CERN SUMMER STUDENT PROJECT

Transcription

1 CERN SUMMER STUDENT PROJECT GENERAL PHYSICS: ANALYSE DATA IN THE FORM OF HISTOGRAM WITH COMPARISON BETWEEN KOLMOGOROV-SMIRNOV TEST AND CHI SQUARE TEST NAME: Ak Muhamad Amirul Irfan Pg Haji Mohd Ariffin COUNTRY: Brunei Darussalam ( University of Brunei Darussalam) SUPERVISOR: Norraphat Srimanobhas DEPARTMENT: PH-UCM DATE:

2 Abstract This paper presents the project that I have been tasked while attending a three-month Summer Programme at CERN. The Project specification is to analyse the result of a weekly data produced by Compact Muon Solenoid (CMS) in the form of histograms. CMS is a detector which is a multi-purpose apparatus use to operate at the Large Hadron Collider (LHC) at CERN. It will yield head-on collisions of two proton (ion) beams of 7 TeV (2.75 TeV per nucleon) each, with a design luminosity of. A comparison of the results is then made using two methods namely Kolmogorov Smirnov Statistic Test and Chi-Squared Test. These tests will be further elaborated in the subsequent paragraphs. To execute this project, I have to firstly study the entire basic computer programming in particular C++ and the ROOT Basic Programmes. This is important to ensure the tasks given can be resolved within the given time. A program is subsequently written to provide output of histogram and calculation of Kolmogorov-Smirnov Test and Chi-square test. 2

3 1. INTRODUCTION The ROOT Basic Programme is a data analysis framework, originally initiated at CERN, which is use to plot many formats of graphical plotting program in C++. It also provides a reach set of mathematical libraries and tools needed for sophisticated data analysis. Moreover, it completes with various number of root libraries and offers users possibility of extending core mathematical functionality by creating plug-ins such as MathMore library and MathCore library. These can import specific types of data into the interface and also provide new tools for manipulating them. With these frameworks, it benefits the users as less code to write is required to plot graphs for analysis based on their own specifications such as fitting and histograms. C++ language is simply the following: Framework: Root V. 5/ Putty Platform: lxplus Language: C++ 2. TASKS UNDERTAKEN 2.1 Lectures Due to lack of experience in programming language, I have to start with basic C++ programming language which is the best way to understand the platform for doing the coding. The topics encompassed C++ Basics, Variables and Fundamental Data Types, Operators, Control Flow including Arrays and loops. In every topic, the theory also incorporates the tutorial for the users to try and apply for writing the programme. Having completed the topics, tutorials were given to test our knowledge on the basic C++. In the tutorial, writing a program for the output to print is the key to tackle the problems and one must be able to include the loops coding. The questions given were as follows: i. Write your own program to print the straight line and circle on the screen. ii. Write a program that can calculate volumes of cone by using concept of integration and including finding the minimum number of loops which can give the difference between numerical value and exact value less than 0.01%. 2.2 Solution The following steps were taken: Starting with, Lxplus service is interactive logon service to Linux at CERN. It consists of public machines for interactive work. In order to access to one of the machines, users have to log on to the service using lxplus.cern.ch with the user s 3

4 username and password. The users get connected to the machine that gives the best response for example least number of login sessions. 2.3 Root Basic Programme Daily use of ROOT applications to analyse their data or to perform simulations such as: 1. Processing data with ROOT (to learn about the interactive and batch use of ROOT) 2. Analysing data with ROOT (about histograms and trees; mathematical and statistical tools) 3. Showing results with ROOT (explaining pads, 2D and 3D graphics) In relation with the basic earlier, ROOT is written mainly in C++ Language. Hence, it is prerequisite to learn and understand C++ beforehand. Chapters that needed to be covered for ROOT were ROOT as Calculator, ROOT as Function Plotter, Plotting Measurements and Histograms. 2.4 Tutorial and solutions Write a macro to visualise a poisson distribution in a histogram for exercise. So here, users need to create a 1D Histogram with maximum number of counts of 15 and using random generator to generate 1000 poissonian counts and mu equal to 4. This follows by creating 2 sets of 1D histogram in one canvas where one of the histograms will show the difference and ratio of two histograms. 4

