Basic Counting Statistics

Basic Counting Statistics

Stat

istics

W. Udo Schröder, 2009

2

Stochastic Nuclear Observables2 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 (e < 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): sx

2 = <x2- <x2> >

Uncertainty and Statistics

Nuclear systems: quantal wave functions yi(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

( , ) | | 1

y

y

y

ii

ii i

t t

dP x t x t idx

NormalizationdP x tP t dx dx x tdx

Transition between states M E

dP x t x e e state disappearsdx

1

2

Partial probability rates 12 for disappearance (decay of 12) can vary over many orders of magnitude no certainty statistics

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istics


4

The Normal Distribution

10 8 6 4 2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

G( )x

G( )s

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

ss

s s

FWHM x

XX

x

x xP x

Continuous function or discrete distribution (over bins)

Normalized probability

1

1

2

22

1( ) exp22 ss

X

x

X

x xP x x dx

2sx

P(x)

P(x)

x

x

<x>

Stat

istics


5

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

s

s

Measured by ensemble sampling expectation values + uncertainties

Sample (Ensemble) = Population instant 236U (0.25mg) source, count a particles emitted during N = 10 time intervals (samples @1 min). =??

2

1

. : / 5

(35496 59) n

9

i: m

s s

pop

n nStd deviation n N

R n nesultSlightly different from sample to sample

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istics


6

Sample Statistics

0 2 4 6 8 10

0.1

0.2

0.3

Normal (Gaussian) Probability

101 105.5 1100

5

10Normally Distributed Events

10

0

xm i

x0

x0 s

x0 s

110101 m300 304.5 3090

5


10

0

xm i

x0

x0 s

x0 s

309300 m

2

22

1( ) exp22

pop

XX

x xP x

vv

x

P(x) Assume true population distribution for

variable x

with true (“population”) mean <x>pop = 5.0, nx =1.0

401 405.5 4100

5


10

0

xm i

x0

x0 s

x0 s

410401 m

Mean arithmetic sample average <x> = (5.11+4.96+4.96)/3 = 5.01Variance of sample averages s2=s2 = [(5.11-5.01)2+2(4.96-5.01)2]/2 = 0.01 s = 0.0075 sx

2 = 0.0075/3 = 0.0025 sx = 0.05 Result: <x>pop 5.01 ± 0.05

<x>-s

<x> =5.11 s =1.11 <x> =4.96 s =0.94

<x>+s<x>

<x> =4.96 s =1.23

3 independent sample measurements (equivalent statistics):

x

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istics


7

Example of Gaussian Population

0 25 500

5


10

0

xm i

x0

x0 s

x0 s

500 m

0 5 100

10

2050MC Events in 10 x bins

ii FRAME

0 5 100

5


xm i

x0

x0 s

x0 s

m

xaver 5.0601( ) xvar 0.8726( )

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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istics


8

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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istics


9

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( )

FHGIKJ 1b g

! ( 1)!( )! 1

N N N m Nm N m mm

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istics


10

Moments and Limits

( ) 1 N mmbinomial

NP m p pm

0 0

1 ( ) 1

N N

N mmbinomial

m m

NormalizationN

P m p pm

Distributions for N=30 and p=0.1 p=0.3Poisson Gaussian

2 (1 )

(1 ) 1

s

s

m

m

Mean and variance

m N p and N p p

N p pm N p N

0 5 10 15 200

0.1

0.2

0.3Binomial Distributions N=30

0.236

0

Pb N m 0.1( )

Pb N m 0.3( )

200 m

222

1lim ( ) exp22binomialN mm

x mP m

ss

Probability for m “successes” out of N trials, individual probability p

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istics


11

Poisson Probability Distribution

0 5 10 15 200

0.2

0.4

0.6

0.8Poisson Distributions

0.607

0

Pp 0.5 m( )

Pp 3 m( )

Pp 5 m( )

Pp 10 m( )

200 m

P m emPoisson

m

( , )!

