CHIWEI: A code of goodness of fit tests for weighted and unweighted histograms

CHIWEI: A co de of go o dness of fit tests for w eigh ted and un w eigh ted histogr a ms N.D. Gagunash vili a, ∗ a University of Akur eyri , Bor gir, v/Nor dursl´ od, IS-600 Akur eyri, Ic e land Abstract A F ortran-77 program for go o dness of fit tests fo r histograms with w eighted en tries as w ell as with unw eigh ted en tries is presen ted. The co de calculates test statistics for case o f histogram with normalized w eigh ts of ev en ts and in case of unnormalized w eigh ts of ev en ts. Keywor ds: c hi-square test generalization, comparison exp e rimen tal and sim ula t ed data, data interpretation, Monte Carlo metho d PR OGRAM SUMM AR Y Pr o gr am Title: CHIWEI Journal R efer enc e: Catalo gue identifier: Lic ensing pr ovisions: none Pr o gr amming language: F ortran-77 Computer: Any Unix/Lin ux workstati on or PC w i th a F ortran-77 compiler Classific ation: 4.13, 11.9, 16.4, 19.4 External r outines/libr aries use d: FPLSOR (M103) fr o m C E RN Prog ram Lib rary Natur e of p r oblem: The program calculate s go o dness of fit test statistics for w eigh ted histograms Solution metho d: Calculation of test s tatistics is done a ccording form ulas p resen ted in Ref. [1] ∗ Corresp onding autho r . E-mail addr ess: nikolai@simnet.is Pr eprint submitt e d to Co mputer Physics Commun i c ations Octob er 29, 2018 References [1] N.G. Gagunash vili, Nu cl. Instrum. Meth. A596 (200 8) 439. 1. I n tro duction A histogram with m bins for a giv en pr o babilit y density function p ( x ) is used to estimate the probabilities p i = Z S i p ( x ) dx, i = 1 , . . . , m (1) that a random ev ent b elongs to bin i . Inte gratio n in (1) is done o v er the bin S i . A histog r a m can b e o btained as a result of a random exp erimen t with probabilit y densit y function p ( x ). Let us denote the num b er of random ev ents b elonging to the i th bin of the histogram as n i . The t o tal n um b er of ev en ts in the histog r am is equal to n = P m i =1 n i . The quantit y ˆ p i = n i /n is a n estimator of p i with exp ectation v a lue E ˆ p i = p i . The problem of go o dness of fit is to test the hypothesis H 0 : p 1 = p 10 , . . . , p m − 1 = p m − 1 , 0 vs. H a : p i 6 = p i 0 for some i, (2) where p i 0 are sp ecified probabilities, and P m i =1 p i 0 = 1. The test is used in a data analysis for comparison theoretical frequencie s np i 0 with the observ ed frequencies n i . The test statistic X 2 = m X i =1 ( n i − np i 0 ) 2 np i 0 (3) w as suggested b y Pearson [2 ]. P earson sho w ed that the statistic (3) has appro ximately a χ 2 m − 1 distribution if the hypothesis H 0 is true. T o define a w eigh ted histogram let us write the probabilit y p i (1) for a giv en probability density function p ( x ) in the form p i = Z S i p ( x ) dx = Z S i w ( x ) g ( x ) dx, (4) where w ( x ) = p ( x ) / g ( x ) (5) 2 is the w eight function and g ( x ) is some other probabilit y densit y function. The function g ( x ) mus t b e > 0 fo r p oints x , where p ( x ) 6 = 0. The w eigh t w ( x ) = 0 if p ( x ) = 0, see Ref. [3]. Because of the condition P i p i = 1 further w e will call the ab ov e defined w eigh ts nor malized w eigh ts as o pp osite to the unnormalized w eigh ts ˇ w ( x ) whic h ar e ˇ w ( x ) = const · w ( x ). The histogram with normalized w eights w as obtained from a random exp eriment with a probabilit y densit y function g ( x ), and the w eigh ts of t he ev ents w ere calculated according to (5). Let us denote the total sum o f the w eigh ts of the eve nts in the i th bin of the histog ram as W i = n i X k =1 w i ( k ) (6) and the total sum of squares of we ights as W 2 i = n i X k =1 w i ( k ) 2 , (7) where n i is the num b er of ev en ts at bin i and w i ( k ) is t he w eigh t of the k th ev ent in the i th bin. The t otal n umber o f ev en ts in the histogram is equal to n = P m i =1 n i , where m is t he num b er of bins. The quantit y ˆ p i = W i /n for the histogram with normalized weigh ts is the estimator of p i with t he exp ectation v alue E ˆ p i = p i . Note tha t in the case where g ( x ) = p ( x ), the w eigh ts of the ev ents a re equal to 1 and the histogr a m with normalized we ights is the usual histogram with un w eighted en tries. F or w eigh ted histograms again the problem of go o dness of fit is to test the h yp othesis