Chi Squared Fit. Chi Squared Fit

Transcription

1 Ch Squared Ft Measured Data (Charge Voxels) Pulse Reconstructon (Charge on one Pad) Ht Reconstructon (Charge n one Pad Row) Track Fndng (Combnng Hts) Track Fndng (Combnng Pulses) Track Fttng: Ch Squared Ft Track Fttng: Global Ft Page 1

2 Least Squares Method N measurements at ponts y (the measurement ponts are gven by the pad rows) Varables x wth error (the x coordnate of the ht s the measurement) Ft functon: f(y; a 1,a,... a M ), a j : parameters to be determned N > M! (more measurement ponts than parameters needed!) For the best values a j, sum S s a mnmum : N S = =1 [ x f y,a j ] S a j =0, j=1... M for S to be a real ch-square, x must be Gaussan dstrbuted wth mean f(y ; a j ) and varance Page

3 Ch Squared Ft Straght Lne: x = f y = a y b a: SlopeX b: InterceptX x So, n ths case: S = a y b and S a = x a y b y =0 and S b = x a y b =0 y x wth: A= y B= 1 C= x D= y E= x y F= x x ths results n E ad ba =0, C aa bb =0 and the parameters a and b are gven by: track a= EB CA DB A and b= DC EA DB A y Page 3

4 Ch Squared Ft nd degree polynomal: x = f y = a y b y c also n rotated coordnate system x Ths leads to S = x a y b y c track (mnmzed numercally) y Radus R = a, Curvature C = 1 R Center x 0, y 0 solveequaton system: x x 0 y y 0 =R for ponts x 1, y 1, x, y Fast ft method, results can be used for crcle ft Page 4

5 Ch Squared Ft Crcle Ft: x x 0 y y 0 =R x also n rotated coordnate system, so the functon s: x = f y = x 0± 1 C y y 0 track Ths leads to S = x x 0± 1 y y C 0 (mnmzed numercally) X y x x 0 y y 0 =R f y = a y b y c Y Page 5

6 Track Object Track Object Collecton of consttuents: A vector (array) of Hts resp. Pulses belongng to the track Track parameters: Intercept (where does t enter the senstve volume) Slope (angle) Curvature Center of Crcle Errors of the track parameters Ch Squared of the track ft (estmate of the ft qualty) Optonal: Number of parameters, de/dx, Page 6

7 Global Ft Measured Data (Charge Voxels) Pulse Reconstructon (Charge on one Pad) Ht Reconstructon (Charge n one Pad Row) Track Fndng (Combnng Hts) Track Fndng (Combnng Pulses) Track Fttng: Ch Squared Ft Track Fttng: Global Ft Page 7

8 Maxmum Lkelhood Method A sample of n ndependent observatons x 1, x,... x n Theoretcal dstrbuton known: f(x a),wth a: Parameter to be estmated Calculate the lkelhood functon: Ths can be recognzed as the probablty for observng the sequence of values x 1, x,... x n Prncple: ths probablty s a maxmum for the observed values So the parameter a must be such, that L s a maxmum. So, a can be found by solvng: In practce: often easer to maxmze the logarthm of L: L a x =f x 1 a f x a...f x n a d L da =0 d ln L da snce: =0 ln y z =ln y ln z ths yelds results whch are equvalent to the above. Page 8

9 Global Ft Assumptons: For each row the track can be descrbed by a straght lne (heght of a pad row much smaller than the radus of the curvature ) curved charge tube of real track Pad assumpton: straght n each row Charge s Gaussan dstrbuted along the track (ths s a vald model for the charge deposton) Varatons of the charge deposton are gnored: assume a constant charge deposton n a row Page 9

10 Global Ft Lkelhood functon descrbng charge deposton per pad: L = p n, wth and n = N G p = : number of prmary e -, and Q exp pads/row Qexp n=1 N (probablty functon) : measured e - G : gan factor Logarthm of product of lkelhood functons of all pads: ln L= Rows Pad, wth Q measured ln [ h Q exp = h Q expected ] Q expected Row w dy dx w wdth of charge dstrbuton ncluded n ft functon as free fttng parameter [ x X 0 cos ysn ] 1 e for detals see: TPC Performance n Magnetc Felds wth GEM and Pad Readout, D. Karlen, P. Poffenberger, G. Rosenbaum, 005 Page 10

11 Ht Postons Ht s defned as the charge deposton n a row The X poston of ths charge deposton s needed for later resoluton calculaton, but the Global Ft n general has no ht reconstructon To get the poston, do a Global Ft n just one lne wth the wdth, angle fxed to the result of the track ft: ths means movng a the charge dstrbuton wth fxed wdth and angle (dependng on curvature) along the x axs untl t fts best to the deposted charge n ths row Page 11

12 Global Ft What to do wth nose Pulses? They are not descrbed by the theoretcal dstrbuton Soluton: assgn a hgher probablty for measurng a sgnal to all pads by ntroducng a constant offset: nose value N p p N 1 N n row ln L= Pad Q measured ln [ Q expected Row Q expected N / 1 N ] Probablty Dstrbuton Expected Sgnal wthout nose value ---- wth nose value N=0.01 Example: pad row wth 10 pads, ptch:.mm Page 1

13 Track Object Track Object Collecton of consttuents: A vector (array) of Hts resp. Pulses belongng to the track Track parameters: Intercept (where does t enter the senstve volume) Slope (angle) Curvature Center of Crcle Wdth of charge dstrbuton Errors of the track parameters Optonal: Number of parameters, de/dx, Page 13

14 Dffuson Parameters From the Global Ft, also the dffuson and defocussng parameters of the setup can be determned σ= D z σ 0 Ft Gaussan to every nterval and get mean values (wth errors) Wdth of charge dstrbuton can be descrbed by: σ= D z σ 0 Ft ths functon wth D and σ 0 as free parameters to the mean values of the ntervals to get the parameter values Page 14

15 Dffuson Parameters Results from smulaton compared to results from Global Ft results Results from Global Ft are n the rght order of magntude but underestmate the coeffcents Among other, possble explanaton: wrong nose factor (1%), further nvestgaton planned Page 15

16 Remarks Global Ft has the advantage that pad response effects are ncorporated n the ft functon Also, mssng nformaton (damaged pad) does not affect the ft too much snce the term n the sum smply vanshes ln L= Q measured [ Q expected ] ln Pad Q expected Row Dsadvantage: the ft s tme consumng wth many pad rows: for 6 pad rows Ch Squared Ft and Global Ft need approxmately the same tme, for 19 rows the Global Ft needs approxmately three tmes longer than the Ch Squared Ft If not many pad rows are used, Global Ft can produce too good resoluton results Wdth can be fxed durng the ft (for a certan Z per row) Ft n YZ plane done wth Ch Squared straght lne ft Page 16

17 Comparson of Ft Methods staggered pad layout staggered pad layout Ch Squared Method: Global Ft wth free σ: 6 rows n comparson too good 8 rows already reasonable 19 rows results show expected shape and are comparable wth Global Ft results for 19 rows 6 rows unreasonably good 8 and 19 rows tend to more reasonable results Global Ft wth fxed σ: results conservatve and scale wth ncreasng number of rows Both flavors comparable at 19 rows Page 17

18 Comparson of Ft Methods Robustness of the ft methods: nfluence of damaged pads (dead channels) tested wth Monte Carlo smulaton for 4T, P5 gas Page 18