Modern Methods of Data Analysis - WS 07/08

Transcription

1 Modern Methods of Data Analysis Lecture VIa ( ) Contents: Uncertainties (II): Re: error propagation Correlated uncertainties Systematic uncertainties

2 Re: Error Propagation (I) x = Vi,j and µi known y(x) is function of first order Taylor expansion...

3 Re: Error Propagation (II)

4 Re: Gaussian error propagation Error estimates for functions of several correlated variables : Additional terms accounting Normal errors for correlations for uncorrelated variables Special case, uncorrelated variables: This is called Gaussian error propagation, however has nothing to do with Gaussian distributions Modern Methods of Data Analysis - WS 07/08

5 And the same in more dimensions (A is Jacobi matrix)

6 Be aware... The approximation using Taylor expansion breaks down if the function is significantly not linear in the region ± 1σ around the mean value. Example: momentum estimate in B field; p ~ 1/κ 10 % momentum bias!

7 2. Order Taylor Expansion

8 Example Systematic Error Measurements are taken with a steel ruler, the ruler was calibrated at 15C, but the measurements were carried out at 22C. This is a systematic mistake (bias) and not a systematic uncertainty! To neglect this effect is a systematic mistake. Effects can be corrected for! If the temperature coefficient and lab temperature is known (exactly), then there is no systematic uncertainty. If we correct for effect, but corrections are not known exactly, then we have to introduce a systematic uncertainty. In practice (unfortunately): often not corrected for such effects, but then just included in sys. errors.

9 Systematic Error Definition: A systematic error denotes the uncertainty in effects caused by systematic mistakes and caused by neglecting systematic mistakes A systematic mistake is not a systematic error. Comments on systematic errors: sys. error do NOT decrease with 1/ N statistical and systematic errors can in general be added in quadrature (if uncorrelated; else include correlations) need to quote them separately in the results, they are often correlated among experiments: m(b0) = ± 0.53 (stat) ± 0.33 (sys)

10 Combing Errors (I) Suppose you have two measurements, with a random (statistical) uncertainties and a common systematic error S. How to make the covariance matrix?

11 Combining Errors (II) Consider and as sum of three random variables: assign according uncertainties More extended case, three measurements with one common systematic uncertainty S, and one systematic uncertainty T common for two of the measurements

12 Example: Pendulum Measure length of bar by measuring period of pendelum. Take two time measurements at different temperature. Compute the difference in length: and associated uncertainties. Given statistical uncertainties on the time measurements, additional common systematic uncertainty on the time measurement ( )and common systematic uncertainty on g ( ).

13 Evaluating Systematic Errors (I) Distinguish systematic errors from known and from unsuspected sources known sources error on factors in the analysis, energy calibration, tracking efficiencies, corrections,... error on external input: theory error, error on branching ratios, masses, fragmentation evaluate systematic uncertainties from known sources s(i) on result R. take several typical assumptions on s(i), compute R for each of them. Compute standard deviation of R take two extreme assumptions, compute R. Take difference of results divided by 12

14 Evaluating Systematic Errors (II) Errors from unsuspected sources need first to be identified repeat the analysis in different form helps to find systematic effects vary the range of data used for extraction of the result, use subset of data change cuts, change histogram binning change parameterizations, change fit techniques look for impossibilities It is clearly wrong to add in quadrature resulting deviations from the check list as systematic error this is misconception Moreover, the more careful you are doing more checks, the bigger should your systematic be??? - No! Modern Methods of Data Analysis - WS 07/08

15 Evaluating Systematic Errors (III) define before the consistency checks a pass/fail criteria. Remember with 20 checks you expect on average one 2σ deviation. However uncertainties are highly correlated! if you do not expect a systematic effect a priori and if the deviation is not significant, then do not add this in the systematic error if there is a deviation, try to understand, where the mistake is in the analysis and fix it! only as a last resort include discrepancy in systematic error

16 Evaluating Systematic Errors (IV) Conservative estimate of uncertainties.. Physicists tend to overestimate their systematics: If we estimate them conservatively, we are save in case we have forgotten to evaluate one source. How can we be sure that this identified source is covered by the conservative uncertainties??!! This is (commonly used) non-sense.