1. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Reconstruction techniques Estimators ML /   Estimator M-Estimator Background in the PDF First look at performance of AartStrategy in low energy events

2. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Introduction to Estimators definitions: [Particle Data Book] A statistic is any function of the data, which does not depend upon any of the unknown parameters An estimator is any statistic whose value is intended as a meaningfull guess for a (fixed) parameter of unknown value. an estimator is a random variable, which is a guess of a fixed parameter of unknown value Data some way of using data to guess what the track is Track estimates & Error estimates True track random process estimator

3. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Introduction to Estimators For each hit, we can calculate the time residual t r = t -t cherenkov (track) and we want to find the track so that the residuals are 'small' PMT muon The 'best' estimator for the track is the Maximum Likelihood Estimator: Calculate P(t i r ) Maximize joint probability of the all hits P tot (event | track) =  P(t i r ) i

4. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Maximum Likelihood Estimator What does P(t i r ) look like ? time residual log(P) 0 background hits (flat) signal hits (peaked with tail) The track that maximises  log(P) is the best estimate for the true track.

5. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Suppose we know the track position and direction and only need to fit the time of the track (this means shifting the time residuals until log(P) is maximal) time residual log(P) best fit situation, corresponding to true track time 0 time residual log(P) best fit will not be found by most minimisation algorithms. 0 signal hits background hit will find local max.

6. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 What if we maximise -    ? -log(   )    time residual) 2 -log(   ) this startinpoint converges to the best fit this is the global maximum, but gives a poor estimate, since too much importance is assigned to the background hit. best fit without background

7. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 ML is best estimator, but practical problem of finding the minimum in 5-dim. space   finds global minimum, but it's at the wrong place, since outliers (background hits) have too much influence  Find some 'pseudo-PDF' that combines good features of both. Let go (for a moment) of the idea that PDF should describe the data Find function that, at high values of the residual, is not too flat (so minimum can be found) is not too steep (so it is robust against background hits)

8. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 L1-L2 M-estimator time residual

9. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 from numerical recipies

10. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Back to the likelihood function all hits final hits pdf hits mest hist prefit hits early hits sel. hits full pdf fit time-pdf fit M-est. fit linear prefit generate start tracks keep best track 'AartStrategy' Overview Other tricks to find the true track linear prefit (improved version) trying different starting points Carefull hit-selections Once we are close to the true track, the ML method will give the best possible estimate of the track. (if the PDF is correct)

11. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 time residual log(P) background hits (flat) signal hits (peaked with tail) Refinements in the pdf Shape of signal PDF depends on hit-amplitude Contribution of background to the PDF is estimated a priori from Hit amplitude (A) angle photon-PMT (a) distance Track-hit (b) intuatively: if a hit is far away from the track and the PMT is looking in the wrong direction, it's more likely to be a background hit

12. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 P sig signal background signal e.g: P(sig|a,b) is given by N sig (a,b)/ (N sig (a,b)+N back (a,b)) a=b= This is done for 5 hit-amplitude bins

13. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 time residual log(P) Background PDF and normalisation To normalise background PDF need to cut off event (at T). It can be shown that: As long as T is large enough to contain all signal hits,the value of T does not alter the place where the likelihood is maximal. Therefore Fit-result is independent of hit-selection, as long as it contains all signal hits In practise the selection of final hits is indeed very loose: (250ns, 300m) Can use different T at PDF-fitting stage and at reconstruction-stage. -T T

14. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 A few words about weighting the hits L =  log P(t i | track ) The (only) correct expression for the likelihood: probability for finding t i given this track L' =  log [ w i P(t i | track ) ] =  log w i  log P(t i | track ) constant factor correct PDF L'' =  w i log [ P(t i | track ) ] =  log [ P(t i | track ) wi ]  PDF is exponentiated: shape is changed W = This may be the desired effect: under the assumption all photons are uncorrelated, hits could be weighted with their amplitude in this way.

15. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Strong points of AartStrategy accurate many events well reconstructed very stringent cut on value of likelihood correct & trustworthy error estimates! E. Carmona before cut before cut after cut  fit = 1.09 pull-distribution showing error estimates are estimating the error

16. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 AartStrategy Performance on low energy events production of 5e10 neutrino interactions (genhen) 5 GeV < E < 10 TeV, generation spec. index = -1.4 geasim v4r5 (no scattering, with hadr. shower) 10 string detector recov4r1 Will show AartStrategy and CarmonaStrategy from recov4r1 Both were devoloped for high-energy reconstruction No special measures were taken (yet) to improve low-energy performance

17. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Effective Volumes effective volume (m 3 ) zenith-error < 10 deg after 'standard' cuts AartStrategy CarmonaStrategy AartStrategy CarmonaStrategy My 'standard' cuts much are much more severe at low energy log10(E )

18. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Zenith angle accuracy for different energy regimes zenith error 0< E  < 10 GeV 50< E  < 100 GeV 25< E  < 50 GeV 10< E  < 25 GeV AartStrategy CarmonaStrategy all this is after cuts

19. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Pull distributions are broader than at low energy (not suprising: PDF was fitted at high-energy, without Hadr. shower) 0< E  < 10 GeV 25< E  < 50 GeV 10< E  < 25 GeV 50< E  < 100 GeV 100< E  < 1000 GeV 1000< E  < 10000 GeV

20. Reconstruction techniques, Aart Heijboer, OWG meeting, Marseille nov. 2001 Conclusions performance AartStrategy at low energy PDF and the algorithm were fitted/tested on high energy events no dramatic drop in performance when going to lower energies Cuts must be optimised compared to E. Carmona's cuts they are very stringent.... But they work: also at low energies wrongly reconstructed events are rejected Error estimates (pull distributions) are getting worse at low E: presumably due to Hadronic Shower... Can be fixed by refitting low-energy PDF. In this production, low statistics for energies < 20 GeV Seems promising to investigate further and compare with Posidonia / Classic