H 0 : p 1 = p 10 , . . . , p m − 1 = p m − 1 , 0 vs. H a : p i 6 = p i 0 for some i, (8) where p i 0 are specified probabilities, and P m i =1 p i 0 = 1. The test statistic that is a generalization of Pe arson’s statistic ( 3 ) w as prop osed in [1] for cases of histograms with normalized w eigh ts of entries as w ell as with unnormalised w eights of en t ries. A code for the calculation of test statistics is presen ted in this a rticle. As shown in [1] if hypothesis H 0 (8) is t r ue then the statistic for a histogram with normalized w eigh ted en tries has appro ximately the χ 2 m − 1 distribution and for a histogram with unnormalized w eigh ted entries has χ 2 m − 2 distribution. 3 Use of the pro p osed test is inappropriate if any exp ected count in bin of histogram is b elo w 1 or if the expected coun t is less than 5 in more than 20% of the bins. This empirical restriction kno wn for the usual c hi-square test [4] is quite reasonable for w eighted histograms. Information for readers. Recen tly , a no ther pap er dedicated to w eighted histograms has b een publishe d in ”Computer Phys ics Communic ation“, see Ref. [6]. T he same author has pr esen ted a program fo r calculating test statistics to compare w eigh ted histogram with un w eigh ted histogram and t w o histograms with w eigh ted entries . The test can b e used fo r the compar- ison of exp erimen tal dat a distributions with sim ulated data distributions as w ell as for the tw o sim ulated data distributions. 2. Computer program CHIWEI is subroutine whic h can b e called from F ortran program for the calculation of test statistics. Usage CALL CHIWEI(P,W 1,W2,N,NCHA,MODE,STAT,NDF,IFAIL) Input D ata P – one dimensional real a r ra y of probabilities p i W1 – one dimensional ar r ay , sum of weigh ts W i in eac h bin W2 – one dimensional ar r ay , sum of squares of weigh ts W 2 i in eac h bin N – n um b er of ev en ts n NCHA – n umber of bins m MODE – mus t b e equ al to 1 for a histog r a m with normalized w eigh ts, and equal 2 for histogram with unnor malized w eigh ts 4 Output data ST A T – test statistic follo wing a chi-sq uare distribution with NDF degrees of freedom if h yp ot hesis H 0 is true NDF – n um b er of degree of freedom (will b e m -MODE) IF AIL – will b e > 0 if calculation is not succes sful. 3. T est run W e tak e a distribution p ( x ) ∝ 2 ( x − 10) 2 + 1 + 1 ( x − 14) 2 + 1 (9) defined on the interv a l [4 , 16] and represen t ing t w o so-called Breit-Wigner p eaks. Tw o cases of the pro babilit y densit y function g ( x ) are considered g 1 ( x ) = p ( x ) (10) g 2 ( x ) ∝ 2 ( x − 9) 2 + 1 + 2 ( x − 15) 2 + 1 (11) Distribution (10) giv es an un w eigh ted histogram and the metho d coin- cides with P earson’s chi square test. Distribution (11 ) has t he same form of parametrization as (9), but with differen t v a lues of the para meters. Three cases of histograms w ere considered: un we ighted histogra m, histogram with w eigh ts p ( x ) /g 2 ( x ) and histogram with unnormalized weigh ts 2 p ( x ) /g 2 ( x ). Histograms with 5 bins w ere created by sim ulatio n 1000 en tries for eac h case. The results of the calculations are presen ted b elo w. Program PR OB(G100) [5] has b een used for calculating p-v alues. T est 1 INPUT P 0.0296 0.1106 0.4460 0.2067 0.2072 W1 26.0000 115.0000 454.0000 183.0000 222.0000 W2 26.0000 115.0000 454.0000 183.0000 222.0000 5 N 1000 NCHA 5 MODE 1 OUTPUT STAT 4.5291 (p-v alue=0.3391 ) NDF 4 IFAIL 0 T est 2 INPUT P 0.0296 0.1106 0.4460 0.2067 0.2072 W1 36.0112 106.1355 458.3037 197.8123 205.7211 W2 28.2698 56.9601 938.7897 363.4649 172.2003 N 1000 NCHA 5 MODE 1 OUTPUT STAT 2.3380 (p-v alue=0.6738 ) NDF 4 IFAIL 0 T est 3 INPUT P 0.0296 0.1106 0.4460 0.2067 0.2072 W1 72.0225 212.2710 916.6075 395.6246 411.4423 W2 113.0790 227.8403 3755.1587 1453.8595 688.8014 N 1000 NCHA 5 MODE 2 OUTPUT 6 STAT 2.2398 (p-v alue=0.5241 ) NDF 3 IFAIL 0 References [1] N.G. G agunash vili, Nucl. Instrum. Meth. A596 (2008) 439. [2] K. Pearson, Phil. Mag. 5th Ser. 50 (1900) 15 7. [3] I. Sobol, A Primer F or The Mon te Carlo Metho d, CRC Press, Bo ca Raton, Florida, 1 994. [4] D.S. Mo o r e, G.P . McCab e, In tro duction to the Practice of Stat istics, W.H. F reeman Publishing Compan y , New Y ork, 2005 . [5] CERN Prog ram Library , http://cernlib.w eb.cern.c h/cernlib/ . [6] N.D. G agunash vili, Comp. Ph ys. Comm. CPC-D- 11-001 9 6R1 7