Probability for observing m events when average is <m> =

m=0,1,2,… = <m> = N·p and s2 = is the mean, the average number of successes in N trials.Observe N counts (events) uncertainty is s= √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 ( , ) ( , )

binomial Poissonpand N

P N m P m

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istics


12

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 10

N mg mgg

np

NProbability for decay decay rate per nucleus

p

corresponds to half life t a

Small probability for process, but many trials (n0 = 6.38·1017)

0< n0· < ∞Statistical process follows a Poisson distribution: n=“random” Different statistical distributions: Binomial, Poisson, Gaussian

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istics


13

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 g: N = 1015 nuclei (=trials for decay)

How many will decay(= activity ) ? =< >= N·p = 8.2·10+5 s-1

Count rate estimate < >=d<N>/dt = (8.2·10+5 ± 905) s-1

estimatedProbability for m actual decays P (,m) =

55 8.52 10(8.52 10 )( , ) ! !m m

Poissone eP m

m m

N

N

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istics


14

Functions of Stochastic Variables

Random independent variable sets {N1}, {N2},….,{Nn } corresponding variances s1

2, s22,….,sn

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 22, 3,.. 1, 3,.. 1, 2,.., 1

....

| | .. |

f nn

N N N N N N Nn

ff fN N N

ff f

s s s s

Further terms if Ni not independent ( correlations)Otherwise, individual component variances (f)2 add.

1

1 ,...,,...,

nn N N

ff N N

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istics


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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+B

N(c1)=B1, N(c2)=B2, Linear background <B> = n(B1+B2)/2Peak Area A:

12 1 2

12 12

2

/ 2/ 4A

A N n B B

N B Bns

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 2s s

Stat

istics


16

Confidence Level

2

22

2(| | ) exp22

s

s

x

popx

x xP x x dx CL

( 1 ) 68.3% ( 2 ) 95.4%( 3 ) 99.7% s s s

CL CLCL

Assume normally distributed observable x:

2

22

1( ) exp22

pop

poppop

x xP x

vv

Sample distribution with data set observed average <x> and std. error s approximate population. Confidence level CL (Central Confidence Interval):

With confidence level CL (probability in %), the true value <xpop> differs by less than = ns from measured average.Trustworthy exptl.

results quote ±3s error bars!

1s

2s

3s

Measured Probability

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istics


17

Setting Confidence LimitsExample: Search for rare decay with decay rate observe no counts within time t. Decay probability law

dP/dt=-dN/Ndt= exp {- ·t}. P(t) = symmetric in and t

0

0

0

00

0 0

0 | 1

( ) 1

1 1ln[1 ( )] ln[1 ] 0

:

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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istics


18

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. 2

11 22

( ,.., ; )1( ,.., ; ) exp22 ss

i m ii m

ii

y f c c xP c c x for every

data point

1 11

2

2211

( ,.., ) ( ,.., ; )

1 exp22 ss

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}: ∂c2/∂ci = 0

2 212

1 2 21 1

( ,.., ; ),.., :c

s s

N Ni i m i

mi ii i

y y f c c xc c

When is c2 as good as can be?

Stat

istics


19

Minimizing c2

2 22

2 21 1

, :cs s

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, si}

Minimize:

22

2 21 1

20 ,N Ni i i i

i ii i

y a bx y a bxa b

a ac

s s

221

20 ,N i i i

i i

x y a bxa b

bc

s

2 2 21 1 1

1N N Ni i

i i ii i ib x ya

s s s

2

2 2 21 1 1

N N Ni i i i

i i ii i ia xbx x y

s s s

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

s ss s

Equivalent to solving system of linear equations

Stat

istics


20

Distribution of Chi-SquaredsDistribution of possible c2 for data sets distributed normally about a theoretical expectation (function) with ndof degrees of freedom:

22 12 22 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

c

c

c

cc

s

c c

(Stirling’s formula)

Reduced c2:2 2 2

2

( 1)

0 1.5 50%

r dof

r

n N mFor

Confidence

c c c

c

22

( , ) ( )dof ndofP n P x dxc

c

P u n 1

2 n2

u2

n2

1

expu

2

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

<c2>ndof=5

u:=c2

Should be P 0.5 for a reasonable fit

Stat

istics


21

CL for c2-Distributions

1-CL

Stat

istics


22

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 s s s

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

s s s

s s

s s s

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

s ss sa s

s

s s s

x

y correlated P(x,y)

a ( , )s xy x x y y P x y dxdy covariance

Stat

istics


23

Correlations in Data Sets

22

: 1 1

;

1 1:1

ss s

s

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 c2 surface

ci

cjuncorrelated c2 surface

ci

cj

correlated c2 surface

({ }) ( )

({ }) ( )

unc i i

i

corr i ii

P c P c

P c P c

Stat

istics


24

Multivariate Correlations

ci

cj

initial guess

search path

Smoothed c2 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,….

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Basic Counting Statistics