of counts expected from background number of counts expected from background is 3, what should the minimum alarm set point is 3, what should the minimum alarm set point
be to produce a false alarm probability of be to produce a false alarm probability of 0.001 or less?0.001 or less?
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Lambda = 3Lambda = 3
Discrete Cumulative
x p(x) ∑p(x)
0 0.04979 0.04979
1 0.14936 0.19915
2 0.22404 0.42319
3 0.22404 0.64723
4 0.16803 0.81526
5 0.10082 0.91608
6 0.05041 0.96649
7 0.02160 0.98810
8 0.00810 0.99620
9 0.00270 0.99890
10 0.00081 0.99971
11 0.00022 0.99993
12 0.00006 0.99998
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Poisson Distribution, Lambda = 3Poisson Distribution, Lambda = 3
Poisson Distribution, Lambda = 3
0%
5%
10%
15%
20%
25%
0 1 2 3 4 5 6 7 8 9 10 11 12
Discrete Value, (x)
Pro
bab
ilit
y
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Poisson Distribution, Lambda = 1.25Poisson Distribution, Lambda = 1.25
Poisson Distribution, Lambda = 1.25
0%
5%
10%
15%
20%
25%
30%
35%
40%
0 1 2 3 4 5 6 7 8 9 10 11 12
Discrete values of x
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Gaussian Distribution FunctionGaussian Distribution Function
dxexPx
2
2
2
1)(
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Gaussian Distribution FunctionGaussian Distribution Function
Is a Density Function, or cumulative Is a Density Function, or cumulative probability (as opposed to discreet).probability (as opposed to discreet).
Can use look-up table or Excel functions to Can use look-up table or Excel functions to applyapply
Scale to data by use of Mean and Standard Scale to data by use of Mean and Standard DeviationDeviation
Single-sided confidence – but can be used Single-sided confidence – but can be used to determine two-sided confidence function to determine two-sided confidence function “Erf(x)”.“Erf(x)”.
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Gaussian Distribution Function
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Excel FunctionExcel Function
F(2) = NORMDIST(2, 0, 1, TRUE) = 0.97725F(2) = NORMDIST(2, 0, 1, TRUE) = 0.97725
2 StdDev2 StdDevMean = 0Mean = 0
StdDev of Data = 1StdDev of Data = 1Cumulative = TrueCumulative = True
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If NORMDIST() set to FALSE…If NORMDIST() set to FALSE…
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Controlling False Alarm ProbabilityControlling False Alarm Probability
Determine expected number of background Determine expected number of background counts that would occur in a single count counts that would occur in a single count cycle.cycle.
Determine the StdDev of that valueDetermine the StdDev of that value Set the alarm setpoint a sufficient number of Set the alarm setpoint a sufficient number of
Standard Deviations above average Standard Deviations above average background counts for an acceptable false background counts for an acceptable false alarm probability.alarm probability.
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False Alarm ProbabilityFalse Alarm Probability
NBFA KFP )(1
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How Many Sigmas?How Many Sigmas?
)1(1 NFAB PFK
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In Excel…In Excel…
KKBB = NORMINV((1-P = NORMINV((1-PFAFA)^(1/N),0,1))^(1/N),0,1)
False Alarm ProbabilityFalse Alarm ProbabilityMeanMean
StdDevStdDev
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Computing Alarm SetpointComputing Alarm Setpoint
BBBBA NNKNN
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T
NK
T
N BB
A
Simplify and Divide by TimeSimplify and Divide by Time
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T
RKR BBMINA )(
……almost!almost!
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Final Form:Final Form:
B
BBBA T
R
T
RKR (min)
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Slight detour … 2-sided distributionSlight detour … 2-sided distribution
±σ
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±1 StdDev = 68%
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±2 StdDev = 95%
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±3 StdDev = 99.7%
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In Excel…In Excel…
Two sided distribution…Two sided distribution… ……=2*(NORMDIST(x, 0, 1, TRUE) – 0.5)=2*(NORMDIST(x, 0, 1, TRUE) – 0.5)
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Getting Back to Alarm Setpoint…Getting Back to Alarm Setpoint…
B
BBBSMAXA T
R
T
REffMDAKEffMDAR
)(
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MDA-Driven Alarm SetpointMDA-Driven Alarm Setpoint
Maximum Alarm Set Point
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““Minimum” Count TimeMinimum” Count Time
Solve for T using the simplified equation below, and round Solve for T using the simplified equation below, and round up to a full no. of seconds:up to a full no. of seconds:
Compute a new value for MDA (see next slide) using the Compute a new value for MDA (see next slide) using the resulting “T” as resulting “T” as
As needed, iteratively, add 1 second to the T and As needed, iteratively, add 1 second to the T and recompute MDA until the result is recompute MDA until the result is << the desired MDA the desired MDA
²
EffMDA
REffMDAKRKT BBSBB
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Computing MDAComputing MDA
Start with MDA=1 for the right side of the following Start with MDA=1 for the right side of the following equation, and compute a new value for MDAequation, and compute a new value for MDA
Substitute the new value on the right hand side and repeat.Substitute the new value on the right hand side and repeat. Continue with the substitution/computation until the value Continue with the substitution/computation until the value
for MDA is sufficiently close to the previous value.for MDA is sufficiently close to the previous value.
Eff
T
R
T
REffMDAK
T
R
T
RK
MDA B
BBBS
B
BBB
Eff
T
R
T
REffMDAK
T
R
T
RK
MDA B
BBBS
B
BBB
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Activity Other than MDAActivity Other than MDAAlarm Probability vs. Activity Level
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0% 5% 10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Percent MDA
Ala
rm P
rob
abil
ity
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Approximation of Nuisance AlarmsApproximation of Nuisance Alarms
Nuisance Alarms
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 5% 10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Percent of MDA
Ala
rm P
rob
abil
ity
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With Extended Count TimeWith Extended Count Time
Nuisance Alarms
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 5% 10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
Percent of MDA
Ala
rm P
rob
abil
ity
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A Look at Q-PASSA Look at Q-PASS1000 cps Background
0
200
400
600
800
1000
1200
1400
1600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Elapsed Time
Clean High Alarm