P445/515 Data Analysis using PAW

Transcription

1 P445/515 Data Analysis using PAW C. McGrew February 10, 2003 Abstract PAW (Physics Analysis Workstation) is a complete physics analysis package developed at CERN to handle high energy physics data. It is well suited to handle most types of physics data (with the exception of image and spectral analysis). A solution to a typical fitting problem is presented. 1 Introduction PAW is a very powerful data analysis package that was developed by CERN as part of they high energy program. It provides an interactive interface to the extensive CERNLIB library of routines that has been developed since the early 1970 s. While CERNLIB is almost entirely Fortran, a language that is falling out of favor, the routines within the library are extensively debugged and are quite reliable. Most of the core functionality has existed in CERNLIB for more than 20 years and the routines that are used in PAW have been tested by large active physics collaborations. Although, PAW is no longer being actively developed, CERN is committed to maintaining it. Further, since it is built on a well debug library, it is quite solid. PAW has the further advantage that it has been developed by physicists to do physics analysis and has many capabilities that are aimed at allowing us to complete a finished, high-quality, analysis. PAW includes a capabilities to Create histograms from very large datasets, small datasets are also easily handled. This is the core functionality. Do interactive data exploration. Do correct error analysis on the bins of histograms. Do complex non-linear fits to data within histograms using MINUIT. There are extensive facilities to all the correct estimation of parameter confidence intervals including the correct treatment of correlated errors. Create publication quality figures. Much more. 1

2 When you read a paper from high energy physics, the plots were probably generated using paw. There is also extensive documentation available describing the algorithms used. One final interesting PAW feature is the inclusion of an Fortran interpreter as well as the ability to dynamically compile and load user functions. 2 Availability PAW is available from CERN at and follow the download link. You will find versions for Windows and Redhat. The Redhat versions are simply tar files of executables and libraries compiled for the specific architecture and since static linking is used, the objects will work an virtually any GNU system that is roughly contemporaneous with a particular Redhat version. I have not personally installed the windows The daring can also attempt to compile the distribution from source. CERN maintains a large library of CERNLIB documentation on the web, most of it can be viewed using a browser, and also be downloaded as postscript. It was not for nothing that the web was invented at CERN. You can reach the documentation via the web addresses given above. 3 Data Analysis with PAW There are several good books which discuss data analysis in detail[1, 2], and this section is not intended as a data analysis tutorial to replace those detailed treatments. The main goal is to demonstrate the capabilities of PAW so that you can take data analysis techniques which you have learned elsewhere and apply them. PAW has extensive built in help and an excellent manual can be downloaded from the web so I will only sketch the command details here. One thing to keep in mind is that there are at least 5 ways to do anything with PAW. I will be presenting one possible solution. The kumac (or macro) file used to generate the plots and fit can be found in Appendix A. To start paw on Unix, the command is paw. This will present you with a command line interface much like shell. You can type any paw command, but the most useful command is one to execute a macro. For instance, to execute the macro given in Appendix A, the command would be exec solution paw.kumac. 3.1 Read the data into PAW PAW has two important ways to read data and manipulate it. The simplest method to read data is using the vector/read command. This command reads the entire table into memory and is convenient for small, one or two column tables, with fewer than entries. Larger tables may exhaust the memory. Line 19 in the example presented in Appendix A shows how to read a file with a single column of data into a vector. The more complicated and versatile way to read data is to construct an ntuple. Ntuples have been the most common way for HEP experiments to process data and are basically a 2

3 multi-column, sequential data base. Immense amounts of data can be handled with an ntuple and PAW includes many facilities to manipulate them. Check the PAW manual for usage details, particularly see the examples. 3.2 Creating a histogram Histograms are the basis for much of the functionality found in PAW. Using a histogram is a three step process: 1. Create or book a histogram. Both one and two dimensional histograms can be created. 2. Fill the histogram with data. 3. Plot the histogram for inspection or publication. When you make plots, you will find that the default style is inexplicably ugly. In particular, the plot title and labels are too small, the lines are too narrow, and the points are too light for publication. The figures (1 and 2) were generated using settings that most journals will accept and which display well during presentations. The example in Appendix A shows how histograms are created and filled. Line 15 shows how to create a new histogram. In PAW, histograms are referred to by number (1000 in this case). The histogram is created with 100 bins with the bottom edge of the lowest bin at 0 GeV, and the high edge of the highest bin at 2.0 GeV. Line 20 shows how to fill a histogram from a vector. The entries in the vector are processed and each is assigned to an appropriate bin in the histogram. Lines show how to plot the histogram. Notice lines 27, 28 and 32 which create an output file, tell paw to fill it with an encapsulated postscript figure, and then close the file after the histogram has been plotted. By default, paw assigns Poissonian errors (σ N = N) to each bin of a histogram which are appropriate for almost all cases. However, it has the capability to accept user calculated errors if required. Further, by setting the appropriate paw command options, it correct will handle error propagation when operations are performed on the histogram data. 3.3 Fitting the data Inspection of Figure 1 shows that the data contains a peak which can be assumed to be Gaussian on top of a non-constant background. I will fit the data using a model with a Gaussian on top of an exponential background. (E p 4 ) 2 N(E) = p1 e p2e + p 3 e 2p 2 5 (1) Where I have chosen a functional form that is very easy to use in PAW. In fact, PAW can be used to fit any arbitrary function. To demonstrate how arbitrary functions can be used, I encode Equation 1 on lines By default, PAW will perform a χ 2 fit of the model to the data using the MINUIT function minimization package. Details of the fitting procedure are can be found in the PAW and 3

