Stochastic Nuclear Observables...How many ways can mevents be chosen out of N ? Binomial coefficient...
Transcript of Stochastic Nuclear Observables...How many ways can mevents be chosen out of N ? Binomial coefficient...
Basic
Counting
Statistics
W. Udo Schröder, 2009
Stat
isti
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Stochastic Nuclear Observables
2 sources of stochastic observables x in nuclear science:
1) Nuclear phenomena are governed by quantal wave functions and inherent statistics2) Detection of processes occurs with imperfect efficiency (ε < 1) and finite resolution distributing sharp events x0 over a range in x. Stochastic observables x have a range of values with frequencies determined by probability distribution P(x).characterize by set of moments of P
<xn> = ∫ xn P (x)dx; n 0, 1, 2,…
Normalization <x0> = 1. First moment (expectation value) of P:
E(x) = <x> = ∫xP (x)dx
second central moment = “variance” of P(x):σx
2 = <x2- <x2> >
Uncertainty and Statistics
Nuclear systems: quantal wave functions ψi(x,…;t)
(x,…;t) = degrees of freedom of system, time.
Probability density (e.g., for x, integrate over other d.o.f.)
( )
( )
( )12
12
2
2
212 12
2121
( , ) | , | 1,2,....
( , )( ) | , | 1
1 2, | | ( )2
( , ) | | 1λ
ψ
ψ
ρπ
ψ
+∞ +∞
−∞ −∞
Γ− ⋅ − ⋅
= =
= = =
→ Γ ≈
−= ∝
∫ ∫
ii
ii i
t t
dP x t x t idx
NormalizationdP x tP t dx dx x t
dx
Transition between states M E
dP x t x e e state disappearsdx
1
2
Partial probability rates λ12 for disappearance (decay of 12) can vary over many orders of magnitude no certainty statistics
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The Normal Distribution
10 8 6 4 2 0 2 4 6 8 100
0. 0 5
0. 1
0. 1 5
0. 2
G( )x
G( )σ
x
30 38 46 54 62 70
0.02
0.04
0.06
0.08
Normal (Gaussian) Probability1 10 1−⋅
10 8−
G xn( )
7030 xn
( )2
22
1( ) exp22
2 2ln 2.352
σπσ
σ σ
− = ⋅ −
= =Γ ⋅⋅FWHM x
XX
x
x xP x
Continuous function or discrete distribution (over bins)
Normalized probability
( )1
1
2
22
1( ) exp22 σπσ −∞
− < = ⋅ −
∫X
x
X
x xP x x dx
2σx
P(x)
P(x)
x
x
<x>
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Experimental Mean Counts and Variance
1
2 2 2
1
22 2
1
:1 ( )
1 ( )1
(" ")
1 ( )( 1)
N
i populationi
N
ii
population
N
n ii
Average count n in a sample
n n n unknownN
Variance of n in the individual samples
s n nN
Variance error of thesample average n n
s n nN N N
σ
σ
=
=
=
= ⋅ ≠
= = ⋅ −−
≠
= = ⋅ −−
∑
∑
∑
Measured by ensemble sampling expectation values + uncertainties
Sample (Ensemble) = Population instant
236U (0.25mg) source, count α particles emitted during N = 10 time intervals (samples @1 min). λ =??
2
1
. : / 5
(35496 59) n
9
i: m
σ σ−
= ≈ < >
≈ = ±
=
pop
n nStd deviation n N
R n nesultSlightly different from sample to sample
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Sample Statistics
0 2 4 6 8 10
1
2
3
101 105.50
5
10Normally Distributed Event
10
0
m i,
σ+
σ−
300 304.50
5
10Normally Distributed Event
10
0
i,
σ+
σ−
( )2
22
1( ) exp22
pop
XX
x xP x
vvπ
− = ⋅ −
x
P(x) Assume true population distribution for variable x
with true (“population”) mean <x>pop = 5.0, νx =1.0
40 1 40 5.50
5
10Normally Distributed Events
10
0
m i,
0
0 σ+
0 σ−
Mean arithmetic sample average <x> = (5.11+4.96+4.96)/3 = 5.01
Variance of sample averages s2=σ2 = [(5.11-5.01)2+2(4.96-5.01)2]/2 = 0.01 s = 0.0075
σx2 = 0.0075/3 = 0.0025 σx = 0.05 Result: <x>pop 5.01 ± 0.05
<x>-σ
<x> =5.11 σ =1.11 <x> =4.96 σ =0.94
<x>+σ<x>
<x> =4.96 σ =1.23
3 independent sample measurements (equivalent statistics):
x
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Example of Gaussian Population
0 25 500
5
10Normally Distributed Events
10
0
xm i,
x0
x0 σ+
x0 σ−
500 m
0 5 100
10
2050MC Events in 10 x bins
ii FRAME:=
0 5 100
5
10Normally Distributed Events
xm i,
x0
x0 σ+
x0 σ−
m
0 5 100
10
2010MC Events in 10 x bins
xm xm
Sample size makes a difference ( weighted average)
n = 10 n = 50
The larger the sample, the narrower the distribution of x values, the more it approaches the true Gaussian (normal) distribution.
