A taste of statisticsA taste of statistics

Normal error (Gaussian) distribution most important in statistical analysis of data, describes the distribution of random observations for many experiments

Poisson distribution generally appropriate for counting experiments related to random processes (e.g., radioactive decay of elementary particles)

Statistical tests: 2 and F-test

Statistical errors and contour plots

Low-count regime: the C-statistic

2

The Gaussian (normal error) The Gaussian (normal error) distributiondistribution

Casual errors are above and below the “true” (most “common”) valuebell-shape distribution if systematic errors are negligible

σσ=standard deviation =standard deviation of the functionof the function

ГГ=half-width at =half-width at half maximum=1.17half maximum=1.17σσ

μμ=mean=mean

FWHMFWHM

3

Gaussian probability functionGaussian probability function

22 2σ/xe− function centered on 0

function centered on μ

normalization factor, so that ∫f(x) dx=1

4

Pro

bab

ilit

yP

rob

ab

ilit

y

σσ

σσ

5

The Poissoin distributionThe Poissoin distribution

Describes experimental results where events are counted and the uncertainty is not related to the measurement but reflects the

intrinsically casual behavior of the process (e.g., radioactive decay of particles, X-ray photons, etc.)

!/)( υμυ υμμ

−=eP

μμ>0: main parameter of the Poisson distribution>0: main parameter of the Poisson distributionνν=observed number of events in a time interval (frequency of events)=observed number of events in a time interval (frequency of events)

∑∑∞

=

−∞

=

===00 υ

υμ

υ

μ μυμυυυυ !/)( eP

μμ=average number of expected events if the experiment is =average number of expected events if the experiment is repeated many timesrepeated many times

average average number number of eventsof events

6

μυμμυ

μυσ

μυ

υ=−=

=⟩−⟨=

−∞

=∑ e

!)(

)(

2

0

22

the Poisson distribution with average counts=the Poisson distribution with average counts=μμ has has standard deviation standard deviation √√μμ

μμ » : » : the Poisson distribution is approximated by the the Poisson distribution is approximated by the Gaussian distributionGaussian distribution

expectation value of the square of the deviations

7

defined by only one parameter defined by only one parameter μμ

8

Statistical tests: Statistical tests: 22

Test to compare the observed distribution of the results with that expected

2

1

2

∑=

−=

n

k k

kk

EEO )(

χ

OOkk=observed values =observed values

EEkk=expected values=expected values

K=number of intervalsK=number of intervals

n≤2

the observed and expected distributions are similar

9

Degrees of freedomDegrees of freedom

Degrees of freedom (d.o.f.)=number of observed data – number of parameters computed from the data and used in the calculation

d.o.f.=n-cd.o.f.=n-cwhere

nn=number of data (e.g., spectral bins)cc=number of parameters which must be computed from the data to

obtain the expected Ek

X-ray spectral fitsX-ray spectral fitsd.o.f.=number of spectral data points – number of free parameters d.o.f.=number of spectral data points – number of free parameters

10

d.o.f.=d.o.f.=channels (PHA bins) – channels (PHA bins) – free parametersfree parameters

.../ fod22

= reduced

11

12

F-testF-test

If two statistics following the 2 distribution have been determined, the ratio of the reduced chi-squares is distributed according to

the F distribution

2

2

2

1

2

121

υ

υυυ/

/),;( =fPf

Example: improvement to a spectral fit due to the Example: improvement to a spectral fit due to the assumption of a different model, with possibly assumption of a different model, with possibly additional terms additional terms see the F-test tables for see the F-test tables for the corresponding probabilities (specific command in XSPEC)the corresponding probabilities (specific command in XSPEC)

k/2

Δ∝with k=number of additional with k=number of additional terms (parameters)terms (parameters)

An application of the F-test

low F value likely low significance

14

Errors within XSPEC: the contour Errors within XSPEC: the contour plotsplots

68%68%90%90%

99%99%

contour plots:contour plots: show the statistical show the statistical uncertainties related uncertainties related to two parametersto two parameters

CONFIDENCE INTERVALSCONFIDENCE INTERVALSConfidence sigma (Delta)chi2

68.3% 1.0 1.00 90.0% 1.6 2.71 95.5% 2.0 4.00 99.0% 2.6 6.63 99.7% 3.0 9.00 see Avni (1976)

Low-counting statistic regime

The fit statistic routinely used is referred to as the 2 statistic

€

S = (Si − Bii

∑ ts / tb −mits)2 /((σ S )i

2 + (σ B )i2)

where Si = src observation counts in the I={1,…,N} data bins withexposure tS, Bi = background counts with exposure tB and mi = model predicted count rate;(σS)2 and (σB)2 = variance on the src and background counts, typically estimated by Si and Bi

BUTBUTthe 2 statistic fails in low-counting regime

(few counts in each data bin)

16

Possible solution: to rebin the data so that each bin contains a large enough number of counts

BUTLoss of information and dependence

upon the rebinning method adopted

Viable solution: to modify S so that it performs better in low-count regime

Estimate the variance for a given data bin by the average counts from surrounding bins

(Churazov et al. 1996)

BUTneed for Monte-Carlo simulations to support the result

17

The Cash statistic

Construct a maximum-likelihood estimator based on the Poisson distribution of the detected counts

(Cash 1979; Wachter et al. 1979) in presence of background counts implemented into XSPEC

Spectrum fitted using the C-stat and then rebinned just for presentation

purposes