4 Figure 1: A histogram of the data in example.data. 4

5 Figure 2: This data contained in the example.data with the fit applied. 5

6 Table 1: Fit results Parameter Name Fitted value True Value p 1 Background Normalization Factor ± p 2 Background Slope ± p 3 Signal Height 569 ± p 4 Signal Mean ± p 5 Signal Width ± χ 2 At Minimum MINUIT manual, and in some books[1]. PAW has many options that can control the way in which data in a histogram is fit, but the defaults are usually satisfactory. The histogram is fit on line 47 and 48 of the example. Part of doing a fit and getting the right answer is to choose the correct starting values for the parameters. The precise values are not important, but they should be close enough to the true minimum that the fit converges. Line 47 specifies first guesses at function parameters (The parameter definitions are the same as in Equation 1. Typically, any non-zero values with the right order of magnitude are close enough to allow the minimization to converge. Line 48 does the actual fit. There is a complex mathematical theory associated with finding the correct minimum, and a mathematician will tell you that it is not possible to tell if you have found the global minimum. However, in physics we have a very practical solution; we plot the function against the data and check that the fit makes sense. Figure 2 shows the overlay of the data and the fit. It is created on lines of the example. Notice that the values of the best fit parameters are plotted on the figure. This is controlled by the option commands on lines 50 and Results Table 1 gives the results of the fit as well as the true values used to generate the distribution. The χ 2 at the minimum is for 95 degrees of freedom. This implies that 60% of distributions drawn from Equation 1 will have a χ 2 further from 95 than this one does so it is a good fit. A The solution Solution to a problem similar to one assigned in P445/515. This can be found in the file solution paw.kumac that can be found on the course web page. 1 ***************************************************************** 2 * This creates two files: 3 * paw_hist.eps -- The histogram of the data without a fit. 6

7 4 * paw_fit.eps -- The histogram of the data with a fit. 5 * Delete all of the existing histograms. 6 histogram/delete * 7 * Same for vectors. 8 vector/delete * 9 ***************************************************************** 10 * Create a histogram to be filled. 11 ***************************************************************** 12 * This create histogram 1000 and a title "eta zero mass 13 * distribution. The histogram has 100 bins between (low edge of 14 * low bin is 0, high edge of high bin is 2.0). 15 histogram/create/1d 1000 [c]^0! mass distribution ***************************************************************** 17 * Read the data and add it to the histogram. 18 ***************************************************************** 19 vector/read data_vector example.data 20 vector/hfill data_vector ***************************************************************** 22 * Make a histogram plot of the data. 23 * 24 * The magic commands fortran/file and graphics/metafile open an 25 * encapsulated postscript file. 26 ***************************************************************** 27 fortran/file 66 paw_hist.eps 28 graphics/metafile histogram/plot 1000 e 30 * Put a title on the axis. 31 atitle Energy (GeV) N (Events / 0.02 GeV) 32 close ***************************************************************** 34 * Fit the histogram 35 ***************************************************************** 36 * Define a fitting function. I could use internal paw functions, 37 * but this shows how to define an arbitrary function. 38 application comis 39 real function func(x) 40 common /pawpar/ par(5) 41 func = par(1)*exp(par(2)*x) 42 func = func + par(3)*exp(-(x-par(4))**2/(2.0*par(5)**2)); 43 return 44 end 45 exit 7

8 46 * Create a vector with initial parameter guesses and then fit. 47 vector/create par(5) R histogram/fit 1000 func q 5 par 49 * Make a histogram plot of the data and fit. 50 option fit 51 set fit fortran/file 66 paw_fit.eps 53 graphics/metafile histogram/plot 1000 e 55 atitle Energy (GeV) N (Events / 0.02 GeV) 56 close 66 References [1] Byron P. Roe. Probability and Statistics in Experimental Physics. Springer, 2 edition, [2] William H. Press et al. Numerical Recipes The Art of Scientific Computing. Cambridge University Press, Cambridge, 1 edition,