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Central-Limit Theorem
Increasing size n of samples: Distribution of sample means Gaussian normal distrib. regardless of form of original (population) distribution.
The means (averages) of different samples in the previous example cluster together closely. general property:
The average of a distribution does not contain information on the shape of the distribution.
The average of any truly random sample of a population is already somewhat close to the true population average.
Many or large samples narrow the choices: smaller Gaussian width Standard error of the mean decreases with incr. sample size
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Integer random variable m = number of events, out of N total, of a given type, e.g., decay of m (from a sample of N ) radioactive nuclei, or detection of m (out of N ) photons arriving at detector. p = probability for a (one) success (decay of one nucleus, detection of one photon)
Choose an arbitrary sample of m trials out of N trialspm = probability for at least m successes(1-p)N- m = probability for N-m failures (survivals, escaping detection)Probability for exactly m successes out of a total of N trials
How many ways can m events be chosen out of N ? Binomial coefficient
Total probability (success rate) for any sample of m events:
Binomial Distribution
( )( ) 1 N mmP m p p −∝ −
P mNm
p pbinomialm N m( ) =
FHG
IKJ − −1b g
! ( 1)!( )! 1
N N N m Nm N m mm
− + ⋅ ⋅ ⋅= = − ⋅ ⋅ ⋅
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Moments and Limits
( )( ) 1 N mmbinomial
NP m p p
m−
= −
( )0 0
1 ( ) 1 −
= =
= = −
∑ ∑
N NN mm
binomialm m
NormalizationN
P m p pm
Distributions for N=30 and p=0.1 p=0.3Poisson Gaussian
2 (1 )
(1 ) 1
σ
σ
= ⋅ = ⋅ −
⋅ −=
⋅
m
m
Mean and variancem N p and N p p
N p pm N p N
0 5 10 15 200
0.1
0.2
0.3Binomial Dis tributions N=30
0.236
0
Pb N m, 0.1,( )
Pb N m, 0.3,( )
200 m
( )222
1lim ( ) exp22
binomialNmm
x mP m
σπσ→∞
− = ⋅ −
Probability for m “successes” out of N trials, individual probability p
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Poisson Probability Distribution
0 5 10 15 200
0.2
0.4
0.6
0.8Po iss on Dist ribu tions
0.607
0
Pp 0.5 m,( )
Pp 3 m,( )
Pp 5 m,( )
Pp 10 m,( )
200 m
P m emPoisson
m
( , )!
µ µ µ
=⋅ −
Probability for observing m events when average is <m> = µ
m=0,1,2,…
µ = <m> = N· p and σ2 = µ
is the mean, the average number of successes in N trials.
Observe N counts (events) uncertainty is σ= √µ
Unlike the binomial distribution, the Poisson distribution does not depend explicitly on p or N !
For large N, p:
Poisson Gaussian (Normal Distribution)
Results from binomial distribution in the limit of small p and large N (N·p > 0)
0lim ( , ) ( , )µ
→→∞
=binomial Poissonpand N
P N m P m
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Moments of Transition Probabilities23
17
4 114 1
17
14 1
71/ 2
6.022 10: 0.25 0.25 6.38 10236
3.5946 10 min 5.6362 10 min6.38 10
( ) :5.6362 10 min
" " 2.34 10λ
−− −
− −
⋅= ⋅ = ⋅
⋅= = = ⋅
⋅
= = ⋅
= ⋅
N mg mgg
np
NProbability for decay decay rate per nucleus
pcorresponds to half life t a
Small probability for process, but many trials (n0 = 6.38·1017)
0< n0·λ < ∞Statistical process follows a Poisson distribution: n=“random” Different statistical distributions: Binomial, Poisson, Gaussian
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Radioactive Decay as Poisson Process
Slow radioactive decay of large sample Sample size N » 1, decay probability p « 1, with 0 < N·p <
137Cs unstable isotope decayt1/2 = 27 years p = ln2/27 = 0.026/a = 8.2·10-10s-1 0Sample of 1 µg: N = 1015 nuclei (=trials for decay)
How many will decay(= activity µ) ? µ =< >= N·p = 8.2·10+5 s-1
Count rate estimate < >=d<N>/dt = (8.2·10+5 ± 905) s-1
estimated
Probability for m actual decays P (µ,m) =
55 8.52 10(8.52 10 )( , )! !