5 3. KOLMOGOROV-SMIRNOV The Kolmogorov-Smirnov test is frequently labelled as goodness-of fit test, i.e. checking if data are normally distributed. In addition, Kolmogorov-Smirnov Test is used to decide if a sample comes from a population with a specific distribution and the test is based on the empirical distribution function (ECDF). The Kolmogorov-Smirnov Test has a unique characteristic where it does not depend on the underlying cumulative distribution function being tested. It is also an exact test which does not depend on adequate sample size to be valid. The Kolmogorov Smirnov, nevertheless, has 3 limitations. First, it only applies to continuous distributions. Secondly, it tends to be more sensitive near the center of distribution. Finally is the distribution parameters must be fully specified. Equations: ( ) { } ( ) ( ) Here, let be an ordered sample and sample comes from a population with cumulative distribution function F(x). Can be seen that it doesn t depend on F but ( ) depends on the sample chosen which result is a random variable. The equation to find the sample data whether it is a good fit or not is ( ) ( ) where is the critical value from the kolmogorov-smirnov table. In order to know it is a good fit must be less than and usually closer to 0. However, in the C++ program, this numbers of steps of calculation above can be skipped as the program can calculate it automatically with parameters assumed to be unknown. In a continuation of two sets of 1D histograms in one canvas, two lines namely gaus_h1 and gaus_h2 can be tested for Kolmogorov-smirnov by writing this in a program cout<<gaus_h1->kolmogorovtest(gaus_h2)<<endl;. Result will be printed in the screen with calculation done automatically. 5

6 4. CHI-SQUARED TEST The chi-square test is used to test if a sample of data came from a population with a specific distribution. The attractive feature that chi-square test has is that it can be applied to any univariate distribution which can calculate the cumulative distribution function. Furthermore it can also be applied to binned data i.e. data put into classes. However, it has limitation where chi-square requires a sufficient sample size in order for the test to be valid. Different from Kolmogorov-Smirnov test, chi-square test can be applied to discrete distributions such as binomial and poisson. Equation: ( ( ) ( )) ( ) From the equations above, F is cumulative distribution function for the distribution being tested, and is the upper limit and lower limit for class respectively. Whereas N is the number of sample size. In Chi-square, the data are divided into k bins for the second equation and also, is the observed frequency for bin and is the expected frequency for bin. Is test statistic then compared to of critical value. of critical value can be found firstly from calculating the degree of freedom where and followed by its under the null hypothesis. Usually, the critical region for the test is of test statistic is greater than the of critical value. Where as in C++, for two binned data, write a program as follows: Given the arrays bins1[1..nbins] and bins2[1..nbins], containing two sets of binned data, and given the number of constraints knstrn (normally 1 or 0), this routine returns the number of degrees of freedom df, the chi-square chsq, and the significance probability. A small value of probability indicates a significant difference between the distributions bins1 and bins2. Note that bins1 and bins2 are both float arrays, although they will normally contain integer values. { float gammq(float a, float x); int j; float temp; *df=nbins-knstrn; *chsq=0.0; for (j=1;j<=nbins;j++) if (bins1[j] == 0.0 && bins2[j] == 0.0) --(*df); else { temp=bins1[j]-bins2[j]; *chsq += temp*temp/(bins1[j]+bins2[j]); } *prob=gammq(0.5*(*df),0.5*(*chsq)); } 6

7 5. FINDINGS/RESULTS The larger the number of loops, the smaller the number of error of rate in Kolmogorov- Smirnov Test i.e. for 10000, the result of error is only whereas for loops, the result of error is For Chi-Squared Test, The number of error rates will decrease as the class increases for each specific loop. Therefore, this shows a noticeable improvement in the reading of error rates. 6. CONCLUSIONS The parameters of the distribution are usually unknown and need to be estimated. When the parameters are estimated from the data, it affects the power of the KS test. Moreover, the KS test does not perform well in the sense of power; when the data are tested with the same type of distribution with a similar mean with less power than the Chi-square test. In additions, the KS test has superior power to the Chi-square test when the sample size is large. In all cases studied, the KS test always has a smaller type I error rate than the Chi-square test. For the Chi-square test, as the number of intervals increased, the type I error rate and power decreased. In general, the computing time of the KS test is longer than that of the Chi-square test. Besides, the limit of this study is empirical and not mathematical. Therefore, with the differences and limitation in each test, this will affect most of the end result as can be seen above. 7

8 REFERENCE: 1) Engineering Statistic Handbook: Chi square goodness of fit test, 2) Engineering Statistic Handbook: Kolmogorov-Smirnov Goodness of fit test, 3) Hsaio-Mei Wang, Comparison of goodness of fit test: Kolmogorov-Smirnov Test and Pearson Chi Square Test. 4) Numerical Recipes in C: The Art Of Scientific Computing, , Cambridge University Press. 5) Unix Tutorial for Beginners, 8