m m
Poissone eP m
m m
µµµ− − ⋅⋅ ⋅ ⋅
= =
N
N
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Functions of Stochastic Variables
Random independent variable sets {N1}, {N2},….,{Nn } corresponding variances σ1
2, σ22,….,σn
2
Function f(N1, N2,….,Nn) defined for any tuple {N1, N2,….,Nn}Expectation value (mean) Gauss’ law of error propagation:
( ) ( ) ( )
1 22 2 22 2 21 2
1 2
2 2 2
2, 3,.. 1, 3,.. 1, 2,.., 1
....
| | .. |
f nn
N N N N N N Nn
f f fN N N
f f f
σ σ σ σ
−
∂ ∂ ∂ = + + + ∂ ∂ ∂
∆ + ∆ + + ∆
Further terms if Ni not independent ( correlations)Otherwise, individual component variances (∆f)2 add.
( )1
1 ,...,,...,
nn N N
f f N N=
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Example: Spectral Analysis
Background B
Peak Area A
B1 B2
Analyze peak in range channels c1– c2: beginning of background left and right of peak n = c1 – c2 +1.
Total area = N12 =A+BN(c1)=B1, N(c2)=B2,
Linear background <B> = n(B1+B2)/2
Peak Area A:
( )( )
12 1 2
12 12
2
/ 2
/ 4A
A N n B B
N B Bnσ
= ⋅
⋅ +
−
+
+
=
( )1 2 11 2 22 1:N N N NN N N NN ± ± ±± = = ± +
Adding or subtracting 2 Poisson distributed numbers N1 and N2:Variances always add
1 2σ σ± ±
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Confidence Level
( )2
22
2(| | ) exp22
δ
σπδ
σ
< >+
< >
− < > − < > ≈ ⋅ − =
< ∫x
popx
x xP x x dx CL
( 1 ) 68.3% ( 2 ) 95.4%( 3 ) 99.7%δ σ δ σδ σ
= = = == =
CL CLCL
Assume normally distributed observable x:
( )2
22
1( ) exp22π
− = ⋅ −
pop
poppop
x xP x
vv
Sample distribution with data set
observed average <x> and std. error σ approximate population. Confidence level CL (Central Confidence Interval):
With confidence level CL (probability in %), the true value <xpop> differs by less than δ = nσ from measured average. Trustworthy exptl.
results quote ±3σerror bars!
1σ±
2σ±
3σ±
Measured Probability
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Setting Confidence Limits
Example: Search for rare decay with decay rate λ, observe no counts within time ∆t. Decay probability law
dP/dt=-dN/Ndt= exp {- λ·t}. P(λ,t) = symmetric in λ and t
( )
0
0
0
00
0 0
0 | 1
( ) 1
1 1ln[1 ( )] ln[1 ] 0
:
λ λ
λλλ
λ λ
λ λ λ
λ λ λ
∞− ⋅∆ − ⋅∆
− ⋅∆− ⋅∆
∆ = ∆ ⋅ ∆ ⋅ =
≤ = ∆ ⋅ = −
− −= ⋅ − ≤ = ⋅ − >
∆ ∆
=
∫
∫
t t
tt
P t e with t e d
P t e d e
P CLt t
CL
normalized P
normalized P
Upper limit
Higher confidence levels CL (0 CL 1) larger upper limits for a given time ∆t inspected. Reduce limit by measuring for longer period.
no counts in ∆t
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Maximum LikelihoodMeasurement of correlations between observables y and x: {xi,yi| i=1-N} Hypothesis: y(x) =f(c1,…,cm; x). Parameters defining f: {c1,…,cm}ndof=N-m degrees of freedom for a “fit” of the data with f.
( )21
1 22
( ,.., ; )1( ,.., ; ) exp22 σπσ
− = ⋅ −
i m ii m
ii
y f c c xP c c x for every
data point
( )
1 11
2
2211
( ,.., ) ( ,.., ; )
1 exp22 σπσ
=
==
= =
∆ = ⋅ −
∏
∑∏
N
m i mi
N Ni
ij ij
P c c P c c x
y
Maximize simultaneous probability
Minimize chi-squared by varying {c1,…,cm}: ∂χ2/∂ci = 0
( ) ( ) ( )2 212
1 2 21 1
( ,.., ; ),.., :χ
σ σ= =
∆ −= =∑ ∑
N Ni i m i
mi ii i
y y f c c xc c
When is χ2 as good as can be?
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Minimizing χ2
( ) ( ) ( )2 22
2 21 1
, :χσ σ= =
∆ − −= =∑ ∑
N Ni i i
i ii i
y y a bxa b
Example: linear fit f(a,b;x) = a + b·x to data set {xi, yi, σi}
Minimize:
( ) ( ) ( )22
2 21 1
20 ,
N Ni i i i
i ii i
y a bx y a bxa b
a aχ
σ σ= =
− − − −∂ ∂= = = −
∂ ∂∑ ∑
( ) ( )22
1
20 ,
Ni i i
i i
x y a bxa b
bχ
σ=
− −∂= = −
∂∑
2 2 21 1 1
1N N Ni i
i i ii i i
bx y
aσ σ σ= = =
+ =∑ ∑ ∑
2
2 2 21 1 1
N N Ni i i i
i i ii i i
ax
bx x yσ σ σ= = =
+ =∑ ∑ ∑
11 12 1
21 22 2
aad d cd db c
b+ =
+ =11 12
21 22
d dD
d d=
1 12 11 1
2 22 21 22
2 22 2
1 1
1 1 1ia b
i i
c d d ca b
c d d cD D
xD D
σ σσ σ
= =
= =∑ ∑
Equivalent to solving system of linear equations
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Distribution of Chi-Squareds
Distribution of possible χ2 for data sets distributed normally about a theoretical expectation (function) with ndof degrees of freedom:
( )( )
( )
22 12 2
2 2
2
2
1 2
2
2
2
e( )
2 2
( ) 1 !
2.507 (1 0.0 )
2 1
833 /
d
ndof
n ndof d
of dof dof
ofdof
n n
P d d
n n n
n
n n
e n n
χ
χ
χ
χχ
σ
χ χ
− −
− −
=Γ
Γ = − =
= +
= =
(Stirling’s formula)
Reduced χ2:2 2 2
2
( 1)
0 1.5 50%
r dof
r
n N mFor
Confidence
χ χ χ
χ
= = − −
≤ < → ≥
2
2( , ) ( )dof ndof
P n P x dxχ
χ∞
= ∫
0 2 4 6 8 100
0.1
0 .2
0 .3
0 .4Chi-Squared Distribution
P u 1,( )
P u 2,( )
P u 3,( )
P u 4,( )
P u 5,( )
u
<χ2>ndof=5
u:=χ2
Should be P 0.5 for a reasonable fit
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CL for χ2-Distributions
1-CL
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Correlations in Data Sets
Correlations within data set. Example: yi small whenever xi small
x
yuncorrel. P(x,y)
( ) ( )2 2
2 22 2
( , ) ( ) ( )
1 exp2 24 σ σπ σ
= ⋅ =
− − = ⋅ − +
unc
x yx
P x y P x P y
x x y y
( ) ( ) ( )( )( )
2 2 2
2 22 2
2 2 2
1( , )2
2exp
2
π σ σ σ
σ σ
σ σ σ
= ⋅−
− + − − − − ⋅ − −
corr
x y xy
x y
x y xy
P x y
x x y y x x y y
( )
( )
22 22 221cot
2 4
1 1
x yx yxy
xy
xy xy x y xyr correlation coefficient r
σ σσ σα σ
σ
σ σ σ
−−
= + +
= − ≤ ≤
x
ycorrelated P(x,y)
α( )( ) ( , )σ = − −∫xy x x y y P x y dxdy covariance
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Correlations in Data Sets
( )( )( ) ( )22
: 1 1
;
1 1:1
σσ σ
σ
− −= − ≤ ≤ +
− ⋅ −
=
= − ⋅ −
∑∑ ∑
∑ ∑ ∑
i i j jij ij
i i j j
ijij
i j
ij i j i j
c c c cr r
c c c c
r
c c c c covariancen n
uncertainties of deduced most likely parameters ci (e.g., a, b for linear fit) depend on depth/shallowness and shape of the χ2 surface
ci
cjuncorrelated χ2 surface
ci
cj
correlated χ2
surface
({ }) ( )
({ }) ( )
=
≠
∏
∏
unc i ii
corr i ii
P c P c
P c P c
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Multivariate Correlations
ci
cj
initial guess
search path
Smoothed χ2 surface Different search strategies:
Steepest gradient, Newton method w or w/odamping of oscillations,
Biased MC:Metropolis MC algorithms,
Simulated annealing (MC derived from metallurgy),
Various software packagesLINFIT, MINUIT,….